dgeqrfp(3)

double

Section 3 liblapack-doc bookworm source

Description

doubleGEcomputational

NAME

doubleGEcomputational - double

SYNOPSIS

Functions

subroutine cgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
CGELQT
recursive subroutine cgelqt3 (M, N, A, LDA, T, LDT, INFO)
CGELQT3
subroutine cgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
CGEMLQT
subroutine dgebak (JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
DGEBAK
subroutine dgebal (JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
DGEBAL
subroutine dgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
subroutine dgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)
DGEBRD
subroutine dgecon (NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
DGECON
subroutine dgeequ (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
DGEEQU
subroutine dgeequb (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
DGEEQUB
subroutine dgehd2 (N, ILO, IHI, A, LDA, TAU, WORK, INFO)
DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
subroutine dgehrd (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
DGEHRD
subroutine dgelq2 (M, N, A, LDA, TAU, WORK, INFO)
DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
subroutine dgelqf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGELQF
subroutine dgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
DGELQT
recursive subroutine dgelqt3 (M, N, A, LDA, T, LDT, INFO)
DGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact WY representation of Q.
subroutine dgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
DGEMLQT
subroutine dgemqrt (SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT, C, LDC, WORK, INFO)
DGEMQRT
subroutine dgeql2 (M, N, A, LDA, TAU, WORK, INFO)
DGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.
subroutine dgeqlf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQLF
subroutine dgeqp3 (M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO)
DGEQP3
subroutine dgeqr2 (M, N, A, LDA, TAU, WORK, INFO)
DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
subroutine dgeqr2p (M, N, A, LDA, TAU, WORK, INFO)
DGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.
subroutine dgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRF
subroutine dgeqrfp (M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRFP
subroutine dgeqrt (M, N, NB, A, LDA, T, LDT, WORK, INFO)
DGEQRT
subroutine dgeqrt2 (M, N, A, LDA, T, LDT, INFO)
DGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
recursive subroutine dgeqrt3 (M, N, A, LDA, T, LDT, INFO)
DGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
subroutine dgerfs (TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DGERFS
subroutine dgerfsx (TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
DGERFSX
subroutine dgerq2 (M, N, A, LDA, TAU, WORK, INFO)
DGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
subroutine dgerqf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGERQF
subroutine dgesvj (JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, LDV, WORK, LWORK, INFO)
DGESVJ
subroutine dgetf2 (M, N, A, LDA, IPIV, INFO)
DGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).
subroutine dgetrf (M, N, A, LDA, IPIV, INFO)
DGETRF
recursive subroutine dgetrf2 (M, N, A, LDA, IPIV, INFO)
DGETRF2
subroutine dgetri (N, A, LDA, IPIV, WORK, LWORK, INFO)
DGETRI
subroutine dgetrs (TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DGETRS
subroutine dhgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
DHGEQZ
subroutine dla_geamv (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds.
double precision function dla_gercond (TRANS, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK)
DLA_GERCOND estimates the Skeel condition number for a general matrix.
subroutine dla_gerfsx_extended (PREC_TYPE, TRANS_TYPE, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERRS_N, ERRS_C, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
DLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
double precision function dla_gerpvgrw (N, NCOLS, A, LDA, AF, LDAF)
DLA_GERPVGRW
subroutine dlaorhr_col_getrfnp (M, N, A, LDA, D, INFO)
DLAORHR_COL_GETRFNP
recursive subroutine dlaorhr_col_getrfnp2 (M, N, A, LDA, D, INFO)
DLAORHR_COL_GETRFNP2
recursive subroutine dlaqz0 (WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, REC, INFO)
DLAQZ0
subroutine dlaqz1 (A, LDA, B, LDB, SR1, SR2, SI, BETA1, BETA2, V)
DLAQZ1
subroutine dlaqz2 (ILQ, ILZ, K, ISTARTM, ISTOPM, IHI, A, LDA, B, LDB, NQ, QSTART, Q, LDQ, NZ, ZSTART, Z, LDZ)
DLAQZ2
recursive subroutine dlaqz3 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B, LDB, Q, LDQ, Z, LDZ, NS, ND, ALPHAR, ALPHAI, BETA, QC, LDQC, ZC, LDZC, WORK, LWORK, REC, INFO)
DLAQZ3
subroutine dlaqz4 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSHIFTS, NBLOCK_DESIRED, SR, SI, SS, A, LDA, B, LDB, Q, LDQ, Z, LDZ, QC, LDQC, ZC, LDZC, WORK, LWORK, INFO)
DLAQZ4
subroutine dtgevc (SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
DTGEVC
subroutine dtgexc (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, WORK, LWORK, INFO)
DTGEXC
subroutine sgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
SGELQT
recursive subroutine sgelqt3 (M, N, A, LDA, T, LDT, INFO)
SGELQT3
subroutine sgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
SGEMLQT
recursive subroutine slaqz0 (WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, REC, INFO)
SLAQZ0
subroutine slaqz1 (A, LDA, B, LDB, SR1, SR2, SI, BETA1, BETA2, V)
SLAQZ1
subroutine slaqz2 (ILQ, ILZ, K, ISTARTM, ISTOPM, IHI, A, LDA, B, LDB, NQ, QSTART, Q, LDQ, NZ, ZSTART, Z, LDZ)
SLAQZ2
recursive subroutine slaqz3 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B, LDB, Q, LDQ, Z, LDZ, NS, ND, ALPHAR, ALPHAI, BETA, QC, LDQC, ZC, LDZC, WORK, LWORK, REC, INFO)
SLAQZ3
subroutine slaqz4 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSHIFTS, NBLOCK_DESIRED, SR, SI, SS, A, LDA, B, LDB, Q, LDQ, Z, LDZ, QC, LDQC, ZC, LDZC, WORK, LWORK, INFO)
SLAQZ4
subroutine zgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
ZGELQT
recursive subroutine zgelqt3 (M, N, A, LDA, T, LDT, INFO)
ZGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact WY representation of Q.
subroutine zgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
ZGEMLQT

Detailed Description

This is the group of double computational functions for GE matrices

Function Documentation

subroutine cgelqt (integer M, integer N, integer MB, complex, dimension( lda,* ) A, integer LDA, complex, dimension( ldt, * ) T, integer LDT, complex,dimension( * ) WORK, integer INFO)

CGELQT

Purpose:

CGELQT computes a blocked LQ factorization of a complex M-by-N matrix A
using the compact WY representation of Q.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

MB is INTEGER
The block size to be used in the blocked QR. MIN(M,N) >= MB >= 1.

A is COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and below the diagonal of the array
contain the M-by-MIN(M,N) lower trapezoidal matrix L (L is
lower triangular if M <= N); the elements above the diagonal
are the rows of V.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

T is COMPLEX array, dimension (LDT,MIN(M,N))
The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See below
for further details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

WORK

WORK is COMPLEX array, dimension (MB*N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix V stores the elementary reflectors H(i) in the i-th row
above the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 v1 v1 v1 v1 )
( 1 v2 v2 v2 )
( 1 v3 v3 )

where the vi’s represent the vectors which define H(i), which are returned
in the matrix A. The 1’s along the diagonal of V are not stored in A.
Let K=MIN(M,N). The number of blocks is B = ceiling(K/MB), where each
block is of order MB except for the last block, which is of order
IB = K - (B-1)*MB. For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB. The MB-by-MB (and IB-by-IB
for the last block) T’s are stored in the MB-by-K matrix T as

T = (T1 T2 ... TB).

recursive subroutine cgelqt3 (integer M, integer N, complex, dimension( lda,* ) A, integer LDA, complex, dimension( ldt, * ) T, integer LDT, integerINFO)

CGELQT3

Purpose:

CGELQT3 recursively computes a LQ factorization of a complex M-by-N
matrix A, using the compact WY representation of Q.

Based on the algorithm of Elmroth and Gustavson,
IBM J. Res. Develop. Vol 44 No. 4 July 2000.

Parameters

M is INTEGER
The number of rows of the matrix A. M =< N.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is COMPLEX array, dimension (LDA,N)
On entry, the complex M-by-N matrix A. On exit, the elements on and
below the diagonal contain the N-by-N lower triangular matrix L; the
elements above the diagonal are the rows of V. See below for
further details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

T is COMPLEX array, dimension (LDT,N)
The N-by-N upper triangular factor of the block reflector.
The elements on and above the diagonal contain the block
reflector T; the elements below the diagonal are not used.
See below for further details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix V stores the elementary reflectors H(i) in the i-th row
above the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 v1 v1 v1 v1 )
( 1 v2 v2 v2 )
( 1 v3 v3 v3 )

where the vi’s represent the vectors which define H(i), which are returned
in the matrix A. The 1’s along the diagonal of V are not stored in A. The
block reflector H is then given by

H = I - V * T * V**T

where V**T is the transpose of V.

For details of the algorithm, see Elmroth and Gustavson (cited above).

subroutine cgemlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer MB, complex, dimension( ldv, * ) V, integer LDV,complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldc, * )C, integer LDC, complex, dimension( * ) WORK, integer INFO)

CGEMLQT

Purpose:

CGEMLQT overwrites the general complex M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q C C Q
TRANS = ’C’: Q**H C C Q**H

where Q is a complex unitary matrix defined as the product of K
elementary reflectors:

Q = H(1) H(2) . . . H(K) = I - V T V**H

generated using the compact WY representation as returned by CGELQT.

Q is of order M if SIDE = ’L’ and of order N if SIDE = ’R’.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**H from the Left;
= ’R’: apply Q or Q**H from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’C’: Conjugate transpose, apply Q**H.

M is INTEGER
The number of rows of the matrix C. M >= 0.

N is INTEGER
The number of columns of the matrix C. N >= 0.

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

MB is INTEGER
The block size used for the storage of T. K >= MB >= 1.
This must be the same value of MB used to generate T
in CGELQT.

V is COMPLEX array, dimension
(LDV,M) if SIDE = ’L’,
(LDV,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
CGELQT in the first K rows of its array argument A.

LDV

LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,K).

T is COMPLEX array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by CGELQT, stored as a MB-by-K matrix.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

C is COMPLEX array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q C, Q**H C, C Q**H or C Q.

LDC

LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).

WORK

WORK is COMPLEX array. The dimension of
WORK is N*MB if SIDE = ’L’, or M*MB if SIDE = ’R’.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgebak (character JOB, character SIDE, integer N, integer ILO,integer IHI, double precision, dimension( * ) SCALE, integer M, doubleprecision, dimension( ldv, * ) V, integer LDV, integer INFO)

DGEBAK

Purpose:

DGEBAK forms the right or left eigenvectors of a real general matrix
by backward transformation on the computed eigenvectors of the
balanced matrix output by DGEBAL.

Parameters

JOB

JOB is CHARACTER*1
Specifies the type of backward transformation required:
= ’N’: do nothing, return immediately;
= ’P’: do backward transformation for permutation only;
= ’S’: do backward transformation for scaling only;
= ’B’: do backward transformations for both permutation and
scaling.
JOB must be the same as the argument JOB supplied to DGEBAL.

SIDE

SIDE is CHARACTER*1
= ’R’: V contains right eigenvectors;
= ’L’: V contains left eigenvectors.

N is INTEGER
The number of rows of the matrix V. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER
The integers ILO and IHI determined by DGEBAL.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

SCALE

SCALE is DOUBLE PRECISION array, dimension (N)
Details of the permutation and scaling factors, as returned
by DGEBAL.

M is INTEGER
The number of columns of the matrix V. M >= 0.

V is DOUBLE PRECISION array, dimension (LDV,M)
On entry, the matrix of right or left eigenvectors to be
transformed, as returned by DHSEIN or DTREVC.
On exit, V is overwritten by the transformed eigenvectors.

LDV

LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,N).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgebal (character JOB, integer N, double precision, dimension(lda, * ) A, integer LDA, integer ILO, integer IHI, double precision,dimension( * ) SCALE, integer INFO)

DGEBAL

Purpose:

DGEBAL balances a general real matrix A. This involves, first,
permuting A by a similarity transformation to isolate eigenvalues
in the first 1 to ILO-1 and last IHI+1 to N elements on the
diagonal; and second, applying a diagonal similarity transformation
to rows and columns ILO to IHI to make the rows and columns as
close in norm as possible. Both steps are optional.

Balancing may reduce the 1-norm of the matrix, and improve the
accuracy of the computed eigenvalues and/or eigenvectors.

Parameters

JOB

JOB is CHARACTER*1
Specifies the operations to be performed on A:
= ’N’: none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
for i = 1,...,N;
= ’P’: permute only;
= ’S’: scale only;
= ’B’: both permute and scale.

N is INTEGER
The order of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the input matrix A.
On exit, A is overwritten by the balanced matrix.
If JOB = ’N’, A is not referenced.
See Further Details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

ILO

ILO is INTEGER

IHI

IHI is INTEGER
ILO and IHI are set to integers such that on exit
A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
If JOB = ’N’ or ’S’, ILO = 1 and IHI = N.

SCALE

SCALE is DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied to
A. If P(j) is the index of the row and column interchanged
with row and column j and D(j) is the scaling factor
applied to row and column j, then
SCALE(j) = P(j) for j = 1,...,ILO-1
= D(j) for j = ILO,...,IHI
= P(j) for j = IHI+1,...,N.
The order in which the interchanges are made is N to IHI+1,
then 1 to ILO-1.

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The permutations consist of row and column interchanges which put
the matrix in the form

( T1 X Y )
P A P = ( 0 B Z )
( 0 0 T2 )

where T1 and T2 are upper triangular matrices whose eigenvalues lie
along the diagonal. The column indices ILO and IHI mark the starting
and ending columns of the submatrix B. Balancing consists of applying
a diagonal similarity transformation inv(D) * B * D to make the
1-norms of each row of B and its corresponding column nearly equal.
The output matrix is

( T1 X*D Y )
( 0 inv(D)*B*D inv(D)*Z ).
( 0 0 T2 )

Information about the permutations P and the diagonal matrix D is
returned in the vector SCALE.

This subroutine is based on the EISPACK routine BALANC.

Modified by Tzu-Yi Chen, Computer Science Division, University of
California at Berkeley, USA

subroutine dgebd2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) D, double precision,dimension( * ) E, double precision, dimension( * ) TAUQ, double precision,dimension( * ) TAUP, double precision, dimension( * ) WORK, integer INFO)

DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:

DGEBD2 reduces a real general m by n matrix A to upper or lower
bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

Parameters

M is INTEGER
The number of rows in the matrix A. M >= 0.

N is INTEGER
The number of columns in the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the m by n general matrix to be reduced.
On exit,
if m >= n, the diagonal and the first superdiagonal are
overwritten with the upper bidiagonal matrix B; the
elements below the diagonal, with the array TAUQ, represent
the orthogonal matrix Q as a product of elementary
reflectors, and the elements above the first superdiagonal,
with the array TAUP, represent the orthogonal matrix P as
a product of elementary reflectors;
if m < n, the diagonal and the first subdiagonal are
overwritten with the lower bidiagonal matrix B; the
elements below the first subdiagonal, with the array TAUQ,
represent the orthogonal matrix Q as a product of
elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the orthogonal matrix P as
a product of elementary reflectors.
See Further Details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

D is DOUBLE PRECISION array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).

E is DOUBLE PRECISION array, dimension (min(M,N)-1)
The off-diagonal elements of the bidiagonal matrix B:
if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

TAUQ

TAUQ is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q. See Further Details.

TAUP

TAUP is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix P. See Further Details.

WORK

WORK is DOUBLE PRECISION array, dimension (max(M,N))

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrices Q and P are represented as products of elementary
reflectors:

If m >= n,

Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real vectors;
v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n,

Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real vectors;
v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
tauq is stored in TAUQ(i) and taup in TAUP(i).

The contents of A on exit are illustrated by the following examples:

m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):

( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )

where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i).

subroutine dgebrd (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) D, double precision,dimension( * ) E, double precision, dimension( * ) TAUQ, double precision,dimension( * ) TAUP, double precision, dimension( * ) WORK, integer LWORK,integer INFO)

DGEBRD

Purpose:

DGEBRD reduces a general real M-by-N matrix A to upper or lower
bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

Parameters

M is INTEGER
The number of rows in the matrix A. M >= 0.

N is INTEGER
The number of columns in the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N general matrix to be reduced.
On exit,
if m >= n, the diagonal and the first superdiagonal are
overwritten with the upper bidiagonal matrix B; the
elements below the diagonal, with the array TAUQ, represent
the orthogonal matrix Q as a product of elementary
reflectors, and the elements above the first superdiagonal,
with the array TAUP, represent the orthogonal matrix P as
a product of elementary reflectors;
if m < n, the diagonal and the first subdiagonal are
overwritten with the lower bidiagonal matrix B; the
elements below the first subdiagonal, with the array TAUQ,
represent the orthogonal matrix Q as a product of
elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the orthogonal matrix P as
a product of elementary reflectors.
See Further Details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

D is DOUBLE PRECISION array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).

TAUQ

TAUQ is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q. See Further Details.

TAUP

TAUP is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix P. See Further Details.

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,M,N).
For optimum performance LWORK >= (M+N)*NB, where NB
is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrices Q and P are represented as products of elementary
reflectors:

If m >= n,

Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T

If m < n,

Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T

The contents of A on exit are illustrated by the following examples:

m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):

where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i).

subroutine dgecon (character NORM, integer N, double precision, dimension(lda, * ) A, integer LDA, double precision ANORM, double precision RCOND,double precision, dimension( * ) WORK, integer, dimension( * ) IWORK,integer INFO)

DGECON

Purpose:

DGECON estimates the reciprocal of the condition number of a general
real matrix A, in either the 1-norm or the infinity-norm, using
the LU factorization computed by DGETRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).

Parameters

NORM

NORM is CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= ’1’ or ’O’: 1-norm;
= ’I’: Infinity-norm.

N is INTEGER
The order of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
The factors L and U from the factorization A = P*L*U
as computed by DGETRF.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

ANORM

ANORM is DOUBLE PRECISION
If NORM = ’1’ or ’O’, the 1-norm of the original matrix A.
If NORM = ’I’, the infinity-norm of the original matrix A.

RCOND

RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).

WORK

WORK is DOUBLE PRECISION array, dimension (4*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgeequ (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) R, double precision,dimension( * ) C, double precision ROWCND, double precision COLCND, doubleprecision AMAX, integer INFO)

DGEEQU

Purpose:

DGEEQU computes row and column scalings intended to equilibrate an
M-by-N matrix A and reduce its condition number. R returns the row
scale factors and C the column scale factors, chosen to try to make
the largest element in each row and column of the matrix B with
elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

R(i) and C(j) are restricted to be between SMLNUM = smallest safe
number and BIGNUM = largest safe number. Use of these scaling
factors is not guaranteed to reduce the condition number of A but
works well in practice.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
The M-by-N matrix whose equilibration factors are
to be computed.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

R is DOUBLE PRECISION array, dimension (M)
If INFO = 0 or INFO > M, R contains the row scale factors
for A.

C is DOUBLE PRECISION array, dimension (N)
If INFO = 0, C contains the column scale factors for A.

ROWCND

ROWCND is DOUBLE PRECISION
If INFO = 0 or INFO > M, ROWCND contains the ratio of the
smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
AMAX is neither too large nor too small, it is not worth
scaling by R.

COLCND

COLCND is DOUBLE PRECISION
If INFO = 0, COLCND contains the ratio of the smallest
C(i) to the largest C(i). If COLCND >= 0.1, it is not
worth scaling by C.

AMAX

AMAX is DOUBLE PRECISION
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= M: the i-th row of A is exactly zero
> M: the (i-M)-th column of A is exactly zero

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgeequb (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) R, double precision,dimension( * ) C, double precision ROWCND, double precision COLCND, doubleprecision AMAX, integer INFO)

DGEEQUB

Purpose:

DGEEQUB computes row and column scalings intended to equilibrate an
M-by-N matrix A and reduce its condition number. R returns the row
scale factors and C the column scale factors, chosen to try to make
the largest element in each row and column of the matrix B with
elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
the radix.

R(i) and C(j) are restricted to be a power of the radix between
SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
of these scaling factors is not guaranteed to reduce the condition
number of A but works well in practice.

This routine differs from DGEEQU by restricting the scaling factors
to a power of the radix. Barring over- and underflow, scaling by
these factors introduces no additional rounding errors. However, the
scaled entries’ magnitudes are no longer approximately 1 but lie
between sqrt(radix) and 1/sqrt(radix).

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
The M-by-N matrix whose equilibration factors are
to be computed.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

R is DOUBLE PRECISION array, dimension (M)
If INFO = 0 or INFO > M, R contains the row scale factors
for A.

C is DOUBLE PRECISION array, dimension (N)
If INFO = 0, C contains the column scale factors for A.

ROWCND

COLCND

COLCND is DOUBLE PRECISION
If INFO = 0, COLCND contains the ratio of the smallest
C(i) to the largest C(i). If COLCND >= 0.1, it is not
worth scaling by C.

AMAX

AMAX is DOUBLE PRECISION
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgehd2 (integer N, integer ILO, integer IHI, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer INFO)

DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation: Q**T * A * Q = H .

Parameters

N is INTEGER
The order of the matrix A. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER

It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to DGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ILO <= IHI <= max(1,N).

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the orthogonal matrix Q as a product of elementary
reflectors. See Further Details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

TAU

TAU is DOUBLE PRECISION array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (N)

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:

on entry, on exit,

( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( a ) ( a )

where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).

subroutine dgehrd (integer N, integer ILO, integer IHI, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGEHRD

Purpose:

DGEHRD reduces a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation: Q**T * A * Q = H .

Parameters

N is INTEGER
The order of the matrix A. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the orthogonal matrix Q as a product of elementary
reflectors. See Further Details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

TAU

TAU is DOUBLE PRECISION array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
zero.

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,N).
For good performance, LWORK should generally be larger.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:

on entry, on exit,

where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).

This file is a slight modification of LAPACK-3.0’s DGEHRD
subroutine incorporating improvements proposed by Quintana-Orti and
Van de Geijn (2006). (See DLAHR2.)

subroutine dgelq2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer INFO)

DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

DGELQ2 computes an LQ factorization of a real m-by-n matrix A:

A = ( L 0 ) * Q

where:

Q is a n-by-n orthogonal matrix;
L is a lower-triangular m-by-m matrix;
0 is a m-by-(n-m) zero matrix, if m < n.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, the elements on and below the diagonal of the array
contain the m by min(m,n) lower trapezoidal matrix L (L is
lower triangular if m <= n); the elements above the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of elementary reflectors (see Further Details).

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

TAU

TAU is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (M)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(k) . . . H(2) H(1), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
and tau in TAU(i).

subroutine dgelqf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

DGELQF

Purpose:

DGELQF computes an LQ factorization of a real M-by-N matrix A:

A = ( L 0 ) * Q

where:

Q is a N-by-N orthogonal matrix;
L is a lower-triangular M-by-M matrix;
0 is a M-by-(N-M) zero matrix, if M < N.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and below the diagonal of the array
contain the m-by-min(m,n) lower trapezoidal matrix L (L is
lower triangular if m <= n); the elements above the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of elementary reflectors (see Further Details).

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

TAU

TAU is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is the
optimal blocksize.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(k) . . . H(2) H(1), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
and tau in TAU(i).

subroutine dgelqt (integer M, integer N, integer MB, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * )T, integer LDT, double precision, dimension( * ) WORK, integer INFO)

DGELQT

Purpose:

DGELQT computes a blocked LQ factorization of a real M-by-N matrix A
using the compact WY representation of Q.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

MB is INTEGER
The block size to be used in the blocked QR. MIN(M,N) >= MB >= 1.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and below the diagonal of the array
contain the M-by-MIN(M,N) lower trapezoidal matrix L (L is
lower triangular if M <= N); the elements above the diagonal
are the rows of V.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

T is DOUBLE PRECISION array, dimension (LDT,MIN(M,N))
The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See below
for further details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

WORK

WORK is DOUBLE PRECISION array, dimension (MB*N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix V stores the elementary reflectors H(i) in the i-th row
above the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 v1 v1 v1 v1 )
( 1 v2 v2 v2 )
( 1 v3 v3 )

T = (T1 T2 ... TB).

recursive subroutine dgelqt3 (integer M, integer N, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * )T, integer LDT, integer INFO)

DGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

DGELQT3 recursively computes a LQ factorization of a real M-by-N
matrix A, using the compact WY representation of Q.

Based on the algorithm of Elmroth and Gustavson,
IBM J. Res. Develop. Vol 44 No. 4 July 2000.

Parameters

M is INTEGER
The number of rows of the matrix A. M =< N.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the real M-by-N matrix A. On exit, the elements on and
below the diagonal contain the N-by-N lower triangular matrix L; the
elements above the diagonal are the rows of V. See below for
further details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

T is DOUBLE PRECISION array, dimension (LDT,N)
The N-by-N upper triangular factor of the block reflector.
The elements on and above the diagonal contain the block
reflector T; the elements below the diagonal are not used.
See below for further details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix V stores the elementary reflectors H(i) in the i-th row
above the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 v1 v1 v1 v1 )
( 1 v2 v2 v2 )
( 1 v3 v3 v3 )

where the vi’s represent the vectors which define H(i), which are returned
in the matrix A. The 1’s along the diagonal of V are not stored in A. The
block reflector H is then given by

H = I - V * T * V**T

where V**T is the transpose of V.

For details of the algorithm, see Elmroth and Gustavson (cited above).

subroutine dgemlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer MB, double precision, dimension( ldv, * ) V, integerLDV, double precision, dimension( ldt, * ) T, integer LDT, doubleprecision, dimension( ldc, * ) C, integer LDC, double precision, dimension(* ) WORK, integer INFO)

DGEMLQT

Purpose:

DGEMLQT overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q C C Q
TRANS = ’T’: Q**T C C Q**T

where Q is a real orthogonal matrix defined as the product of K
elementary reflectors:

Q = H(1) H(2) . . . H(K) = I - V T V**T

generated using the compact WY representation as returned by DGELQT.

Q is of order M if SIDE = ’L’ and of order N if SIDE = ’R’.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’C’: Transpose, apply Q**T.

M is INTEGER
The number of rows of the matrix C. M >= 0.

N is INTEGER
The number of columns of the matrix C. N >= 0.

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

MB is INTEGER
The block size used for the storage of T. K >= MB >= 1.
This must be the same value of MB used to generate T
in DGELQT.

V is DOUBLE PRECISION array, dimension
(LDV,M) if SIDE = ’L’,
(LDV,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
DGELQT in the first K rows of its array argument A.

LDV

LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,K).

T is DOUBLE PRECISION array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by DGELQT, stored as a MB-by-K matrix.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q C, Q**T C, C Q**T or C Q.

LDC

LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).

WORK

WORK is DOUBLE PRECISION array. The dimension of
WORK is N*MB if SIDE = ’L’, or M*MB if SIDE = ’R’.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgemqrt (character SIDE, character TRANS, integer M, integer N,integer K, integer NB, double precision, dimension( ldv, * ) V, integerLDV, double precision, dimension( ldt, * ) T, integer LDT, doubleprecision, dimension( ldc, * ) C, integer LDC, double precision, dimension(* ) WORK, integer INFO)

DGEMQRT

Purpose:

DGEMQRT overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q C C Q
TRANS = ’T’: Q**T C C Q**T

where Q is a real orthogonal matrix defined as the product of K
elementary reflectors:

Q = H(1) H(2) . . . H(K) = I - V T V**T

generated using the compact WY representation as returned by DGEQRT.

Q is of order M if SIDE = ’L’ and of order N if SIDE = ’R’.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’C’: Transpose, apply Q**T.

M is INTEGER
The number of rows of the matrix C. M >= 0.

N is INTEGER
The number of columns of the matrix C. N >= 0.

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

NB is INTEGER
The block size used for the storage of T. K >= NB >= 1.
This must be the same value of NB used to generate T
in DGEQRT.

V is DOUBLE PRECISION array, dimension (LDV,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
DGEQRT in the first K columns of its array argument A.

LDV

LDV is INTEGER
The leading dimension of the array V.
If SIDE = ’L’, LDA >= max(1,M);
if SIDE = ’R’, LDA >= max(1,N).

T is DOUBLE PRECISION array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by DGEQRT, stored as a NB-by-N matrix.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= NB.

C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q C, Q**T C, C Q**T or C Q.

LDC

LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).

WORK

WORK is DOUBLE PRECISION array. The dimension of
WORK is N*NB if SIDE = ’L’, or M*NB if SIDE = ’R’.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgeql2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer INFO)

DGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

DGEQL2 computes a QL factorization of a real m by n matrix A:
A = Q * L.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, if m >= n, the lower triangle of the subarray
A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
if m <= n, the elements on and below the (n-m)-th
superdiagonal contain the m by n lower trapezoidal matrix L;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of elementary reflectors
(see Further Details).

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

TAU

TAU is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(k) . . . H(2) H(1), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
A(1:m-k+i-1,n-k+i), and tau in TAU(i).

subroutine dgeqlf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

DGEQLF

Purpose:

DGEQLF computes a QL factorization of a real M-by-N matrix A:
A = Q * L.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
if m >= n, the lower triangle of the subarray
A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
if m <= n, the elements on and below the (n-m)-th
superdiagonal contain the M-by-N lower trapezoidal matrix L;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of elementary reflectors
(see Further Details).

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

TAU

TAU is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is the
optimal blocksize.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(k) . . . H(2) H(1), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
A(1:m-k+i-1,n-k+i), and tau in TAU(i).

subroutine dgeqp3 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, integer, dimension( ) JPVT, double precision,dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK,integer INFO)

DGEQP3

Purpose:

DGEQP3 computes a QR factorization with column pivoting of a
matrix A: A*P = Q*R using Level 3 BLAS.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the upper triangle of the array contains the
min(M,N)-by-N upper trapezoidal matrix R; the elements below
the diagonal, together with the array TAU, represent the
orthogonal matrix Q as a product of min(M,N) elementary
reflectors.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

JPVT

JPVT is INTEGER array, dimension (N)
On entry, if JPVT(J).ne.0, the J-th column of A is permuted
to the front of A*P (a leading column); if JPVT(J)=0,
the J-th column of A is a free column.
On exit, if JPVT(J)=K, then the J-th column of A*P was the
the K-th column of A.

TAU

TAU is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors.

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO=0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= 3*N+1.
For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
is the optimal blocksize.

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real/complex vector
with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
A(i+1:m,i), and tau in TAU(i).

Contributors:

G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA

subroutine dgeqr2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer INFO)

DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

DGEQR2 computes a QR factorization of a real m-by-n matrix A:

A = Q * ( R ),
( 0 )

where:

Q is a m-by-m orthogonal matrix;
R is an upper-triangular n-by-n matrix;
0 is a (m-n)-by-n zero matrix, if m > n.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(m,n) by n upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of elementary reflectors (see Further Details).

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

TAU

TAU is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).

subroutine dgeqr2p (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer INFO)

DGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.

Purpose:

DGEQR2P computes a QR factorization of a real m-by-n matrix A:

A = Q * ( R ),
( 0 )

where:

Q is a m-by-m orthogonal matrix;
R is an upper-triangular n-by-n matrix with nonnegative diagonal
entries;
0 is a (m-n)-by-n zero matrix, if m > n.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(m,n) by n upper trapezoidal matrix R (R is
upper triangular if m >= n). The diagonal entries of R are
nonnegative; the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of elementary reflectors (see Further Details).

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

TAU

TAU is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).

See Lapack Working Note 203 for details

subroutine dgeqrf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

DGEQRF

Purpose:

DGEQRF computes a QR factorization of a real M-by-N matrix A:

A = Q * ( R ),
( 0 )

where:

Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix;
0 is a (M-N)-by-N zero matrix, if M > N.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details).

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

TAU

TAU is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK.
LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
For optimum performance LWORK >= N*NB, where NB is
the optimal blocksize.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).

subroutine dgeqrfp (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

DGEQRFP

Purpose:

DGEQR2P computes a QR factorization of a real M-by-N matrix A:

A = Q * ( R ),
( 0 )

where:

Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix with nonnegative diagonal
entries;
0 is a (M-N)-by-N zero matrix, if M > N.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n). The diagonal entries of R
are nonnegative; the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details).

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

TAU

TAU is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is
the optimal blocksize.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).

See Lapack Working Note 203 for details

subroutine dgeqrt (integer M, integer N, integer NB, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * )T, integer LDT, double precision, dimension( * ) WORK, integer INFO)

DGEQRT

Purpose:

DGEQRT computes a blocked QR factorization of a real M-by-N matrix A
using the compact WY representation of Q.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

NB is INTEGER
The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

T is DOUBLE PRECISION array, dimension (LDT,MIN(M,N))
The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See below
for further details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= NB.

WORK

WORK is DOUBLE PRECISION array, dimension (NB*N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )

where the vi’s represent the vectors which define H(i), which are returned
in the matrix A. The 1’s along the diagonal of V are not stored in A.

Let K=MIN(M,N). The number of blocks is B = ceiling(K/NB), where each
block is of order NB except for the last block, which is of order
IB = K - (B-1)*NB. For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB
for the last block) T’s are stored in the NB-by-K matrix T as

T = (T1 T2 ... TB).

subroutine dgeqrt2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ldt, ) T, integer LDT,integer INFO)

DGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

DGEQRT2 computes a QR factorization of a real M-by-N matrix A,
using the compact WY representation of Q.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= N.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the real M-by-N matrix A. On exit, the elements on and
above the diagonal contain the N-by-N upper triangular matrix R; the
elements below the diagonal are the columns of V. See below for
further details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )

where the vi’s represent the vectors which define H(i), which are returned
in the matrix A. The 1’s along the diagonal of V are not stored in A. The
block reflector H is then given by

H = I - V * T * V**T

where V**T is the transpose of V.

recursive subroutine dgeqrt3 (integer M, integer N, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * )T, integer LDT, integer INFO)

DGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

DGEQRT3 recursively computes a QR factorization of a real M-by-N
matrix A, using the compact WY representation of Q.

Based on the algorithm of Elmroth and Gustavson,
IBM J. Res. Develop. Vol 44 No. 4 July 2000.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= N.

N is INTEGER
The number of columns of the matrix A. N >= 0.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )

where the vi’s represent the vectors which define H(i), which are returned
in the matrix A. The 1’s along the diagonal of V are not stored in A. The
block reflector H is then given by

H = I - V * T * V**T

where V**T is the transpose of V.

For details of the algorithm, see Elmroth and Gustavson (cited above).

subroutine dgerfs (character TRANS, integer N, integer NRHS, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision,dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * )X, integer LDX, double precision, dimension( * ) FERR, double precision,dimension( * ) BERR, double precision, dimension( * ) WORK, integer,dimension( * ) IWORK, integer INFO)

DGERFS

Purpose:

DGERFS improves the computed solution to a system of linear
equations and provides error bounds and backward error estimates for
the solution.

Parameters

TRANS

TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)

N is INTEGER
The order of the matrix A. N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
The original N-by-N matrix A.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

AF is DOUBLE PRECISION array, dimension (LDAF,N)
The factors L and U from the factorization A = P*L*U
as computed by DGETRF.

LDAF

LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
The pivot indices from DGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
The right hand side matrix B.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

X is DOUBLE PRECISION array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by DGETRS.
On exit, the improved solution matrix X.

LDX

LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).

FERR

FERR is DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.

BERR

BERR is DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Internal Parameters:

ITMAX is the maximum number of steps of iterative refinement.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgerfsx (character TRANS, character EQUED, integer N, integerNRHS, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * )IPIV, double precision, dimension( * ) R, double precision, dimension( * )C, double precision, dimension( ldb, * ) B, integer LDB, double precision,dimension( ldx , * ) X, integer LDX, double precision RCOND, doubleprecision, dimension( * ) BERR, integer N_ERR_BNDS, double precision,dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * )ERR_BNDS_COMP, integer NPARAMS, double precision, dimension( * ) PARAMS,double precision, dimension( * ) WORK, integer, dimension( * ) IWORK,integer INFO)

DGERFSX

Purpose:

DGERFSX improves the computed solution to a system of linear
equations and provides error bounds and backward error estimates
for the solution. In addition to normwise error bound, the code
provides maximum componentwise error bound if possible. See
comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
error bounds.

The original system of linear equations may have been equilibrated
before calling this routine, as described by arguments EQUED, R
and C below. In this case, the solution and error bounds returned
are for the original unequilibrated system.

Some optional parameters are bundled in the PARAMS array. These
settings determine how refinement is performed, but often the
defaults are acceptable. If the defaults are acceptable, users
can pass NPARAMS = 0 which prevents the source code from accessing
the PARAMS argument.

Parameters

TRANS

TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)

EQUED

EQUED is CHARACTER*1
Specifies the form of equilibration that was done to A
before calling this routine. This is needed to compute
the solution and error bounds correctly.
= ’N’: No equilibration
= ’R’: Row equilibration, i.e., A has been premultiplied by
diag(R).
= ’C’: Column equilibration, i.e., A has been postmultiplied
by diag(C).
= ’B’: Both row and column equilibration, i.e., A has been
replaced by diag(R) * A * diag(C).
The right hand side B has been changed accordingly.

N is INTEGER
The order of the matrix A. N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
The original N-by-N matrix A.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

AF is DOUBLE PRECISION array, dimension (LDAF,N)
The factors L and U from the factorization A = P*L*U
as computed by DGETRF.

LDAF

LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
The pivot indices from DGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).

R is DOUBLE PRECISION array, dimension (N)
The row scale factors for A. If EQUED = ’R’ or ’B’, A is
multiplied on the left by diag(R); if EQUED = ’N’ or ’C’, R
is not accessed.
If R is accessed, each element of R should be a power of the radix
to ensure a reliable solution and error estimates. Scaling by
powers of the radix does not cause rounding errors unless the
result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.

C is DOUBLE PRECISION array, dimension (N)
The column scale factors for A. If EQUED = ’C’ or ’B’, A is
multiplied on the right by diag(C); if EQUED = ’N’ or ’R’, C
is not accessed.
If C is accessed, each element of C should be a power of the radix
to ensure a reliable solution and error estimates. Scaling by
powers of the radix does not cause rounding errors unless the
result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
The right hand side matrix B.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

X is DOUBLE PRECISION array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by DGETRS.
On exit, the improved solution matrix X.

LDX

LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).

RCOND

RCOND is DOUBLE PRECISION
Reciprocal scaled condition number. This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done). If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision. Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned.

BERR

BERR is DOUBLE PRECISION array, dimension (NRHS)
Componentwise relative backward error. This is the
componentwise relative backward error of each solution vector X(j)
(i.e., the smallest relative change in any element of A or B that
makes X(j) an exact solution).

N_ERR_BNDS

N_ERR_BNDS is INTEGER
Number of error bounds to return for each right hand side
and each type (normwise or componentwise). See ERR_BNDS_NORM and
ERR_BNDS_COMP below.

ERR_BNDS_NORM

ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))

The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 ’Trust/don’t trust’ boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * dlamch(’Epsilon’).

err = 2 ’Guaranteed’ error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * dlamch(’Epsilon’). This error bound should only
be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number. Compared with the threshold
sqrt(n) * dlamch(’Epsilon’) to determine if the error
estimate is ’guaranteed’. These reciprocal condition
numbers are 1 / (norm(Zˆ{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra
cautions.

ERR_BNDS_COMP

ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))

The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 ’Trust/don’t trust’ boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * dlamch(’Epsilon’).

err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number. Compared with the threshold
sqrt(n) * dlamch(’Epsilon’) to determine if the error
estimate is ’guaranteed’. These reciprocal condition
numbers are 1 / (norm(Zˆ{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra
cautions.

NPARAMS

NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS. If <= 0, the
PARAMS array is never referenced and default values are used.

PARAMS

PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
Specifies algorithm parameters. If an entry is < 0.0, then
that entry will be filled with default value used for that
parameter. Only positions up to NPARAMS are accessed; defaults
are used for higher-numbered parameters.

PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
refinement or not.
Default: 1.0D+0
= 0.0: No refinement is performed, and no error bounds are
computed.
= 1.0: Use the double-precision refinement algorithm,
possibly with doubled-single computations if the
compilation environment does not support DOUBLE
PRECISION.
(other values are reserved for future use)

PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
computations allowed for refinement.
Default: 10
Aggressive: Set to 100 to permit convergence using approximate
factorizations or factorizations other than LU. If
the factorization uses a technique other than
Gaussian elimination, the guarantees in
err_bnds_norm and err_bnds_comp may no longer be
trustworthy.

PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
will attempt to find a solution with small componentwise
relative error in the double-precision algorithm. Positive
is true, 0.0 is false.
Default: 1.0 (attempt componentwise convergence)

WORK

WORK is DOUBLE PRECISION array, dimension (4*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

INFO is INTEGER
= 0: Successful exit. The solution to every right-hand side is
guaranteed.
< 0: If INFO = -i, the i-th argument had an illegal value
> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
has been completed, but the factor U is exactly singular, so
the solution and error bounds could not be computed. RCOND = 0
is returned.
= N+J: The solution corresponding to the Jth right-hand side is
not guaranteed. The solutions corresponding to other right-
hand sides K with K > J may not be guaranteed as well, but
only the first such right-hand side is reported. If a small
componentwise error is not requested (PARAMS(3) = 0.0) then
the Jth right-hand side is the first with a normwise error
bound that is not guaranteed (the smallest J such
that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
the Jth right-hand side is the first with either a normwise or
componentwise error bound that is not guaranteed (the smallest
J such that either ERR_BNDS_NORM(J,1) = 0.0 or
ERR_BNDS_COMP(J,1) = 0.0). See the definition of
ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
about all of the right-hand sides check ERR_BNDS_NORM or
ERR_BNDS_COMP.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgerq2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer INFO)

DGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

DGERQ2 computes an RQ factorization of a real m by n matrix A:
A = R * Q.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
if m >= n, the elements on and above the (m-n)-th subdiagonal
contain the m by n upper trapezoidal matrix R; the remaining
elements, with the array TAU, represent the orthogonal matrix
Q as a product of elementary reflectors (see Further
Details).

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

TAU

TAU is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (M)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and tau in TAU(i).

subroutine dgerqf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

DGERQF

Purpose:

DGERQF computes an RQ factorization of a real M-by-N matrix A:
A = R * Q.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
if m >= n, the elements on and above the (m-n)-th subdiagonal
contain the M-by-N upper trapezoidal matrix R;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of min(m,n) elementary
reflectors (see Further Details).

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

TAU

TAU is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK.
LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise.
For optimum performance LWORK >= M*NB, where NB is
the optimal blocksize.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and tau in TAU(i).

subroutine dgesvj (character1 JOBA, character1 JOBU, character1 JOBV,integer M, integer N, double precision, dimension( lda, ) A, integer LDA,double precision, dimension( n ) SVA, integer MV, double precision,dimension( ldv, * ) V, integer LDV, double precision, dimension( lwork )WORK, integer LWORK, integer INFO)

DGESVJ

Purpose:

DGESVJ computes the singular value decomposition (SVD) of a real
M-by-N matrix A, where M >= N. The SVD of A is written as
[++] [xx] [x0] [xx]
A = U * SIGMA * Vˆt, [++] = [xx] * [ox] * [xx]
[++] [xx]
where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
of SIGMA are the singular values of A. The columns of U and V are the
left and the right singular vectors of A, respectively.
DGESVJ can sometimes compute tiny singular values and their singular vectors much
more accurately than other SVD routines, see below under Further Details.

Parameters

JOBA

JOBA is CHARACTER*1
Specifies the structure of A.
= ’L’: The input matrix A is lower triangular;
= ’U’: The input matrix A is upper triangular;
= ’G’: The input matrix A is general M-by-N matrix, M >= N.

JOBU

JOBU is CHARACTER*1
Specifies whether to compute the left singular vectors
(columns of U):
= ’U’: The left singular vectors corresponding to the nonzero
singular values are computed and returned in the leading
columns of A. See more details in the description of A.
The default numerical orthogonality threshold is set to
approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH(’E’).
= ’C’: Analogous to JOBU=’U’, except that user can control the
level of numerical orthogonality of the computed left
singular vectors. TOL can be set to TOL = CTOL*EPS, where
CTOL is given on input in the array WORK.
No CTOL smaller than ONE is allowed. CTOL greater
than 1 / EPS is meaningless. The option ’C’
can be used if M*EPS is satisfactory orthogonality
of the computed left singular vectors, so CTOL=M could
save few sweeps of Jacobi rotations.
See the descriptions of A and WORK(1).
= ’N’: The matrix U is not computed. However, see the
description of A.

JOBV

JOBV is CHARACTER*1
Specifies whether to compute the right singular vectors, that
is, the matrix V:
= ’V’: the matrix V is computed and returned in the array V
= ’A’: the Jacobi rotations are applied to the MV-by-N
array V. In other words, the right singular vector
matrix V is not computed explicitly, instead it is
applied to an MV-by-N matrix initially stored in the
first MV rows of V.
= ’N’: the matrix V is not computed and the array V is not
referenced

M is INTEGER
The number of rows of the input matrix A. 1/DLAMCH(’E’) > M >= 0.

N is INTEGER
The number of columns of the input matrix A.
M >= N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit :
If JOBU = ’U’ .OR. JOBU = ’C’ :
If INFO = 0 :
RANKA orthonormal columns of U are returned in the
leading RANKA columns of the array A. Here RANKA <= N
is the number of computed singular values of A that are
above the underflow threshold DLAMCH(’S’). The singular
vectors corresponding to underflowed or zero singular
values are not computed. The value of RANKA is returned
in the array WORK as RANKA=NINT(WORK(2)). Also see the
descriptions of SVA and WORK. The computed columns of U
are mutually numerically orthogonal up to approximately
TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = ’C’),
see the description of JOBU.
If INFO > 0 :
the procedure DGESVJ did not converge in the given number
of iterations (sweeps). In that case, the computed
columns of U may not be orthogonal up to TOL. The output
U (stored in A), SIGMA (given by the computed singular
values in SVA(1:N)) and V is still a decomposition of the
input matrix A in the sense that the residual
||A-SCALE*U*SIGMA*VˆT||_2 / ||A||_2 is small.

If JOBU = ’N’ :
If INFO = 0 :
Note that the left singular vectors are ’for free’ in the
one-sided Jacobi SVD algorithm. However, if only the
singular values are needed, the level of numerical
orthogonality of U is not an issue and iterations are
stopped when the columns of the iterated matrix are
numerically orthogonal up to approximately M*EPS. Thus,
on exit, A contains the columns of U scaled with the
corresponding singular values.
If INFO > 0 :
the procedure DGESVJ did not converge in the given number
of iterations (sweeps).

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

SVA

SVA is DOUBLE PRECISION array, dimension (N)
On exit :
If INFO = 0 :
depending on the value SCALE = WORK(1), we have:
If SCALE = ONE :
SVA(1:N) contains the computed singular values of A.
During the computation SVA contains the Euclidean column
norms of the iterated matrices in the array A.
If SCALE .NE. ONE :
The singular values of A are SCALE*SVA(1:N), and this
factored representation is due to the fact that some of the
singular values of A might underflow or overflow.
If INFO > 0 :
the procedure DGESVJ did not converge in the given number of
iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.

MV is INTEGER
If JOBV = ’A’, then the product of Jacobi rotations in DGESVJ
is applied to the first MV rows of V. See the description of JOBV.

V is DOUBLE PRECISION array, dimension (LDV,N)
If JOBV = ’V’, then V contains on exit the N-by-N matrix of
the right singular vectors;
If JOBV = ’A’, then V contains the product of the computed right
singular vector matrix and the initial matrix in
the array V.
If JOBV = ’N’, then V is not referenced.

LDV

LDV is INTEGER
The leading dimension of the array V, LDV >= 1.
If JOBV = ’V’, then LDV >= max(1,N).
If JOBV = ’A’, then LDV >= max(1,MV) .

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)
On entry :
If JOBU = ’C’ :
WORK(1) = CTOL, where CTOL defines the threshold for convergence.
The process stops if all columns of A are mutually
orthogonal up to CTOL*EPS, EPS=DLAMCH(’E’).
It is required that CTOL >= ONE, i.e. it is not
allowed to force the routine to obtain orthogonality
below EPS.
On exit :
WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
are the computed singular values of A.
(See description of SVA().)
WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
singular values.
WORK(3) = NINT(WORK(3)) is the number of the computed singular
values that are larger than the underflow threshold.
WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
rotations needed for numerical convergence.
WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
This is useful information in cases when DGESVJ did
not converge, as it can be used to estimate whether
the output is still useful and for post festum analysis.
WORK(6) = the largest absolute value over all sines of the
Jacobi rotation angles in the last sweep. It can be
useful for a post festum analysis.

LWORK

LWORK is INTEGER
length of WORK, WORK >= MAX(6,M+N)

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, then the i-th argument had an illegal value
> 0: DGESVJ did not converge in the maximal allowed number (30)
of sweeps. The output may still be useful. See the
description of WORK.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
rotations. The rotations are implemented as fast scaled rotations of
Anda and Park [1]. In the case of underflow of the Jacobi angle, a
modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
column interchanges of de Rijk [2]. The relative accuracy of the computed
singular values and the accuracy of the computed singular vectors (in
angle metric) is as guaranteed by the theory of Demmel and Veselic [3].
The condition number that determines the accuracy in the full rank case
is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
spectral condition number. The best performance of this Jacobi SVD
procedure is achieved if used in an accelerated version of Drmac and
Veselic [5,6], and it is the kernel routine in the SIGMA library [7].
Some tuning parameters (marked with [TP]) are available for the
implementer.
The computational range for the nonzero singular values is the machine
number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
denormalized singular values can be computed with the corresponding
gradual loss of accurate digits.

Contributors:

============

Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

References:

[1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
[2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
singular value decomposition on a vector computer.
SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
[3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
[4] Z. Drmac: Implementation of Jacobi rotations for accurate singular
value computation in floating point arithmetic.
SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
[5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
LAPACK Working note 169.
[6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
LAPACK Working note 170.
[7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
QSVD, (H,K)-SVD computations.
Department of Mathematics, University of Zagreb, 2008.

Bugs, examples and comments:

===========================
Please report all bugs and send interesting test examples and comments to
drmac@math.hr. Thank you.

subroutine dgetf2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, integer INFO)

DGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).

Purpose:

DGETF2 computes an LU factorization of a general m-by-n matrix A
using partial pivoting with row interchanges.

The factorization has the form
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).

This is the right-looking Level 2 BLAS version of the algorithm.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the m by n matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

IPIV

IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, U(k,k) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgetrf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, integer INFO)

DGETRF DGETRF VARIANT: iterative version of Sivan Toledo’s recursive LU algorithm

DGETRF VARIANT: left-looking Level 3 BLAS version of the algorithm.

Purpose:

DGETRF computes an LU factorization of a general M-by-N matrix A
using partial pivoting with row interchanges.

This is the right-looking Level 3 BLAS version of the algorithm.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

IPIV

IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Purpose:

DGETRF computes an LU factorization of a general M-by-N matrix A
using partial pivoting with row interchanges.

This is the left-looking Level 3 BLAS version of the algorithm.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

IPIV

IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

Purpose:

DGETRF computes an LU factorization of a general M-by-N matrix A
using partial pivoting with row interchanges.

This code implements an iterative version of Sivan Toledo’s recursive
LU algorithm[1]. For square matrices, this iterative versions should
be within a factor of two of the optimum number of memory transfers.

The pattern is as follows, with the large blocks of U being updated
in one call to DTRSM, and the dotted lines denoting sections that
have had all pending permutations applied:

1 2 3 4 5 6 7 8
+-+-+---+-------+------
| |1| | |
|.+-+ 2 | |
| | | | |
|.|.+-+-+ 4 |
| | | |1| |
| | |.+-+ |
| | | | | |
|.|.|.|.+-+-+---+ 8
| | | | | |1| |
| | | | |.+-+ 2 |
| | | | | | | |
| | | | |.|.+-+-+
| | | | | | | |1|
| | | | | | |.+-+
| | | | | | | | |
|.|.|.|.|.|.|.|.+-----
| | | | | | | | |

The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
the binary expansion of the current column. Each Schur update is
applied as soon as the necessary portion of U is available.

[1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
1065-1081. http://dx.doi.org/10.1137/S0895479896297744

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

IPIV

IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

recursive subroutine dgetrf2 (integer M, integer N, double precision,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integerINFO)

DGETRF2

Purpose:

DGETRF2 computes an LU factorization of a general M-by-N matrix A
using partial pivoting with row interchanges.

This is the recursive version of the algorithm. It divides
the matrix into four submatrices:

[ A11 | A12 ] where A11 is n1 by n1 and A22 is n2 by n2
A = [ -----|----- ] with n1 = min(m,n)/2
[ A21 | A22 ] n2 = n-n1

[ A11 ]
The subroutine calls itself to factor [ --- ],
[ A12 ]
[ A12 ]
do the swaps on [ --- ], solve A12, update A22,
[ A22 ]

then calls itself to factor A22 and do the swaps on A21.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

IPIV

IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgetri (integer N, double precision, dimension( lda, * ) A,integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * )WORK, integer LWORK, integer INFO)

DGETRI

Purpose:

DGETRI computes the inverse of a matrix using the LU factorization
computed by DGETRF.

This method inverts U and then computes inv(A) by solving the system
inv(A)*L = inv(U) for inv(A).

Parameters

N is INTEGER
The order of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the factors L and U from the factorization
A = P*L*U as computed by DGETRF.
On exit, if INFO = 0, the inverse of the original matrix A.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
The pivot indices from DGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO=0, then WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimal performance LWORK >= N*NB, where NB is
the optimal blocksize returned by ILAENV.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero; the matrix is
singular and its inverse could not be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgetrs (character TRANS, integer N, integer NRHS, doubleprecision, dimension( lda, * ) A, integer LDA, integer, dimension( * )IPIV, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

DGETRS

Purpose:

DGETRS solves a system of linear equations
A * X = B or A**T * X = B
with a general N-by-N matrix A using the LU factorization computed
by DGETRF.

Parameters

TRANS

TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T* X = B (Transpose)
= ’C’: A**T* X = B (Conjugate transpose = Transpose)

N is INTEGER
The order of the matrix A. N >= 0.

NRHS

NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
The factors L and U from the factorization A = P*L*U
as computed by DGETRF.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
The pivot indices from DGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dhgeqz (character JOB, character COMPQ, character COMPZ, integerN, integer ILO, integer IHI, double precision, dimension( ldh, * ) H,integer LDH, double precision, dimension( ldt, * ) T, integer LDT, doubleprecision, dimension( * ) ALPHAR, double precision, dimension( * ) ALPHAI,double precision, dimension( * ) BETA, double precision, dimension( ldq, ) Q, integer LDQ, double precision, dimension( ldz, ) Z, integer LDZ,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DHGEQZ

Purpose:

DHGEQZ computes the eigenvalues of a real matrix pair (H,T),
where H is an upper Hessenberg matrix and T is upper triangular,
using the double-shift QZ method.
Matrix pairs of this type are produced by the reduction to
generalized upper Hessenberg form of a real matrix pair (A,B):

A = Q1*H*Z1**T, B = Q1*T*Z1**T,

as computed by DGGHRD.

If JOB=’S’, then the Hessenberg-triangular pair (H,T) is
also reduced to generalized Schur form,

H = Q*S*Z**T, T = Q*P*Z**T,

where Q and Z are orthogonal matrices, P is an upper triangular
matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
diagonal blocks.

The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
(H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
eigenvalues.

Additionally, the 2-by-2 upper triangular diagonal blocks of P
corresponding to 2-by-2 blocks of S are reduced to positive diagonal
form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
P(j,j) > 0, and P(j+1,j+1) > 0.

Optionally, the orthogonal matrix Q from the generalized Schur
factorization may be postmultiplied into an input matrix Q1, and the
orthogonal matrix Z may be postmultiplied into an input matrix Z1.
If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
the matrix pair (A,B) to generalized upper Hessenberg form, then the
output matrices Q1*Q and Z1*Z are the orthogonal factors from the
generalized Schur factorization of (A,B):

A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.

To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
complex and beta real.
If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
generalized nonsymmetric eigenvalue problem (GNEP)
A*x = lambda*B*x
and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
alternate form of the GNEP
mu*A*y = B*y.
Real eigenvalues can be read directly from the generalized Schur
form:
alpha = S(i,i), beta = P(i,i).

Ref: C.B. Moler & G.W. Stewart, ’An Algorithm for Generalized Matrix
Eigenvalue Problems’, SIAM J. Numer. Anal., 10(1973),
pp. 241--256.

Parameters

JOB

JOB is CHARACTER*1
= ’E’: Compute eigenvalues only;
= ’S’: Compute eigenvalues and the Schur form.

COMPQ

COMPQ is CHARACTER*1
= ’N’: Left Schur vectors (Q) are not computed;
= ’I’: Q is initialized to the unit matrix and the matrix Q
of left Schur vectors of (H,T) is returned;
= ’V’: Q must contain an orthogonal matrix Q1 on entry and
the product Q1*Q is returned.

COMPZ

COMPZ is CHARACTER*1
= ’N’: Right Schur vectors (Z) are not computed;
= ’I’: Z is initialized to the unit matrix and the matrix Z
of right Schur vectors of (H,T) is returned;
= ’V’: Z must contain an orthogonal matrix Z1 on entry and
the product Z1*Z is returned.

N is INTEGER
The order of the matrices H, T, Q, and Z. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER
ILO and IHI mark the rows and columns of H which are in
Hessenberg form. It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.

H is DOUBLE PRECISION array, dimension (LDH, N)
On entry, the N-by-N upper Hessenberg matrix H.
On exit, if JOB = ’S’, H contains the upper quasi-triangular
matrix S from the generalized Schur factorization.
If JOB = ’E’, the diagonal blocks of H match those of S, but
the rest of H is unspecified.

LDH

LDH is INTEGER
The leading dimension of the array H. LDH >= max( 1, N ).

T is DOUBLE PRECISION array, dimension (LDT, N)
On entry, the N-by-N upper triangular matrix T.
On exit, if JOB = ’S’, T contains the upper triangular
matrix P from the generalized Schur factorization;
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
are reduced to positive diagonal form, i.e., if H(j+1,j) is
non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
T(j+1,j+1) > 0.
If JOB = ’E’, the diagonal blocks of T match those of P, but
the rest of T is unspecified.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= max( 1, N ).

ALPHAR

ALPHAR is DOUBLE PRECISION array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of GNEP.

ALPHAI

ALPHAI is DOUBLE PRECISION array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of GNEP.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

BETA

BETA is DOUBLE PRECISION array, dimension (N)
The scalars beta that define the eigenvalues of GNEP.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the matrix
pair (A,B), in one of the forms lambda = alpha/beta or
mu = beta/alpha. Since either lambda or mu may overflow,
they should not, in general, be computed.

Q is DOUBLE PRECISION array, dimension (LDQ, N)
On entry, if COMPQ = ’V’, the orthogonal matrix Q1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = ’I’, the orthogonal matrix of left Schur
vectors of (H,T), and if COMPQ = ’V’, the orthogonal matrix
of left Schur vectors of (A,B).
Not referenced if COMPQ = ’N’.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ=’V’ or ’I’, then LDQ >= N.

Z is DOUBLE PRECISION array, dimension (LDZ, N)
On entry, if COMPZ = ’V’, the orthogonal matrix Z1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = ’I’, the orthogonal matrix of
right Schur vectors of (H,T), and if COMPZ = ’V’, the
orthogonal matrix of right Schur vectors of (A,B).
Not referenced if COMPZ = ’N’.

LDZ

LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ=’V’ or ’I’, then LDZ >= N.

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (H,T) is not
in Schur form, but ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO+1,...,N should be correct.
= N+1,...,2*N: the shift calculation failed. (H,T) is not
in Schur form, but ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO-N+1,...,N should be correct.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

Iteration counters:

JITER -- counts iterations.
IITER -- counts iterations run since ILAST was last
changed. This is therefore reset only when a 1-by-1 or
2-by-2 block deflates off the bottom.

subroutine dla_geamv (integer TRANS, integer M, integer N, double precisionALPHA, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) X, integer INCX, double precision BETA, doubleprecision, dimension( * ) Y, integer INCY)

DLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds.

Purpose:

DLA_GEAMV performs one of the matrix-vector operations

y := alpha*abs(A)*abs(x) + beta*abs(y),
or y := alpha*abs(A)**T*abs(x) + beta*abs(y),

where alpha and beta are scalars, x and y are vectors and A is an
m by n matrix.

This function is primarily used in calculating error bounds.
To protect against underflow during evaluation, components in
the resulting vector are perturbed away from zero by (N+1)
times the underflow threshold. To prevent unnecessarily large
errors for block-structure embedded in general matrices,
’symbolically’ zero components are not perturbed. A zero
entry is considered ’symbolic’ if all multiplications involved
in computing that entry have at least one zero multiplicand.

Parameters

TRANS

TRANS is INTEGER
On entry, TRANS specifies the operation to be performed as
follows:

BLAS_NO_TRANS y := alpha*abs(A)*abs(x) + beta*abs(y)
BLAS_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)
BLAS_CONJ_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)

Unchanged on exit.

M is INTEGER
On entry, M specifies the number of rows of the matrix A.
M must be at least zero.
Unchanged on exit.

N is INTEGER
On entry, N specifies the number of columns of the matrix A.
N must be at least zero.
Unchanged on exit.

ALPHA

ALPHA is DOUBLE PRECISION
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.

A is DOUBLE PRECISION array, dimension ( LDA, n )
Before entry, the leading m by n part of the array A must
contain the matrix of coefficients.
Unchanged on exit.

LDA

LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
max( 1, m ).
Unchanged on exit.

X is DOUBLE PRECISION array, dimension
( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = ’N’ or ’n’
and at least
( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
Before entry, the incremented array X must contain the
vector x.
Unchanged on exit.

INCX

INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.

BETA

BETA is DOUBLE PRECISION
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.
Unchanged on exit.

Y is DOUBLE PRECISION array,
dimension at least
( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = ’N’ or ’n’
and at least
( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
Before entry with BETA non-zero, the incremented array Y
must contain the vector y. On exit, Y is overwritten by the
updated vector y.

INCY

INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.
Unchanged on exit.

Level 2 Blas routine.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

double precision function dla_gercond (character TRANS, integer N, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, integer CMODE,double precision, dimension( * ) C, integer INFO, double precision,dimension( * ) WORK, integer, dimension( * ) IWORK)

DLA_GERCOND estimates the Skeel condition number for a general matrix.

Purpose:

DLA_GERCOND estimates the Skeel condition number of op(A) * op2(C)
where op2 is determined by CMODE as follows
CMODE = 1 op2(C) = C
CMODE = 0 op2(C) = I
CMODE = -1 op2(C) = inv(C)
The Skeel condition number cond(A) = norminf( |inv(A)||A| )
is computed by computing scaling factors R such that
diag(R)*A*op2(C) is row equilibrated and computing the standard
infinity-norm condition number.

Parameters

TRANS

TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate Transpose = Transpose)

N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

AF is DOUBLE PRECISION array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by DGETRF.

LDAF

LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by DGETRF; row i of the matrix was interchanged
with row IPIV(i).

CMODE

CMODE is INTEGER
Determines op2(C) in the formula op(A) * op2(C) as follows:
CMODE = 1 op2(C) = C
CMODE = 0 op2(C) = I
CMODE = -1 op2(C) = inv(C)

C is DOUBLE PRECISION array, dimension (N)
The vector C in the formula op(A) * op2(C).

INFO

INFO is INTEGER
= 0: Successful exit.
i > 0: The ith argument is invalid.

WORK

WORK is DOUBLE PRECISION array, dimension (3*N).
Workspace.

IWORK

IWORK is INTEGER array, dimension (N).
Workspace.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dla_gerfsx_extended (integer PREC_TYPE, integer TRANS_TYPE,integer N, integer NRHS, double precision, dimension( lda, * ) A, integerLDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer,dimension( * ) IPIV, logical COLEQU, double precision, dimension( * ) C,double precision, dimension( ldb, * ) B, integer LDB, double precision,dimension( ldy, * ) Y, integer LDY, double precision, dimension( * )BERR_OUT, integer N_NORMS, double precision, dimension( nrhs, * ) ERRS_N,double precision, dimension( nrhs, * ) ERRS_C, double precision, dimension(* ) RES, double precision, dimension( * ) AYB, double precision, dimension(* ) DY, double precision, dimension( * ) Y_TAIL, double precision RCOND,integer ITHRESH, double precision RTHRESH, double precision DZ_UB, logicalIGNORE_CWISE, integer INFO)

DLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:

DLA_GERFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extra-precise iterative refinement
and provides error bounds and backward error estimates for the solution.
This subroutine is called by DGERFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERRS_N
and ERRS_C for details of the error bounds. Note that this
subroutine is only responsible for setting the second fields of
ERRS_N and ERRS_C.

Parameters

PREC_TYPE

PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement.
The value is defined by ILAPREC(P) where P is a CHARACTER and P
= ’S’: Single
= ’D’: Double
= ’I’: Indigenous
= ’X’ or ’E’: Extra

TRANS_TYPE

TRANS_TYPE is INTEGER
Specifies the transposition operation on A.
The value is defined by ILATRANS(T) where T is a CHARACTER and T
= ’N’: No transpose
= ’T’: Transpose
= ’C’: Conjugate transpose

N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.

NRHS

NRHS is INTEGER
The number of right-hand-sides, i.e., the number of columns of the
matrix B.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

AF is DOUBLE PRECISION array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by DGETRF.

LDAF

LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by DGETRF; row i of the matrix was interchanged
with row IPIV(i).

COLEQU

COLEQU is LOGICAL
If .TRUE. then column equilibration was done to A before calling
this routine. This is needed to compute the solution and error
bounds correctly.

C is DOUBLE PRECISION array, dimension (N)
The column scale factors for A. If COLEQU = .FALSE., C
is not accessed. If C is input, each element of C should be a power
of the radix to ensure a reliable solution and error estimates.
Scaling by powers of the radix does not cause rounding errors unless
the result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
The right-hand-side matrix B.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

Y is DOUBLE PRECISION array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by DGETRS.
On exit, the improved solution matrix Y.

LDY

LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).

BERR_OUT

BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
On exit, BERR_OUT(j) contains the componentwise relative backward
error for right-hand-side j from the formula
max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. This is computed by DLA_LIN_BERR.

N_NORMS

N_NORMS is INTEGER
Determines which error bounds to return (see ERRS_N
and ERRS_C).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.

ERRS_N

ERRS_N is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))

The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.

The first index in ERRS_N(i,:) corresponds to the ith
right-hand side.

The second index in ERRS_N(:,err) contains the following
three fields:
err = 1 ’Trust/don’t trust’ boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch(’Epsilon’).

err = 2 ’Guaranteed’ error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch(’Epsilon’). This error bound should only
be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch(’Epsilon’) to determine if the error
estimate is ’guaranteed’. These reciprocal condition
numbers are 1 / (norm(Zˆ{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1.

This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.

ERRS_C

ERRS_C is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))

The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERRS_C is not accessed. If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned.

The first index in ERRS_C(i,:) corresponds to the ith
right-hand side.

The second index in ERRS_C(:,err) contains the following
three fields:
err = 1 ’Trust/don’t trust’ boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch(’Epsilon’).

err = 2 ’Guaranteed’ error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch(’Epsilon’). This error bound should only
be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch(’Epsilon’) to determine if the error
estimate is ’guaranteed’. These reciprocal condition
numbers are 1 / (norm(Zˆ{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.

This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.

RES

RES is DOUBLE PRECISION array, dimension (N)
Workspace to hold the intermediate residual.

AYB

AYB is DOUBLE PRECISION array, dimension (N)
Workspace. This can be the same workspace passed for Y_TAIL.

DY is DOUBLE PRECISION array, dimension (N)
Workspace to hold the intermediate solution.

Y_TAIL

Y_TAIL is DOUBLE PRECISION array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution.

RCOND

ITHRESH

ITHRESH is INTEGER
The maximum number of residual computations allowed for
refinement. The default is 10. For ’aggressive’ set to 100 to
permit convergence using approximate factorizations or
factorizations other than LU. If the factorization uses a
technique other than Gaussian elimination, the guarantees in
ERRS_N and ERRS_C may no longer be trustworthy.

RTHRESH

RTHRESH is DOUBLE PRECISION
Determines when to stop refinement if the error estimate stops
decreasing. Refinement will stop when the next solution no longer
satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
default value is 0.5. For ’aggressive’ set to 0.9 to permit
convergence on extremely ill-conditioned matrices. See LAWN 165
for more details.

DZ_UB

DZ_UB is DOUBLE PRECISION
Determines when to start considering componentwise convergence.
Componentwise convergence is only considered after each component
of the solution Y is stable, which we define as the relative
change in each component being less than DZ_UB. The default value
is 0.25, requiring the first bit to be stable. See LAWN 165 for
more details.

IGNORE_CWISE

IGNORE_CWISE is LOGICAL
If .TRUE. then ignore componentwise convergence. Default value
is .FALSE..

INFO

INFO is INTEGER
= 0: Successful exit.
< 0: if INFO = -i, the ith argument to DGETRS had an illegal
value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

double precision function dla_gerpvgrw (integer N, integer NCOLS, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldaf, * ) AF, integer LDAF)

DLA_GERPVGRW

Purpose:

DLA_GERPVGRW computes the reciprocal pivot growth factor
norm(A)/norm(U). The ’max absolute element’ norm is used. If this is
much less than 1, the stability of the LU factorization of the
(equilibrated) matrix A could be poor. This also means that the
solution X, estimated condition numbers, and error bounds could be
unreliable.

Parameters

N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.

NCOLS

NCOLS is INTEGER
The number of columns of the matrix A. NCOLS >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

AF is DOUBLE PRECISION array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by DGETRF.

LDAF

LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dlaorhr_col_getrfnp (integer M, integer N, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D,integer INFO)

DLAORHR_COL_GETRFNP

Purpose:

DLAORHR_COL_GETRFNP computes the modified LU factorization without
pivoting of a real general M-by-N matrix A. The factorization has
the form:

A - S = L * U,

where:
S is a m-by-n diagonal sign matrix with the diagonal D, so that
D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
i-1 steps of Gaussian elimination. This means that the diagonal
element at each step of ’modified’ Gaussian elimination is
at least one in absolute value (so that division-by-zero not
not possible during the division by the diagonal element);

L is a M-by-N lower triangular matrix with unit diagonal elements
(lower trapezoidal if M > N);

and U is a M-by-N upper triangular matrix
(upper trapezoidal if M < N).

This routine is an auxiliary routine used in the Householder
reconstruction routine DORHR_COL. In DORHR_COL, this routine is
applied to an M-by-N matrix A with orthonormal columns, where each
element is bounded by one in absolute value. With the choice of
the matrix S above, one can show that the diagonal element at each
step of Gaussian elimination is the largest (in absolute value) in
the column on or below the diagonal, so that no pivoting is required
for numerical stability [1].

For more details on the Householder reconstruction algorithm,
including the modified LU factorization, see [1].

This is the blocked right-looking version of the algorithm,
calling Level 3 BLAS to update the submatrix. To factorize a block,
this routine calls the recursive routine DLAORHR_COL_GETRFNP2.

[1] ’Reconstructing Householder vectors from tall-skinny QR’,
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A-S=L*U; the unit diagonal elements of L are not stored.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

D is DOUBLE PRECISION array, dimension min(M,N)
The diagonal elements of the diagonal M-by-N sign matrix S,
D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
be only plus or minus one.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

recursive subroutine dlaorhr_col_getrfnp2 (integer M, integer N, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(* ) D, integer INFO)

DLAORHR_COL_GETRFNP2

Purpose:

DLAORHR_COL_GETRFNP2 computes the modified LU factorization without
pivoting of a real general M-by-N matrix A. The factorization has
the form:

A - S = L * U,

where:
S is a m-by-n diagonal sign matrix with the diagonal D, so that
D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
i-1 steps of Gaussian elimination. This means that the diagonal
element at each step of ’modified’ Gaussian elimination is at
least one in absolute value (so that division-by-zero not
possible during the division by the diagonal element);

L is a M-by-N lower triangular matrix with unit diagonal elements
(lower trapezoidal if M > N);

and U is a M-by-N upper triangular matrix
(upper trapezoidal if M < N).

For more details on the Householder reconstruction algorithm,
including the modified LU factorization, see [1].

This is the recursive version of the LU factorization algorithm.
Denote A - S by B. The algorithm divides the matrix B into four
submatrices:

[ B11 | B12 ] where B11 is n1 by n1,
B = [ -----|----- ] B21 is (m-n1) by n1,
[ B21 | B22 ] B12 is n1 by n2,
B22 is (m-n1) by n2,
with n1 = min(m,n)/2, n2 = n-n1.

The subroutine calls itself to factor B11, solves for B21,
solves for B12, updates B22, then calls itself to factor B22.

For more details on the recursive LU algorithm, see [2].

DLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked
routine DLAORHR_COL_GETRFNP, which uses blocked code calling
Level 3 BLAS to update the submatrix. However, DLAORHR_COL_GETRFNP2
is self-sufficient and can be used without DLAORHR_COL_GETRFNP.

[1] ’Reconstructing Householder vectors from tall-skinny QR’,
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.

[2] ’Recursion leads to automatic variable blocking for dense linear
algebra algorithms’, F. Gustavson, IBM J. of Res. and Dev.,
vol. 41, no. 6, pp. 737-755, 1997.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A-S=L*U; the unit diagonal elements of L are not stored.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

D is DOUBLE PRECISION array, dimension min(M,N)
The diagonal elements of the diagonal M-by-N sign matrix S,
D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
be only plus or minus one.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

recursive subroutine dlaqz0 (character, intent(in) WANTS, character,intent(in) WANTQ, character, intent(in) WANTZ, integer, intent(in) N,integer, intent(in) ILO, integer, intent(in) IHI, double precision,dimension( lda, * ), intent(inout) A, integer, intent(in) LDA, doubleprecision, dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB,double precision, dimension( * ), intent(inout) ALPHAR, double precision,dimension( * ), intent(inout) ALPHAI, double precision, dimension( * ),intent(inout) BETA, double precision, dimension( ldq, * ), intent(inout) Q,integer, intent(in) LDQ, double precision, dimension( ldz, * ),intent(inout) Z, integer, intent(in) LDZ, double precision, dimension( * ),intent(inout) WORK, integer, intent(in) LWORK, integer, intent(in) REC,integer, intent(out) INFO)

DLAQZ0

Purpose:

DLAQZ0 computes the eigenvalues of a real matrix pair (H,T),
where H is an upper Hessenberg matrix and T is upper triangular,
using the double-shift QZ method.
Matrix pairs of this type are produced by the reduction to
generalized upper Hessenberg form of a real matrix pair (A,B):

A = Q1*H*Z1**T, B = Q1*T*Z1**T,

as computed by DGGHRD.

If JOB=’S’, then the Hessenberg-triangular pair (H,T) is
also reduced to generalized Schur form,

H = Q*S*Z**T, T = Q*P*Z**T,

where Q and Z are orthogonal matrices, P is an upper triangular
matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
diagonal blocks.

The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
(H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
eigenvalues.

A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.

Ref: C.B. Moler & G.W. Stewart, ’An Algorithm for Generalized Matrix
Eigenvalue Problems’, SIAM J. Numer. Anal., 10(1973),
pp. 241--256.

Ref: B. Kagstrom, D. Kressner, ’Multishift Variants of the QZ
Algorithm with Aggressive Early Deflation’, SIAM J. Numer.
Anal., 29(2006), pp. 199--227.

Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril ’A multishift,
multipole rational QZ method with agressive early deflation’

Parameters

WANTS

WANTS is CHARACTER*1
= ’E’: Compute eigenvalues only;
= ’S’: Compute eigenvalues and the Schur form.

WANTQ

WANTQ is CHARACTER*1
= ’N’: Left Schur vectors (Q) are not computed;
= ’I’: Q is initialized to the unit matrix and the matrix Q
of left Schur vectors of (A,B) is returned;
= ’V’: Q must contain an orthogonal matrix Q1 on entry and
the product Q1*Q is returned.

WANTZ

WANTZ is CHARACTER*1
= ’N’: Right Schur vectors (Z) are not computed;
= ’I’: Z is initialized to the unit matrix and the matrix Z
of right Schur vectors of (A,B) is returned;
= ’V’: Z must contain an orthogonal matrix Z1 on entry and
the product Z1*Z is returned.

N is INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER
ILO and IHI mark the rows and columns of A which are in
Hessenberg form. It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.

A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the N-by-N upper Hessenberg matrix A.
On exit, if JOB = ’S’, A contains the upper quasi-triangular
matrix S from the generalized Schur factorization.
If JOB = ’E’, the diagonal blocks of A match those of S, but
the rest of A is unspecified.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).

B is DOUBLE PRECISION array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
On exit, if JOB = ’S’, B contains the upper triangular
matrix P from the generalized Schur factorization;
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
are reduced to positive diagonal form, i.e., if A(j+1,j) is
non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and
B(j+1,j+1) > 0.
If JOB = ’E’, the diagonal blocks of B match those of P, but
the rest of B is unspecified.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).

ALPHAR

ALPHAR is DOUBLE PRECISION array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of GNEP.

ALPHAI

BETA

Q is DOUBLE PRECISION array, dimension (LDQ, N)
On entry, if COMPQ = ’V’, the orthogonal matrix Q1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = ’I’, the orthogonal matrix of left Schur
vectors of (A,B), and if COMPQ = ’V’, the orthogonal matrix
of left Schur vectors of (A,B).
Not referenced if COMPQ = ’N’.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ=’V’ or ’I’, then LDQ >= N.

LDZ

LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ=’V’ or ’I’, then LDZ >= N.

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).

REC

REC is INTEGER
REC indicates the current recursion level. Should be set
to 0 on first call.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (A,B) is not
in Schur form, but ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO+1,...,N should be correct.

Author

Thijs Steel, KU Leuven

Date

May 2020

subroutine dlaqz1 (double precision, dimension( lda, * ), intent(in) A,integer, intent(in) LDA, double precision, dimension( ldb, * ), intent(in)B, integer, intent(in) LDB, double precision, intent(in) SR1, doubleprecision, intent(in) SR2, double precision, intent(in) SI, doubleprecision, intent(in) BETA1, double precision, intent(in) BETA2, doubleprecision, dimension( * ), intent(out) V)

DLAQZ1

Purpose:

Given a 3-by-3 matrix pencil (A,B), DLAQZ1 sets v to a
scalar multiple of the first column of the product

(*) K = (A - (beta2*sr2 - i*si)*B)*Bˆ(-1)*(beta1*A - (sr2 + i*si2)*B)*Bˆ(-1).

It is assumed that either

1) sr1 = sr2
or
2) si = 0.

This is useful for starting double implicit shift bulges
in the QZ algorithm.

Parameters

A is DOUBLE PRECISION array, dimension (LDA,N)
The 3-by-3 matrix A in (*).

LDA

LDA is INTEGER
The leading dimension of A as declared in
the calling procedure.

B is DOUBLE PRECISION array, dimension (LDB,N)
The 3-by-3 matrix B in (*).

LDB

LDB is INTEGER
The leading dimension of B as declared in
the calling procedure.

SR1

SR1 is DOUBLE PRECISION

SR2

SR2 is DOUBLE PRECISION

SI is DOUBLE PRECISION

BETA1

BETA1 is DOUBLE PRECISION

BETA2

BETA2 is DOUBLE PRECISION

V is DOUBLE PRECISION array, dimension (N)
A scalar multiple of the first column of the
matrix K in (*).

Author

Thijs Steel, KU Leuven

Date

May 2020

subroutine dlaqz2 (logical, intent(in) ILQ, logical, intent(in) ILZ, integer,intent(in) K, integer, intent(in) ISTARTM, integer, intent(in) ISTOPM,integer, intent(in) IHI, double precision, dimension( lda, * ) A, integer,intent(in) LDA, double precision, dimension( ldb, * ) B, integer,intent(in) LDB, integer, intent(in) NQ, integer, intent(in) QSTART, doubleprecision, dimension( ldq, * ) Q, integer, intent(in) LDQ, integer,intent(in) NZ, integer, intent(in) ZSTART, double precision, dimension(ldz, * ) Z, integer, intent(in) LDZ)

DLAQZ2

Purpose:

DLAQZ2 chases a 2x2 shift bulge in a matrix pencil down a single position

Parameters

ILQ

ILQ is LOGICAL
Determines whether or not to update the matrix Q

ILZ

ILZ is LOGICAL
Determines whether or not to update the matrix Z

K is INTEGER
Index indicating the position of the bulge.
On entry, the bulge is located in
(A(k+1:k+2,k:k+1),B(k+1:k+2,k:k+1)).
On exit, the bulge is located in
(A(k+2:k+3,k+1:k+2),B(k+2:k+3,k+1:k+2)).

ISTARTM

ISTARTM is INTEGER

ISTOPM

ISTOPM is INTEGER
Updates to (A,B) are restricted to
(istartm:k+3,k:istopm). It is assumed
without checking that istartm <= k+1 and
k+2 <= istopm

IHI

IHI is INTEGER

A is DOUBLE PRECISION array, dimension (LDA,N)

LDA

LDA is INTEGER
The leading dimension of A as declared in
the calling procedure.

B is DOUBLE PRECISION array, dimension (LDB,N)

LDB

LDB is INTEGER
The leading dimension of B as declared in
the calling procedure.

NQ is INTEGER
The order of the matrix Q

QSTART

QSTART is INTEGER
Start index of the matrix Q. Rotations are applied
To columns k+2-qStart:k+4-qStart of Q.

Q is DOUBLE PRECISION array, dimension (LDQ,NQ)

LDQ

LDQ is INTEGER
The leading dimension of Q as declared in
the calling procedure.

NZ is INTEGER
The order of the matrix Z

ZSTART

ZSTART is INTEGER
Start index of the matrix Z. Rotations are applied
To columns k+1-qStart:k+3-qStart of Z.

Z is DOUBLE PRECISION array, dimension (LDZ,NZ)

LDZ

LDZ is INTEGER
The leading dimension of Q as declared in
the calling procedure.

Author

Thijs Steel, KU Leuven

Date

May 2020

recursive subroutine dlaqz3 (logical, intent(in) ILSCHUR, logical, intent(in)ILQ, logical, intent(in) ILZ, integer, intent(in) N, integer, intent(in)ILO, integer, intent(in) IHI, integer, intent(in) NW, double precision,dimension( lda, * ), intent(inout) A, integer, intent(in) LDA, doubleprecision, dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB,double precision, dimension( ldq, * ), intent(inout) Q, integer, intent(in)LDQ, double precision, dimension( ldz, * ), intent(inout) Z, integer,intent(in) LDZ, integer, intent(out) NS, integer, intent(out) ND, doubleprecision, dimension( * ), intent(inout) ALPHAR, double precision,dimension( * ), intent(inout) ALPHAI, double precision, dimension( * ),intent(inout) BETA, double precision, dimension( ldqc, * ) QC, integer,intent(in) LDQC, double precision, dimension( ldzc, * ) ZC, integer,intent(in) LDZC, double precision, dimension( * ) WORK, integer, intent(in)LWORK, integer, intent(in) REC, integer, intent(out) INFO)

DLAQZ3

Purpose:

DLAQZ3 performs AED

Parameters

ILSCHUR

ILSCHUR is LOGICAL
Determines whether or not to update the full Schur form

ILQ

ILQ is LOGICAL
Determines whether or not to update the matrix Q

ILZ

ILZ is LOGICAL
Determines whether or not to update the matrix Z

N is INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER
ILO and IHI mark the rows and columns of (A,B) which
are to be normalized

NW is INTEGER
The desired size of the deflation window.

A is DOUBLE PRECISION array, dimension (LDA, N)

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).

B is DOUBLE PRECISION array, dimension (LDB, N)

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).

Q is DOUBLE PRECISION array, dimension (LDQ, N)

LDQ

LDQ is INTEGER

Z is DOUBLE PRECISION array, dimension (LDZ, N)

LDZ

LDZ is INTEGER

NS is INTEGER
The number of unconverged eigenvalues available to
use as shifts.

ND is INTEGER
The number of converged eigenvalues found.

ALPHAR

ALPHAR is DOUBLE PRECISION array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of GNEP.

ALPHAI

BETA

QC is DOUBLE PRECISION array, dimension (LDQC, NW)

LDQC

LDQC is INTEGER

ZC is DOUBLE PRECISION array, dimension (LDZC, NW)

LDZC

LDZ is INTEGER

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).

REC

REC is INTEGER
REC indicates the current recursion level. Should be set
to 0 on first call.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Thijs Steel, KU Leuven

Date

May 2020

subroutine dlaqz4 (logical, intent(in) ILSCHUR, logical, intent(in) ILQ,logical, intent(in) ILZ, integer, intent(in) N, integer, intent(in) ILO,integer, intent(in) IHI, integer, intent(in) NSHIFTS, integer, intent(in)NBLOCK_DESIRED, double precision, dimension( * ), intent(inout) SR, doubleprecision, dimension( * ), intent(inout) SI, double precision, dimension( ), intent(inout) SS, double precision, dimension( lda, ), intent(inout)A, integer, intent(in) LDA, double precision, dimension( ldb, * ),intent(inout) B, integer, intent(in) LDB, double precision, dimension( ldq,* ), intent(inout) Q, integer, intent(in) LDQ, double precision, dimension(ldz, * ), intent(inout) Z, integer, intent(in) LDZ, double precision,dimension( ldqc, * ), intent(inout) QC, integer, intent(in) LDQC, doubleprecision, dimension( ldzc, * ), intent(inout) ZC, integer, intent(in)LDZC, double precision, dimension( * ), intent(inout) WORK, integer,intent(in) LWORK, integer, intent(out) INFO)

DLAQZ4

Purpose:

DLAQZ4 Executes a single multishift QZ sweep

Parameters

ILSCHUR

ILSCHUR is LOGICAL
Determines whether or not to update the full Schur form

ILQ

ILQ is LOGICAL
Determines whether or not to update the matrix Q

ILZ

ILZ is LOGICAL
Determines whether or not to update the matrix Z

N is INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER

NSHIFTS

NSHIFTS is INTEGER
The desired number of shifts to use

NBLOCK_DESIRED

NBLOCK_DESIRED is INTEGER
The desired size of the computational windows

SR is DOUBLE PRECISION array. SR contains
the real parts of the shifts to use.

SI is DOUBLE PRECISION array. SI contains
the imaginary parts of the shifts to use.

SS is DOUBLE PRECISION array. SS contains
the scale of the shifts to use.

A is DOUBLE PRECISION array, dimension (LDA, N)

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).

B is DOUBLE PRECISION array, dimension (LDB, N)

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).

Q is DOUBLE PRECISION array, dimension (LDQ, N)

LDQ

LDQ is INTEGER

Z is DOUBLE PRECISION array, dimension (LDZ, N)

LDZ

LDZ is INTEGER

QC is DOUBLE PRECISION array, dimension (LDQC, NBLOCK_DESIRED)

LDQC

LDQC is INTEGER

ZC is DOUBLE PRECISION array, dimension (LDZC, NBLOCK_DESIRED)

LDZC

LDZ is INTEGER

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Thijs Steel, KU Leuven

Date

May 2020

subroutine dtgevc (character SIDE, character HOWMNY, logical, dimension( * )SELECT, integer N, double precision, dimension( lds, * ) S, integer LDS,double precision, dimension( ldp, * ) P, integer LDP, double precision,dimension( ldvl, * ) VL, integer LDVL, double precision, dimension( ldvr, ) VR, integer LDVR, integer MM, integer M, double precision, dimension( )WORK, integer INFO)

DTGEVC

Purpose:

DTGEVC computes some or all of the right and/or left eigenvectors of
a pair of real matrices (S,P), where S is a quasi-triangular matrix
and P is upper triangular. Matrix pairs of this type are produced by
the generalized Schur factorization of a matrix pair (A,B):

A = Q*S*Z**T, B = Q*P*Z**T

as computed by DGGHRD + DHGEQZ.

The right eigenvector x and the left eigenvector y of (S,P)
corresponding to an eigenvalue w are defined by:

S*x = w*P*x, (y**H)*S = w*(y**H)*P,

where y**H denotes the conjugate tranpose of y.
The eigenvalues are not input to this routine, but are computed
directly from the diagonal blocks of S and P.

This routine returns the matrices X and/or Y of right and left
eigenvectors of (S,P), or the products Z*X and/or Q*Y,
where Z and Q are input matrices.
If Q and Z are the orthogonal factors from the generalized Schur
factorization of a matrix pair (A,B), then Z*X and Q*Y
are the matrices of right and left eigenvectors of (A,B).

Parameters

SIDE

SIDE is CHARACTER*1
= ’R’: compute right eigenvectors only;
= ’L’: compute left eigenvectors only;
= ’B’: compute both right and left eigenvectors.

HOWMNY

HOWMNY is CHARACTER*1
= ’A’: compute all right and/or left eigenvectors;
= ’B’: compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= ’S’: compute selected right and/or left eigenvectors,
specified by the logical array SELECT.

SELECT

SELECT is LOGICAL array, dimension (N)
If HOWMNY=’S’, SELECT specifies the eigenvectors to be
computed. If w(j) is a real eigenvalue, the corresponding
real eigenvector is computed if SELECT(j) is .TRUE..
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector
is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
set to .FALSE..
Not referenced if HOWMNY = ’A’ or ’B’.

N is INTEGER
The order of the matrices S and P. N >= 0.

S is DOUBLE PRECISION array, dimension (LDS,N)
The upper quasi-triangular matrix S from a generalized Schur
factorization, as computed by DHGEQZ.

LDS

LDS is INTEGER
The leading dimension of array S. LDS >= max(1,N).

P is DOUBLE PRECISION array, dimension (LDP,N)
The upper triangular matrix P from a generalized Schur
factorization, as computed by DHGEQZ.
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
of S must be in positive diagonal form.

LDP

LDP is INTEGER
The leading dimension of array P. LDP >= max(1,N).

VL is DOUBLE PRECISION array, dimension (LDVL,MM)
On entry, if SIDE = ’L’ or ’B’ and HOWMNY = ’B’, VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of left Schur vectors returned by DHGEQZ).
On exit, if SIDE = ’L’ or ’B’, VL contains:
if HOWMNY = ’A’, the matrix Y of left eigenvectors of (S,P);
if HOWMNY = ’B’, the matrix Q*Y;
if HOWMNY = ’S’, the left eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of
VL, in the same order as their eigenvalues.

A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.

Not referenced if SIDE = ’R’.

LDVL

LDVL is INTEGER
The leading dimension of array VL. LDVL >= 1, and if
SIDE = ’L’ or ’B’, LDVL >= N.

VR is DOUBLE PRECISION array, dimension (LDVR,MM)
On entry, if SIDE = ’R’ or ’B’ and HOWMNY = ’B’, VR must
contain an N-by-N matrix Z (usually the orthogonal matrix Z
of right Schur vectors returned by DHGEQZ).

On exit, if SIDE = ’R’ or ’B’, VR contains:
if HOWMNY = ’A’, the matrix X of right eigenvectors of (S,P);
if HOWMNY = ’B’ or ’b’, the matrix Z*X;
if HOWMNY = ’S’ or ’s’, the right eigenvectors of (S,P)
specified by SELECT, stored consecutively in the
columns of VR, in the same order as their
eigenvalues.

A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.

Not referenced if SIDE = ’L’.

LDVR

LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1, and if
SIDE = ’R’ or ’B’, LDVR >= N.

MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.

M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors. If HOWMNY = ’A’ or ’B’, M
is set to N. Each selected real eigenvector occupies one
column and each selected complex eigenvector occupies two
columns.

WORK

WORK is DOUBLE PRECISION array, dimension (6*N)

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: the 2-by-2 block (INFO:INFO+1) does not have a complex
eigenvalue.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

Allocation of workspace:
---------- -- ---------

WORK( j ) = 1-norm of j-th column of A, above the diagonal
WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
WORK( 2*N+1:3*N ) = real part of eigenvector
WORK( 3*N+1:4*N ) = imaginary part of eigenvector
WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector

Rowwise vs. columnwise solution methods:
------- -- ---------- -------- -------

Finding a generalized eigenvector consists basically of solving the
singular triangular system

(A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left)

Consider finding the i-th right eigenvector (assume all eigenvalues
are real). The equation to be solved is:
n i
0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1
k=j k=j

where C = (A - w B) (The components v(i+1:n) are 0.)

The ’rowwise’ method is:

(1) v(i) := 1
for j = i-1,. . .,1:
i
(2) compute s = - sum C(j,k) v(k) and
k=j+1

(3) v(j) := s / C(j,j)

Step 2 is sometimes called the ’dot product’ step, since it is an
inner product between the j-th row and the portion of the eigenvector
that has been computed so far.

The ’columnwise’ method consists basically in doing the sums
for all the rows in parallel. As each v(j) is computed, the
contribution of v(j) times the j-th column of C is added to the
partial sums. Since FORTRAN arrays are stored columnwise, this has
the advantage that at each step, the elements of C that are accessed
are adjacent to one another, whereas with the rowwise method, the
elements accessed at a step are spaced LDS (and LDP) words apart.

When finding left eigenvectors, the matrix in question is the
transpose of the one in storage, so the rowwise method then
actually accesses columns of A and B at each step, and so is the
preferred method.

subroutine dtgexc (logical WANTQ, logical WANTZ, integer N, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * )B, integer LDB, double precision, dimension( ldq, * ) Q, integer LDQ,double precision, dimension( ldz, * ) Z, integer LDZ, integer IFST, integerILST, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DTGEXC

Purpose:

DTGEXC reorders the generalized real Schur decomposition of a real
matrix pair (A,B) using an orthogonal equivalence transformation

(A, B) = Q * (A, B) * Z**T,

so that the diagonal block of (A, B) with row index IFST is moved
to row ILST.

(A, B) must be in generalized real Schur canonical form (as returned
by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
diagonal blocks. B is upper triangular.

Optionally, the matrices Q and Z of generalized Schur vectors are
updated.

Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T

Parameters

WANTQ

WANTQ is LOGICAL
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q.

WANTZ

WANTZ is LOGICAL
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z.

N is INTEGER
The order of the matrices A and B. N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the matrix A in generalized real Schur canonical
form.
On exit, the updated matrix A, again in generalized
real Schur canonical form.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the matrix B in generalized real Schur canonical
form (A,B).
On exit, the updated matrix B, again in generalized
real Schur canonical form (A,B).

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
On exit, the updated matrix Q.
If WANTQ = .FALSE., Q is not referenced.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If WANTQ = .TRUE., LDQ >= N.

Z is DOUBLE PRECISION array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., the orthogonal matrix Z.
On exit, the updated matrix Z.
If WANTZ = .FALSE., Z is not referenced.

LDZ

LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If WANTZ = .TRUE., LDZ >= N.

IFST

IFST is INTEGER

ILST

ILST is INTEGER
Specify the reordering of the diagonal blocks of (A, B).
The block with row index IFST is moved to row ILST, by a
sequence of swapping between adjacent blocks.
On exit, if IFST pointed on entry to the second row of
a 2-by-2 block, it is changed to point to the first row;
ILST always points to the first row of the block in its
final position (which may differ from its input value by
+1 or -1). 1 <= IFST, ILST <= N.

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK.
LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.

INFO

INFO is INTEGER
=0: successful exit.
<0: if INFO = -i, the i-th argument had an illegal value.
=1: The transformed matrix pair (A, B) would be too far
from generalized Schur form; the problem is ill-
conditioned. (A, B) may have been partially reordered,
and ILST points to the first row of the current
position of the block being moved.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

subroutine sgelqt (integer M, integer N, integer MB, real, dimension( lda, ) A, integer LDA, real, dimension( ldt, ) T, integer LDT, real,dimension( * ) WORK, integer INFO)

SGELQT

Purpose:

DGELQT computes a blocked LQ factorization of a real M-by-N matrix A
using the compact WY representation of Q.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

MB is INTEGER
The block size to be used in the blocked QR. MIN(M,N) >= MB >= 1.

A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and below the diagonal of the array
contain the M-by-MIN(M,N) lower trapezoidal matrix L (L is
lower triangular if M <= N); the elements above the diagonal
are the rows of V.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

T is REAL array, dimension (LDT,MIN(M,N))
The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See below
for further details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

WORK

WORK is REAL array, dimension (MB*N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix V stores the elementary reflectors H(i) in the i-th row
above the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 v1 v1 v1 v1 )
( 1 v2 v2 v2 )
( 1 v3 v3 )

T = (T1 T2 ... TB).

recursive subroutine sgelqt3 (integer M, integer N, real, dimension( lda, * )A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO)

SGELQT3

Purpose:

SGELQT3 recursively computes a LQ factorization of a real M-by-N
matrix A, using the compact WY representation of Q.

Based on the algorithm of Elmroth and Gustavson,
IBM J. Res. Develop. Vol 44 No. 4 July 2000.

Parameters

M is INTEGER
The number of rows of the matrix A. M =< N.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is REAL array, dimension (LDA,N)
On entry, the real M-by-N matrix A. On exit, the elements on and
below the diagonal contain the N-by-N lower triangular matrix L; the
elements above the diagonal are the rows of V. See below for
further details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

T is REAL array, dimension (LDT,N)
The N-by-N upper triangular factor of the block reflector.
The elements on and above the diagonal contain the block
reflector T; the elements below the diagonal are not used.
See below for further details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix V stores the elementary reflectors H(i) in the i-th row
above the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 v1 v1 v1 v1 )
( 1 v2 v2 v2 )
( 1 v3 v3 v3 )

where the vi’s represent the vectors which define H(i), which are returned
in the matrix A. The 1’s along the diagonal of V are not stored in A. The
block reflector H is then given by

H = I - V * T * V**T

where V**T is the transpose of V.

For details of the algorithm, see Elmroth and Gustavson (cited above).

subroutine sgemlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer MB, real, dimension( ldv, * ) V, integer LDV, real,dimension( ldt, * ) T, integer LDT, real, dimension( ldc, * ) C, integerLDC, real, dimension( * ) WORK, integer INFO)

SGEMLQT

Purpose:

DGEMLQT overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q C C Q
TRANS = ’T’: Q**T C C Q**T

where Q is a real orthogonal matrix defined as the product of K
elementary reflectors:

Q = H(1) H(2) . . . H(K) = I - V T V**T

generated using the compact WY representation as returned by SGELQT.

Q is of order M if SIDE = ’L’ and of order N if SIDE = ’R’.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’C’: Transpose, apply Q**T.

M is INTEGER
The number of rows of the matrix C. M >= 0.

N is INTEGER
The number of columns of the matrix C. N >= 0.

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

MB is INTEGER
The block size used for the storage of T. K >= MB >= 1.
This must be the same value of MB used to generate T
in SGELQT.

V is REAL array, dimension
(LDV,M) if SIDE = ’L’,
(LDV,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
SGELQT in the first K rows of its array argument A.

LDV

LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,K).

T is REAL array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by SGELQT, stored as a MB-by-K matrix.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q C, Q**T C, C Q**T or C Q.

LDC

LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).

WORK

WORK is REAL array. The dimension of
WORK is N*MB if SIDE = ’L’, or M*MB if SIDE = ’R’.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

recursive subroutine slaqz0 (character, intent(in) WANTS, character,intent(in) WANTQ, character, intent(in) WANTZ, integer, intent(in) N,integer, intent(in) ILO, integer, intent(in) IHI, real, dimension( lda, ), intent(inout) A, integer, intent(in) LDA, real, dimension( ldb, ),intent(inout) B, integer, intent(in) LDB, real, dimension( * ),intent(inout) ALPHAR, real, dimension( * ), intent(inout) ALPHAI, real,dimension( * ), intent(inout) BETA, real, dimension( ldq, * ),intent(inout) Q, integer, intent(in) LDQ, real, dimension( ldz, * ),intent(inout) Z, integer, intent(in) LDZ, real, dimension( * ),intent(inout) WORK, integer, intent(in) LWORK, integer, intent(in) REC,integer, intent(out) INFO)

SLAQZ0

Purpose:

SLAQZ0 computes the eigenvalues of a real matrix pair (H,T),
where H is an upper Hessenberg matrix and T is upper triangular,
using the double-shift QZ method.
Matrix pairs of this type are produced by the reduction to
generalized upper Hessenberg form of a real matrix pair (A,B):

A = Q1*H*Z1**T, B = Q1*T*Z1**T,

as computed by SGGHRD.

If JOB=’S’, then the Hessenberg-triangular pair (H,T) is
also reduced to generalized Schur form,

H = Q*S*Z**T, T = Q*P*Z**T,

where Q and Z are orthogonal matrices, P is an upper triangular
matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
diagonal blocks.

The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
(H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
eigenvalues.

Optionally, the orthogonal matrix Q from the generalized Schur
factorization may be postmultiplied into an input matrix Q1, and the
orthogonal matrix Z may be postmultiplied into an input matrix Z1.
If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced
the matrix pair (A,B) to generalized upper Hessenberg form, then the
output matrices Q1*Q and Z1*Z are the orthogonal factors from the
generalized Schur factorization of (A,B):

A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.

Ref: C.B. Moler & G.W. Stewart, ’An Algorithm for Generalized Matrix
Eigenvalue Problems’, SIAM J. Numer. Anal., 10(1973),
pp. 241--256.

Ref: B. Kagstrom, D. Kressner, ’Multishift Variants of the QZ
Algorithm with Aggressive Early Deflation’, SIAM J. Numer.
Anal., 29(2006), pp. 199--227.

Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril ’A multishift,
multipole rational QZ method with agressive early deflation’

Parameters

WANTS

WANTS is CHARACTER*1
= ’E’: Compute eigenvalues only;
= ’S’: Compute eigenvalues and the Schur form.

WANTQ

WANTZ

N is INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.

ILO

ILO is INTEGER

IHI

A is REAL array, dimension (LDA, N)
On entry, the N-by-N upper Hessenberg matrix A.
On exit, if JOB = ’S’, A contains the upper quasi-triangular
matrix S from the generalized Schur factorization.
If JOB = ’E’, the diagonal blocks of A match those of S, but
the rest of A is unspecified.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).

B is REAL array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
On exit, if JOB = ’S’, B contains the upper triangular
matrix P from the generalized Schur factorization;
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
are reduced to positive diagonal form, i.e., if A(j+1,j) is
non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and
B(j+1,j+1) > 0.
If JOB = ’E’, the diagonal blocks of B match those of P, but
the rest of B is unspecified.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).

ALPHAR

ALPHAR is REAL array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of GNEP.

ALPHAI

ALPHAI is REAL array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of GNEP.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

BETA

BETA is REAL array, dimension (N)
The scalars beta that define the eigenvalues of GNEP.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the matrix
pair (A,B), in one of the forms lambda = alpha/beta or
mu = beta/alpha. Since either lambda or mu may overflow,
they should not, in general, be computed.

Q is REAL array, dimension (LDQ, N)
On entry, if COMPQ = ’V’, the orthogonal matrix Q1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = ’I’, the orthogonal matrix of left Schur
vectors of (A,B), and if COMPQ = ’V’, the orthogonal matrix
of left Schur vectors of (A,B).
Not referenced if COMPQ = ’N’.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ=’V’ or ’I’, then LDQ >= N.

Z is REAL array, dimension (LDZ, N)
On entry, if COMPZ = ’V’, the orthogonal matrix Z1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = ’I’, the orthogonal matrix of
right Schur vectors of (H,T), and if COMPZ = ’V’, the
orthogonal matrix of right Schur vectors of (A,B).
Not referenced if COMPZ = ’N’.

LDZ

LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ=’V’ or ’I’, then LDZ >= N.

WORK

WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).

REC

REC is INTEGER
REC indicates the current recursion level. Should be set
to 0 on first call.

INFO

Author

Thijs Steel, KU Leuven

Date

May 2020

subroutine slaqz1 (real, dimension( lda, * ), intent(in) A, integer,intent(in) LDA, real, dimension( ldb, * ), intent(in) B, integer,intent(in) LDB, real, intent(in) SR1, real, intent(in) SR2, real,intent(in) SI, real, intent(in) BETA1, real, intent(in) BETA2, real,dimension( * ), intent(out) V)

SLAQZ1

Purpose:

Given a 3-by-3 matrix pencil (A,B), SLAQZ1 sets v to a
scalar multiple of the first column of the product

(*) K = (A - (beta2*sr2 - i*si)*B)*Bˆ(-1)*(beta1*A - (sr2 + i*si2)*B)*Bˆ(-1).

It is assumed that either

1) sr1 = sr2
or
2) si = 0.

This is useful for starting double implicit shift bulges
in the QZ algorithm.

Parameters

A is REAL array, dimension (LDA,N)
The 3-by-3 matrix A in (*).

LDA

LDA is INTEGER
The leading dimension of A as declared in
the calling procedure.

B is REAL array, dimension (LDB,N)
The 3-by-3 matrix B in (*).

LDB

LDB is INTEGER
The leading dimension of B as declared in
the calling procedure.

SR1

SR1 is REAL

SR2

SR2 is REAL

SI is REAL

BETA1

BETA1 is REAL

BETA2

BETA2 is REAL

V is REAL array, dimension (N)
A scalar multiple of the first column of the
matrix K in (*).

Author

Thijs Steel, KU Leuven

Date

May 2020

subroutine slaqz2 (logical, intent(in) ILQ, logical, intent(in) ILZ, integer,intent(in) K, integer, intent(in) ISTARTM, integer, intent(in) ISTOPM,integer, intent(in) IHI, real, dimension( lda, * ) A, integer, intent(in)LDA, real, dimension( ldb, * ) B, integer, intent(in) LDB, integer,intent(in) NQ, integer, intent(in) QSTART, real, dimension( ldq, * ) Q,integer, intent(in) LDQ, integer, intent(in) NZ, integer, intent(in)ZSTART, real, dimension( ldz, * ) Z, integer, intent(in) LDZ)

SLAQZ2

Purpose:

SLAQZ2 chases a 2x2 shift bulge in a matrix pencil down a single position

Parameters

ILQ

ILQ is LOGICAL
Determines whether or not to update the matrix Q

ILZ

ILZ is LOGICAL
Determines whether or not to update the matrix Z

ISTARTM

ISTARTM is INTEGER

ISTOPM

ISTOPM is INTEGER
Updates to (A,B) are restricted to
(istartm:k+3,k:istopm). It is assumed
without checking that istartm <= k+1 and
k+2 <= istopm

IHI

IHI is INTEGER

A is REAL array, dimension (LDA,N)

LDA

LDA is INTEGER
The leading dimension of A as declared in
the calling procedure.

B is REAL array, dimension (LDB,N)

LDB

LDB is INTEGER
The leading dimension of B as declared in
the calling procedure.

NQ is INTEGER
The order of the matrix Q

QSTART

QSTART is INTEGER
Start index of the matrix Q. Rotations are applied
To columns k+2-qStart:k+4-qStart of Q.

Q is REAL array, dimension (LDQ,NQ)

LDQ

LDQ is INTEGER
The leading dimension of Q as declared in
the calling procedure.

NZ is INTEGER
The order of the matrix Z

ZSTART

ZSTART is INTEGER
Start index of the matrix Z. Rotations are applied
To columns k+1-qStart:k+3-qStart of Z.

Z is REAL array, dimension (LDZ,NZ)

LDZ

LDZ is INTEGER
The leading dimension of Q as declared in
the calling procedure.

Author

Thijs Steel, KU Leuven

Date

May 2020

recursive subroutine slaqz3 (logical, intent(in) ILSCHUR, logical, intent(in)ILQ, logical, intent(in) ILZ, integer, intent(in) N, integer, intent(in)ILO, integer, intent(in) IHI, integer, intent(in) NW, real, dimension( lda,* ), intent(inout) A, integer, intent(in) LDA, real, dimension( ldb, * ),intent(inout) B, integer, intent(in) LDB, real, dimension( ldq, * ),intent(inout) Q, integer, intent(in) LDQ, real, dimension( ldz, * ),intent(inout) Z, integer, intent(in) LDZ, integer, intent(out) NS, integer,intent(out) ND, real, dimension( * ), intent(inout) ALPHAR, real,dimension( * ), intent(inout) ALPHAI, real, dimension( * ), intent(inout)BETA, real, dimension( ldqc, * ) QC, integer, intent(in) LDQC, real,dimension( ldzc, * ) ZC, integer, intent(in) LDZC, real, dimension( * )WORK, integer, intent(in) LWORK, integer, intent(in) REC, integer,intent(out) INFO)

SLAQZ3

Purpose:

SLAQZ3 performs AED

Parameters

ILSCHUR

ILSCHUR is LOGICAL
Determines whether or not to update the full Schur form

ILQ

ILQ is LOGICAL
Determines whether or not to update the matrix Q

ILZ

ILZ is LOGICAL
Determines whether or not to update the matrix Z

N is INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER
ILO and IHI mark the rows and columns of (A,B) which
are to be normalized

NW is INTEGER
The desired size of the deflation window.

A is REAL array, dimension (LDA, N)

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).

B is REAL array, dimension (LDB, N)

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).

Q is REAL array, dimension (LDQ, N)

LDQ

LDQ is INTEGER

Z is REAL array, dimension (LDZ, N)

LDZ

LDZ is INTEGER

NS is INTEGER
The number of unconverged eigenvalues available to
use as shifts.

ND is INTEGER
The number of converged eigenvalues found.

ALPHAR

ALPHAR is REAL array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of GNEP.

ALPHAI

BETA

QC is REAL array, dimension (LDQC, NW)

LDQC

LDQC is INTEGER

ZC is REAL array, dimension (LDZC, NW)

LDZC

LDZ is INTEGER

WORK

WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).

REC

REC is INTEGER
REC indicates the current recursion level. Should be set
to 0 on first call.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Thijs Steel, KU Leuven

Date

May 2020

subroutine slaqz4 (logical, intent(in) ILSCHUR, logical, intent(in) ILQ,logical, intent(in) ILZ, integer, intent(in) N, integer, intent(in) ILO,integer, intent(in) IHI, integer, intent(in) NSHIFTS, integer, intent(in)NBLOCK_DESIRED, real, dimension( * ), intent(inout) SR, real, dimension( ), intent(inout) SI, real, dimension( ), intent(inout) SS, real,dimension( lda, * ), intent(inout) A, integer, intent(in) LDA, real,dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB, real,dimension( ldq, * ), intent(inout) Q, integer, intent(in) LDQ, real,dimension( ldz, * ), intent(inout) Z, integer, intent(in) LDZ, real,dimension( ldqc, * ), intent(inout) QC, integer, intent(in) LDQC, real,dimension( ldzc, * ), intent(inout) ZC, integer, intent(in) LDZC, real,dimension( * ), intent(inout) WORK, integer, intent(in) LWORK, integer,intent(out) INFO)

SLAQZ4

Purpose:

SLAQZ4 Executes a single multishift QZ sweep

Parameters

ILSCHUR

ILSCHUR is LOGICAL
Determines whether or not to update the full Schur form

ILQ

ILQ is LOGICAL
Determines whether or not to update the matrix Q

ILZ

ILZ is LOGICAL
Determines whether or not to update the matrix Z

N is INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER

NSHIFTS

NSHIFTS is INTEGER
The desired number of shifts to use

NBLOCK_DESIRED

NBLOCK_DESIRED is INTEGER
The desired size of the computational windows

SR is REAL array. SR contains
the real parts of the shifts to use.

SI is REAL array. SI contains
the imaginary parts of the shifts to use.

SS is REAL array. SS contains
the scale of the shifts to use.

A is REAL array, dimension (LDA, N)

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).

B is REAL array, dimension (LDB, N)

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).

Q is REAL array, dimension (LDQ, N)

LDQ

LDQ is INTEGER

Z is REAL array, dimension (LDZ, N)

LDZ

LDZ is INTEGER

QC is REAL array, dimension (LDQC, NBLOCK_DESIRED)

LDQC

LDQC is INTEGER

ZC is REAL array, dimension (LDZC, NBLOCK_DESIRED)

LDZC

LDZ is INTEGER

WORK

WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Thijs Steel, KU Leuven

Date

May 2020

subroutine zgelqt (integer M, integer N, integer MB, complex16, dimension(lda, ) A, integer LDA, complex16, dimension( ldt, ) T, integer LDT,complex16, dimension( ) WORK, integer INFO)

ZGELQT

Purpose:

ZGELQT computes a blocked LQ factorization of a complex M-by-N matrix A
using the compact WY representation of Q.

Parameters

M is INTEGER
The number of rows of the matrix A. M >= 0.

N is INTEGER
The number of columns of the matrix A. N >= 0.

MB is INTEGER
The block size to be used in the blocked QR. MIN(M,N) >= MB >= 1.

A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and below the diagonal of the array
contain the M-by-MIN(M,N) lower trapezoidal matrix L (L is
lower triangular if M <= N); the elements above the diagonal
are the rows of V.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

T is COMPLEX*16 array, dimension (LDT,MIN(M,N))
The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See below
for further details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

WORK

WORK is COMPLEX*16 array, dimension (MB*N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix V stores the elementary reflectors H(i) in the i-th row
above the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 v1 v1 v1 v1 )
( 1 v2 v2 v2 )
( 1 v3 v3 )

T = (T1 T2 ... TB).

recursive subroutine zgelqt3 (integer M, integer N, complex16, dimension(lda, ) A, integer LDA, complex16, dimension( ldt, ) T, integer LDT,integer INFO)

ZGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

ZGELQT3 recursively computes a LQ factorization of a complex M-by-N
matrix A, using the compact WY representation of Q.

Based on the algorithm of Elmroth and Gustavson,
IBM J. Res. Develop. Vol 44 No. 4 July 2000.

Parameters

M is INTEGER
The number of rows of the matrix A. M =< N.

N is INTEGER
The number of columns of the matrix A. N >= 0.

A is COMPLEX*16 array, dimension (LDA,N)
On entry, the complex M-by-N matrix A. On exit, the elements on and
below the diagonal contain the N-by-N lower triangular matrix L; the
elements above the diagonal are the rows of V. See below for
further details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

T is COMPLEX*16 array, dimension (LDT,N)
The N-by-N upper triangular factor of the block reflector.
The elements on and above the diagonal contain the block
reflector T; the elements below the diagonal are not used.
See below for further details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix V stores the elementary reflectors H(i) in the i-th row
above the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 v1 v1 v1 v1 )
( 1 v2 v2 v2 )
( 1 v3 v3 v3 )

where the vi’s represent the vectors which define H(i), which are returned
in the matrix A. The 1’s along the diagonal of V are not stored in A. The
block reflector H is then given by

H = I - V * T * V**T

where V**T is the transpose of V.

For details of the algorithm, see Elmroth and Gustavson (cited above).

subroutine zgemlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer MB, complex16, dimension( ldv, ) V, integer LDV,complex16, dimension( ldt, ) T, integer LDT, complex16, dimension( ldc, ) C, integer LDC, complex16, dimension( ) WORK, integer INFO)

ZGEMLQT

Purpose:

ZGEMLQT overwrites the general complex M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q C C Q
TRANS = ’C’: Q**H C C Q**H

where Q is a complex unitary matrix defined as the product of K
elementary reflectors:

Q = H(1) H(2) . . . H(K) = I - V T V**H

generated using the compact WY representation as returned by ZGELQT.

Q is of order M if SIDE = ’L’ and of order N if SIDE = ’R’.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**H from the Left;
= ’R’: apply Q or Q**H from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’C’: Conjugate transpose, apply Q**H.

M is INTEGER
The number of rows of the matrix C. M >= 0.

N is INTEGER
The number of columns of the matrix C. N >= 0.

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

MB is INTEGER
The block size used for the storage of T. K >= MB >= 1.
This must be the same value of MB used to generate T
in ZGELQT.

V is COMPLEX*16 array, dimension
(LDV,M) if SIDE = ’L’,
(LDV,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
ZGELQT in the first K rows of its array argument A.

LDV

LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,K).

T is COMPLEX*16 array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by ZGELQT, stored as a MB-by-K matrix.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

C is COMPLEX*16 array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q C, Q**H C, C Q**H or C Q.

LDC

LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).

WORK

WORK is COMPLEX*16 array. The dimension of
WORK is N*MB if SIDE = ’L’, or M*MB if SIDE = ’R’.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Description

doubleGEcomputational

NAME

SYNOPSIS

Functions

Detailed Description

Function Documentation

subroutine cgelqt (integer M, integer N, integer MB, complex, dimension( lda,* ) A, integer LDA, complex, dimension( ldt, * ) T, integer LDT, complex,dimension( * ) WORK, integer INFO)

recursive subroutine cgelqt3 (integer M, integer N, complex, dimension( lda,* ) A, integer LDA, complex, dimension( ldt, * ) T, integer LDT, integerINFO)

subroutine cgemlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer MB, complex, dimension( ldv, * ) V, integer LDV,complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldc, * )C, integer LDC, complex, dimension( * ) WORK, integer INFO)

subroutine dgebak (character JOB, character SIDE, integer N, integer ILO,integer IHI, double precision, dimension( * ) SCALE, integer M, doubleprecision, dimension( ldv, * ) V, integer LDV, integer INFO)

subroutine dgebal (character JOB, integer N, double precision, dimension(lda, * ) A, integer LDA, integer ILO, integer IHI, double precision,dimension( * ) SCALE, integer INFO)

subroutine dgebd2 (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) D, double precision,dimension( * ) E, double precision, dimension( * ) TAUQ, double precision,dimension( * ) TAUP, double precision, dimension( * ) WORK, integer INFO)

subroutine dgecon (character NORM, integer N, double precision, dimension(lda, * ) A, integer LDA, double precision ANORM, double precision RCOND,double precision, dimension( * ) WORK, integer, dimension( * ) IWORK,integer INFO)

subroutine dgeequ (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) R, double precision,dimension( * ) C, double precision ROWCND, double precision COLCND, doubleprecision AMAX, integer INFO)

subroutine dgeequb (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) R, double precision,dimension( * ) C, double precision ROWCND, double precision COLCND, doubleprecision AMAX, integer INFO)

subroutine dgehd2 (integer N, integer ILO, integer IHI, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer INFO)

subroutine dgehrd (integer N, integer ILO, integer IHI, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dgelq2 (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) TAU, double precision,dimension( * ) WORK, integer INFO)

subroutine dgelqf (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dgelqt (integer M, integer N, integer MB, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * )T, integer LDT, double precision, dimension( * ) WORK, integer INFO)

recursive subroutine dgelqt3 (integer M, integer N, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * )T, integer LDT, integer INFO)

subroutine dgeql2 (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) TAU, double precision,dimension( * ) WORK, integer INFO)

subroutine dgeqlf (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dgeqp3 (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, integer, dimension( * ) JPVT, double precision,dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK,integer INFO)

subroutine dgeqr2 (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) TAU, double precision,dimension( * ) WORK, integer INFO)

subroutine dgeqr2p (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) TAU, double precision,dimension( * ) WORK, integer INFO)

subroutine dgeqrf (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dgeqrfp (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dgeqrt (integer M, integer N, integer NB, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * )T, integer LDT, double precision, dimension( * ) WORK, integer INFO)

subroutine dgeqrt2 (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( ldt, * ) T, integer LDT,integer INFO)

recursive subroutine dgeqrt3 (integer M, integer N, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * )T, integer LDT, integer INFO)

subroutine dgerq2 (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) TAU, double precision,dimension( * ) WORK, integer INFO)

subroutine dgerqf (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, double precision, dimension( * ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dgetf2 (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

subroutine dgetrf (integer M, integer N, double precision, dimension( lda, *) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

recursive subroutine dgetrf2 (integer M, integer N, double precision,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integerINFO)

subroutine dgetri (integer N, double precision, dimension( lda, * ) A,integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * )WORK, integer LWORK, integer INFO)

subroutine dgetrs (character TRANS, integer N, integer NRHS, doubleprecision, dimension( lda, * ) A, integer LDA, integer, dimension( * )IPIV, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

subroutine dla_geamv (integer TRANS, integer M, integer N, double precisionALPHA, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) X, integer INCX, double precision BETA, doubleprecision, dimension( * ) Y, integer INCY)

double precision function dla_gerpvgrw (integer N, integer NCOLS, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldaf, * ) AF, integer LDAF)

subroutine dlaorhr_col_getrfnp (integer M, integer N, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D,integer INFO)

recursive subroutine dlaorhr_col_getrfnp2 (integer M, integer N, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(* ) D, integer INFO)

subroutine sgelqt (integer M, integer N, integer MB, real, dimension( lda, *) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, real,dimension( * ) WORK, integer INFO)

recursive subroutine sgelqt3 (integer M, integer N, real, dimension( lda, * )A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO)

subroutine sgemlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer MB, real, dimension( ldv, * ) V, integer LDV, real,dimension( ldt, * ) T, integer LDT, real, dimension( ldc, * ) C, integerLDC, real, dimension( * ) WORK, integer INFO)

subroutine slaqz1 (real, dimension( lda, * ), intent(in) A, integer,intent(in) LDA, real, dimension( ldb, * ), intent(in) B, integer,intent(in) LDB, real, intent(in) SR1, real, intent(in) SR2, real,intent(in) SI, real, intent(in) BETA1, real, intent(in) BETA2, real,dimension( * ), intent(out) V)

subroutine zgelqt (integer M, integer N, integer MB, complex*16, dimension(lda, * ) A, integer LDA, complex*16, dimension( ldt, * ) T, integer LDT,complex*16, dimension( * ) WORK, integer INFO)

recursive subroutine zgelqt3 (integer M, integer N, complex*16, dimension(lda, * ) A, integer LDA, complex*16, dimension( ldt, * ) T, integer LDT,integer INFO)

subroutine zgemlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer MB, complex*16, dimension( ldv, * ) V, integer LDV,complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldc,* ) C, integer LDC, complex*16, dimension( * ) WORK, integer INFO)

Author

subroutine dgebd2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) D, double precision,dimension( * ) E, double precision, dimension( * ) TAUQ, double precision,dimension( * ) TAUP, double precision, dimension( * ) WORK, integer INFO)

subroutine dgeequ (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) R, double precision,dimension( * ) C, double precision ROWCND, double precision COLCND, doubleprecision AMAX, integer INFO)

subroutine dgeequb (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) R, double precision,dimension( * ) C, double precision ROWCND, double precision COLCND, doubleprecision AMAX, integer INFO)

subroutine dgelq2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer INFO)

subroutine dgelqf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dgeql2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer INFO)

subroutine dgeqlf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dgeqp3 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, integer, dimension( ) JPVT, double precision,dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK,integer INFO)

subroutine dgeqr2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer INFO)

subroutine dgeqr2p (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer INFO)

subroutine dgeqrf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dgeqrfp (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dgeqrt2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ldt, ) T, integer LDT,integer INFO)

subroutine dgerq2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer INFO)

subroutine dgerqf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dgetf2 (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, integer INFO)

subroutine dgetrf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, integer INFO)

subroutine sgelqt (integer M, integer N, integer MB, real, dimension( lda, ) A, integer LDA, real, dimension( ldt, ) T, integer LDT, real,dimension( * ) WORK, integer INFO)

subroutine zgelqt (integer M, integer N, integer MB, complex16, dimension(lda, ) A, integer LDA, complex16, dimension( ldt, ) T, integer LDT,complex16, dimension( ) WORK, integer INFO)

recursive subroutine zgelqt3 (integer M, integer N, complex16, dimension(lda, ) A, integer LDA, complex16, dimension( ldt, ) T, integer LDT,integer INFO)

subroutine zgemlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer MB, complex16, dimension( ldv, ) V, integer LDV,complex16, dimension( ldt, ) T, integer LDT, complex16, dimension( ldc, ) C, integer LDC, complex16, dimension( ) WORK, integer INFO)