csytri_rook(3)

complex

Section 3 liblapack-doc bookworm source

Description

complexSYcomputational

NAME

complexSYcomputational - complex

SYNOPSIS

Functions

subroutine chesv_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, WORK, LWORK, INFO)
CHESV_AA_2STAGE computes the solution to system of linear equations A * X = B for HE matrices
subroutine chetrf_aa_2stage (UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2, WORK, LWORK, INFO)
CHETRF_AA_2STAGE
subroutine chetrs_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, INFO)
CHETRS_AA_2STAGE
subroutine cla_syamv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.
real function cla_syrcond_c (UPLO, N, A, LDA, AF, LDAF, IPIV, C, CAPPLY, INFO, WORK, RWORK)
CLA_SYRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices.
real function cla_syrcond_x (UPLO, N, A, LDA, AF, LDAF, IPIV, X, INFO, WORK, RWORK)
CLA_SYRCOND_X computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices.
subroutine cla_syrfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
CLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
real function cla_syrpvgrw (UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)
CLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
subroutine clahef_aa (UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK)
CLAHEF_AA
subroutine clasyf (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
CLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
subroutine clasyf_aa (UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK)
CLASYF_AA
subroutine clasyf_rk (UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, INFO)
CLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
subroutine clasyf_rook (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
CLASYF_ROOK computes a partial factorization of a complex symmetric matrix using the bounded Bunch-Kaufman (’rook’) diagonal pivoting method.
subroutine csycon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
CSYCON
subroutine csycon_3 (UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, WORK, INFO)
CSYCON_3
subroutine csycon_rook (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
CSYCON_ROOK
subroutine csyconv (UPLO, WAY, N, A, LDA, IPIV, E, INFO)
CSYCONV
subroutine csyconvf (UPLO, WAY, N, A, LDA, E, IPIV, INFO)
CSYCONVF
subroutine csyconvf_rook (UPLO, WAY, N, A, LDA, E, IPIV, INFO)
CSYCONVF_ROOK
subroutine csyequb (UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)
CSYEQUB
subroutine csyrfs (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CSYRFS
subroutine csyrfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
CSYRFSX
subroutine csysv_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, WORK, LWORK, INFO)
CSYSV_AA_2STAGE computes the solution to system of linear equations A * X = B for SY matrices
subroutine csytf2 (UPLO, N, A, LDA, IPIV, INFO)
CSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
subroutine csytf2_rk (UPLO, N, A, LDA, E, IPIV, INFO)
CSYTF2_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
subroutine csytf2_rook (UPLO, N, A, LDA, IPIV, INFO)
CSYTF2_ROOK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (’rook’) diagonal pivoting method (unblocked algorithm).
subroutine csytrf (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
CSYTRF
subroutine csytrf_aa (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
CSYTRF_AA
subroutine csytrf_aa_2stage (UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2, WORK, LWORK, INFO)
CSYTRF_AA_2STAGE
subroutine csytrf_rk (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)
CSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
subroutine csytrf_rook (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
CSYTRF_ROOK
subroutine csytri (UPLO, N, A, LDA, IPIV, WORK, INFO)
CSYTRI
subroutine csytri2 (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
CSYTRI2
subroutine csytri2x (UPLO, N, A, LDA, IPIV, WORK, NB, INFO)
CSYTRI2X
subroutine csytri_3 (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)
CSYTRI_3
subroutine csytri_3x (UPLO, N, A, LDA, E, IPIV, WORK, NB, INFO)
CSYTRI_3X
subroutine csytri_rook (UPLO, N, A, LDA, IPIV, WORK, INFO)
CSYTRI_ROOK
subroutine csytrs (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CSYTRS
subroutine csytrs2 (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO)
CSYTRS2
subroutine csytrs_3 (UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)
CSYTRS_3
subroutine csytrs_aa (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO)
CSYTRS_AA
subroutine csytrs_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, INFO)
CSYTRS_AA_2STAGE
subroutine csytrs_rook (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CSYTRS_ROOK
subroutine ctgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
CTGSYL
subroutine ctrsyl (TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO)
CTRSYL
subroutine ctrsyl3 (TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, SWORK, LDSWORK, INFO)
CTRSYL3

Detailed Description

This is the group of complex computational functions for SY matrices

Function Documentation

subroutine chesv_aa_2stage (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, complex, dimension( * ) TB, integerLTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex,dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, integerLWORK, integer INFO)

CHESV_AA_2STAGE computes the solution to system of linear equations A * X = B for HE matrices

Purpose:

CHESV_AA_2STAGE computes the solution to a complex system of
linear equations
A * X = B,
where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
matrices.

Aasen’s 2-stage algorithm is used to factor A as
A = U**H * T * U, if UPLO = ’U’, or
A = L * T * L**H, if UPLO = ’L’,
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and T is Hermitian and band. The matrix T is
then LU-factored with partial pivoting. The factored form of A
is then used to solve the system of equations A * X = B.

This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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 COMPLEX array, dimension (LDA,N)
On entry, the hermitian matrix A. If UPLO = ’U’, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.

On exit, L is stored below (or above) the subdiaonal blocks,
when UPLO is ’L’ (or ’U’).

LDA

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

TB is COMPLEX array, dimension (LTB)
On exit, details of the LU factorization of the band matrix.

LTB

LTB is INTEGER
The size of the array TB. LTB >= 4*N, internally
used to select NB such that LTB >= (3*NB+1)*N.

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

IPIV

IPIV is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of A were interchanged with the
row and column IPIV(k).

IPIV2

IPIV2 is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of T were interchanged with the
row and column IPIV(k).

B is COMPLEX 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).

WORK

WORK is COMPLEX workspace of size LWORK

LWORK

LWORK is INTEGER
The size of WORK. LWORK >= N, internally used to select NB
such that LWORK >= N*NB.

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.
> 0: if INFO = i, band LU factorization failed on i-th column

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine chetrf_aa_2stage (character UPLO, integer N, complex, dimension(lda, * ) A, integer LDA, complex, dimension( * ) TB, integer LTB, integer,dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex, dimension( * )WORK, integer LWORK, integer INFO)

CHETRF_AA_2STAGE

Purpose:

CHETRF_AA_2STAGE computes the factorization of a real hermitian matrix A
using the Aasen’s algorithm. The form of the factorization is

A = U**T*T*U or A = L*T*L**T

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and T is a hermitian band matrix with the
bandwidth of NB (NB is internally selected and stored in TB( 1 ), and T is
LU factorized with partial pivoting).

This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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

On exit, L is stored below (or above) the subdiaonal blocks,
when UPLO is ’L’ (or ’U’).

LDA

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

TB is COMPLEX array, dimension (LTB)
On exit, details of the LU factorization of the band matrix.

LTB

LTB is INTEGER
The size of the array TB. LTB >= 4*N, internally
used to select NB such that LTB >= (3*NB+1)*N.

IPIV

IPIV is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of A were interchanged with the
row and column IPIV(k).

IPIV2

IPIV2 is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of T were interchanged with the
row and column IPIV(k).

WORK

WORK is COMPLEX workspace of size LWORK

LWORK

LWORK is INTEGER
The size of WORK. LWORK >= N, internally used to select NB
such that LWORK >= N*NB.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, band LU factorization failed on i-th column

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine chetrs_aa_2stage (character UPLO, integer N, integer NRHS,complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TB,integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2,complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CHETRS_AA_2STAGE

Purpose:

CHETRS_AA_2STAGE solves a system of linear equations A*X = B with a real
hermitian matrix A using the factorization A = U**T*T*U or
A = L*T*L**T computed by CHETRF_AA_2STAGE.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= ’U’: Upper triangular, form is A = U**T*T*U;
= ’L’: Lower triangular, form is A = L*T*L**T.

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 COMPLEX array, dimension (LDA,N)
Details of factors computed by CHETRF_AA_2STAGE.

LDA

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

TB is COMPLEX array, dimension (LTB)
Details of factors computed by CHETRF_AA_2STAGE.

LTB

LTB is INTEGER
The size of the array TB. LTB >= 4*N.

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges as computed by
CHETRF_AA_2STAGE.

IPIV2

IPIV2 is INTEGER array, dimension (N)
Details of the interchanges as computed by
CHETRF_AA_2STAGE.

B is COMPLEX 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 cla_syamv (integer UPLO, integer N, real ALPHA, complex,dimension( lda, * ) A, integer LDA, complex, dimension( * ) X, integerINCX, real BETA, real, dimension( * ) Y, integer INCY)

CLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.

Purpose:

CLA_SYAMV performs the matrix-vector operation

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

where alpha and beta are scalars, x and y are vectors and A is an
n by n symmetric 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

UPLO

UPLO is INTEGER
On entry, UPLO specifies whether the upper or lower
triangular part of the array A is to be referenced as
follows:

UPLO = BLAS_UPPER Only the upper triangular part of A
is to be referenced.

UPLO = BLAS_LOWER Only the lower triangular part of A
is to be referenced.

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 REAL .
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.

A is COMPLEX 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, n ).
Unchanged on exit.

X is COMPLEX array, dimension
( 1 + ( n - 1 )*abs( INCX ) )
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 REAL .
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 REAL array, dimension
( 1 + ( n - 1 )*abs( INCY ) )
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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

Level 2 Blas routine.

-- Written on 22-October-1986.
Jack Dongarra, Argonne National Lab.
Jeremy Du Croz, Nag Central Office.
Sven Hammarling, Nag Central Office.
Richard Hanson, Sandia National Labs.
-- Modified for the absolute-value product, April 2006
Jason Riedy, UC Berkeley

real function cla_syrcond_c (character UPLO, integer N, complex, dimension(lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF,integer, dimension( * ) IPIV, real, dimension( * ) C, logical CAPPLY,integer INFO, complex, dimension( * ) WORK, real, dimension( * ) RWORK)

CLA_SYRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices.

Purpose:

CLA_SYRCOND_C Computes the infinity norm condition number of
op(A) * inv(diag(C)) where C is a REAL vector.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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

A is COMPLEX 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 COMPLEX array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CSYTRF.

LDAF

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

C is REAL array, dimension (N)
The vector C in the formula op(A) * inv(diag(C)).

CAPPLY

CAPPLY is LOGICAL
If .TRUE. then access the vector C in the formula above.

INFO

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

WORK

WORK is COMPLEX array, dimension (2*N).
Workspace.

RWORK

RWORK is REAL array, dimension (N).
Workspace.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

real function cla_syrcond_x (character UPLO, integer N, complex, dimension(lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF,integer, dimension( * ) IPIV, complex, dimension( * ) X, integer INFO,complex, dimension( * ) WORK, real, dimension( * ) RWORK)

CLA_SYRCOND_X computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices.

Purpose:

CLA_SYRCOND_X Computes the infinity norm condition number of
op(A) * diag(X) where X is a COMPLEX vector.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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

A is COMPLEX 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 COMPLEX array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CSYTRF.

LDAF

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

X is COMPLEX array, dimension (N)
The vector X in the formula op(A) * diag(X).

INFO

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

WORK

WORK is COMPLEX array, dimension (2*N).
Workspace.

RWORK

RWORK is REAL array, dimension (N).
Workspace.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine cla_syrfsx_extended (integer PREC_TYPE, character UPLO, integer N,integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex,dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV,logical COLEQU, real, dimension( * ) C, complex, dimension( ldb, * ) B,integer LDB, complex, dimension( ldy, * ) Y, integer LDY, real, dimension(* ) BERR_OUT, integer N_NORMS, real, dimension( nrhs, * ) ERR_BNDS_NORM,real, dimension( nrhs, * ) ERR_BNDS_COMP, complex, dimension( * ) RES,real, dimension( * ) AYB, complex, dimension( * ) DY, complex, dimension( *) Y_TAIL, real RCOND, integer ITHRESH, real RTHRESH, real DZ_UB, logicalIGNORE_CWISE, integer INFO)

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

Purpose:

CLA_SYRFSX_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 CSYRFSX to perform iterative refinement.
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. Note that this
subroutine is only responsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP.

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

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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 COMPLEX 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 COMPLEX array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CSYTRF.

LDAF

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

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 REAL 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 COMPLEX 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 COMPLEX array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by CSYTRS.
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 REAL 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 CLA_LIN_BERR.

N_NORMS

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

ERR_BNDS_NORM

ERR_BNDS_NORM is REAL 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) * 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.

ERR_BNDS_COMP

ERR_BNDS_COMP is REAL 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) * slamch(’Epsilon’).

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 COMPLEX array, dimension (N)
Workspace to hold the intermediate residual.

AYB

AYB is REAL array, dimension (N)
Workspace.

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

Y_TAIL

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

RCOND

RCOND is REAL
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.

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
ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.

RTHRESH

RTHRESH is REAL
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 REAL
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 CLA_SYRFSX_EXTENDED had an illegal
value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

real function cla_syrpvgrw (character1 UPLO, integer N, integer INFO,complex, dimension( lda, ) A, integer LDA, complex, dimension( ldaf, * )AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) WORK)

CLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.

Purpose:

CLA_SYRPVGRW 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

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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

INFO

INFO is INTEGER
The value of INFO returned from CSYTRF, .i.e., the pivot in
column INFO is exactly 0.

A is COMPLEX 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 COMPLEX array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CSYTRF.

LDAF

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

WORK

WORK is REAL array, dimension (2*N)

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine clahef_aa (character UPLO, integer J1, integer M, integer NB,complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) WORK)

CLAHEF_AA

Purpose:

CLAHEF_AA factorizes a panel of a complex hermitian matrix A using
the Aasen’s algorithm. The panel consists of a set of NB rows of A
when UPLO is U, or a set of NB columns when UPLO is L.

In order to factorize the panel, the Aasen’s algorithm requires the
last row, or column, of the previous panel. The first row, or column,
of A is set to be the first row, or column, of an identity matrix,
which is used to factorize the first panel.

The resulting J-th row of U, or J-th column of L, is stored in the
(J-1)-th row, or column, of A (without the unit diagonals), while
the diagonal and subdiagonal of A are overwritten by those of T.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

J1 is INTEGER
The location of the first row, or column, of the panel
within the submatrix of A, passed to this routine, e.g.,
when called by CHETRF_AA, for the first panel, J1 is 1,
while for the remaining panels, J1 is 2.

M is INTEGER
The dimension of the submatrix. M >= 0.

NB is INTEGER
The dimension of the panel to be facotorized.

A is COMPLEX array, dimension (LDA,M) for
the first panel, while dimension (LDA,M+1) for the
remaining panels.

On entry, A contains the last row, or column, of
the previous panel, and the trailing submatrix of A
to be factorized, except for the first panel, only
the panel is passed.

On exit, the leading panel is factorized.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the row and column interchanges,
the row and column k were interchanged with the row and
column IPIV(k).

H is COMPLEX workspace, dimension (LDH,NB).

LDH

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

WORK

WORK is COMPLEX workspace, dimension (M).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine clasyf (character UPLO, integer N, integer NB, integer KB,complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,complex, dimension( ldw, * ) W, integer LDW, integer INFO)

CLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.

Purpose:

CLASYF computes a partial factorization of a complex symmetric matrix
A using the Bunch-Kaufman diagonal pivoting method. The partial
factorization has the form:

A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = ’U’, or:
( 0 U22 ) ( 0 D ) ( U12**T U22**T )

A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = ’L’
( L21 I ) ( 0 A22 ) ( 0 I )

where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
Note that U**T denotes the transpose of U.

CLASYF is an auxiliary routine called by CSYTRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = ’U’) or
A22 (if UPLO = ’L’).

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ’U’: Upper triangular
= ’L’: Lower triangular

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

NB is INTEGER
The maximum number of columns of the matrix A that should be
factored. NB should be at least 2 to allow for 2-by-2 pivot
blocks.

KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.

A is COMPLEX array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = ’U’, the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.

If UPLO = ’U’:
Only the last KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.

If UPLO = ’L’:
Only the first KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.

W is COMPLEX array, dimension (LDW,NB)

LDW

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

INFO

INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2013, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine clasyf_aa (character UPLO, integer J1, integer M, integer NB,complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) WORK)

CLASYF_AA

Purpose:

DLATRF_AA factorizes a panel of a complex symmetric matrix A using
the Aasen’s algorithm. The panel consists of a set of NB rows of A
when UPLO is U, or a set of NB columns when UPLO is L.

The resulting J-th row of U, or J-th column of L, is stored in the
(J-1)-th row, or column, of A (without the unit diagonals), while
the diagonal and subdiagonal of A are overwritten by those of T.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

J1 is INTEGER
The location of the first row, or column, of the panel
within the submatrix of A, passed to this routine, e.g.,
when called by CSYTRF_AA, for the first panel, J1 is 1,
while for the remaining panels, J1 is 2.

M is INTEGER
The dimension of the submatrix. M >= 0.

NB is INTEGER
The dimension of the panel to be facotorized.

A is COMPLEX array, dimension (LDA,M) for
the first panel, while dimension (LDA,M+1) for the
remaining panels.

On entry, A contains the last row, or column, of
the previous panel, and the trailing submatrix of A
to be factorized, except for the first panel, only
the panel is passed.

On exit, the leading panel is factorized.

LDA

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

IPIV

IPIV is INTEGER array, dimension (M)
Details of the row and column interchanges,
the row and column k were interchanged with the row and
column IPIV(k).

H is COMPLEX workspace, dimension (LDH,NB).

LDH

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

WORK

WORK is COMPLEX workspace, dimension (M).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine clasyf_rk (character UPLO, integer N, integer NB, integer KB,complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E,integer, dimension( * ) IPIV, complex, dimension( ldw, * ) W, integer LDW,integer INFO)

CLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.

Purpose:

CLASYF_RK computes a partial factorization of a complex symmetric
matrix A using the bounded Bunch-Kaufman (rook) diagonal
pivoting method. The partial factorization has the form:

A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = ’U’, or:
( 0 U22 ) ( 0 D ) ( U12**T U22**T )

A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = ’L’,
( L21 I ) ( 0 A22 ) ( 0 I )

where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.

CLASYF_RK is an auxiliary routine called by CSYTRF_RK. It uses
blocked code (calling Level 3 BLAS) to update the submatrix
A11 (if UPLO = ’U’) or A22 (if UPLO = ’L’).

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ’U’: Upper triangular
= ’L’: Lower triangular

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

NB is INTEGER
The maximum number of columns of the matrix A that should be
factored. NB should be at least 2 to allow for 2-by-2 pivot
blocks.

KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.

A is COMPLEX array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = ’U’: the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.

If UPLO = ’L’: the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.

On exit, contains:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = ’U’: factor U in the superdiagonal part of A.
If UPLO = ’L’: factor L in the subdiagonal part of A.

LDA

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

E is COMPLEX array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = ’U’: E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = ’L’: E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = ’U’ or UPLO = ’L’ cases.

IPIV

IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step.

If UPLO = ’U’,
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the submatrix A(1:N,N-KB+1:N);
If IPIV(k) = k, no interchange occurred.

b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,N-KB+1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the submatrix A(1:N,N-KB+1:N).
If -IPIV(k-1) = k-1, no interchange occurred.

c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.

d) NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = ’L’,
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the submatrix A(1:N,1:KB).
If IPIV(k) = k, no interchange occurred.

b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the submatrix A(1:N,1:KB).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the submatrix A(1:N,1:KB).
If -IPIV(k+1) = k+1, no interchange occurred.

c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.

d) NOTE: Any entry IPIV(k) is always NONZERO on output.

W is COMPLEX array, dimension (LDW,NB)

LDW

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

INFO

INFO is INTEGER
= 0: successful exit

< 0: If INFO = -k, the k-th argument had an illegal value

> 0: If INFO = k, the matrix A is singular, because:
If UPLO = ’U’: column k in the upper
triangular part of A contains all zeros.
If UPLO = ’L’: column k in the lower
triangular part of A contains all zeros.

Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.

NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine clasyf_rook (character UPLO, integer N, integer NB, integer KB,complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,complex, dimension( ldw, * ) W, integer LDW, integer INFO)

CLASYF_ROOK computes a partial factorization of a complex symmetric matrix using the bounded Bunch-Kaufman (’rook’) diagonal pivoting method.

Purpose:

CLASYF_ROOK computes a partial factorization of a complex symmetric
matrix A using the bounded Bunch-Kaufman (’rook’) diagonal
pivoting method. The partial factorization has the form:

A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = ’U’, or:
( 0 U22 ) ( 0 D ) ( U12**T U22**T )

A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = ’L’
( L21 I ) ( 0 A22 ) ( 0 I )

where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.

CLASYF_ROOK is an auxiliary routine called by CSYTRF_ROOK. It uses
blocked code (calling Level 3 BLAS) to update the submatrix
A11 (if UPLO = ’U’) or A22 (if UPLO = ’L’).

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ’U’: Upper triangular
= ’L’: Lower triangular

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

NB is INTEGER
The maximum number of columns of the matrix A that should be
factored. NB should be at least 2 to allow for 2-by-2 pivot
blocks.

KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.

If UPLO = ’U’:
Only the last KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k-1 and -IPIV(k-1) were inerchaged,
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If UPLO = ’L’:
Only the first KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k+1 and -IPIV(k+1) were inerchaged,
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

W is COMPLEX array, dimension (LDW,NB)

LDW

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

INFO

INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2013, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine csycon (character UPLO, integer N, complex, dimension( lda, * ) A,integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, complex,dimension( * ) WORK, integer INFO)

CSYCON

Purpose:

CSYCON estimates the reciprocal of the condition number (in the
1-norm) of a complex symmetric matrix A using the factorization
A = U*D*U**T or A = L*D*L**T computed by CSYTRF.

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

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= ’U’: Upper triangular, form is A = U*D*U**T;
= ’L’: Lower triangular, form is A = L*D*L**T.

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

A is COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CSYTRF.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

ANORM

ANORM is REAL
The 1-norm of the original matrix A.

RCOND

RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.

WORK

WORK is COMPLEX array, dimension (2*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 csycon_3 (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,real ANORM, real RCOND, complex, dimension( * ) WORK, integer INFO)

CSYCON_3

Purpose:

CSYCON_3 estimates the reciprocal of the condition number (in the
1-norm) of a complex symmetric matrix A using the factorization
computed by CSYTRF_RK or CSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
This routine uses BLAS3 solver CSYTRS_3.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix:
= ’U’: Upper triangular, form is A = P*U*D*(U**T)*(P**T);
= ’L’: Lower triangular, form is A = P*L*D*(L**T)*(P**T).

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

A is COMPLEX array, dimension (LDA,N)
Diagonal of the block diagonal matrix D and factors U or L
as computed by CSYTRF_RK and CSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = ’U’: factor U in the superdiagonal part of A.
If UPLO = ’L’: factor L in the subdiagonal part of A.

LDA

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

E is COMPLEX array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = ’U’: E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = ’L’: E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = ’U’ or UPLO = ’L’ cases.

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF_RK or CSYTRF_BK.

ANORM

ANORM is REAL
The 1-norm of the original matrix A.

RCOND

RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.

WORK

WORK is COMPLEX array, dimension (2*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.

Contributors:

June 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine csycon_rook (character UPLO, integer N, complex, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, real ANORM, real RCOND,complex, dimension( * ) WORK, integer INFO)

CSYCON_ROOK

Purpose:

CSYCON_ROOK estimates the reciprocal of the condition number (in the
1-norm) of a complex symmetric matrix A using the factorization
A = U*D*U**T or A = L*D*L**T computed by CSYTRF_ROOK.

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

Parameters

UPLO

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

A is COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CSYTRF_ROOK.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF_ROOK.

ANORM

ANORM is REAL
The 1-norm of the original matrix A.

RCOND

RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.

WORK

WORK is COMPLEX array, dimension (2*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.

Contributors:

April 2012, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine csyconv (character UPLO, character WAY, integer N, complex,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex,dimension( * ) E, integer INFO)

CSYCONV

Purpose:

CSYCONV convert A given by TRF into L and D and vice-versa.
Get Non-diag elements of D (returned in workspace) and
apply or reverse permutation done in TRF.

Parameters

UPLO

WAY

WAY is CHARACTER*1
= ’C’: Convert
= ’R’: Revert

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

A is COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CSYTRF.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

E is COMPLEX array, dimension (N)
E stores the supdiagonal/subdiagonal of the symmetric 1-by-1
or 2-by-2 block diagonal matrix D in LDLT.

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 csyconvf (character UPLO, character WAY, integer N, complex,dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer,dimension( * ) IPIV, integer INFO)

CSYCONVF

Purpose:

If parameter WAY = ’C’:
CSYCONVF converts the factorization output format used in
CSYTRF provided on entry in parameter A into the factorization
output format used in CSYTRF_RK (or CSYTRF_BK) that is stored
on exit in parameters A and E. It also converts in place details of
the intechanges stored in IPIV from the format used in CSYTRF into
the format used in CSYTRF_RK (or CSYTRF_BK).

If parameter WAY = ’R’:
CSYCONVF performs the conversion in reverse direction, i.e.
converts the factorization output format used in CSYTRF_RK
(or CSYTRF_BK) provided on entry in parameters A and E into
the factorization output format used in CSYTRF that is stored
on exit in parameter A. It also converts in place details of
the intechanges stored in IPIV from the format used in CSYTRF_RK
(or CSYTRF_BK) into the format used in CSYTRF.

CSYCONVF can also convert in Hermitian matrix case, i.e. between
formats used in CHETRF and CHETRF_RK (or CHETRF_BK).

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix A.
= ’U’: Upper triangular
= ’L’: Lower triangular

WAY

WAY is CHARACTER*1
= ’C’: Convert
= ’R’: Revert

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

A is COMPLEX array, dimension (LDA,N)

1) If WAY =’C’:

On entry, contains factorization details in format used in
CSYTRF:
a) all elements of the symmetric block diagonal
matrix D on the diagonal of A and on superdiagonal
(or subdiagonal) of A, and
b) If UPLO = ’U’: multipliers used to obtain factor U
in the superdiagonal part of A.
If UPLO = ’L’: multipliers used to obtain factor L
in the superdiagonal part of A.

On exit, contains factorization details in format used in
CSYTRF_RK or CSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = ’U’: factor U in the superdiagonal part of A.
If UPLO = ’L’: factor L in the subdiagonal part of A.

2) If WAY = ’R’:

On entry, contains factorization details in format used in
CSYTRF_RK or CSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = ’U’: factor U in the superdiagonal part of A.
If UPLO = ’L’: factor L in the subdiagonal part of A.

On exit, contains factorization details in format used in
CSYTRF:
a) all elements of the symmetric block diagonal
matrix D on the diagonal of A and on superdiagonal
(or subdiagonal) of A, and
b) If UPLO = ’U’: multipliers used to obtain factor U
in the superdiagonal part of A.
If UPLO = ’L’: multipliers used to obtain factor L
in the superdiagonal part of A.

LDA

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

E is COMPLEX array, dimension (N)

1) If WAY =’C’:

On entry, just a workspace.

On exit, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = ’U’: E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = ’L’: E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

2) If WAY = ’R’:

On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = ’U’: E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = ’L’: E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

On exit, is not changed

IPIV

IPIV is INTEGER array, dimension (N)

1) If WAY =’C’:
On entry, details of the interchanges and the block
structure of D in the format used in CSYTRF.
On exit, details of the interchanges and the block
structure of D in the format used in CSYTRF_RK
( or CSYTRF_BK).

1) If WAY =’R’:
On entry, details of the interchanges and the block
structure of D in the format used in CSYTRF_RK
( or CSYTRF_BK).
On exit, details of the interchanges and the block
structure of D in the format used in CSYTRF.

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 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine csyconvf_rook (character UPLO, character WAY, integer N, complex,dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer,dimension( * ) IPIV, integer INFO)

CSYCONVF_ROOK

Purpose:

If parameter WAY = ’C’:
CSYCONVF_ROOK converts the factorization output format used in
CSYTRF_ROOK provided on entry in parameter A into the factorization
output format used in CSYTRF_RK (or CSYTRF_BK) that is stored
on exit in parameters A and E. IPIV format for CSYTRF_ROOK and
CSYTRF_RK (or CSYTRF_BK) is the same and is not converted.

If parameter WAY = ’R’:
CSYCONVF_ROOK performs the conversion in reverse direction, i.e.
converts the factorization output format used in CSYTRF_RK
(or CSYTRF_BK) provided on entry in parameters A and E into
the factorization output format used in CSYTRF_ROOK that is stored
on exit in parameter A. IPIV format for CSYTRF_ROOK and
CSYTRF_RK (or CSYTRF_BK) is the same and is not converted.

CSYCONVF_ROOK can also convert in Hermitian matrix case, i.e. between
formats used in CHETRF_ROOK and CHETRF_RK (or CHETRF_BK).

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix A.
= ’U’: Upper triangular
= ’L’: Lower triangular

WAY

WAY is CHARACTER*1
= ’C’: Convert
= ’R’: Revert

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

A is COMPLEX array, dimension (LDA,N)

1) If WAY =’C’:

On entry, contains factorization details in format used in
CSYTRF_ROOK:
a) all elements of the symmetric block diagonal
matrix D on the diagonal of A and on superdiagonal
(or subdiagonal) of A, and
b) If UPLO = ’U’: multipliers used to obtain factor U
in the superdiagonal part of A.
If UPLO = ’L’: multipliers used to obtain factor L
in the superdiagonal part of A.

2) If WAY = ’R’:

On exit, contains factorization details in format used in
CSYTRF_ROOK:
a) all elements of the symmetric block diagonal
matrix D on the diagonal of A and on superdiagonal
(or subdiagonal) of A, and
b) If UPLO = ’U’: multipliers used to obtain factor U
in the superdiagonal part of A.
If UPLO = ’L’: multipliers used to obtain factor L
in the superdiagonal part of A.

LDA

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

E is COMPLEX array, dimension (N)

1) If WAY =’C’:

On entry, just a workspace.

2) If WAY = ’R’:

On exit, is not changed

IPIV

IPIV is INTEGER array, dimension (N)
On entry, details of the interchanges and the block
structure of D as determined:
1) by CSYTRF_ROOK, if WAY =’C’;
2) by CSYTRF_RK (or CSYTRF_BK), if WAY =’R’.
The IPIV format is the same for all these routines.

On exit, is not changed.

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 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine csyequb (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, complex,dimension( * ) WORK, integer INFO)

CSYEQUB

Purpose:

CSYEQUB computes row and column scalings intended to equilibrate a
symmetric matrix A (with respect to the Euclidean norm) and reduce
its condition number. The scale factors S are computed by the BIN
algorithm (see references) so that the scaled matrix B with elements
B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of
the smallest possible condition number over all possible diagonal
scalings.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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

A is COMPLEX array, dimension (LDA,N)
The N-by-N symmetric matrix whose scaling factors are to be
computed.

LDA

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

S is REAL array, dimension (N)
If INFO = 0, S contains the scale factors for A.

SCOND

SCOND is REAL
If INFO = 0, S contains the ratio of the smallest S(i) to
the largest S(i). If SCOND >= 0.1 and AMAX is neither too
large nor too small, it is not worth scaling by S.

AMAX

AMAX is REAL
Largest absolute value of any matrix element. If AMAX is
very close to overflow or very close to underflow, the
matrix should be scaled.

WORK

WORK is COMPLEX array, dimension (2*N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

References:

Livne, O.E. and Golub, G.H., ’Scaling by Binormalization’,
Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
DOI 10.1023/B:NUMA.0000016606.32820.69
Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679

subroutine csyrfs (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF,integer LDAF, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B,integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension(* ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real,dimension( * ) RWORK, integer INFO)

CSYRFS

Purpose:

CSYRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefinite, and
provides error bounds and backward error estimates for the solution.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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 COMPLEX array, dimension (LDA,N)
The symmetric matrix A. If UPLO = ’U’, the leading N-by-N
upper triangular part of A contains the upper triangular part
of the matrix A, and the strictly lower triangular part of A
is not referenced. If UPLO = ’L’, the leading N-by-N lower
triangular part of A contains the lower triangular part of
the matrix A, and the strictly upper triangular part of A is
not referenced.

LDA

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

AF is COMPLEX array, dimension (LDAF,N)
The factored form of the matrix A. AF contains the block
diagonal matrix D and the multipliers used to obtain the
factor U or L from the factorization A = U*D*U**T or
A = L*D*L**T as computed by CSYTRF.

LDAF

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

B is COMPLEX 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 COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by CSYTRS.
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 REAL 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 REAL 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 COMPLEX array, dimension (2*N)

RWORK

RWORK is REAL 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 csyrfsx (character UPLO, character EQUED, integer N, integer NRHS,complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * )AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) S,complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * )X, integer LDX, real RCOND, real, dimension( * ) BERR, integer N_ERR_BNDS,real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * )ERR_BNDS_COMP, integer NPARAMS, real, dimension( * ) PARAMS, complex,dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CSYRFSX

Purpose:

CSYRFSX improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefinite, 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 and S
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

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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
= ’Y’: Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S).
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 COMPLEX array, dimension (LDA,N)
The symmetric matrix A. If UPLO = ’U’, the leading N-by-N
upper triangular part of A contains the upper triangular
part of the matrix A, and the strictly lower triangular
part of A is not referenced. If UPLO = ’L’, the leading
N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.

LDA

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

LDAF

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

S is REAL array, dimension (N)
The scale factors for A. If EQUED = ’Y’, A is multiplied on
the left and right by diag(S). S is an input argument if FACT =
’F’; otherwise, S is an output argument. If FACT = ’F’ and EQUED
= ’Y’, each element of S must be positive. If S is output, each
element of S is a power of the radix. If S is input, each element
of S 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 COMPLEX 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 COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by CGETRS.
On exit, the improved solution matrix X.

LDX

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

RCOND

BERR

BERR is REAL 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

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.

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

ERR_BNDS_COMP

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

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

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 REAL 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: 1.0
= 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 COMPLEX array, dimension (2*N)

RWORK

RWORK is REAL array, dimension (2*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 csysv_aa_2stage (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, complex, dimension( * ) TB, integerLTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex,dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, integerLWORK, integer INFO)

CSYSV_AA_2STAGE computes the solution to system of linear equations A * X = B for SY matrices

Purpose:

CSYSV_AA_2STAGE computes the solution to a complex system of
linear equations
A * X = B,
where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
matrices.

Aasen’s 2-stage algorithm is used to factor A as
A = U**T * T * U, if UPLO = ’U’, or
A = L * T * L**T, if UPLO = ’L’,
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and T is symmetric and band. The matrix T is
then LU-factored with partial pivoting. The factored form of A
is then used to solve the system of equations A * X = B.

This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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 COMPLEX array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = ’U’, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.

On exit, L is stored below (or above) the subdiaonal blocks,
when UPLO is ’L’ (or ’U’).

LDA

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

TB is COMPLEX array, dimension (LTB)
On exit, details of the LU factorization of the band matrix.

LTB

LTB is INTEGER
The size of the array TB. LTB >= 4*N, internally
used to select NB such that LTB >= (3*NB+1)*N.

IPIV

IPIV is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of A were interchanged with the
row and column IPIV(k).

IPIV2

IPIV2 is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of T were interchanged with the
row and column IPIV(k).

B is COMPLEX 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).

WORK

WORK is COMPLEX workspace of size LWORK

LWORK

LWORK is INTEGER
The size of WORK. LWORK >= N, internally used to select NB
such that LWORK >= N*NB.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, band LU factorization failed on i-th column

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine csytf2 (character UPLO, integer N, complex, dimension( lda, * ) A,integer LDA, integer, dimension( * ) IPIV, integer INFO)

CSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

Purpose:

CSYTF2 computes the factorization of a complex symmetric matrix A
using the Bunch-Kaufman diagonal pivoting method:

A = U*D*U**T or A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, U**T is the transpose of U, and D is symmetric and
block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the unblocked version of the algorithm, calling Level 2 BLAS.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ’U’: Upper triangular
= ’L’: Lower triangular

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

On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.

If UPLO = ’U’:
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.

If UPLO = ’L’:
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D 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.

Further Details:

If UPLO = ’U’, then A = U*D*U**T, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = ’L’, then A = L*D*L**T, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:

09-29-06 - patch from
Bobby Cheng, MathWorks

Replace l.209 and l.377
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
by
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN

1-96 - Based on modifications by J. Lewis, Boeing Computer Services
Company

subroutine csytf2_rk (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,integer INFO)

CSYTF2_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).

Purpose:

CSYTF2_RK computes the factorization of a complex symmetric matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

This is the unblocked version of the algorithm, calling Level 2 BLAS.
For more information see Further Details section.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ’U’: Upper triangular
= ’L’: Lower triangular

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

If UPLO = ’L’: the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.

LDA

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

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = ’U’ or UPLO = ’L’ cases.

IPIV

If UPLO = ’U’,
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.

b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.

c) In both cases a) and b), always ABS( IPIV(k) ) <= k.

d) NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = ’L’,
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.

b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.

c) In both cases a) and b), always ABS( IPIV(k) ) >= k.

d) NOTE: Any entry IPIV(k) is always NONZERO on output.

INFO

INFO is INTEGER
= 0: successful exit

< 0: If INFO = -k, the k-th argument had an illegal value

NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

TODO: put further details

Contributors:

December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept.,
Univ. of Tenn., Knoxville abd , USA

subroutine csytf2_rook (character UPLO, integer N, complex, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, integer INFO)

CSYTF2_ROOK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (’rook’) diagonal pivoting method (unblocked algorithm).

Purpose:

CSYTF2_ROOK computes the factorization of a complex symmetric matrix A
using the bounded Bunch-Kaufman (’rook’) diagonal pivoting method:

A = U*D*U**T or A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, U**T is the transpose of U, and D is symmetric and
block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the unblocked version of the algorithm, calling Level 2 BLAS.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ’U’: Upper triangular
= ’L’: Lower triangular

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

On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.

If UPLO = ’U’:
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k-1 and -IPIV(k-1) were inerchaged,
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If UPLO = ’L’:
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k+1 and -IPIV(k+1) were inerchaged,
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).

( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:

November 2013, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA

subroutine csytrf (character UPLO, integer N, complex, dimension( lda, * ) A,integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK,integer LWORK, integer INFO)

CSYTRF

Purpose:

CSYTRF computes the factorization of a complex symmetric matrix A
using the Bunch-Kaufman diagonal pivoting method. The form of the
factorization is

A = U*D*U**T or A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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

On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = ’U’ and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = ’L’ and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

WORK

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

LWORK

LWORK is INTEGER
The length of WORK. LWORK >=1. For best performance
LWORK >= N*NB, where NB is the block size returned by ILAENV.

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
> 0: if INFO = i, D(i,i) is exactly zero. The factorization
has been completed, but the block diagonal matrix D 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.

Further Details:

( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).

( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

subroutine csytrf_aa (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK,integer LWORK, integer INFO)

CSYTRF_AA

Purpose:

CSYTRF_AA computes the factorization of a complex symmetric matrix A
using the Aasen’s algorithm. The form of the factorization is

A = U**T*T*U or A = L*T*L**T

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and T is a complex symmetric tridiagonal matrix.

This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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

On exit, the tridiagonal matrix is stored in the diagonals
and the subdiagonals of A just below (or above) the diagonals,
and L is stored below (or above) the subdiaonals, when UPLO
is ’L’ (or ’U’).

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of A were interchanged with the
row and column IPIV(k).

WORK

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

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= MAX(1,2*N). For optimum performance
LWORK >= N*(1+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.

subroutine csytrf_aa_2stage (character UPLO, integer N, complex, dimension(lda, * ) A, integer LDA, complex, dimension( * ) TB, integer LTB, integer,dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex, dimension( * )WORK, integer LWORK, integer INFO)

CSYTRF_AA_2STAGE

Purpose:

CSYTRF_AA_2STAGE computes the factorization of a complex symmetric matrix A
using the Aasen’s algorithm. The form of the factorization is

A = U**T*T*U or A = L*T*L**T

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and T is a complex symmetric band matrix with the
bandwidth of NB (NB is internally selected and stored in TB( 1 ), and T is
LU factorized with partial pivoting).

This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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

On exit, L is stored below (or above) the subdiaonal blocks,
when UPLO is ’L’ (or ’U’).

LDA

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

TB is COMPLEX array, dimension (LTB)
On exit, details of the LU factorization of the band matrix.

LTB

LTB is INTEGER
The size of the array TB. LTB >= 4*N, internally
used to select NB such that LTB >= (3*NB+1)*N.

IPIV

IPIV is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of A were interchanged with the
row and column IPIV(k).

IPIV2

IPIV2 is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of T were interchanged with the
row and column IPIV(k).

WORK

WORK is COMPLEX workspace of size LWORK

LWORK

LWORK is INTEGER
The size of WORK. LWORK >= N, internally used to select NB
such that LWORK >= N*NB.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, band LU factorization failed on i-th column

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine csytrf_rk (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,complex, dimension( * ) WORK, integer LWORK, integer INFO)

CSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

Purpose:

CSYTRF_RK computes the factorization of a complex symmetric matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

This is the blocked version of the algorithm, calling Level 3 BLAS.
For more information see Further Details section.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ’U’: Upper triangular
= ’L’: Lower triangular

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

If UPLO = ’L’: the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.

LDA

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

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = ’U’ or UPLO = ’L’ cases.

IPIV

If UPLO = ’U’,
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.

b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.

c) In both cases a) and b), always ABS( IPIV(k) ) <= k.

d) NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = ’L’,
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.

b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.

c) In both cases a) and b), always ABS( IPIV(k) ) >= k.

d) NOTE: Any entry IPIV(k) is always NONZERO on output.

WORK

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

LWORK

LWORK is INTEGER
The length of WORK. LWORK >=1. For best performance
LWORK >= N*NB, where NB is the block size returned
by ILAENV.

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 = -k, the k-th argument had an illegal value

NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

TODO: put correct description

Contributors:

December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine csytrf_rook (character UPLO, integer N, complex, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, complex, dimension( * )WORK, integer LWORK, integer INFO)

CSYTRF_ROOK

Purpose:

CSYTRF_ROOK computes the factorization of a complex symmetric matrix A
using the bounded Bunch-Kaufman (’rook’) diagonal pivoting method.
The form of the factorization is

A = U*D*U**T or A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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

On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.

If UPLO = ’U’:
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k-1 and -IPIV(k-1) were inerchaged,
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If UPLO = ’L’:
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k+1 and -IPIV(k+1) were inerchaged,
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

WORK

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

LWORK

LWORK is INTEGER
The length of WORK. LWORK >=1. For best performance
LWORK >= N*NB, where NB is the block size returned by ILAENV.

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).

( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:

June 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine csytri (character UPLO, integer N, complex, dimension( lda, * ) A,integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK,integer INFO)

CSYTRI

Purpose:

CSYTRI computes the inverse of a complex symmetric indefinite matrix
A using the factorization A = U*D*U**T or A = L*D*L**T computed by
CSYTRF.

Parameters

UPLO

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

A is COMPLEX array, dimension (LDA,N)
On entry, the block diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by CSYTRF.

On exit, if INFO = 0, the (symmetric) inverse of the original
matrix. If UPLO = ’U’, the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = ’L’ the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

WORK

WORK is COMPLEX array, dimension (2*N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; 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 csytri2 (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK,integer LWORK, integer INFO)

CSYTRI2

Purpose:

CSYTRI2 computes the inverse of a COMPLEX symmetric indefinite matrix
A using the factorization A = U*D*U**T or A = L*D*L**T computed by
CSYTRF. CSYTRI2 sets the LEADING DIMENSION of the workspace
before calling CSYTRI2X that actually computes the inverse.

Parameters

UPLO

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

A is COMPLEX array, dimension (LDA,N)
On entry, the block diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by CSYTRF.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

WORK

WORK is COMPLEX array, dimension (N+NB+1)*(NB+3)

LWORK

LWORK is INTEGER
The dimension of the array WORK.
WORK is size >= (N+NB+1)*(NB+3)
If LWORK = -1, then a workspace query is assumed; the routine
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
> 0: if INFO = i, D(i,i) = 0; 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 csytri2x (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( n+nb+1,*) WORK, integer NB, integer INFO)

CSYTRI2X

Purpose:

CSYTRI2X computes the inverse of a real symmetric indefinite matrix
A using the factorization A = U*D*U**T or A = L*D*L**T computed by
CSYTRF.

Parameters

UPLO

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

A is COMPLEX array, dimension (LDA,N)
On entry, the NNB diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by CSYTRF.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the NNB structure of D
as determined by CSYTRF.

WORK

WORK is COMPLEX array, dimension (N+NB+1,NB+3)

NB is INTEGER
Block size

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; 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 csytri_3 (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,complex, dimension( * ) WORK, integer LWORK, integer INFO)

CSYTRI_3

Purpose:

CSYTRI_3 computes the inverse of a complex symmetric indefinite
matrix A using the factorization computed by CSYTRF_RK or CSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

CSYTRI_3 sets the leading dimension of the workspace before calling
CSYTRI_3X that actually computes the inverse. This is the blocked
version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix.
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

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

A is COMPLEX array, dimension (LDA,N)
On entry, diagonal of the block diagonal matrix D and
factors U or L as computed by CSYTRF_RK and CSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = ’U’: factor U in the superdiagonal part of A.
If UPLO = ’L’: factor L in the subdiagonal part of A.

On exit, if INFO = 0, the symmetric inverse of the original
matrix.
If UPLO = ’U’: the upper triangular part of the inverse
is formed and the part of A below the diagonal is not
referenced;
If UPLO = ’L’: the lower triangular part of the inverse
is formed and the part of A above the diagonal is not
referenced.

LDA

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

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = ’U’ or UPLO = ’L’ cases.

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF_RK or CSYTRF_BK.

WORK

WORK is COMPLEX array, dimension (N+NB+1)*(NB+3).
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= (N+NB+1)*(NB+3).

If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of 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
> 0: if INFO = i, D(i,i) = 0; 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.

Contributors:

November 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine csytri_3x (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,complex, dimension( n+nb+1, * ) WORK, integer NB, integer INFO)

CSYTRI_3X

Purpose:

CSYTRI_3X computes the inverse of a complex symmetric indefinite
matrix A using the factorization computed by CSYTRF_RK or CSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO

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

LDA

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

E is COMPLEX array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = ’U’: E(i) = D(i-1,i), i=2:N, E(1) not referenced;
If UPLO = ’L’: E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = ’U’ or UPLO = ’L’ cases.

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF_RK or CSYTRF_BK.

WORK

WORK is COMPLEX array, dimension (N+NB+1,NB+3).

NB is INTEGER
Block size.

INFO

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

Contributors:

June 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine csytri_rook (character UPLO, integer N, complex, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, complex, dimension( * )WORK, integer INFO)

CSYTRI_ROOK

Purpose:

CSYTRI_ROOK computes the inverse of a complex symmetric
matrix A using the factorization A = U*D*U**T or A = L*D*L**T
computed by CSYTRF_ROOK.

Parameters

UPLO

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

A is COMPLEX array, dimension (LDA,N)
On entry, the block diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by CSYTRF_ROOK.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF_ROOK.

WORK

WORK is COMPLEX array, dimension (N)

INFO

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

Contributors:

December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine csytrs (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex,dimension( ldb, * ) B, integer LDB, integer INFO)

CSYTRS

Purpose:

CSYTRS solves a system of linear equations A*X = B with a complex
symmetric matrix A using the factorization A = U*D*U**T or
A = L*D*L**T computed by CSYTRF.

Parameters

UPLO

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 COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CSYTRF.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

B is COMPLEX 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 csytrs2 (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex,dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, integerINFO)

CSYTRS2

Purpose:

CSYTRS2 solves a system of linear equations A*X = B with a complex
symmetric matrix A using the factorization A = U*D*U**T or
A = L*D*L**T computed by CSYTRF and converted by CSYCONV.

Parameters

UPLO

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 COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CSYTRF.
Note that A is input / output. This might be counter-intuitive,
and one may think that A is input only. A is input / output. This
is because, at the start of the subroutine, we permute A in a
’better’ form and then we permute A back to its original form at
the end.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.

B is COMPLEX 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).

WORK

WORK is COMPLEX 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 csytrs_3 (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer,dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integerINFO)

CSYTRS_3

Purpose:

CSYTRS_3 solves a system of linear equations A * X = B with a complex
symmetric matrix A using the factorization computed
by CSYTRF_RK or CSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

This algorithm is using Level 3 BLAS.

Parameters

UPLO

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.

LDA

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

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = ’U’ or UPLO = ’L’ cases.

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF_RK or CSYTRF_BK.

B is COMPLEX 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.

Contributors:

June 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine csytrs_aa (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex,dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, integerLWORK, integer INFO)

CSYTRS_AA

Purpose:

CSYTRS_AA solves a system of linear equations A*X = B with a complex
symmetric matrix A using the factorization A = U**T*T*U or
A = L*T*L**T computed by CSYTRF_AA.

Parameters

UPLO

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 COMPLEX array, dimension (LDA,N)
Details of factors computed by CSYTRF_AA.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges as computed by CSYTRF_AA.

B is COMPLEX 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).

WORK

WORK is COMPLEX array, dimension (MAX(1,LWORK))

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,3*N-2).

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 csytrs_aa_2stage (character UPLO, integer N, integer NRHS,complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TB,integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2,complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CSYTRS_AA_2STAGE

Purpose:

CSYTRS_AA_2STAGE solves a system of linear equations A*X = B with a complex
symmetric matrix A using the factorization A = U**T*T*U or
A = L*T*L**T computed by CSYTRF_AA_2STAGE.

Parameters

UPLO

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 COMPLEX array, dimension (LDA,N)
Details of factors computed by CSYTRF_AA_2STAGE.

LDA

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

TB is COMPLEX array, dimension (LTB)
Details of factors computed by CSYTRF_AA_2STAGE.

LTB

LTB is INTEGER
The size of the array TB. LTB >= 4*N.

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges as computed by
CSYTRF_AA_2STAGE.

IPIV2

IPIV2 is INTEGER array, dimension (N)
Details of the interchanges as computed by
CSYTRF_AA_2STAGE.

B is COMPLEX 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 csytrs_rook (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex,dimension( ldb, * ) B, integer LDB, integer INFO)

CSYTRS_ROOK

Purpose:

CSYTRS_ROOK solves a system of linear equations A*X = B with
a complex symmetric matrix A using the factorization A = U*D*U**T or
A = L*D*L**T computed by CSYTRF_ROOK.

Parameters

UPLO

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 COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CSYTRF_ROOK.

LDA

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

IPIV

IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF_ROOK.

B is COMPLEX 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.

Contributors:

December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine ctgsyl (character TRANS, integer IJOB, integer M, integer N,complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * )B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, complex,dimension( ldd, * ) D, integer LDD, complex, dimension( lde, * ) E, integerLDE, complex, dimension( ldf, * ) F, integer LDF, real SCALE, real DIF,complex, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK,integer INFO)

CTGSYL

Purpose:

CTGSYL solves the generalized Sylvester equation:

A * R - L * B = scale * C (1)
D * R - L * E = scale * F

where R and L are unknown m-by-n matrices, (A, D), (B, E) and
(C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,
respectively, with complex entries. A, B, D and E are upper
triangular (i.e., (A,D) and (B,E) in generalized Schur form).

The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1
is an output scaling factor chosen to avoid overflow.

In matrix notation (1) is equivalent to solve Zx = scale*b, where Z
is defined as

Z = [ kron(In, A) -kron(B**H, Im) ] (2)
[ kron(In, D) -kron(E**H, Im) ],

Here Ix is the identity matrix of size x and X**H is the conjugate
transpose of X. Kron(X, Y) is the Kronecker product between the
matrices X and Y.

If TRANS = ’C’, y in the conjugate transposed system Z**H *y = scale*b
is solved for, which is equivalent to solve for R and L in

A**H * R + D**H * L = scale * C (3)
R * B**H + L * E**H = scale * -F

This case (TRANS = ’C’) is used to compute an one-norm-based estimate
of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
and (B,E), using CLACON.

If IJOB >= 1, CTGSYL computes a Frobenius norm-based estimate of
Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
reciprocal of the smallest singular value of Z.

This is a level-3 BLAS algorithm.

Parameters

TRANS

TRANS is CHARACTER*1
= ’N’: solve the generalized sylvester equation (1).
= ’C’: solve the ’conjugate transposed’ system (3).

IJOB

IJOB is INTEGER
Specifies what kind of functionality to be performed.
=0: solve (1) only.
=1: The functionality of 0 and 3.
=2: The functionality of 0 and 4.
=3: Only an estimate of Dif[(A,D), (B,E)] is computed.
(look ahead strategy is used).
=4: Only an estimate of Dif[(A,D), (B,E)] is computed.
(CGECON on sub-systems is used).
Not referenced if TRANS = ’C’.

M is INTEGER
The order of the matrices A and D, and the row dimension of
the matrices C, F, R and L.

N is INTEGER
The order of the matrices B and E, and the column dimension
of the matrices C, F, R and L.

A is COMPLEX array, dimension (LDA, M)
The upper triangular matrix A.

LDA

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

B is COMPLEX array, dimension (LDB, N)
The upper triangular matrix B.

LDB

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

C is COMPLEX array, dimension (LDC, N)
On entry, C contains the right-hand-side of the first matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, C has been overwritten by
the solution R. If IJOB = 3 or 4 and TRANS = ’N’, C holds R,
the solution achieved during the computation of the
Dif-estimate.

LDC

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

D is COMPLEX array, dimension (LDD, M)
The upper triangular matrix D.

LDD

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

E is COMPLEX array, dimension (LDE, N)
The upper triangular matrix E.

LDE

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

F is COMPLEX array, dimension (LDF, N)
On entry, F contains the right-hand-side of the second matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, F has been overwritten by
the solution L. If IJOB = 3 or 4 and TRANS = ’N’, F holds L,
the solution achieved during the computation of the
Dif-estimate.

LDF

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

DIF

DIF is REAL
On exit DIF is the reciprocal of a lower bound of the
reciprocal of the Dif-function, i.e. DIF is an upper bound of
Dif[(A,D), (B,E)] = sigma-min(Z), where Z as in (2).
IF IJOB = 0 or TRANS = ’C’, DIF is not referenced.

SCALE

SCALE is REAL
On exit SCALE is the scaling factor in (1) or (3).
If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
to a slightly perturbed system but the input matrices A, B,
D and E have not been changed. If SCALE = 0, R and L will
hold the solutions to the homogeneous system with C = F = 0.

WORK

WORK is COMPLEX 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 IJOB = 1 or 2 and TRANS = ’N’, LWORK >= max(1,2*M*N).

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.

IWORK

IWORK is INTEGER array, dimension (M+N+2)

INFO

INFO is INTEGER
=0: successful exit
<0: If INFO = -i, the i-th argument had an illegal value.
>0: (A, D) and (B, E) have common or very close
eigenvalues.

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 and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal. Appl., 15(4):1045-1060, 1994.
[3] B. Kagstrom and L. Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

subroutine ctrsyl (character TRANA, character TRANB, integer ISGN, integer M,integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension(ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, realSCALE, integer INFO)

CTRSYL

Purpose:

CTRSYL solves the complex Sylvester matrix equation:

op(A)*X + X*op(B) = scale*C or
op(A)*X - X*op(B) = scale*C,

where op(A) = A or A**H, and A and B are both upper triangular. A is
M-by-M and B is N-by-N; the right hand side C and the solution X are
M-by-N; and scale is an output scale factor, set <= 1 to avoid
overflow in X.

Parameters

TRANA

TRANA is CHARACTER*1
Specifies the option op(A):
= ’N’: op(A) = A (No transpose)
= ’C’: op(A) = A**H (Conjugate transpose)

TRANB

TRANB is CHARACTER*1
Specifies the option op(B):
= ’N’: op(B) = B (No transpose)
= ’C’: op(B) = B**H (Conjugate transpose)

ISGN

ISGN is INTEGER
Specifies the sign in the equation:
= +1: solve op(A)*X + X*op(B) = scale*C
= -1: solve op(A)*X - X*op(B) = scale*C

M is INTEGER
The order of the matrix A, and the number of rows in the
matrices X and C. M >= 0.

N is INTEGER
The order of the matrix B, and the number of columns in the
matrices X and C. N >= 0.

A is COMPLEX array, dimension (LDA,M)
The upper triangular matrix A.

LDA

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

B is COMPLEX array, dimension (LDB,N)
The upper triangular matrix B.

LDB

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

C is COMPLEX array, dimension (LDC,N)
On entry, the M-by-N right hand side matrix C.
On exit, C is overwritten by the solution matrix X.

LDC

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

SCALE

SCALE is REAL
The scale factor, scale, set <= 1 to avoid overflow in X.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: A and B have common or very close eigenvalues; perturbed
values were used to solve the equation (but the matrices
A and B are unchanged).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine ctrsyl3 (character TRANA, character TRANB, integer ISGN, integerM, integer N, complex, dimension( lda, * ) A, integer LDA, complex,dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integerLDC, real SCALE, real, dimension( ldswork, * ) SWORK, integer LDSWORK,integer INFO)

CTRSYL3

Purpose

CTRSYL3 solves the complex Sylvester matrix equation:

op(A)*X + X*op(B) = scale*C or
op(A)*X - X*op(B) = scale*C,

This is the block version of the algorithm.

Parameters

TRANA

TRANA is CHARACTER*1
Specifies the option op(A):
= ’N’: op(A) = A (No transpose)
= ’C’: op(A) = A**H (Conjugate transpose)

TRANB

TRANB is CHARACTER*1
Specifies the option op(B):
= ’N’: op(B) = B (No transpose)
= ’C’: op(B) = B**H (Conjugate transpose)

ISGN

ISGN is INTEGER
Specifies the sign in the equation:
= +1: solve op(A)*X + X*op(B) = scale*C
= -1: solve op(A)*X - X*op(B) = scale*C

M is INTEGER
The order of the matrix A, and the number of rows in the
matrices X and C. M >= 0.

N is INTEGER
The order of the matrix B, and the number of columns in the
matrices X and C. N >= 0.

A is COMPLEX array, dimension (LDA,M)
The upper triangular matrix A.

LDA

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

B is COMPLEX array, dimension (LDB,N)
The upper triangular matrix B.

LDB

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

C is COMPLEX array, dimension (LDC,N)
On entry, the M-by-N right hand side matrix C.
On exit, C is overwritten by the solution matrix X.

LDC

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

SCALE

SCALE is REAL
The scale factor, scale, set <= 1 to avoid overflow in X.

SWORK

SWORK is REAL array, dimension (MAX(2, ROWS), MAX(1,COLS)).
On exit, if INFO = 0, SWORK(1) returns the optimal value ROWS
and SWORK(2) returns the optimal COLS.

LDSWORK

LDSWORK is INTEGER
LDSWORK >= MAX(2,ROWS), where ROWS = ((M + NB - 1) / NB + 1)
and NB is the optimal block size.

If LDSWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal dimensions of the SWORK matrix,
returns these values as the first and second entry of the SWORK
matrix, and no error message related LWORK is issued by XERBLA.

INFO

Author

Generated automatically by Doxygen for LAPACK from the source code.

Description

complexSYcomputational

NAME

SYNOPSIS

Functions

Detailed Description

Function Documentation

subroutine chetrf_aa_2stage (character UPLO, integer N, complex, dimension(lda, * ) A, integer LDA, complex, dimension( * ) TB, integer LTB, integer,dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex, dimension( * )WORK, integer LWORK, integer INFO)

subroutine chetrs_aa_2stage (character UPLO, integer N, integer NRHS,complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TB,integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2,complex, dimension( ldb, * ) B, integer LDB, integer INFO)

subroutine cla_syamv (integer UPLO, integer N, real ALPHA, complex,dimension( lda, * ) A, integer LDA, complex, dimension( * ) X, integerINCX, real BETA, real, dimension( * ) Y, integer INCY)

real function cla_syrcond_c (character UPLO, integer N, complex, dimension(lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF,integer, dimension( * ) IPIV, real, dimension( * ) C, logical CAPPLY,integer INFO, complex, dimension( * ) WORK, real, dimension( * ) RWORK)

real function cla_syrcond_x (character UPLO, integer N, complex, dimension(lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF,integer, dimension( * ) IPIV, complex, dimension( * ) X, integer INFO,complex, dimension( * ) WORK, real, dimension( * ) RWORK)

real function cla_syrpvgrw (character*1 UPLO, integer N, integer INFO,complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * )AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) WORK)

subroutine clahef_aa (character UPLO, integer J1, integer M, integer NB,complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) WORK)

subroutine clasyf (character UPLO, integer N, integer NB, integer KB,complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,complex, dimension( ldw, * ) W, integer LDW, integer INFO)

subroutine clasyf_aa (character UPLO, integer J1, integer M, integer NB,complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) WORK)

subroutine clasyf_rk (character UPLO, integer N, integer NB, integer KB,complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E,integer, dimension( * ) IPIV, complex, dimension( ldw, * ) W, integer LDW,integer INFO)

subroutine clasyf_rook (character UPLO, integer N, integer NB, integer KB,complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,complex, dimension( ldw, * ) W, integer LDW, integer INFO)

subroutine csycon (character UPLO, integer N, complex, dimension( lda, * ) A,integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, complex,dimension( * ) WORK, integer INFO)

subroutine csycon_3 (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,real ANORM, real RCOND, complex, dimension( * ) WORK, integer INFO)

subroutine csycon_rook (character UPLO, integer N, complex, dimension( lda, *) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND,complex, dimension( * ) WORK, integer INFO)

subroutine csyconv (character UPLO, character WAY, integer N, complex,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex,dimension( * ) E, integer INFO)

subroutine csyconvf (character UPLO, character WAY, integer N, complex,dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer,dimension( * ) IPIV, integer INFO)

subroutine csyconvf_rook (character UPLO, character WAY, integer N, complex,dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer,dimension( * ) IPIV, integer INFO)

subroutine csyequb (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, complex,dimension( * ) WORK, integer INFO)

subroutine csytf2 (character UPLO, integer N, complex, dimension( lda, * ) A,integer LDA, integer, dimension( * ) IPIV, integer INFO)

subroutine csytf2_rk (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,integer INFO)

subroutine csytf2_rook (character UPLO, integer N, complex, dimension( lda, *) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

subroutine csytrf (character UPLO, integer N, complex, dimension( lda, * ) A,integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK,integer LWORK, integer INFO)

subroutine csytrf_aa (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK,integer LWORK, integer INFO)

subroutine csytrf_aa_2stage (character UPLO, integer N, complex, dimension(lda, * ) A, integer LDA, complex, dimension( * ) TB, integer LTB, integer,dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex, dimension( * )WORK, integer LWORK, integer INFO)

subroutine csytrf_rk (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,complex, dimension( * ) WORK, integer LWORK, integer INFO)

subroutine csytrf_rook (character UPLO, integer N, complex, dimension( lda, *) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * )WORK, integer LWORK, integer INFO)

subroutine csytri (character UPLO, integer N, complex, dimension( lda, * ) A,integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK,integer INFO)

subroutine csytri2 (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK,integer LWORK, integer INFO)

subroutine csytri2x (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( n+nb+1,*) WORK, integer NB, integer INFO)

subroutine csytri_3 (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,complex, dimension( * ) WORK, integer LWORK, integer INFO)

subroutine csytri_3x (character UPLO, integer N, complex, dimension( lda, * )A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,complex, dimension( n+nb+1, * ) WORK, integer NB, integer INFO)

subroutine csytri_rook (character UPLO, integer N, complex, dimension( lda, *) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * )WORK, integer INFO)

subroutine csytrs (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex,dimension( ldb, * ) B, integer LDB, integer INFO)

subroutine csytrs2 (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex,dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, integerINFO)

subroutine csytrs_3 (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer,dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integerINFO)

subroutine csytrs_aa (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex,dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, integerLWORK, integer INFO)

subroutine csytrs_aa_2stage (character UPLO, integer N, integer NRHS,complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TB,integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2,complex, dimension( ldb, * ) B, integer LDB, integer INFO)

subroutine csytrs_rook (character UPLO, integer N, integer NRHS, complex,dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex,dimension( ldb, * ) B, integer LDB, integer INFO)

subroutine ctrsyl (character TRANA, character TRANB, integer ISGN, integer M,integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension(ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, realSCALE, integer INFO)

Author

real function cla_syrpvgrw (character1 UPLO, integer N, integer INFO,complex, dimension( lda, ) A, integer LDA, complex, dimension( ldaf, * )AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) WORK)

subroutine csycon_rook (character UPLO, integer N, complex, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, real ANORM, real RCOND,complex, dimension( * ) WORK, integer INFO)

subroutine csytf2_rook (character UPLO, integer N, complex, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, integer INFO)

subroutine csytrf_rook (character UPLO, integer N, complex, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, complex, dimension( * )WORK, integer LWORK, integer INFO)

subroutine csytri_rook (character UPLO, integer N, complex, dimension( lda, ) A, integer LDA, integer, dimension( ) IPIV, complex, dimension( * )WORK, integer INFO)