Man page - hesv_rk(3)

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Manual

hesv_rk

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine chesv_rk (character uplo, integer n, integer nrhs, complex,dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer,dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb,complex, dimension( * ) work, integer lwork, integer info)
subroutine csysv_rk (character uplo, integer n, integer nrhs, complex,dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer,dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb,complex, dimension( * ) work, integer lwork, integer info)
subroutine dsysv_rk (character uplo, integer n, integer nrhs, doubleprecision, dimension( lda, * ) a, integer lda, double precision,dimension( * ) e, integer, dimension( * ) ipiv, double precision,dimension( ldb, * ) b, integer ldb, double precision, dimension( * )work, integer lwork, integer info)
subroutine ssysv_rk (character uplo, integer n, integer nrhs, real,dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer,dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, real,dimension( * ) work, integer lwork, integer info)
subroutine zhesv_rk (character uplo, integer n, integer nrhs, complex*16,dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e,integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b,integer ldb, complex*16, dimension( * ) work, integer lwork, integerinfo)
subroutine zsysv_rk (character uplo, integer n, integer nrhs, complex*16,dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e,integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b,integer ldb, complex*16, dimension( * ) work, integer lwork, integerinfo)
Author

NAME

hesv_rk - {he,sy}sv_rk: rook (v3)

SYNOPSIS

Functions

subroutine chesv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)
CHESV_RK computes the solution to system of linear equations A * X = B for SY matrices
subroutine csysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)
CSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices
subroutine dsysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)
DSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices
subroutine ssysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)
SSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices
subroutine zhesv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)
ZHESV_RK computes the solution to system of linear equations A * X = B for SY matrices
subroutine zsysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)
ZSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices

Detailed Description

Function Documentation

subroutine chesv_rk (character uplo, integer n, integer nrhs, complex,dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer,dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb,complex, dimension( * ) work, integer lwork, integer info)

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

Purpose:

CHESV_RK 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.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
to factor A as
A = P*U*D*(U**H)*(P**T), if UPLO = ā€™Uā€™, or
A = P*L*D*(L**H)*(P**T), if UPLO = ā€™Lā€™,
where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

CHETRF_RK is called to compute the factorization of a complex
Hermitian matrix. The factored form of A is then used to solve
the system of equations A * X = B by calling BLAS3 routine CHETRS_3.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

N

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. NRHS >= 0.

A

A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A.
If UPLO = ā€™Uā€™: 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 = ā€™Lā€™: 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, if INFO = 0, diagonal of the block diagonal
matrix D and factors U or L as computed by CHETRF_RK:
a) ONLY diagonal elements of the Hermitian 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 = ā€™Uā€™: factor U in the superdiagonal part of A.
If UPLO = ā€™Lā€™: factor L in the subdiagonal part of A.

For more info see the description of CHETRF_RK routine.

LDA

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

E

E is COMPLEX array, dimension (N)
On exit, contains the output computed by the factorization
routine CHETRF_RK, i.e. the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = ā€™Uā€™: E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = ā€™Lā€™: 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 = ā€™Uā€™ or UPLO = ā€™Lā€™ cases.

For more info see the description of CHETRF_RK routine.

IPIV

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

For more info see the description of CHETRF_RK routine.

B

B is COMPLEX array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS 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) ).
Work array used in the factorization stage.
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= 1. For best performance
of factorization stage LWORK >= max(1,N*NB), where NB is
the optimal blocksize for CHETRF_RK.

If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the WORK
array for factorization stage, 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

> 0: If INFO = k, the matrix A is singular, because:
If UPLO = ā€™Uā€™: column k in the upper
triangular part of A contains all zeros.
If UPLO = ā€™Lā€™: 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 csysv_rk (character uplo, integer n, integer nrhs, complex,dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer,dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb,complex, dimension( * ) work, integer lwork, integer info)

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

Purpose:

CSYSV_RK 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.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
to factor A as
A = P*U*D*(U**T)*(P**T), if UPLO = ā€™Uā€™, or
A = P*L*D*(L**T)*(P**T), if UPLO = ā€™Lā€™,
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.

CSYTRF_RK is called to compute the factorization of a complex
symmetric matrix. The factored form of A is then used to solve
the system of equations A * X = B by calling BLAS3 routine CSYTRS_3.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

N

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. NRHS >= 0.

A

A is COMPLEX array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = ā€™Uā€™: 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 = ā€™Lā€™: 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, if INFO = 0, diagonal of the block diagonal
matrix D and factors U or L as computed by CSYTRF_RK:
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 = ā€™Uā€™: factor U in the superdiagonal part of A.
If UPLO = ā€™Lā€™: factor L in the subdiagonal part of A.

For more info see the description of CSYTRF_RK routine.

LDA

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

E

E is COMPLEX array, dimension (N)
On exit, contains the output computed by the factorization
routine CSYTRF_RK, i.e. 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 = ā€™Uā€™: E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = ā€™Lā€™: 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 = ā€™Uā€™ or UPLO = ā€™Lā€™ cases.

For more info see the description of CSYTRF_RK routine.

IPIV

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

For more info see the description of CSYTRF_RK routine.

B

B is COMPLEX array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS 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) ).
Work array used in the factorization stage.
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= 1. For best performance
of factorization stage LWORK >= max(1,N*NB), where NB is
the optimal blocksize for CSYTRF_RK.

If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the WORK
array for factorization stage, 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

> 0: If INFO = k, the matrix A is singular, because:
If UPLO = ā€™Uā€™: column k in the upper
triangular part of A contains all zeros.
If UPLO = ā€™Lā€™: 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 dsysv_rk (character uplo, integer n, integer nrhs, doubleprecision, dimension( lda, * ) a, integer lda, double precision,dimension( * ) e, integer, dimension( * ) ipiv, double precision,dimension( ldb, * ) b, integer ldb, double precision, dimension( * )work, integer lwork, integer info)

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

Purpose:

DSYSV_RK computes the solution to a real 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.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
to factor A as
A = P*U*D*(U**T)*(P**T), if UPLO = ā€™Uā€™, or
A = P*L*D*(L**T)*(P**T), if UPLO = ā€™Lā€™,
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.

DSYTRF_RK is called to compute the factorization of a real
symmetric matrix. The factored form of A is then used to solve
the system of equations A * X = B by calling BLAS3 routine DSYTRS_3.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

N

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. NRHS >= 0.

A

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = ā€™Uā€™: 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 = ā€™Lā€™: 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, if INFO = 0, diagonal of the block diagonal
matrix D and factors U or L as computed by DSYTRF_RK:
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 = ā€™Uā€™: factor U in the superdiagonal part of A.
If UPLO = ā€™Lā€™: factor L in the subdiagonal part of A.

For more info see the description of DSYTRF_RK routine.

LDA

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

E

E is DOUBLE PRECISION array, dimension (N)
On exit, contains the output computed by the factorization
routine DSYTRF_RK, i.e. 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 = ā€™Uā€™: E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = ā€™Lā€™: 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 = ā€™Uā€™ or UPLO = ā€™Lā€™ cases.

For more info see the description of DSYTRF_RK routine.

IPIV

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

For more info see the description of DSYTRF_RK routine.

B

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS solution matrix X.

LDB

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

WORK

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

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= 1. For best performance
of factorization stage LWORK >= max(1,N*NB), where NB is
the optimal blocksize for DSYTRF_RK.

If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the WORK
array for factorization stage, 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

> 0: If INFO = k, the matrix A is singular, because:
If UPLO = ā€™Uā€™: column k in the upper
triangular part of A contains all zeros.
If UPLO = ā€™Lā€™: 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 ssysv_rk (character uplo, integer n, integer nrhs, real,dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer,dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, real,dimension( * ) work, integer lwork, integer info)

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

Purpose:

SSYSV_RK computes the solution to a real 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.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
to factor A as
A = P*U*D*(U**T)*(P**T), if UPLO = ā€™Uā€™, or
A = P*L*D*(L**T)*(P**T), if UPLO = ā€™Lā€™,
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.

SSYTRF_RK is called to compute the factorization of a real
symmetric matrix. The factored form of A is then used to solve
the system of equations A * X = B by calling BLAS3 routine SSYTRS_3.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

N

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. NRHS >= 0.

A

A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = ā€™Uā€™: 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 = ā€™Lā€™: 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, if INFO = 0, diagonal of the block diagonal
matrix D and factors U or L as computed by SSYTRF_RK:
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 = ā€™Uā€™: factor U in the superdiagonal part of A.
If UPLO = ā€™Lā€™: factor L in the subdiagonal part of A.

For more info see the description of DSYTRF_RK routine.

LDA

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

E

E is REAL array, dimension (N)
On exit, contains the output computed by the factorization
routine DSYTRF_RK, i.e. 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 = ā€™Uā€™: E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = ā€™Lā€™: 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 = ā€™Uā€™ or UPLO = ā€™Lā€™ cases.

For more info see the description of DSYTRF_RK routine.

IPIV

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

For more info see the description of DSYTRF_RK routine.

B

B is REAL array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS solution matrix X.

LDB

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

WORK

WORK is REAL array, dimension ( MAX(1,LWORK) ).
Work array used in the factorization stage.
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= 1. For best performance
of factorization stage LWORK >= max(1,N*NB), where NB is
the optimal blocksize for DSYTRF_RK.

If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the WORK
array for factorization stage, 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

> 0: If INFO = k, the matrix A is singular, because:
If UPLO = ā€™Uā€™: column k in the upper
triangular part of A contains all zeros.
If UPLO = ā€™Lā€™: 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 zhesv_rk (character uplo, integer n, integer nrhs, complex*16,dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e,integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b,integer ldb, complex*16, dimension( * ) work, integer lwork, integerinfo)

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

Purpose:

ZHESV_RK 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.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
to factor A as
A = P*U*D*(U**H)*(P**T), if UPLO = ā€™Uā€™, or
A = P*L*D*(L**H)*(P**T), if UPLO = ā€™Lā€™,
where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

ZHETRF_RK is called to compute the factorization of a complex
Hermitian matrix. The factored form of A is then used to solve
the system of equations A * X = B by calling BLAS3 routine ZHETRS_3.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

N

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. NRHS >= 0.

A

A is COMPLEX*16 array, dimension (LDA,N)
On entry, the Hermitian matrix A.
If UPLO = ā€™Uā€™: 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 = ā€™Lā€™: 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, if INFO = 0, diagonal of the block diagonal
matrix D and factors U or L as computed by ZHETRF_RK:
a) ONLY diagonal elements of the Hermitian 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 = ā€™Uā€™: factor U in the superdiagonal part of A.
If UPLO = ā€™Lā€™: factor L in the subdiagonal part of A.

For more info see the description of ZHETRF_RK routine.

LDA

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

E

E is COMPLEX*16 array, dimension (N)
On exit, contains the output computed by the factorization
routine ZHETRF_RK, i.e. the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = ā€™Uā€™: E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = ā€™Lā€™: 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 = ā€™Uā€™ or UPLO = ā€™Lā€™ cases.

For more info see the description of ZHETRF_RK routine.

IPIV

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

For more info see the description of ZHETRF_RK routine.

B

B is COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS solution matrix X.

LDB

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

WORK

WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ).
Work array used in the factorization stage.
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= 1. For best performance
of factorization stage LWORK >= max(1,N*NB), where NB is
the optimal blocksize for ZHETRF_RK.

If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the WORK
array for factorization stage, 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

> 0: If INFO = k, the matrix A is singular, because:
If UPLO = ā€™Uā€™: column k in the upper
triangular part of A contains all zeros.
If UPLO = ā€™Lā€™: 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 zsysv_rk (character uplo, integer n, integer nrhs, complex*16,dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e,integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b,integer ldb, complex*16, dimension( * ) work, integer lwork, integerinfo)

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

Purpose:

ZSYSV_RK 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.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
to factor A as
A = P*U*D*(U**T)*(P**T), if UPLO = ā€™Uā€™, or
A = P*L*D*(L**T)*(P**T), if UPLO = ā€™Lā€™,
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.

ZSYTRF_RK is called to compute the factorization of a complex
symmetric matrix. The factored form of A is then used to solve
the system of equations A * X = B by calling BLAS3 routine ZSYTRS_3.

Parameters

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

N

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. NRHS >= 0.

A

A is COMPLEX*16 array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = ā€™Uā€™: 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 = ā€™Lā€™: 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, if INFO = 0, diagonal of the block diagonal
matrix D and factors U or L as computed by ZSYTRF_RK:
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 = ā€™Uā€™: factor U in the superdiagonal part of A.
If UPLO = ā€™Lā€™: factor L in the subdiagonal part of A.

For more info see the description of ZSYTRF_RK routine.

LDA

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

E

E is COMPLEX*16 array, dimension (N)
On exit, contains the output computed by the factorization
routine ZSYTRF_RK, i.e. 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 = ā€™Uā€™: E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = ā€™Lā€™: 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 = ā€™Uā€™ or UPLO = ā€™Lā€™ cases.

For more info see the description of ZSYTRF_RK routine.

IPIV

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

For more info see the description of ZSYTRF_RK routine.

B

B is COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS solution matrix X.

LDB

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

WORK

WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ).
Work array used in the factorization stage.
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of WORK. LWORK >= 1. For best performance
of factorization stage LWORK >= max(1,N*NB), where NB is
the optimal blocksize for ZSYTRF_RK.

If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the WORK
array for factorization stage, 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

> 0: If INFO = k, the matrix A is singular, because:
If UPLO = ā€™Uā€™: column k in the upper
triangular part of A contains all zeros.
If UPLO = ā€™Lā€™: 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

Author

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