hesv_rk(3)

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 = 'U', or
    !>    A = P*L*D*(L**H)*(P**T),  if UPLO = 'L',
    !> 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:
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  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 = '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, 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 = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': 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 = '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.
    !>
    !>          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 = '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 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 = 'U', or
    !>    A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',
    !> 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:
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  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 = '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, 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 = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': 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 = '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.
    !>
    !>          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 = '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 dsysv_rk (character uplo, integer n, integer nrhs, double precision, 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 = 'U', or
    !>    A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',
    !> 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:
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  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 = '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, 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 = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': 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 = '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.
    !>
    !>          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 = '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 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 = 'U', or
    !>    A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',
    !> 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:
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  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 = '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, 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 = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': 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 = '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.
    !>
    !>          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 = '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 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, integer info)

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 = 'U', or
    !>    A = P*L*D*(L**H)*(P**T),  if UPLO = 'L',
    !> 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:
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  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 = '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, 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 = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': 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 = '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.
    !>
    !>          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 = '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 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, integer info)

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 = 'U', or
    !>    A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',
    !> 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:
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  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 = '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, 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 = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': 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 = '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.
    !>
    !>          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 = '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
    !>
    !>

Author

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