hpsv(3)

NAME
hpsv - {hp,sp}sv: factor and solve

SYNOPSIS

  Functions

subroutine chpsv (uplo, n, nrhs, ap, ipiv, b, ldb, info)
CHPSV computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine cspsv (uplo, n, nrhs, ap, ipiv, b, ldb, info)
CSPSV computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine dspsv (uplo, n, nrhs, ap, ipiv, b, ldb, info)
DSPSV computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine sspsv (uplo, n, nrhs, ap, ipiv, b, ldb, info)
SSPSV computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine zhpsv (uplo, n, nrhs, ap, ipiv, b, ldb, info)
ZHPSV computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine zspsv (uplo, n, nrhs, ap, ipiv, b, ldb, info)
ZSPSV computes the solution to system of linear equations A * X = B for OTHER matrices

Detailed Description

Function Documentation

  subroutine chpsv (character uplo, integer n, integer nrhs, complex, dimension( * ) ap, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, integer info)

CHPSV computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

    !>
    !> CHPSV computes the solution to a complex system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N Hermitian matrix stored in packed format and X
    !> and B are N-by-NRHS matrices.
    !>
    !> The diagonal pivoting method is used to factor A as
    !>    A = U * D * U**H,  if UPLO = 'U', or
    !>    A = L * D * L**H,  if UPLO = 'L',
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, D is Hermitian and block diagonal with 1-by-1
    !> and 2-by-2 diagonal blocks.  The factored form of A is then used to
    !> solve the system of equations A * X = B.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = '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.
    !>

    AP

    !>          AP is COMPLEX array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the Hermitian matrix
    !>          A, packed columnwise in a linear array.  The j-th column of A
    !>          is stored in the array AP as follows:
    !>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>
    !>          On exit, the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L from the factorization
    !>          A = U*D*U**H or A = L*D*L**H as computed by CHPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>

    IPIV

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

    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).
    !>

    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, so the solution could not be
    !>                computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The packed storage scheme is illustrated by the following example
    !>  when N = 4, UPLO = 'U':
    !>
    !>  Two-dimensional storage of the Hermitian matrix A:
    !>
    !>     a11 a12 a13 a14
    !>         a22 a23 a24
    !>             a33 a34     (aij = conjg(aji))
    !>                 a44
    !>
    !>  Packed storage of the upper triangle of A:
    !>
    !>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
    !>

  subroutine cspsv (character uplo, integer n, integer nrhs, complex, dimension( * ) ap, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, integer info)

CSPSV computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

    !>
    !> CSPSV computes the solution to a complex system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N symmetric matrix stored in packed format and X
    !> and B are N-by-NRHS matrices.
    !>
    !> The diagonal pivoting method is used to factor A as
    !>    A = U * D * U**T,  if UPLO = 'U', or
    !>    A = L * D * L**T,  if UPLO = 'L',
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, D is symmetric and block diagonal with 1-by-1
    !> and 2-by-2 diagonal blocks.  The factored form of A is then used to
    !> solve the system of equations A * X = B.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = '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.
    !>

    AP

    !>          AP is COMPLEX array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the symmetric matrix
    !>          A, packed columnwise in a linear array.  The j-th column of A
    !>          is stored in the array AP as follows:
    !>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>
    !>          On exit, 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 CSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>

    IPIV

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

    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).
    !>

    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, so the solution could not be
    !>                computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The packed storage scheme is illustrated by the following example
    !>  when N = 4, UPLO = 'U':
    !>
    !>  Two-dimensional storage of the symmetric matrix A:
    !>
    !>     a11 a12 a13 a14
    !>         a22 a23 a24
    !>             a33 a34     (aij = aji)
    !>                 a44
    !>
    !>  Packed storage of the upper triangle of A:
    !>
    !>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
    !>

  subroutine dspsv (character uplo, integer n, integer nrhs, double precision, dimension( * ) ap, integer, dimension( * ) ipiv, double precision, dimension( ldb, * ) b, integer ldb, integer info)

DSPSV computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

    !>
    !> DSPSV computes the solution to a real system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N symmetric matrix stored in packed format and X
    !> and B are N-by-NRHS matrices.
    !>
    !> The diagonal pivoting method is used to factor A as
    !>    A = U * D * U**T,  if UPLO = 'U', or
    !>    A = L * D * L**T,  if UPLO = 'L',
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, D is symmetric and block diagonal with 1-by-1
    !> and 2-by-2 diagonal blocks.  The factored form of A is then used to
    !> solve the system of equations A * X = B.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = '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.
    !>

    AP

    !>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the symmetric matrix
    !>          A, packed columnwise in a linear array.  The j-th column of A
    !>          is stored in the array AP as follows:
    !>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>
    !>          On exit, 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 DSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>

    IPIV

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

    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).
    !>

    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, so the solution could not be
    !>                computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The packed storage scheme is illustrated by the following example
    !>  when N = 4, UPLO = 'U':
    !>
    !>  Two-dimensional storage of the symmetric matrix A:
    !>
    !>     a11 a12 a13 a14
    !>         a22 a23 a24
    !>             a33 a34     (aij = aji)
    !>                 a44
    !>
    !>  Packed storage of the upper triangle of A:
    !>
    !>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
    !>

  subroutine sspsv (character uplo, integer n, integer nrhs, real, dimension( * ) ap, integer, dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, integer info)

SSPSV computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

    !>
    !> SSPSV computes the solution to a real system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N symmetric matrix stored in packed format and X
    !> and B are N-by-NRHS matrices.
    !>
    !> The diagonal pivoting method is used to factor A as
    !>    A = U * D * U**T,  if UPLO = 'U', or
    !>    A = L * D * L**T,  if UPLO = 'L',
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, D is symmetric and block diagonal with 1-by-1
    !> and 2-by-2 diagonal blocks.  The factored form of A is then used to
    !> solve the system of equations A * X = B.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = '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.
    !>

    AP

    !>          AP is REAL array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the symmetric matrix
    !>          A, packed columnwise in a linear array.  The j-th column of A
    !>          is stored in the array AP as follows:
    !>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>
    !>          On exit, 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 SSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>

    IPIV

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

    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).
    !>

    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, so the solution could not be
    !>                computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The packed storage scheme is illustrated by the following example
    !>  when N = 4, UPLO = 'U':
    !>
    !>  Two-dimensional storage of the symmetric matrix A:
    !>
    !>     a11 a12 a13 a14
    !>         a22 a23 a24
    !>             a33 a34     (aij = aji)
    !>                 a44
    !>
    !>  Packed storage of the upper triangle of A:
    !>
    !>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
    !>

  subroutine zhpsv (character uplo, integer n, integer nrhs, complex*16, dimension( * ) ap, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

ZHPSV computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

    !>
    !> ZHPSV computes the solution to a complex system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N Hermitian matrix stored in packed format and X
    !> and B are N-by-NRHS matrices.
    !>
    !> The diagonal pivoting method is used to factor A as
    !>    A = U * D * U**H,  if UPLO = 'U', or
    !>    A = L * D * L**H,  if UPLO = 'L',
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, D is Hermitian and block diagonal with 1-by-1
    !> and 2-by-2 diagonal blocks.  The factored form of A is then used to
    !> solve the system of equations A * X = B.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = '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.
    !>

    AP

    !>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the Hermitian matrix
    !>          A, packed columnwise in a linear array.  The j-th column of A
    !>          is stored in the array AP as follows:
    !>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>
    !>          On exit, the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L from the factorization
    !>          A = U*D*U**H or A = L*D*L**H as computed by ZHPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>

    IPIV

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

    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).
    !>

    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, so the solution could not be
    !>                computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The packed storage scheme is illustrated by the following example
    !>  when N = 4, UPLO = 'U':
    !>
    !>  Two-dimensional storage of the Hermitian matrix A:
    !>
    !>     a11 a12 a13 a14
    !>         a22 a23 a24
    !>             a33 a34     (aij = conjg(aji))
    !>                 a44
    !>
    !>  Packed storage of the upper triangle of A:
    !>
    !>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
    !>

  subroutine zspsv (character uplo, integer n, integer nrhs, complex*16, dimension( * ) ap, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

ZSPSV computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

    !>
    !> ZSPSV computes the solution to a complex system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N symmetric matrix stored in packed format and X
    !> and B are N-by-NRHS matrices.
    !>
    !> The diagonal pivoting method is used to factor A as
    !>    A = U * D * U**T,  if UPLO = 'U', or
    !>    A = L * D * L**T,  if UPLO = 'L',
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, D is symmetric and block diagonal with 1-by-1
    !> and 2-by-2 diagonal blocks.  The factored form of A is then used to
    !> solve the system of equations A * X = B.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = '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.
    !>

    AP

    !>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
    !>          On entry, the upper or lower triangle of the symmetric matrix
    !>          A, packed columnwise in a linear array.  The j-th column of A
    !>          is stored in the array AP as follows:
    !>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    !>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>
    !>          On exit, 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 ZSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>

    IPIV

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

    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).
    !>

    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, so the solution could not be
    !>                computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The packed storage scheme is illustrated by the following example
    !>  when N = 4, UPLO = 'U':
    !>
    !>  Two-dimensional storage of the symmetric matrix A:
    !>
    !>     a11 a12 a13 a14
    !>         a22 a23 a24
    !>             a33 a34     (aij = aji)
    !>                 a44
    !>
    !>  Packed storage of the upper triangle of A:
    !>
    !>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
    !>

Author

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