hpsvx(3)

NAME
hpsvx - {hp,sp}svx: factor and solve, expert

SYNOPSIS

  Functions

subroutine chpsvx (fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
CHPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine cspsvx (fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
CSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine dspsvx (fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
DSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine sspsvx (fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
SSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine zhpsvx (fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
ZHPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine zspsvx (fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
ZSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Detailed Description

Function Documentation

  subroutine chpsvx (character fact, character uplo, integer n, integer nrhs, complex, dimension( * ) ap, complex, dimension( * ) afp, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

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

Purpose:

    !>
    !> CHPSVX uses the diagonal pivoting factorization A = U*D*U**H or
    !> A = L*D*L**H to compute 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.
    !>
    !> Error bounds on the solution and a condition estimate are also
    !> provided.
    !>

Description:

    !>
    !> The following steps are performed:
    !>
    !> 1. If FACT = 'N', 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 and D is Hermitian and block diagonal with
    !>    1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
    !>    returns with INFO = i. Otherwise, the factored form of A is used
    !>    to estimate the condition number of the matrix A.  If the
    !>    reciprocal of the condition number is less than machine precision,
    !>    INFO = N+1 is returned as a warning, but the routine still goes on
    !>    to solve for X and compute error bounds as described below.
    !>
    !> 3. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 4. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of A has been
    !>          supplied on entry.
    !>          = 'F':  On entry, AFP and IPIV contain the factored form of
    !>                  A.  AFP and IPIV will not be modified.
    !>          = 'N':  The matrix A will be copied to AFP and factored.
    !>

    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 matrices B and X.  NRHS >= 0.
    !>

    AP

    !>          AP is COMPLEX array, dimension (N*(N+1)/2)
    !>          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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>

    AFP

    !>          AFP is COMPLEX array, dimension (N*(N+1)/2)
    !>          If FACT = 'F', then AFP is an input argument and on entry
    !>          contains 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.
    !>
    !>          If FACT = 'N', then AFP is an output argument and on exit
    !>          contains 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)
    !>          If FACT = 'F', then IPIV is an input argument and on entry
    !>          contains 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.
    !>
    !>          If FACT = 'N', then IPIV is an output argument and on exit
    !>          contains details of the interchanges and the block structure
    !>          of D, as determined by CHPTRF.
    !>

    B

    !>          B is COMPLEX array, dimension (LDB,NRHS)
    !>          The N-by-NRHS right hand side matrix B.
    !>

    LDB

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

    X

    !>          X is COMPLEX array, dimension (LDX,NRHS)
    !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
    !>

    LDX

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

    RCOND

    !>          RCOND is REAL
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A.  If RCOND is less than the machine precision (in
    !>          particular, if RCOND = 0), the matrix is singular to working
    !>          precision.  This condition is indicated by a return code of
    !>          INFO > 0.
    !>

    FERR

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

    BERR

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

    WORK

    !>          WORK is COMPLEX array, dimension (2*N)
    !>

    RWORK

    !>          RWORK is REAL array, dimension (N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO = i, and i is
    !>                <= N:  D(i,i) is exactly zero.  The factorization
    !>                       has been completed but the factor D is exactly
    !>                       singular, so the solution and error bounds could
    !>                       not be computed. RCOND = 0 is returned.
    !>                = N+1: D is nonsingular, but RCOND is less than machine
    !>                       precision, meaning that the matrix is singular
    !>                       to working precision.  Nevertheless, the
    !>                       solution and error bounds are computed because
    !>                       there are a number of situations where the
    !>                       computed solution can be more accurate than the
    !>                       value of RCOND would suggest.
    !>

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 cspsvx (character fact, character uplo, integer n, integer nrhs, complex, dimension( * ) ap, complex, dimension( * ) afp, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

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

Purpose:

    !>
    !> CSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
    !> A = L*D*L**T to compute 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.
    !>
    !> Error bounds on the solution and a condition estimate are also
    !> provided.
    !>

Description:

    !>
    !> The following steps are performed:
    !>
    !> 1. If FACT = 'N', 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 and D is symmetric and block diagonal with
    !>    1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
    !>    returns with INFO = i. Otherwise, the factored form of A is used
    !>    to estimate the condition number of the matrix A.  If the
    !>    reciprocal of the condition number is less than machine precision,
    !>    INFO = N+1 is returned as a warning, but the routine still goes on
    !>    to solve for X and compute error bounds as described below.
    !>
    !> 3. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 4. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of A has been
    !>          supplied on entry.
    !>          = 'F':  On entry, AFP and IPIV contain the factored form
    !>                  of A.  AP, AFP and IPIV will not be modified.
    !>          = 'N':  The matrix A will be copied to AFP and factored.
    !>

    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 matrices B and X.  NRHS >= 0.
    !>

    AP

    !>          AP is COMPLEX array, dimension (N*(N+1)/2)
    !>          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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>

    AFP

    !>          AFP is COMPLEX array, dimension (N*(N+1)/2)
    !>          If FACT = 'F', then AFP is an input argument and on entry
    !>          contains the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L from the factorization
    !>          A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>
    !>          If FACT = 'N', then AFP is an output argument and on exit
    !>          contains the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L from the factorization
    !>          A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          If FACT = 'F', then IPIV is an input argument and on entry
    !>          contains 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.
    !>
    !>          If FACT = 'N', then IPIV is an output argument and on exit
    !>          contains details of the interchanges and the block structure
    !>          of D, as determined by CSPTRF.
    !>

    B

    !>          B is COMPLEX array, dimension (LDB,NRHS)
    !>          The N-by-NRHS right hand side matrix B.
    !>

    LDB

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

    X

    !>          X is COMPLEX array, dimension (LDX,NRHS)
    !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
    !>

    LDX

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

    RCOND

    !>          RCOND is REAL
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A.  If RCOND is less than the machine precision (in
    !>          particular, if RCOND = 0), the matrix is singular to working
    !>          precision.  This condition is indicated by a return code of
    !>          INFO > 0.
    !>

    FERR

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

    BERR

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

    WORK

    !>          WORK is COMPLEX array, dimension (2*N)
    !>

    RWORK

    !>          RWORK is REAL array, dimension (N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO = i, and i is
    !>                <= N:  D(i,i) is exactly zero.  The factorization
    !>                       has been completed but the factor D is exactly
    !>                       singular, so the solution and error bounds could
    !>                       not be computed. RCOND = 0 is returned.
    !>                = N+1: D is nonsingular, but RCOND is less than machine
    !>                       precision, meaning that the matrix is singular
    !>                       to working precision.  Nevertheless, the
    !>                       solution and error bounds are computed because
    !>                       there are a number of situations where the
    !>                       computed solution can be more accurate than the
    !>                       value of RCOND would suggest.
    !>

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 dspsvx (character fact, character uplo, integer n, integer nrhs, double precision, dimension( * ) ap, double precision, dimension( * ) afp, integer, dimension( * ) ipiv, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

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

Purpose:

    !>
    !> DSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
    !> A = L*D*L**T to compute 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.
    !>
    !> Error bounds on the solution and a condition estimate are also
    !> provided.
    !>

Description:

    !>
    !> The following steps are performed:
    !>
    !> 1. If FACT = 'N', 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 and D is symmetric and block diagonal with
    !>    1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
    !>    returns with INFO = i. Otherwise, the factored form of A is used
    !>    to estimate the condition number of the matrix A.  If the
    !>    reciprocal of the condition number is less than machine precision,
    !>    INFO = N+1 is returned as a warning, but the routine still goes on
    !>    to solve for X and compute error bounds as described below.
    !>
    !> 3. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 4. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of A has been
    !>          supplied on entry.
    !>          = 'F':  On entry, AFP and IPIV contain the factored form of
    !>                  A.  AP, AFP and IPIV will not be modified.
    !>          = 'N':  The matrix A will be copied to AFP and factored.
    !>

    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 matrices B and X.  NRHS >= 0.
    !>

    AP

    !>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
    !>          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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>

    AFP

    !>          AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
    !>          If FACT = 'F', then AFP is an input argument and on entry
    !>          contains the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L from the factorization
    !>          A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>
    !>          If FACT = 'N', then AFP is an output argument and on exit
    !>          contains the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L from the factorization
    !>          A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          If FACT = 'F', then IPIV is an input argument and on entry
    !>          contains 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.
    !>
    !>          If FACT = 'N', then IPIV is an output argument and on exit
    !>          contains details of the interchanges and the block structure
    !>          of D, as determined by DSPTRF.
    !>

    B

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

    LDB

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

    X

    !>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
    !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
    !>

    LDX

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

    RCOND

    !>          RCOND is DOUBLE PRECISION
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A.  If RCOND is less than the machine precision (in
    !>          particular, if RCOND = 0), the matrix is singular to working
    !>          precision.  This condition is indicated by a return code of
    !>          INFO > 0.
    !>

    FERR

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

    BERR

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

    WORK

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

    IWORK

    !>          IWORK is INTEGER array, dimension (N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO = i, and i is
    !>                <= N:  D(i,i) is exactly zero.  The factorization
    !>                       has been completed but the factor D is exactly
    !>                       singular, so the solution and error bounds could
    !>                       not be computed. RCOND = 0 is returned.
    !>                = N+1: D is nonsingular, but RCOND is less than machine
    !>                       precision, meaning that the matrix is singular
    !>                       to working precision.  Nevertheless, the
    !>                       solution and error bounds are computed because
    !>                       there are a number of situations where the
    !>                       computed solution can be more accurate than the
    !>                       value of RCOND would suggest.
    !>

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 sspsvx (character fact, character uplo, integer n, integer nrhs, real, dimension( * ) ap, real, dimension( * ) afp, integer, dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

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

Purpose:

    !>
    !> SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
    !> A = L*D*L**T to compute 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.
    !>
    !> Error bounds on the solution and a condition estimate are also
    !> provided.
    !>

Description:

    !>
    !> The following steps are performed:
    !>
    !> 1. If FACT = 'N', 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 and D is symmetric and block diagonal with
    !>    1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
    !>    returns with INFO = i. Otherwise, the factored form of A is used
    !>    to estimate the condition number of the matrix A.  If the
    !>    reciprocal of the condition number is less than machine precision,
    !>    INFO = N+1 is returned as a warning, but the routine still goes on
    !>    to solve for X and compute error bounds as described below.
    !>
    !> 3. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 4. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of A has been
    !>          supplied on entry.
    !>          = 'F':  On entry, AFP and IPIV contain the factored form of
    !>                  A.  AP, AFP and IPIV will not be modified.
    !>          = 'N':  The matrix A will be copied to AFP and factored.
    !>

    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 matrices B and X.  NRHS >= 0.
    !>

    AP

    !>          AP is REAL array, dimension (N*(N+1)/2)
    !>          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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>

    AFP

    !>          AFP is REAL array, dimension (N*(N+1)/2)
    !>          If FACT = 'F', then AFP is an input argument and on entry
    !>          contains the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L from the factorization
    !>          A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>
    !>          If FACT = 'N', then AFP is an output argument and on exit
    !>          contains the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L from the factorization
    !>          A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          If FACT = 'F', then IPIV is an input argument and on entry
    !>          contains 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.
    !>
    !>          If FACT = 'N', then IPIV is an output argument and on exit
    !>          contains details of the interchanges and the block structure
    !>          of D, as determined by SSPTRF.
    !>

    B

    !>          B is REAL array, dimension (LDB,NRHS)
    !>          The N-by-NRHS right hand side matrix B.
    !>

    LDB

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

    X

    !>          X is REAL array, dimension (LDX,NRHS)
    !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
    !>

    LDX

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

    RCOND

    !>          RCOND is REAL
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A.  If RCOND is less than the machine precision (in
    !>          particular, if RCOND = 0), the matrix is singular to working
    !>          precision.  This condition is indicated by a return code of
    !>          INFO > 0.
    !>

    FERR

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

    BERR

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

    WORK

    !>          WORK is REAL array, dimension (3*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO = i, and i is
    !>                <= N:  D(i,i) is exactly zero.  The factorization
    !>                       has been completed but the factor D is exactly
    !>                       singular, so the solution and error bounds could
    !>                       not be computed. RCOND = 0 is returned.
    !>                = N+1: D is nonsingular, but RCOND is less than machine
    !>                       precision, meaning that the matrix is singular
    !>                       to working precision.  Nevertheless, the
    !>                       solution and error bounds are computed because
    !>                       there are a number of situations where the
    !>                       computed solution can be more accurate than the
    !>                       value of RCOND would suggest.
    !>

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 zhpsvx (character fact, character uplo, integer n, integer nrhs, complex*16, dimension( * ) ap, complex*16, dimension( * ) afp, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

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

Purpose:

    !>
    !> ZHPSVX uses the diagonal pivoting factorization A = U*D*U**H or
    !> A = L*D*L**H to compute 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.
    !>
    !> Error bounds on the solution and a condition estimate are also
    !> provided.
    !>

Description:

    !>
    !> The following steps are performed:
    !>
    !> 1. If FACT = 'N', 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 and D is Hermitian and block diagonal with
    !>    1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
    !>    returns with INFO = i. Otherwise, the factored form of A is used
    !>    to estimate the condition number of the matrix A.  If the
    !>    reciprocal of the condition number is less than machine precision,
    !>    INFO = N+1 is returned as a warning, but the routine still goes on
    !>    to solve for X and compute error bounds as described below.
    !>
    !> 3. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 4. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of A has been
    !>          supplied on entry.
    !>          = 'F':  On entry, AFP and IPIV contain the factored form of
    !>                  A.  AFP and IPIV will not be modified.
    !>          = 'N':  The matrix A will be copied to AFP and factored.
    !>

    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 matrices B and X.  NRHS >= 0.
    !>

    AP

    !>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
    !>          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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>

    AFP

    !>          AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
    !>          If FACT = 'F', then AFP is an input argument and on entry
    !>          contains 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.
    !>
    !>          If FACT = 'N', then AFP is an output argument and on exit
    !>          contains 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)
    !>          If FACT = 'F', then IPIV is an input argument and on entry
    !>          contains 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.
    !>
    !>          If FACT = 'N', then IPIV is an output argument and on exit
    !>          contains details of the interchanges and the block structure
    !>          of D, as determined by ZHPTRF.
    !>

    B

    !>          B is COMPLEX*16 array, dimension (LDB,NRHS)
    !>          The N-by-NRHS right hand side matrix B.
    !>

    LDB

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

    X

    !>          X is COMPLEX*16 array, dimension (LDX,NRHS)
    !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
    !>

    LDX

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

    RCOND

    !>          RCOND is DOUBLE PRECISION
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A.  If RCOND is less than the machine precision (in
    !>          particular, if RCOND = 0), the matrix is singular to working
    !>          precision.  This condition is indicated by a return code of
    !>          INFO > 0.
    !>

    FERR

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

    BERR

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

    WORK

    !>          WORK is COMPLEX*16 array, dimension (2*N)
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO = i, and i is
    !>                <= N:  D(i,i) is exactly zero.  The factorization
    !>                       has been completed but the factor D is exactly
    !>                       singular, so the solution and error bounds could
    !>                       not be computed. RCOND = 0 is returned.
    !>                = N+1: D is nonsingular, but RCOND is less than machine
    !>                       precision, meaning that the matrix is singular
    !>                       to working precision.  Nevertheless, the
    !>                       solution and error bounds are computed because
    !>                       there are a number of situations where the
    !>                       computed solution can be more accurate than the
    !>                       value of RCOND would suggest.
    !>

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 zspsvx (character fact, character uplo, integer n, integer nrhs, complex*16, dimension( * ) ap, complex*16, dimension( * ) afp, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

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

Purpose:

    !>
    !> ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
    !> A = L*D*L**T to compute 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.
    !>
    !> Error bounds on the solution and a condition estimate are also
    !> provided.
    !>

Description:

    !>
    !> The following steps are performed:
    !>
    !> 1. If FACT = 'N', 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 and D is symmetric and block diagonal with
    !>    1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
    !>    returns with INFO = i. Otherwise, the factored form of A is used
    !>    to estimate the condition number of the matrix A.  If the
    !>    reciprocal of the condition number is less than machine precision,
    !>    INFO = N+1 is returned as a warning, but the routine still goes on
    !>    to solve for X and compute error bounds as described below.
    !>
    !> 3. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 4. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of A has been
    !>          supplied on entry.
    !>          = 'F':  On entry, AFP and IPIV contain the factored form
    !>                  of A.  AP, AFP and IPIV will not be modified.
    !>          = 'N':  The matrix A will be copied to AFP and factored.
    !>

    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 matrices B and X.  NRHS >= 0.
    !>

    AP

    !>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
    !>          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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    !>          See below for further details.
    !>

    AFP

    !>          AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
    !>          If FACT = 'F', then AFP is an input argument and on entry
    !>          contains the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L from the factorization
    !>          A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>
    !>          If FACT = 'N', then AFP is an output argument and on exit
    !>          contains the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L from the factorization
    !>          A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as
    !>          a packed triangular matrix in the same storage format as A.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          If FACT = 'F', then IPIV is an input argument and on entry
    !>          contains 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.
    !>
    !>          If FACT = 'N', then IPIV is an output argument and on exit
    !>          contains details of the interchanges and the block structure
    !>          of D, as determined by ZSPTRF.
    !>

    B

    !>          B is COMPLEX*16 array, dimension (LDB,NRHS)
    !>          The N-by-NRHS right hand side matrix B.
    !>

    LDB

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

    X

    !>          X is COMPLEX*16 array, dimension (LDX,NRHS)
    !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
    !>

    LDX

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

    RCOND

    !>          RCOND is DOUBLE PRECISION
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A.  If RCOND is less than the machine precision (in
    !>          particular, if RCOND = 0), the matrix is singular to working
    !>          precision.  This condition is indicated by a return code of
    !>          INFO > 0.
    !>

    FERR

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

    BERR

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

    WORK

    !>          WORK is COMPLEX*16 array, dimension (2*N)
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO = i, and i is
    !>                <= N:  D(i,i) is exactly zero.  The factorization
    !>                       has been completed but the factor D is exactly
    !>                       singular, so the solution and error bounds could
    !>                       not be computed. RCOND = 0 is returned.
    !>                = N+1: D is nonsingular, but RCOND is less than machine
    !>                       precision, meaning that the matrix is singular
    !>                       to working precision.  Nevertheless, the
    !>                       solution and error bounds are computed because
    !>                       there are a number of situations where the
    !>                       computed solution can be more accurate than the
    !>                       value of RCOND would suggest.
    !>

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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