ppsvx(3)

NAME
ppsvx - ppsvx: factor and solve, expert

SYNOPSIS

  Functions
subroutine cppsvx (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
CPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine dppsvx (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
DPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine sppsvx (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine zppsvx (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
ZPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Detailed Description

Function Documentation

  subroutine cppsvx (character fact, character uplo, integer n, integer nrhs, complex, dimension( * ) ap, complex, dimension( * ) afp, character equed, real, dimension( * ) s, 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)

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

Purpose:

    !>
    !> CPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
    !> compute the solution to a complex system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N Hermitian positive definite 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 = 'E', real scaling factors are computed to equilibrate
    !>    the system:
    !>       diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
    !>    Whether or not the system will be equilibrated depends on the
    !>    scaling of the matrix A, but if equilibration is used, A is
    !>    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
    !>
    !> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
    !>    factor the matrix A (after equilibration if FACT = 'E') as
    !>       A = U**H * U ,  if UPLO = 'U', or
    !>       A = L * L**H,  if UPLO = 'L',
    !>    where U is an upper triangular matrix, L is a lower triangular
    !>    matrix, and **H indicates conjugate transpose.
    !>
    !> 3. If the leading principal minor of order i is not positive,
    !>    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.
    !>
    !> 4. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 5. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>
    !> 6. If equilibration was used, the matrix X is premultiplied by
    !>    diag(S) so that it solves the original system before
    !>    equilibration.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of the matrix A is
    !>          supplied on entry, and if not, whether the matrix A should be
    !>          equilibrated before it is factored.
    !>          = 'F':  On entry, AFP contains the factored form of A.
    !>                  If EQUED = 'Y', the matrix A has been equilibrated
    !>                  with scaling factors given by S.  AP and AFP will not
    !>                  be modified.
    !>          = 'N':  The matrix A will be copied to AFP and factored.
    !>          = 'E':  The matrix A will be equilibrated if necessary, then
    !>                  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)
    !>          On entry, the upper or lower triangle of the Hermitian matrix
    !>          A, packed columnwise in a linear array, except if FACT = 'F'
    !>          and EQUED = 'Y', then A must contain the equilibrated matrix
    !>          diag(S)*A*diag(S).  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.  A is not modified if
    !>          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
    !>
    !>          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
    !>          diag(S)*A*diag(S).
    !>

    AFP

    !>          AFP is COMPLEX array, dimension (N*(N+1)/2)
    !>          If FACT = 'F', then AFP is an input argument and on entry
    !>          contains the triangular factor U or L from the Cholesky
    !>          factorization A = U**H*U or A = L*L**H, in the same storage
    !>          format as A.  If EQUED .ne. 'N', then AFP is the factored
    !>          form of the equilibrated matrix A.
    !>
    !>          If FACT = 'N', then AFP is an output argument and on exit
    !>          returns the triangular factor U or L from the Cholesky
    !>          factorization A = U**H * U or A = L * L**H of the original
    !>          matrix A.
    !>
    !>          If FACT = 'E', then AFP is an output argument and on exit
    !>          returns the triangular factor U or L from the Cholesky
    !>          factorization A = U**H*U or A = L*L**H of the equilibrated
    !>          matrix A (see the description of AP for the form of the
    !>          equilibrated matrix).
    !>

    EQUED

    !>          EQUED is CHARACTER*1
    !>          Specifies the form of equilibration that was done.
    !>          = 'N':  No equilibration (always true if FACT = 'N').
    !>          = 'Y':  Equilibration was done, i.e., A has been replaced by
    !>                  diag(S) * A * diag(S).
    !>          EQUED is an input argument if FACT = 'F'; otherwise, it is an
    !>          output argument.
    !>

    S

    !>          S is REAL array, dimension (N)
    !>          The scale factors for A; not accessed if EQUED = 'N'.  S is
    !>          an input argument if FACT = 'F'; otherwise, S is an output
    !>          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
    !>          must be positive.
    !>

    B

    !>          B is COMPLEX array, dimension (LDB,NRHS)
    !>          On entry, the N-by-NRHS right hand side matrix B.
    !>          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
    !>          B is overwritten by diag(S) * 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 to
    !>          the original system of equations.  Note that if EQUED = 'Y',
    !>          A and B are modified on exit, and the solution to the
    !>          equilibrated system is inv(diag(S))*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 after equilibration (if done).  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:  the leading principal minor of order i of A
    !>                       is not positive, so the factorization could not
    !>                       be completed, and the solution has not been
    !>                       computed. RCOND = 0 is returned.
    !>                = N+1: U 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 ]
    !>

Definition at line 309 of file cppsvx.f.

  subroutine dppsvx (character fact, character uplo, integer n, integer nrhs, double precision, dimension( * ) ap, double precision, dimension( * ) afp, character equed, double precision, dimension( * ) s, 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)

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

Purpose:

    !>
    !> DPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
    !> compute the solution to a real system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N symmetric positive definite 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 = 'E', real scaling factors are computed to equilibrate
    !>    the system:
    !>       diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
    !>    Whether or not the system will be equilibrated depends on the
    !>    scaling of the matrix A, but if equilibration is used, A is
    !>    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
    !>
    !> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
    !>    factor the matrix A (after equilibration if FACT = 'E') as
    !>       A = U**T* U,  if UPLO = 'U', or
    !>       A = L * L**T,  if UPLO = 'L',
    !>    where U is an upper triangular matrix and L is a lower triangular
    !>    matrix.
    !>
    !> 3. If the leading principal minor of order i is not positive,
    !>    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.
    !>
    !> 4. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 5. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>
    !> 6. If equilibration was used, the matrix X is premultiplied by
    !>    diag(S) so that it solves the original system before
    !>    equilibration.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of the matrix A is
    !>          supplied on entry, and if not, whether the matrix A should be
    !>          equilibrated before it is factored.
    !>          = 'F':  On entry, AFP contains the factored form of A.
    !>                  If EQUED = 'Y', the matrix A has been equilibrated
    !>                  with scaling factors given by S.  AP and AFP will not
    !>                  be modified.
    !>          = 'N':  The matrix A will be copied to AFP and factored.
    !>          = 'E':  The matrix A will be equilibrated if necessary, then
    !>                  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)
    !>          On entry, the upper or lower triangle of the symmetric matrix
    !>          A, packed columnwise in a linear array, except if FACT = 'F'
    !>          and EQUED = 'Y', then A must contain the equilibrated matrix
    !>          diag(S)*A*diag(S).  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.  A is not modified if
    !>          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
    !>
    !>          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
    !>          diag(S)*A*diag(S).
    !>

    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 triangular factor U or L from the Cholesky
    !>          factorization A = U**T*U or A = L*L**T, in the same storage
    !>          format as A.  If EQUED .ne. 'N', then AFP is the factored
    !>          form of the equilibrated matrix A.
    !>
    !>          If FACT = 'N', then AFP is an output argument and on exit
    !>          returns the triangular factor U or L from the Cholesky
    !>          factorization A = U**T * U or A = L * L**T of the original
    !>          matrix A.
    !>
    !>          If FACT = 'E', then AFP is an output argument and on exit
    !>          returns the triangular factor U or L from the Cholesky
    !>          factorization A = U**T * U or A = L * L**T of the equilibrated
    !>          matrix A (see the description of AP for the form of the
    !>          equilibrated matrix).
    !>

    EQUED

    !>          EQUED is CHARACTER*1
    !>          Specifies the form of equilibration that was done.
    !>          = 'N':  No equilibration (always true if FACT = 'N').
    !>          = 'Y':  Equilibration was done, i.e., A has been replaced by
    !>                  diag(S) * A * diag(S).
    !>          EQUED is an input argument if FACT = 'F'; otherwise, it is an
    !>          output argument.
    !>

    S

    !>          S is DOUBLE PRECISION array, dimension (N)
    !>          The scale factors for A; not accessed if EQUED = 'N'.  S is
    !>          an input argument if FACT = 'F'; otherwise, S is an output
    !>          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
    !>          must be positive.
    !>

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
    !>          On entry, the N-by-NRHS right hand side matrix B.
    !>          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
    !>          B is overwritten by diag(S) * 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 to
    !>          the original system of equations.  Note that if EQUED = 'Y',
    !>          A and B are modified on exit, and the solution to the
    !>          equilibrated system is inv(diag(S))*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 after equilibration (if done).  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:  the leading principal minor of order i of A
    !>                       is not positive, so the factorization could not
    !>                       be completed, and the solution has not been
    !>                       computed. RCOND = 0 is returned.
    !>                = N+1: U 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 = conjg(aji))
    !>                 a44
    !>
    !>  Packed storage of the upper triangle of A:
    !>
    !>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
    !>

Definition at line 309 of file dppsvx.f.

  subroutine sppsvx (character fact, character uplo, integer n, integer nrhs, real, dimension( * ) ap, real, dimension( * ) afp, character equed, real, dimension( * ) s, 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)

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

Purpose:

    !>
    !> SPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
    !> compute the solution to a real system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N symmetric positive definite 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 = 'E', real scaling factors are computed to equilibrate
    !>    the system:
    !>       diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
    !>    Whether or not the system will be equilibrated depends on the
    !>    scaling of the matrix A, but if equilibration is used, A is
    !>    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
    !>
    !> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
    !>    factor the matrix A (after equilibration if FACT = 'E') as
    !>       A = U**T* U,  if UPLO = 'U', or
    !>       A = L * L**T,  if UPLO = 'L',
    !>    where U is an upper triangular matrix and L is a lower triangular
    !>    matrix.
    !>
    !> 3. If the leading principal minor of order i is not positive,
    !>    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.
    !>
    !> 4. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 5. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>
    !> 6. If equilibration was used, the matrix X is premultiplied by
    !>    diag(S) so that it solves the original system before
    !>    equilibration.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of the matrix A is
    !>          supplied on entry, and if not, whether the matrix A should be
    !>          equilibrated before it is factored.
    !>          = 'F':  On entry, AFP contains the factored form of A.
    !>                  If EQUED = 'Y', the matrix A has been equilibrated
    !>                  with scaling factors given by S.  AP and AFP will not
    !>                  be modified.
    !>          = 'N':  The matrix A will be copied to AFP and factored.
    !>          = 'E':  The matrix A will be equilibrated if necessary, then
    !>                  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)
    !>          On entry, the upper or lower triangle of the symmetric matrix
    !>          A, packed columnwise in a linear array, except if FACT = 'F'
    !>          and EQUED = 'Y', then A must contain the equilibrated matrix
    !>          diag(S)*A*diag(S).  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.  A is not modified if
    !>          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
    !>
    !>          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
    !>          diag(S)*A*diag(S).
    !>

    AFP

    !>          AFP is REAL array, dimension (N*(N+1)/2)
    !>          If FACT = 'F', then AFP is an input argument and on entry
    !>          contains the triangular factor U or L from the Cholesky
    !>          factorization A = U**T*U or A = L*L**T, in the same storage
    !>          format as A.  If EQUED .ne. 'N', then AFP is the factored
    !>          form of the equilibrated matrix A.
    !>
    !>          If FACT = 'N', then AFP is an output argument and on exit
    !>          returns the triangular factor U or L from the Cholesky
    !>          factorization A = U**T * U or A = L * L**T of the original
    !>          matrix A.
    !>
    !>          If FACT = 'E', then AFP is an output argument and on exit
    !>          returns the triangular factor U or L from the Cholesky
    !>          factorization A = U**T * U or A = L * L**T of the equilibrated
    !>          matrix A (see the description of AP for the form of the
    !>          equilibrated matrix).
    !>

    EQUED

    !>          EQUED is CHARACTER*1
    !>          Specifies the form of equilibration that was done.
    !>          = 'N':  No equilibration (always true if FACT = 'N').
    !>          = 'Y':  Equilibration was done, i.e., A has been replaced by
    !>                  diag(S) * A * diag(S).
    !>          EQUED is an input argument if FACT = 'F'; otherwise, it is an
    !>          output argument.
    !>

    S

    !>          S is REAL array, dimension (N)
    !>          The scale factors for A; not accessed if EQUED = 'N'.  S is
    !>          an input argument if FACT = 'F'; otherwise, S is an output
    !>          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
    !>          must be positive.
    !>

    B

    !>          B is REAL array, dimension (LDB,NRHS)
    !>          On entry, the N-by-NRHS right hand side matrix B.
    !>          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
    !>          B is overwritten by diag(S) * 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 to
    !>          the original system of equations.  Note that if EQUED = 'Y',
    !>          A and B are modified on exit, and the solution to the
    !>          equilibrated system is inv(diag(S))*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 after equilibration (if done).  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:  the leading principal minor of order i of A
    !>                       is not positive, so the factorization could not
    !>                       be completed, and the solution has not been
    !>                       computed. RCOND = 0 is returned.
    !>                = N+1: U 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 = conjg(aji))
    !>                 a44
    !>
    !>  Packed storage of the upper triangle of A:
    !>
    !>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
    !>

Definition at line 309 of file sppsvx.f.

  subroutine zppsvx (character fact, character uplo, integer n, integer nrhs, complex*16, dimension( * ) ap, complex*16, dimension( * ) afp, character equed, double precision, dimension( * ) s, 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)

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

Purpose:

    !>
    !> ZPPSVX uses the Cholesky factorization A = U**H * U or A = L * L**H to
    !> compute the solution to a complex system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N Hermitian positive definite 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 = 'E', real scaling factors are computed to equilibrate
    !>    the system:
    !>       diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
    !>    Whether or not the system will be equilibrated depends on the
    !>    scaling of the matrix A, but if equilibration is used, A is
    !>    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
    !>
    !> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
    !>    factor the matrix A (after equilibration if FACT = 'E') as
    !>       A = U**H * U ,  if UPLO = 'U', or
    !>       A = L * L**H,  if UPLO = 'L',
    !>    where U is an upper triangular matrix, L is a lower triangular
    !>    matrix, and **H indicates conjugate transpose.
    !>
    !> 3. If the leading principal minor of order i is not positive,
    !>    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.
    !>
    !> 4. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 5. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>
    !> 6. If equilibration was used, the matrix X is premultiplied by
    !>    diag(S) so that it solves the original system before
    !>    equilibration.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of the matrix A is
    !>          supplied on entry, and if not, whether the matrix A should be
    !>          equilibrated before it is factored.
    !>          = 'F':  On entry, AFP contains the factored form of A.
    !>                  If EQUED = 'Y', the matrix A has been equilibrated
    !>                  with scaling factors given by S.  AP and AFP will not
    !>                  be modified.
    !>          = 'N':  The matrix A will be copied to AFP and factored.
    !>          = 'E':  The matrix A will be equilibrated if necessary, then
    !>                  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)
    !>          On entry, the upper or lower triangle of the Hermitian matrix
    !>          A, packed columnwise in a linear array, except if FACT = 'F'
    !>          and EQUED = 'Y', then A must contain the equilibrated matrix
    !>          diag(S)*A*diag(S).  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.  A is not modified if
    !>          FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
    !>
    !>          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
    !>          diag(S)*A*diag(S).
    !>

    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 triangular factor U or L from the Cholesky
    !>          factorization A = U**H*U or A = L*L**H, in the same storage
    !>          format as A.  If EQUED .ne. 'N', then AFP is the factored
    !>          form of the equilibrated matrix A.
    !>
    !>          If FACT = 'N', then AFP is an output argument and on exit
    !>          returns the triangular factor U or L from the Cholesky
    !>          factorization A = U**H * U or A = L * L**H of the original
    !>          matrix A.
    !>
    !>          If FACT = 'E', then AFP is an output argument and on exit
    !>          returns the triangular factor U or L from the Cholesky
    !>          factorization A = U**H * U or A = L * L**H of the equilibrated
    !>          matrix A (see the description of AP for the form of the
    !>          equilibrated matrix).
    !>

    EQUED

    !>          EQUED is CHARACTER*1
    !>          Specifies the form of equilibration that was done.
    !>          = 'N':  No equilibration (always true if FACT = 'N').
    !>          = 'Y':  Equilibration was done, i.e., A has been replaced by
    !>                  diag(S) * A * diag(S).
    !>          EQUED is an input argument if FACT = 'F'; otherwise, it is an
    !>          output argument.
    !>

    S

    !>          S is DOUBLE PRECISION array, dimension (N)
    !>          The scale factors for A; not accessed if EQUED = 'N'.  S is
    !>          an input argument if FACT = 'F'; otherwise, S is an output
    !>          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
    !>          must be positive.
    !>

    B

    !>          B is COMPLEX*16 array, dimension (LDB,NRHS)
    !>          On entry, the N-by-NRHS right hand side matrix B.
    !>          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
    !>          B is overwritten by diag(S) * 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 to
    !>          the original system of equations.  Note that if EQUED = 'Y',
    !>          A and B are modified on exit, and the solution to the
    !>          equilibrated system is inv(diag(S))*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 after equilibration (if done).  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:  the leading principal minor of order i of A
    !>                       is not positive, so the factorization could not
    !>                       be completed, and the solution has not been
    !>                       computed. RCOND = 0 is returned.
    !>                = N+1: U 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 ]
    !>

Definition at line 309 of file zppsvx.f.

Author

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