posvx(3)

NAME
posvx - posvx: factor and solve, expert

SYNOPSIS

  Functions

subroutine cposvx (fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
CPOSVX computes the solution to system of linear equations A * X = B for PO matrices
subroutine dposvx (fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
DPOSVX computes the solution to system of linear equations A * X = B for PO matrices
subroutine sposvx (fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
SPOSVX computes the solution to system of linear equations A * X = B for PO matrices
subroutine zposvx (fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
ZPOSVX computes the solution to system of linear equations A * X = B for PO matrices

Detailed Description

Function Documentation

  subroutine cposvx (character fact, character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, 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)

CPOSVX computes the solution to system of linear equations A * X = B for PO matrices

Purpose:

    !>
    !> CPOSVX 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 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 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, AF contains the factored form of A.
    !>                  If EQUED = 'Y', the matrix A has been equilibrated
    !>                  with scaling factors given by S.  A and AF will not
    !>                  be modified.
    !>          = 'N':  The matrix A will be copied to AF and factored.
    !>          = 'E':  The matrix A will be equilibrated if necessary, then
    !>                  copied to AF 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.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>          On entry, the Hermitian matrix A, except if FACT = 'F' and
    !>          EQUED = 'Y', then A must contain the equilibrated matrix
    !>          diag(S)*A*diag(S).  If UPLO = 'U', the leading
    !>          N-by-N upper triangular part of A contains the upper
    !>          triangular part of the matrix A, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading N-by-N lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.  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).
    !>

    LDA

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

    AF

    !>          AF is COMPLEX array, dimension (LDAF,N)
    !>          If FACT = 'F', then AF 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 AF is the factored form
    !>          of the equilibrated matrix diag(S)*A*diag(S).
    !>
    !>          If FACT = 'N', then AF 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 AF 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 A for the form of the
    !>          equilibrated matrix).
    !>

    LDAF

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

    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 righthand 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.

  subroutine dposvx (character fact, character uplo, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldaf, * ) af, integer ldaf, 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)

DPOSVX computes the solution to system of linear equations A * X = B for PO matrices

Purpose:

    !>
    !> DPOSVX 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 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, AF contains the factored form of A.
    !>                  If EQUED = 'Y', the matrix A has been equilibrated
    !>                  with scaling factors given by S.  A and AF will not
    !>                  be modified.
    !>          = 'N':  The matrix A will be copied to AF and factored.
    !>          = 'E':  The matrix A will be equilibrated if necessary, then
    !>                  copied to AF 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.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>          On entry, the symmetric matrix A, except if FACT = 'F' and
    !>          EQUED = 'Y', then A must contain the equilibrated matrix
    !>          diag(S)*A*diag(S).  If UPLO = 'U', the leading
    !>          N-by-N upper triangular part of A contains the upper
    !>          triangular part of the matrix A, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading N-by-N lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.  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).
    !>

    LDA

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

    AF

    !>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
    !>          If FACT = 'F', then AF 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 AF is the factored form
    !>          of the equilibrated matrix diag(S)*A*diag(S).
    !>
    !>          If FACT = 'N', then AF 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 AF 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 A for the form of the
    !>          equilibrated matrix).
    !>

    LDAF

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

    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.

  subroutine sposvx (character fact, character uplo, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldaf, * ) af, integer ldaf, 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)

SPOSVX computes the solution to system of linear equations A * X = B for PO matrices

Purpose:

    !>
    !> SPOSVX 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 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, AF contains the factored form of A.
    !>                  If EQUED = 'Y', the matrix A has been equilibrated
    !>                  with scaling factors given by S.  A and AF will not
    !>                  be modified.
    !>          = 'N':  The matrix A will be copied to AF and factored.
    !>          = 'E':  The matrix A will be equilibrated if necessary, then
    !>                  copied to AF 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.
    !>

    A

    !>          A is REAL array, dimension (LDA,N)
    !>          On entry, the symmetric matrix A, except if FACT = 'F' and
    !>          EQUED = 'Y', then A must contain the equilibrated matrix
    !>          diag(S)*A*diag(S).  If UPLO = 'U', the leading
    !>          N-by-N upper triangular part of A contains the upper
    !>          triangular part of the matrix A, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading N-by-N lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.  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).
    !>

    LDA

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

    AF

    !>          AF is REAL array, dimension (LDAF,N)
    !>          If FACT = 'F', then AF 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 AF is the factored form
    !>          of the equilibrated matrix diag(S)*A*diag(S).
    !>
    !>          If FACT = 'N', then AF 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 AF 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 A for the form of the
    !>          equilibrated matrix).
    !>

    LDAF

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

    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.

  subroutine zposvx (character fact, character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, 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)

ZPOSVX computes the solution to system of linear equations A * X = B for PO matrices

Purpose:

    !>
    !> ZPOSVX 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 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 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, AF contains the factored form of A.
    !>                  If EQUED = 'Y', the matrix A has been equilibrated
    !>                  with scaling factors given by S.  A and AF will not
    !>                  be modified.
    !>          = 'N':  The matrix A will be copied to AF and factored.
    !>          = 'E':  The matrix A will be equilibrated if necessary, then
    !>                  copied to AF 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.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA,N)
    !>          On entry, the Hermitian matrix A, except if FACT = 'F' and
    !>          EQUED = 'Y', then A must contain the equilibrated matrix
    !>          diag(S)*A*diag(S).  If UPLO = 'U', the leading
    !>          N-by-N upper triangular part of A contains the upper
    !>          triangular part of the matrix A, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading N-by-N lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.  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).
    !>

    LDA

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

    AF

    !>          AF is COMPLEX*16 array, dimension (LDAF,N)
    !>          If FACT = 'F', then AF 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 AF is the factored form
    !>          of the equilibrated matrix diag(S)*A*diag(S).
    !>
    !>          If FACT = 'N', then AF 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 AF 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 A for the form of the
    !>          equilibrated matrix).
    !>

    LDAF

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

    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 righthand 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.

Author

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