ppsvx - Man Page

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

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.

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.

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.

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.