Loading...
Searching...
No Matches

## zppsvx()

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

Download ZPPSVX + dependencies [TGZ] [ZIP] [TXT]

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

  [in]

  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.
      !>

  [in]

  UPLO

      !>          UPLO is CHARACTER*1
      !>          = 'U':  Upper triangle of A is stored;
      !>          = 'L':  Lower triangle of A is stored.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The number of linear equations, i.e., the order of the
      !>          matrix A.  N >= 0.
      !>

  [in]

  NRHS

      !>          NRHS is INTEGER
      !>          The number of right hand sides, i.e., the number of columns
      !>          of the matrices B and X.  NRHS >= 0.
      !>

  [in,out]

  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).
      !>

  [in,out]

  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).
      !>

  [in,out]

  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.
      !>

  [in,out]

  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.
      !>

  [in,out]

  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.
      !>

  [in]

  LDB

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

  [out]

  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.
      !>

  [in]

  LDX

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

  [out]

  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.
      !>

  [out]

  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.
      !>

  [out]

  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).
      !>

  [out]

  WORK

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

  [out]

  RWORK

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

  [out]

  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 ]
      !>