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## dposvx()

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

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

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

  [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, 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.
      !>

  [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]

  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).
      !>

  [in]

  LDA

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

  [in,out]

  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).
      !>

  [in]

  LDAF

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

  [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 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.
      !>

  [in]

  LDB

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

  [out]

  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.
      !>

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

  [out]

  IWORK

      !>          IWORK is INTEGER 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.