SSPSVX (3lapack)

use the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices

SYNOPSIS

  SUBROUTINE SSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
                     RCOND, FERR, BERR, WORK, IWORK, INFO )

      CHARACTER      FACT, UPLO

      INTEGER        INFO, LDB, LDX, N, NRHS

      REAL           RCOND

      INTEGER        IPIV( * ), IWORK( * )

      REAL           AFP( * ), AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
                     WORK( * ), X( LDX, * )

PURPOSE

  SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A =
  L*D*L**T to compute the solution to a real system of linear equations A * X
  = B, where A is an N-by-N symmetric 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 = 'N', the diagonal pivoting method is used to factor A as
        A = U * D * U**T,  if UPLO = 'U', or
        A = L * D * L**T,  if UPLO = 'L',
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices and D is symmetric and block diagonal with
     1-by-1 and 2-by-2 diagonal blocks.

  2. 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, steps 3 and 4 are skipped.

  3. The system of equations is solved for X using the factored form
     of A.

  4. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

ARGUMENTS

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of A has been supplied
          on entry.  = 'F':  On entry, AFP and IPIV contain the factored form
          of A.  AP, AFP and IPIV will not be modified.  = 'N':  The matrix A
          will be copied to AFP and factored.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The number of linear equations, i.e., the order of the matrix A.  N
          >= 0.

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

  AP      (input) REAL array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  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)*(2*n-j)/2) = A(i,j) for
          j<=i<=n.  See below for further details.

  AFP     (input or output) REAL array, dimension
          (N*(N+1)/2) If FACT = 'F', then AFP is an input argument and on
          entry contains the block diagonal matrix D and the multipliers used
          to obtain the factor U or L from the factorization A = U*D*U**T or
          A = L*D*L**T as computed by SSPTRF, stored as a packed triangular
          matrix in the same storage format as A.

          If FACT = 'N', then AFP is an output argument and on exit contains
          the block diagonal matrix D and the multipliers used to obtain the
          factor U or L from the factorization A = U*D*U**T or A = L*D*L**T
          as computed by SSPTRF, stored as a packed triangular matrix in the
          same storage format as A.

  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry contains
          details of the interchanges and the block structure of D, as
          determined by SSPTRF.  If IPIV(k) > 0, then rows and columns k and
          IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns
          k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
          diagonal block.  If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then
          rows and columns k+1 and -IPIV(k) were interchanged and
          D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

          If FACT = 'N', then IPIV is an output argument and on exit contains
          details of the interchanges and the block structure of D, as
          determined by SSPTRF.

  B       (input) REAL array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

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

  X       (output) REAL array, dimension (LDX,NRHS)
          If INFO = 0, the N-by-NRHS solution matrix X.

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

  RCOND   (output) REAL
          The estimate of the reciprocal condition number of the matrix A.
          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, and the
          solution and error bounds are not computed.

  FERR    (output) 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    (output) 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    (workspace) REAL array, dimension (3*N)

  IWORK   (workspace) INTEGER array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0 and <= N: if INFO = i, D(i,i) is exactly zero.  The
          factorization has been completed, but the block diagonal matrix D
          is exactly singular, so the solution and error bounds could not be
          computed.  = N+1: the block diagonal matrix D is nonsingular, but
          RCOND is less than machine precision.  The factorization has been
          completed, but the matrix is singular to working precision, so the
          solution and error bounds have not been computed.

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 = aji)
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]