CSPSV (3lapack)

compute the solution to a complex system of linear equations A * X = B,

SYNOPSIS

  SUBROUTINE CSPSV( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )

      CHARACTER     UPLO

      INTEGER       INFO, LDB, N, NRHS

      INTEGER       IPIV( * )

      COMPLEX       AP( * ), B( LDB, * )

PURPOSE

  CSPSV computes the solution to a complex 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.

  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, D is symmetric and block diagonal with 1-by-1 and 2-
  by-2 diagonal blocks.  The factored form of A is then used to solve the
  system of equations A * X = B.

ARGUMENTS

  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
          matrix B.  NRHS >= 0.

  AP      (input/output) COMPLEX array, dimension (N*(N+1)/2)
          On entry, 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)*(2n-j)/2) =
          A(i,j) for j<=i<=n.  See below for further details.

          On exit, 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 CSPTRF, stored as a packed triangular
          matrix in the same storage format as A.

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D, as
          determined by CSPTRF.  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.

  B       (input/output) COMPLEX array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.  On exit, if INFO
          = 0, the N-by-NRHS solution matrix X.

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

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  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 could not be 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 ]