CSPSV(3)       LAPACK routine of NEC Numeric Library Collection       CSPSV(3)



NAME
       CSPSV

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



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

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

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

           AP        (input/output)
                     AP is 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)
                     IPIV is 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)
                     B is 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)
                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           INFO      (output)
                     INFO is 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 ]



LAPACK routine                  31 October 2017                       CSPSV(3)