zpbsv(3S)

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

SYNOPSIS
     SUBROUTINE ZPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )

         CHARACTER     UPLO

         INTEGER       INFO, KD, LDAB, LDB, N, NRHS

         COMPLEX*16    AB( LDAB, * ), B( LDB, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     ZPBSV computes the solution to a complex system of linear equations A * X
     = B, where A is an N-by-N Hermitian positive definite band matrix and X
     and B are N-by-NRHS matrices.

     The Cholesky decomposition is used to factor A as
        A = U**H * U,  if UPLO = 'U', or
        A = L * L**H,  if UPLO = 'L',
     where U is an upper triangular band matrix, and L is a lower triangular
     band matrix, with the same number of superdiagonals or subdiagonals as A.
     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.

     KD      (input) INTEGER
             The number of superdiagonals of the matrix A if UPLO = 'U', or
             the number of subdiagonals if UPLO = 'L'.  KD >= 0.
     NRHS    (input) INTEGER
             The number of right hand sides, i.e., the number of columns of
             the matrix B.  NRHS >= 0.

     AB      (input/output) COMPLEX*16 array, dimension (LDAB,N)
             On entry, the upper or lower triangle of the Hermitian band
             matrix A, stored in the first KD+1 rows of the array.  The j-th
             column of A is stored in the j-th column of the array AB as
             follows:  if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-
             KD)<=i<=j; if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for
             j<=i<=min(N,j+KD).  See below for further details.

             On exit, if INFO = 0, the triangular factor U or L from the
             Cholesky factorization A = U**H*U or A = L*L**H of the band
             matrix A, in the same storage format as A.

     LDAB    (input) INTEGER
             The leading dimension of the array AB.  LDAB >= KD+1.

     B       (input/output) COMPLEX*16 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, the leading minor of order i of A is not
             positive definite, so the factorization could not be completed,
             and the solution has not been computed.

FURTHER DETAILS
     The band storage scheme is illustrated by the following example, when N =
     6, KD = 2, and UPLO = 'U':

     On entry:                       On exit:

         *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
         *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
        a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

     Similarly, if UPLO = 'L' the format of A is as follows:

     On entry:                       On exit:

        a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
        a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
        a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

     Array elements marked * are not used by the routine.
SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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