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



NAME
       CPBSV

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



PURPOSE
            CPBSV 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)
                     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.

           KD        (input)
                     KD is INTEGER
                     The number of superdiagonals of the matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

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

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

           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, 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.



LAPACK routine                  31 October 2017                       CPBSV(3)