ZPBSV (3lapack)

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, * )

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.