CGBSV (3lapack)

compute the solution to a complex system of linear equations A * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices

SYNOPSIS

  SUBROUTINE CGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )

      INTEGER       INFO, KL, KU, LDAB, LDB, N, NRHS

      INTEGER       IPIV( * )

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

PURPOSE

  CGBSV computes the solution to a complex system of linear equations A * X =
  B, where A is a band matrix of order N with KL subdiagonals and KU
  superdiagonals, and X and B are N-by-NRHS matrices.

  The LU decomposition with partial pivoting and row interchanges is used to
  factor A as A = L * U, where L is a product of permutation and unit lower
  triangular matrices with KL subdiagonals, and U is upper triangular with
  KL+KU superdiagonals.  The factored form of A is then used to solve the
  system of equations A * X = B.

ARGUMENTS

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

  KL      (input) INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.

  KU      (input) INTEGER
          The number of superdiagonals within the band of A.  KU >= 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 array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1;
          rows 1 to KL of the array need not be set.  The j-th column of A is
          stored in the j-th column of the array AB as follows:
          AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL) On exit,
          details of the factorization: U is stored as an upper triangular
          band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the
          multipliers used during the factorization are stored in rows
          KL+KU+2 to 2*KL+KU+1.  See below for further details.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

  IPIV    (output) INTEGER array, dimension (N)
          The pivot indices that define the permutation matrix P; row i of
          the matrix was interchanged with row IPIV(i).

  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, U(i,i) is exactly zero.  The factorization has
          been completed, but the factor U is exactly singular, and the
          solution has not been computed.

FURTHER DETAILS

  The band storage scheme is illustrated by the following example, when M = N
  = 6, KL = 2, KU = 1:

  On entry:                       On exit:

      *    *    *    +    +    +       *    *    *   u14  u25  u36
      *    *    +    +    +    +       *    *   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
     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

  Array elements marked * are not used by the routine; elements marked + need
  not be set on entry, but are required by the routine to store elements of U
  because of fill-in resulting from the row interchanges.