CXML

CPBSVX (3lapack)


SYNOPSIS

  SUBROUTINE CPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S,
                     B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )

      CHARACTER      EQUED, FACT, UPLO

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

      REAL           RCOND

      REAL           BERR( * ), FERR( * ), RWORK( * ), S( * )

      COMPLEX        AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( * ),
                     X( LDX, * )

PURPOSE

  CPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute
  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.

  Error bounds on the solution and a condition estimate are also provided.

DESCRIPTION

  The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
     Whether or not the system will be equilibrated depends on the
     scaling of the matrix A, but if equilibration is used, A is
     overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

  2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
     factor the matrix A (after equilibration if FACT = 'E') 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.

  3. The factored form of A is used to estimate the condition number
     of the matrix A.  If the reciprocal of the condition number is
     less than machine precision, steps 4-6 are skipped.

  4. The system of equations is solved for X using the factored form
     of A.

  5. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

  6. If equilibration was used, the matrix X is premultiplied by
     diag(S) so that it solves the original system before
     equilibration.

ARGUMENTS

  FACT    (input) CHARACTER*1
          Specifies whether or not the factored form of the matrix A is
          supplied on entry, and if not, whether the matrix A should be
          equilibrated before it is factored.  = 'F':  On entry, AFB contains
          the factored form of A.  If EQUED = 'Y', the matrix A has been
          equilibrated with scaling factors given by S.  AB and AFB will not
          be modified.  = 'N':  The matrix A will be copied to AFB and
          factored.
          = 'E':  The matrix A will be equilibrated if necessary, then copied
          to AFB and factored.

  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
          matrices B and X.  NRHS >= 0.

  AB      (input/output) 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, except if FACT = 'F'
          and EQUED = 'Y', then A must contain the equilibrated matrix
          diag(S)*A*diag(S).  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 FACT = 'E' and EQUED = 'Y', A is overwritten by
          diag(S)*A*diag(S).

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

  AFB     (input or output) COMPLEX array, dimension (LDAFB,N)
          If FACT = 'F', then AFB is an input argument and on entry contains
          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 (see AB).  If EQUED = 'Y', then AFB is the factored
          form of the equilibrated matrix A.

          If FACT = 'N', then AFB is an output argument and on exit returns
          the triangular factor U or L from the Cholesky factorization A =
          U**H*U or A = L*L**H.

          If FACT = 'E', then AFB is an output argument and on exit returns
          the triangular factor U or L from the Cholesky factorization A =
          U**H*U or A = L*L**H of the equilibrated matrix A (see the
          description of A for the form of the equilibrated matrix).

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

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.  = 'N':  No
          equilibration (always true if FACT = 'N').
          = 'Y':  Equilibration was done, i.e., A has been replaced by
          diag(S) * A * diag(S).  EQUED is an input argument if FACT = 'F';
          otherwise, it is an output argument.

  S       (input or output) REAL array, dimension (N)
          The scale factors for A; not accessed if EQUED = 'N'.  S is an
          input argument if FACT = 'F'; otherwise, S is an output argument.
          If FACT = 'F' and EQUED = 'Y', each element of S must be positive.

  B       (input/output) COMPLEX array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand side matrix B.  On exit, if
          EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by
          diag(S) * B.

  LDB     (input) INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

  X       (output) COMPLEX array, dimension (LDX,NRHS)
          If INFO = 0, the N-by-NRHS solution matrix X to the original system
          of equations.  Note that if EQUED = 'Y', A and B are modified on
          exit, and the solution to the equilibrated system is
          inv(diag(S))*X.

  LDX     (input) INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

  RCOND   (output) REAL
          The estimate of the reciprocal condition number of the matrix A
          after equilibration (if done).  If RCOND is less than the machine
          precision (in particular, if RCOND = 0), the matrix is singular to
          working precision.  This condition is indicated by a return code of
          INFO > 0, and the solution and error bounds are not computed.

  FERR    (output) REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector X(j)
          (the j-th column of the solution matrix X).  If XTRUE is the true
          solution corresponding to X(j), FERR(j) is an estimated upper bound
          for the magnitude of the largest element in (X(j) - XTRUE) divided
          by the magnitude of the largest element in X(j).  The estimate is
          as reliable as the estimate for RCOND, and is almost always a
          slight overestimate of the true error.

  BERR    (output) REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution vector
          X(j) (i.e., the smallest relative change in any element of A or B
          that makes X(j) an exact solution).

  WORK    (workspace) COMPLEX array, dimension (2*N)

  RWORK   (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, and i is
          <= N: 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.  = N+1: RCOND is less than machine precision.
          The factorization has been completed, but the matrix is singular to
          working precision, and the solution and error bounds have not been
          computed.

FURTHER DETAILS

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

  Two-dimensional storage of the Hermitian matrix A:

     a11  a12  a13
          a22  a23  a24
               a33  a34  a35
                    a44  a45  a46
                         a55  a56
     (aij=conjg(aji))         a66

  Band storage of the upper triangle of A:

      *    *   a13  a24  a35  a46
      *   a12  a23  a34  a45  a56
     a11  a22  a33  a44  a55  a66

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

     a11  a22  a33  a44  a55  a66
     a21  a32  a43  a54  a65   *
     a31  a42  a53  a64   *    *

  Array elements marked * are not used by the routine.

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