spbsvx.f(3)

NAME
spbsvx.f -

SYNOPSIS

  Functions/Subroutines
subroutine SPBSVX (FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
SPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Function/Subroutine Documentation

  subroutine SPBSVX (CHARACTERFACT, CHARACTERUPLO, INTEGERN, INTEGERKD, INTEGERNRHS, REAL, dimension( ldab, * )AB, INTEGERLDAB, REAL, dimension( ldafb, * )AFB, INTEGERLDAFB, CHARACTEREQUED, REAL, dimension( * )S, REAL, dimension( ldb, * )B, INTEGERLDB, REAL, dimension( ldx, * )X, INTEGERLDX, REALRCOND, REAL, dimension( * )FERR, REAL, dimension( * )BERR, REAL, dimension( * )WORK, INTEGER, dimension( * )IWORK, INTEGERINFO)

SPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

     SPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
     compute the solution to a real system of linear equations
        A * X = B,
     where A is an N-by-N symmetric 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**T * U,  if UPLO = 'U', or
           A = L * L**T,  if UPLO = 'L',
        where U is an upper triangular band matrix, and L is a lower
        triangular band matrix.
     3. If the leading i-by-i principal minor is not positive definite,
        then the routine returns with INFO = i. Otherwise, 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, INFO = N+1 is returned as a warning, but the routine
        still goes on to solve for X and compute error bounds as
        described below.
     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.

Parameters:
    FACT

              FACT is 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

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    KD

              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

              NRHS is INTEGER
              The number of right-hand sides, i.e., the number of columns
              of the matrices B and X.  NRHS >= 0.

    AB

              AB is REAL array, dimension (LDAB,N)
              On entry, the upper or lower triangle of the symmetric 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

              LDAB is INTEGER
              The leading dimension of the array A.  LDAB >= KD+1.

    AFB

              AFB is REAL 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**T*U or A = L*L**T 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**T*U or A = L*L**T.
              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**T*U or A = L*L**T of the equilibrated
              matrix A (see the description of A for the form of the
              equilibrated matrix).

    LDAFB

              LDAFB is INTEGER
              The leading dimension of the array AFB.  LDAFB >= KD+1.

    EQUED

              EQUED is 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

              S is 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

              B is REAL 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

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

    X

              X is REAL array, dimension (LDX,NRHS)
              If INFO = 0 or INFO = N+1, 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

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

    RCOND

              RCOND is 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.

    FERR

              FERR is 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

              BERR is 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

              WORK is REAL array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is 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. RCOND = 0 is returned.
                    = N+1: U is nonsingular, but RCOND is less than machine
                           precision, meaning that the matrix is singular
                           to working precision.  Nevertheless, the
                           solution and error bounds are computed because
                           there are a number of situations where the
                           computed solution can be more accurate than the
                           value of RCOND would suggest.

Author:
    Univ'. of Tennessee

    Univ'. of California Berkeley

    Univ'. of Colorado Denver

    NAG Ltd'.

Date:
    April 2012

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 symmetric 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.

Definition at line 342 of file spbsvx.f'.

Author

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