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dpbsvx


 NAME
      DPBSVX - use 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,

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

          CHARACTER      EQUED, FACT, UPLO

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

          DOUBLE         PRECISION RCOND

          INTEGER        IWORK( * )

          DOUBLE         PRECISION AB( LDAB, * ), AFB( LDAFB, * ),
                         B( LDB, * ), BERR( * ), FERR( * ), S( *
                         ), WORK( * ), X( LDX, * )

 PURPOSE
      DPBSVX 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 defin-
      ite 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. 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 fac-
      tored 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 fac-
              tored 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 neces-
              sary, 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) DOUBLE PRECISION array, dimension (LDAB,N)
              On entry, the upper or lower triangle of the sym-
              metric 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.

 (LDAFB,N)
      AFB     (input or output) DOUBLE PRECISION array, dimension
              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   (input) INTEGER
              The leading dimension of the array AFB.  LDAFB >=
              KD+1.

      EQUED   (input/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 variable if FACT = 'F'; otherwise, it is an

              output variable.

      S       (input/output) DOUBLE PRECISION array, dimension (N)
              The scale factors for A; not accessed if EQUED =
              'N'.  S is an input variable if FACT = 'F'; other-
              wise, S is an output variable.  If FACT = 'F' and
              EQUED = 'Y', each element of S must be positive.

 (LDB,NRHS)
      B       (input/output) DOUBLE PRECISION array, dimension
              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) DOUBLE PRECISION 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) DOUBLE PRECISION
              The estimate of the reciprocal condition number of
              the matrix A after equilibration (if done).  If
              RCOND is less than the machine precision (in partic-
              ular, if RCOND = 0), the matrix is singular to work-
              ing precision.  This condition is indicated by a
              return code of INFO > 0, and the solution and error
              bounds are not computed.

      FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
              The estimated forward error bounds for each solution
              vector X(j) (the j-th column of the solution matrix
              X).  If XTRUE is the true solution, FERR(j) bounds
              the magnitude of the largest entry in (X(j) - XTRUE)
              divided by the magnitude of the largest entry in
              X(j).  The quality of the error bound depends on the
              quality of the estimate of norm(inv(A)) computed in
              the code; if the estimate of norm(inv(A)) is accu-
              rate, the error bound is guaranteed.

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

              exact solution).

      WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

      IWORK   (workspace) INTEGER 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: 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 com-
              puted.  = 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 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.