SGBSVX (3lapack)

use the LU factorization to compute the solution to a real system of linear equations A * X = B, A**T * X = B, or A**H * X = B,

SYNOPSIS

  SUBROUTINE SGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
                     IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
                     WORK, IWORK, INFO )

      CHARACTER      EQUED, FACT, TRANS

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

      REAL           RCOND

      INTEGER        IPIV( * ), IWORK( * )

      REAL           AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( * ),
                     C( * ), FERR( * ), R( * ), WORK( * ), X( LDX, * )

PURPOSE

  SGBSVX uses the LU factorization to compute the solution to a real system
  of linear equations A * X = B, A**T * X = B, or A**H * 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.

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

DESCRIPTION

  The following steps are performed by this subroutine:

  1. If FACT = 'E', real scaling factors are computed to equilibrate
     the system:
        TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
        TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
        TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
     or diag(C)*B (if TRANS = 'T' or 'C').

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

  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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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 and IPIV
          contain the factored form of A.  If EQUED is not 'N', the matrix A
          has been equilibrated with scaling factors given by R and C.  AB,
          AFB, and IPIV are not 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.

  TRANS   (input) CHARACTER*1
          Specifies the form of the system of equations.  = 'N':  A * X = B
          (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Transpose)

  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
          matrices B and X.  NRHS >= 0.

  AB      (input/output) REAL array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.  The
          j-th column of A is stored in the j-th column of the array AB as
          follows: AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

          If FACT = 'F' and EQUED is not 'N', then A must have been
          equilibrated by the scaling factors in R and/or C.  AB is not
          modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on
          exit.

          On exit, if EQUED .ne. 'N', A is scaled as follows: EQUED = 'R':  A
          := diag(R) * A
          EQUED = 'C':  A := A * diag(C)
          EQUED = 'B':  A := diag(R) * A * diag(C).

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

  AFB     (input or output) REAL array, dimension (LDAFB,N)
          If FACT = 'F', then AFB is an input argument and on entry contains
          details of the LU factorization of the band matrix A, as computed
          by SGBTRF.  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.
          If EQUED .ne. 'N', then AFB is the factored form of the
          equilibrated matrix A.

          If FACT = 'N', then AFB is an output argument and on exit returns
          details of the LU factorization of A.

          If FACT = 'E', then AFB is an output argument and on exit returns
          details of the LU factorization of the equilibrated matrix A (see
          the description of AB for the form of the equilibrated matrix).

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

  IPIV    (input or output) INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry contains
          the pivot indices from the factorization A = L*U as computed by
          SGBTRF; row i of the matrix was interchanged with row IPIV(i).

          If FACT = 'N', then IPIV is an output argument and on exit contains
          the pivot indices from the factorization A = L*U of the original
          matrix A.

          If FACT = 'E', then IPIV is an output argument and on exit contains
          the pivot indices from the factorization A = L*U of the
          equilibrated matrix A.

  EQUED   (input or output) CHARACTER*1
          Specifies the form of equilibration that was done.  = 'N':  No
          equilibration (always true if FACT = 'N').
          = 'R':  Row equilibration, i.e., A has been premultiplied by
          diag(R).  = 'C':  Column equilibration, i.e., A has been
          postmultiplied by diag(C).  = 'B':  Both row and column
          equilibration, i.e., A has been replaced by diag(R) * A * diag(C).
          EQUED is an input argument if FACT = 'F'; otherwise, it is an
          output argument.

  R       (input or output) REAL array, dimension (N)
          The row scale factors for A.  If EQUED = 'R' or 'B', A is
          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not
          accessed.  R is an input argument if FACT = 'F'; otherwise, R is an
          output argument.  If FACT = 'F' and EQUED = 'R' or 'B', each
          element of R must be positive.

  C       (input or output) REAL array, dimension (N)
          The column scale factors for A.  If EQUED = 'C' or 'B', A is
          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not
          accessed.  C is an input argument if FACT = 'F'; otherwise, C is an
          output argument.  If FACT = 'F' and EQUED = 'C' or 'B', each
          element of C must be positive.

  B       (input/output) REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.  On exit, if EQUED = 'N', B
          is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is
          overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED = 'C' or
          'B', B is overwritten by diag(C)*B.

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

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

  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/output) REAL array, dimension (3*N)
          On exit, WORK(1) contains the reciprocal pivot growth factor
          norm(A)/norm(U). The "max absolute element" norm is used. If
          WORK(1) is much less than 1, then the stability of the LU
          factorization of the (equilibrated) matrix A could be poor. This
          also means that the solution X, condition estimator RCOND, and
          forward error bound FERR could be unreliable. If factorization
          fails with 0<INFO<=N, then WORK(1) contains the reciprocal pivot
          growth factor for the leading INFO columns of A.

  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:  U(i,i) is exactly zero.  The factorization has been
          completed, but the factor U is exactly singular, so the solution
          and error bounds could not be computed.  = N+1: RCOND is less than
          machine precision.  The factorization has been completed, but the
          matrix A is singular to working precision, and the solution and
          error bounds have not been computed.