SRC/sgbsvx.f(3)

NAME
SRC/sgbsvx.f

SYNOPSIS

  Functions/Subroutines
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)
SGBSVX computes the solution to system of linear equations A * X = B for GB matrices

Function/Subroutine Documentation

  subroutine sgbsvx (character fact, character trans, integer n, integer kl, integer ku, integer nrhs, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, character equed, real, dimension( * ) r, real, dimension( * ) c, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SGBSVX computes the solution to system of linear equations A * X = B for GB matrices

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. If some U(i,i)=0, so that U is exactly singular, 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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 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

    !>          TRANS is 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

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

    KL

    !>          KL is INTEGER
    !>          The number of subdiagonals within the band of A.  KL >= 0.
    !>

    KU

    !>          KU is INTEGER
    !>          The number of superdiagonals within the band of A.  KU >= 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 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

    !>          LDAB is INTEGER
    !>          The leading dimension of the array AB.  LDAB >= KL+KU+1.
    !>

    AFB

    !>          AFB is 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

    !>          LDAFB is INTEGER
    !>          The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
    !>

    IPIV

    !>          IPIV is 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

    !>          EQUED is 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

    !>          R is 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

    !>          C is 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

    !>          B is 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

    !>          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 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 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
    !>          and EQUED = 'R' or 'B'.
    !>

    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 (MAX(1,3*N))
    !>          On exit, WORK(1) contains the reciprocal pivot growth
    !>          factor norm(A)/norm(U). The  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

    !>          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:  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. 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
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 365 of file sgbsvx.f.

Author

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