SRC/sgtsvx.f(3)

NAME
SRC/sgtsvx.f

SYNOPSIS

  Functions/Subroutines
subroutine sgtsvx (fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
SGTSVX computes the solution to system of linear equations A * X = B for GT matrices

Function/Subroutine Documentation

  subroutine sgtsvx (character fact, character trans, integer n, integer nrhs, real, dimension( * ) dl, real, dimension( * ) d, real, dimension( * ) du, real, dimension( * ) dlf, real, dimension( * ) df, real, dimension( * ) duf, real, dimension( * ) du2, integer, dimension( * ) ipiv, 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)

SGTSVX computes the solution to system of linear equations A * X = B for GT matrices

Purpose:

    !>
    !> SGTSVX uses the LU factorization to compute the solution to a real
    !> system of linear equations A * X = B or A**T * X = B,
    !> where A is a tridiagonal matrix of order N 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 = 'N', the LU decomposition is used to factor the matrix A
    !>    as A = L * U, where L is a product of permutation and unit lower
    !>    bidiagonal matrices and U is upper triangular with nonzeros in
    !>    only the main diagonal and first two superdiagonals.
    !>
    !> 2. 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.
    !>
    !> 3. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 4. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of A has been
    !>          supplied on entry.
    !>          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored
    !>                  form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
    !>                  will not be modified.
    !>          = 'N':  The matrix will be copied to DLF, DF, and DUF
    !>                  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  (Conjugate transpose = Transpose)
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    NRHS

    !>          NRHS is INTEGER
    !>          The number of right hand sides, i.e., the number of columns
    !>          of the matrix B.  NRHS >= 0.
    !>

    DL

    !>          DL is REAL array, dimension (N-1)
    !>          The (n-1) subdiagonal elements of A.
    !>

    D

    !>          D is REAL array, dimension (N)
    !>          The n diagonal elements of A.
    !>

    DU

    !>          DU is REAL array, dimension (N-1)
    !>          The (n-1) superdiagonal elements of A.
    !>

    DLF

    !>          DLF is REAL array, dimension (N-1)
    !>          If FACT = 'F', then DLF is an input argument and on entry
    !>          contains the (n-1) multipliers that define the matrix L from
    !>          the LU factorization of A as computed by SGTTRF.
    !>
    !>          If FACT = 'N', then DLF is an output argument and on exit
    !>          contains the (n-1) multipliers that define the matrix L from
    !>          the LU factorization of A.
    !>

    DF

    !>          DF is REAL array, dimension (N)
    !>          If FACT = 'F', then DF is an input argument and on entry
    !>          contains the n diagonal elements of the upper triangular
    !>          matrix U from the LU factorization of A.
    !>
    !>          If FACT = 'N', then DF is an output argument and on exit
    !>          contains the n diagonal elements of the upper triangular
    !>          matrix U from the LU factorization of A.
    !>

    DUF

    !>          DUF is REAL array, dimension (N-1)
    !>          If FACT = 'F', then DUF is an input argument and on entry
    !>          contains the (n-1) elements of the first superdiagonal of U.
    !>
    !>          If FACT = 'N', then DUF is an output argument and on exit
    !>          contains the (n-1) elements of the first superdiagonal of U.
    !>

    DU2

    !>          DU2 is REAL array, dimension (N-2)
    !>          If FACT = 'F', then DU2 is an input argument and on entry
    !>          contains the (n-2) elements of the second superdiagonal of
    !>          U.
    !>
    !>          If FACT = 'N', then DU2 is an output argument and on exit
    !>          contains the (n-2) elements of the second superdiagonal of
    !>          U.
    !>

    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 LU factorization of A as
    !>          computed by SGTTRF.
    !>
    !>          If FACT = 'N', then IPIV is an output argument and on exit
    !>          contains the pivot indices from the LU factorization of A;
    !>          row i of the matrix was interchanged with row IPIV(i).
    !>          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
    !>          a row interchange was not required.
    !>

    B

    !>          B is REAL array, dimension (LDB,NRHS)
    !>          The N-by-NRHS right hand side matrix 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.
    !>

    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.  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:  U(i,i) is exactly zero.  The factorization
    !>                       has not been completed unless i = N, 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
    !>                       value of RCOND would suggest.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 290 of file sgtsvx.f.

Author

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