realGTsolve(3)

NAME
realGTsolve - real

SYNOPSIS

  Functions
subroutine sgtsv (n, nrhs, dl, d, du, b, ldb, info)
SGTSV computes the solution to system of linear equations A * X = B for GT matrices
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

Detailed Description

This is the group of real solve driver functions for GT matrices

Function Documentation

  subroutine sgtsv (integer n, integer nrhs, real, dimension( * ) dl, real, dimension( * ) d, real, dimension( * ) du, real, dimension( ldb, * ) b, integer ldb, integer info)

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

Purpose:

     SGTSV  solves the equation
        A*X = B,
     where A is an n by n tridiagonal matrix, by Gaussian elimination with
     partial pivoting.
     Note that the equation  A**T*X = B  may be solved by interchanging the
     order of the arguments DU and DL.

Parameters
    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)
              On entry, DL must contain the (n-1) sub-diagonal elements of
              A.
              On exit, DL is overwritten by the (n-2) elements of the
              second super-diagonal of the upper triangular matrix U from
              the LU factorization of A, in DL(1), ..., DL(n-2).

    D

              D is REAL array, dimension (N)
              On entry, D must contain the diagonal elements of A.
              On exit, D is overwritten by the n diagonal elements of U.

    DU

              DU is REAL array, dimension (N-1)
              On entry, DU must contain the (n-1) super-diagonal elements
              of A.
              On exit, DU is overwritten by the (n-1) elements of the first
              super-diagonal of U.

    B

              B is REAL array, dimension (LDB,NRHS)
              On entry, the N by NRHS matrix of right hand side matrix B.
              On exit, if INFO = 0, the N by NRHS solution matrix X.

    LDB

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

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, U(i,i) is exactly zero, and the solution
                   has not been computed.  The factorization has not been
                   completed unless i = N.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 126 of file sgtsv.f.

  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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