GTSVX - 5.2 English - 68552

GTSVX uses the LU factorization to compute the solution to a real system of linear equations.

Purpose:

    GTSVX 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.

* \b 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 – [in]

  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 – [in]

  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 – [in]

  N is INTEGER

  The order of the matrix A. N >= 0.

• NRHS – [in]

  NRHS is INTEGER

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

• DL – [in]

  DL is REAL array, dimension (N-1)

  The (n-1) subdiagonal elements of A.

• D – [in]

  D is REAL array, dimension (N)

  The n diagonal elements of A.

• DU – [in]

  DU is REAL array, dimension (N-1)

  The (n-1) superdiagonal elements of A.

• DLF – [inout]

  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 – [inout]

  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 – [inout]

  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 – [inout]

  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 – [inout]

  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 – [in]

  B is REAL array, dimension (LDB,NRHS)

  The N-by-NRHS right hand side matrix B.

• LDB – [in]

  LDB is INTEGER

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

• X – [out]

  X is REAL array, dimension (LDX,NRHS)

  If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

• LDX – [in]

  LDX is INTEGER

  The leading dimension of the array X. LDX >= fla_max(1,N).

• RCOND – [out]

  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 – [out]

  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 – [out]

  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 – [out] WORK is REAL array, dimension (3*N)

• IWORK – [out] IWORK is INTEGER array, dimension (N)

• INFO – [out]

  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.