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## sgtsvx()

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

Download SGTSVX + dependencies [TGZ] [ZIP] [TXT]

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

  [in]

  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.
      !>

  [in]

  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)
      !>

  [in]

  N

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

  [in]

  NRHS

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

  [in]

  DL

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

  [in]

  D

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

  [in]

  DU

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

  [in,out]

  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.
      !>

  [in,out]

  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.
      !>

  [in,out]

  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.
      !>

  [in,out]

  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.
      !>

  [in,out]

  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.
      !>

  [in]

  B

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

  [in]

  LDB

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

  [out]

  X

      !>          X is REAL array, dimension (LDX,NRHS)
      !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
      !>

  [in]

  LDX

      !>          LDX is INTEGER
      !>          The leading dimension of the array X.  LDX >= max(1,N).
      !>

  [out]

  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.
      !>

  [out]

  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.
      !>

  [out]

  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).
      !>

  [out]

  WORK

      !>          WORK is REAL array, dimension (3*N)
      !>

  [out]

  IWORK

      !>          IWORK is INTEGER array, dimension (N)
      !>

  [out]

  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.