realPTcomputational(3)

NAME
realPTcomputational

SYNOPSIS

  Functions
subroutine sptcon (N, D, E, ANORM, RCOND, WORK, INFO)
SPTCON
subroutine spteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
SPTEQR
subroutine sptrfs (N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO)
SPTRFS
subroutine spttrs (N, NRHS, D, E, B, LDB, INFO)
SPTTRS
subroutine sptts2 (N, NRHS, D, E, B, LDB)
SPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Detailed Description

This is the group of real computational functions for PT matrices

Function Documentation

  subroutine sptcon (integer N, real, dimension( * ) D, real, dimension( * ) E, real ANORM, real RCOND, real, dimension( * ) WORK, integer INFO)

SPTCON

Purpose:

     SPTCON computes the reciprocal of the condition number (in the
     1-norm) of a real symmetric positive definite tridiagonal matrix
     using the factorization A = L*D*L**T or A = U**T*D*U computed by
     SPTTRF.
     Norm(inv(A)) is computed by a direct method, and the reciprocal of
     the condition number is computed as
                  RCOND = 1 / (ANORM * norm(inv(A))).

Parameters:
    N

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

    D

              D is REAL array, dimension (N)
              The n diagonal elements of the diagonal matrix D from the
              factorization of A, as computed by SPTTRF.

    E

              E is REAL array, dimension (N-1)
              The (n-1) off-diagonal elements of the unit bidiagonal factor
              U or L from the factorization of A,  as computed by SPTTRF.

    ANORM

              ANORM is REAL
              The 1-norm of the original matrix A.

    RCOND

              RCOND is REAL
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
              1-norm of inv(A) computed in this routine.

    WORK

              WORK is REAL array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Further Details:

      The method used is described in Nicholas J. Higham, "Efficient
      Algorithms for Computing the Condition Number of a Tridiagonal
      Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.

Definition at line 120 of file sptcon.f.

  subroutine spteqr (character COMPZ, integer N, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer INFO)

SPTEQR

Purpose:

     SPTEQR computes all eigenvalues and, optionally, eigenvectors of a
     symmetric positive definite tridiagonal matrix by first factoring the
     matrix using SPTTRF, and then calling SBDSQR to compute the singular
     values of the bidiagonal factor.
     This routine computes the eigenvalues of the positive definite
     tridiagonal matrix to high relative accuracy.  This means that if the
     eigenvalues range over many orders of magnitude in size, then the
     small eigenvalues and corresponding eigenvectors will be computed
     more accurately than, for example, with the standard QR method.
     The eigenvectors of a full or band symmetric positive definite matrix
     can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to
     reduce this matrix to tridiagonal form. (The reduction to tridiagonal
     form, however, may preclude the possibility of obtaining high
     relative accuracy in the small eigenvalues of the original matrix, if
     these eigenvalues range over many orders of magnitude.)

Parameters:
    COMPZ

              COMPZ is CHARACTER*1
              = 'N':  Compute eigenvalues only.
              = 'V':  Compute eigenvectors of original symmetric
                      matrix also.  Array Z contains the orthogonal
                      matrix used to reduce the original matrix to
                      tridiagonal form.
              = 'I':  Compute eigenvectors of tridiagonal matrix also.

    N

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

    D

              D is REAL array, dimension (N)
              On entry, the n diagonal elements of the tridiagonal
              matrix.
              On normal exit, D contains the eigenvalues, in descending
              order.

    E

              E is REAL array, dimension (N-1)
              On entry, the (n-1) subdiagonal elements of the tridiagonal
              matrix.
              On exit, E has been destroyed.

    Z

              Z is REAL array, dimension (LDZ, N)
              On entry, if COMPZ = 'V', the orthogonal matrix used in the
              reduction to tridiagonal form.
              On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
              original symmetric matrix;
              if COMPZ = 'I', the orthonormal eigenvectors of the
              tridiagonal matrix.
              If INFO > 0 on exit, Z contains the eigenvectors associated
              with only the stored eigenvalues.
              If  COMPZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              COMPZ = 'V' or 'I', LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (4*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  the Cholesky factorization of the matrix could
                          not be performed because the i-th principal minor
                          was not positive definite.
                    > N   the SVD algorithm failed to converge;
                          if INFO = N+i, i off-diagonal elements of the
                          bidiagonal factor did not converge to zero.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 147 of file spteqr.f.

  subroutine sptrfs (integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) DF, real, dimension( * ) EF, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer INFO)

SPTRFS

Purpose:

     SPTRFS improves the computed solution to a system of linear
     equations when the coefficient matrix is symmetric positive definite
     and tridiagonal, and provides error bounds and backward error
     estimates for the solution.

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.

    D

              D is REAL array, dimension (N)
              The n diagonal elements of the tridiagonal matrix A.

    E

              E is REAL array, dimension (N-1)
              The (n-1) subdiagonal elements of the tridiagonal matrix A.

    DF

              DF is REAL array, dimension (N)
              The n diagonal elements of the diagonal matrix D from the
              factorization computed by SPTTRF.

    EF

              EF is REAL array, dimension (N-1)
              The (n-1) subdiagonal elements of the unit bidiagonal factor
              L from the factorization computed by SPTTRF.

    B

              B is REAL array, dimension (LDB,NRHS)
              The 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)
              On entry, the solution matrix X, as computed by SPTTRS.
              On exit, the improved solution matrix X.

    LDX

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

    FERR

              FERR is REAL array, dimension (NRHS)
              The 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).

    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 (2*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Internal Parameters:

      ITMAX is the maximum number of steps of iterative refinement.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 165 of file sptrfs.f.

  subroutine spttrs (integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldb, * ) B, integer LDB, integer INFO)

SPTTRS

Purpose:

     SPTTRS solves a tridiagonal system of the form
        A * X = B
     using the L*D*L**T factorization of A computed by SPTTRF.  D is a
     diagonal matrix specified in the vector D, L is a unit bidiagonal
     matrix whose subdiagonal is specified in the vector E, and X and B
     are N by NRHS matrices.

Parameters:
    N

              N is INTEGER
              The order of the tridiagonal 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.

    D

              D is REAL array, dimension (N)
              The n diagonal elements of the diagonal matrix D from the
              L*D*L**T factorization of A.

    E

              E is REAL array, dimension (N-1)
              The (n-1) subdiagonal elements of the unit bidiagonal factor
              L from the L*D*L**T factorization of A.  E can also be regarded
              as the superdiagonal of the unit bidiagonal factor U from the
              factorization A = U**T*D*U.

    B

              B is REAL array, dimension (LDB,NRHS)
              On entry, the right hand side vectors B for the system of
              linear equations.
              On exit, the solution vectors, 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 = -k, the k-th argument had an illegal value

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 111 of file spttrs.f.

  subroutine sptts2 (integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldb, * ) B, integer LDB)

SPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose:

     SPTTS2 solves a tridiagonal system of the form
        A * X = B
     using the L*D*L**T factorization of A computed by SPTTRF.  D is a
     diagonal matrix specified in the vector D, L is a unit bidiagonal
     matrix whose subdiagonal is specified in the vector E, and X and B
     are N by NRHS matrices.

Parameters:
    N

              N is INTEGER
              The order of the tridiagonal 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.

    D

              D is REAL array, dimension (N)
              The n diagonal elements of the diagonal matrix D from the
              L*D*L**T factorization of A.

    E

              E is REAL array, dimension (N-1)
              The (n-1) subdiagonal elements of the unit bidiagonal factor
              L from the L*D*L**T factorization of A.  E can also be regarded
              as the superdiagonal of the unit bidiagonal factor U from the
              factorization A = U**T*D*U.

    B

              B is REAL array, dimension (LDB,NRHS)
              On entry, the right hand side vectors B for the system of
              linear equations.
              On exit, the solution vectors, X.

    LDB

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

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 104 of file sptts2.f.

Author

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