doublePTcomputational(3)

NAME
doublePTcomputational - double

SYNOPSIS

  Functions
subroutine dptcon (n, d, e, anorm, rcond, work, info)
DPTCON
subroutine dpteqr (compz, n, d, e, z, ldz, work, info)
DPTEQR
subroutine dptrfs (n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info)
DPTRFS
subroutine dpttrf (n, d, e, info)
DPTTRF
subroutine dpttrs (n, nrhs, d, e, b, ldb, info)
DPTTRS
subroutine dptts2 (n, nrhs, d, e, b, ldb)
DPTTS2 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 double computational functions for PT matrices

Function Documentation

  subroutine dptcon (integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision anorm, double precision rcond, double precision, dimension( * ) work, integer info)

DPTCON

Purpose:

     DPTCON 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
     DPTTRF.
     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 DOUBLE PRECISION array, dimension (N)
              The n diagonal elements of the diagonal matrix D from the
              factorization of A, as computed by DPTTRF.

    E

              E is DOUBLE PRECISION 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 DPTTRF.

    ANORM

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

    RCOND

              RCOND is DOUBLE PRECISION
              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 DOUBLE PRECISION 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.

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 117 of file dptcon.f.

  subroutine dpteqr (character compz, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer info)

DPTEQR

Purpose:

     DPTEQR computes all eigenvalues and, optionally, eigenvectors of a
     symmetric positive definite tridiagonal matrix by first factoring the
     matrix using DPTTRF, and then calling DBDSQR 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 DSYTRD, DSPTRD, or DSBTRD 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N-1)
              On entry, the (n-1) subdiagonal elements of the tridiagonal
              matrix.
              On exit, E has been destroyed.

    Z

              Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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.

Definition at line 144 of file dpteqr.f.

  subroutine dptrfs (integer n, integer nrhs, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) df, double precision, dimension( * ) ef, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer info)

DPTRFS

Purpose:

     DPTRFS 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 DOUBLE PRECISION array, dimension (N)
              The n diagonal elements of the tridiagonal matrix A.

    E

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

    DF

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

    EF

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

    B

              B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by DPTTRS.
              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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.

Definition at line 161 of file dptrfs.f.

  subroutine dpttrf (integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, integer info)

DPTTRF

Purpose:

     DPTTRF computes the L*D*L**T factorization of a real symmetric
     positive definite tridiagonal matrix A.  The factorization may also
     be regarded as having the form A = U**T*D*U.

Parameters
    N

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

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the n diagonal elements of the tridiagonal matrix
              A.  On exit, the n diagonal elements of the diagonal matrix
              D from the L*D*L**T factorization of A.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              On entry, the (n-1) subdiagonal elements of the tridiagonal
              matrix A.  On exit, 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 U**T*D*U factorization of A.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -k, the k-th argument had an illegal value
              > 0: if INFO = k, the leading minor of order k is not
                   positive definite; if k < N, the factorization could not
                   be completed, while if k = N, the factorization was
                   completed, but D(N) <= 0.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 90 of file dpttrf.f.

  subroutine dpttrs (integer n, integer nrhs, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldb, * ) b, integer ldb, integer info)

DPTTRS

Purpose:

     DPTTRS solves a tridiagonal system of the form
        A * X = B
     using the L*D*L**T factorization of A computed by DPTTRF.  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 DOUBLE PRECISION array, dimension (N)
              The n diagonal elements of the diagonal matrix D from the
              L*D*L**T factorization of A.

    E

              E is DOUBLE PRECISION 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 DOUBLE PRECISION 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.

Definition at line 108 of file dpttrs.f.

  subroutine dptts2 (integer n, integer nrhs, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldb, * ) b, integer ldb)

DPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose:

     DPTTS2 solves a tridiagonal system of the form
        A * X = B
     using the L*D*L**T factorization of A computed by DPTTRF.  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 DOUBLE PRECISION array, dimension (N)
              The n diagonal elements of the diagonal matrix D from the
              L*D*L**T factorization of A.

    E

              E is DOUBLE PRECISION 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 DOUBLE PRECISION 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.

Definition at line 101 of file dptts2.f.

Author

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