Provided by: liblapack-doc_3.8.0-2_all bug

NAME

       doublePTcomputational

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.

       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.

   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.

       Date:
           December 2016

   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.

       Date:
           December 2016

   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.

       Date:
           December 2016

   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.

       Date:
           December 2016

   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.

       Date:
           December 2016

Author

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