Provided by: liblapack-doc_3.11.0-2build1_all bug

NAME

       realPTcomputational - real

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.

       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 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.

   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.

   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.

   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.

Author

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