complex16PTcomputational(3)

NAME
complex16PTcomputational

SYNOPSIS

  Functions
subroutine zptcon (N, D, E, ANORM, RCOND, RWORK, INFO)
ZPTCON
subroutine zpteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
ZPTEQR
subroutine zptrfs (UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZPTRFS
subroutine zpttrf (N, D, E, INFO)
ZPTTRF
subroutine zpttrs (UPLO, N, NRHS, D, E, B, LDB, INFO)
ZPTTRS
subroutine zptts2 (IUPLO, N, NRHS, D, E, B, LDB)
ZPTTS2 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 complex16 computational functions for PT matrices

Function Documentation

  subroutine zptcon (integer N, double precision, dimension( * ) D, complex*16, dimension( * ) E, double precision ANORM, double precision RCOND, double precision, dimension( * ) RWORK, integer INFO)

ZPTCON

Purpose:

     ZPTCON computes the reciprocal of the condition number (in the
     1-norm) of a complex Hermitian positive definite tridiagonal matrix
     using the factorization A = L*D*L**H or A = U**H*D*U computed by
     ZPTTRF.
     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 ZPTTRF.

    E

              E is COMPLEX*16 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 ZPTTRF.

    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.

    RWORK

              RWORK 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.

Definition at line 121 of file zptcon.f.

  subroutine zpteqr (character COMPZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)

ZPTEQR

Purpose:

     ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a
     symmetric positive definite tridiagonal matrix by first factoring the
     matrix using DPTTRF and then calling ZBDSQR 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 positive definite Hermitian matrix
     can also be found if ZHETRD, ZHPTRD, or ZHBTRD 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 Hermitian
                      matrix also.  Array Z contains the unitary 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 COMPLEX*16 array, dimension (LDZ, N)
              On entry, if COMPZ = 'V', the unitary matrix used in the
              reduction to tridiagonal form.
              On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
              original Hermitian 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

Definition at line 147 of file zpteqr.f.

  subroutine zptrfs (character UPLO, integer N, integer NRHS, double precision, dimension( * ) D, complex*16, dimension( * ) E, double precision, dimension( * ) DF, complex*16, dimension( * ) EF, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)

ZPTRFS

Purpose:

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

Parameters:
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the superdiagonal or the subdiagonal of the
              tridiagonal matrix A is stored and the form of the
              factorization:
              = 'U':  E is the superdiagonal of A, and A = U**H*D*U;
              = 'L':  E is the subdiagonal of A, and A = L*D*L**H.
              (The two forms are equivalent if A is real.)

    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 real diagonal elements of the tridiagonal matrix A.

    E

              E is COMPLEX*16 array, dimension (N-1)
              The (n-1) off-diagonal elements of the tridiagonal matrix A
              (see UPLO).

    DF

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

    EF

              EF is COMPLEX*16 array, dimension (N-1)
              The (n-1) off-diagonal elements of the unit bidiagonal
              factor U or L from the factorization computed by ZPTTRF
              (see UPLO).

    B

              B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by ZPTTRS.
              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 COMPLEX*16 array, dimension (N)

    RWORK

              RWORK 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

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 185 of file zptrfs.f.

  subroutine zpttrf (integer N, double precision, dimension( * ) D, complex*16, dimension( * ) E, integer INFO)

ZPTTRF

Purpose:

     ZPTTRF computes the L*D*L**H factorization of a complex Hermitian
     positive definite tridiagonal matrix A.  The factorization may also
     be regarded as having the form A = U**H *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**H factorization of A.

    E

              E is COMPLEX*16 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**H factorization of A.
              E can also be regarded as the superdiagonal of the unit
              bidiagonal factor U from the U**H *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

Definition at line 94 of file zpttrf.f.

  subroutine zpttrs (character UPLO, integer N, integer NRHS, double precision, dimension( * ) D, complex*16, dimension( * ) E, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)

ZPTTRS

Purpose:

     ZPTTRS solves a tridiagonal system of the form
        A * X = B
     using the factorization A = U**H *D* U or A = L*D*L**H computed by ZPTTRF.
     D is a diagonal matrix specified in the vector D, U (or L) is a unit
     bidiagonal matrix whose superdiagonal (subdiagonal) is specified in
     the vector E, and X and B are N by NRHS matrices.

Parameters:
    UPLO

              UPLO is CHARACTER*1
              Specifies the form of the factorization and whether the
              vector E is the superdiagonal of the upper bidiagonal factor
              U or the subdiagonal of the lower bidiagonal factor L.
              = 'U':  A = U**H *D*U, E is the superdiagonal of U
              = 'L':  A = L*D*L**H, E is the subdiagonal of L

    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
              factorization A = U**H *D*U or A = L*D*L**H.

    E

              E is COMPLEX*16 array, dimension (N-1)
              If UPLO = 'U', the (n-1) superdiagonal elements of the unit
              bidiagonal factor U from the factorization A = U**H*D*U.
              If UPLO = 'L', the (n-1) subdiagonal elements of the unit
              bidiagonal factor L from the factorization A = L*D*L**H.

    B

              B is COMPLEX*16 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:
    June 2016

Definition at line 123 of file zpttrs.f.

  subroutine zptts2 (integer IUPLO, integer N, integer NRHS, double precision, dimension( * ) D, complex*16, dimension( * ) E, complex*16, dimension( ldb, * ) B, integer LDB)

ZPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose:

     ZPTTS2 solves a tridiagonal system of the form
        A * X = B
     using the factorization A = U**H *D*U or A = L*D*L**H computed by ZPTTRF.
     D is a diagonal matrix specified in the vector D, U (or L) is a unit
     bidiagonal matrix whose superdiagonal (subdiagonal) is specified in
     the vector E, and X and B are N by NRHS matrices.

Parameters:
    IUPLO

              IUPLO is INTEGER
              Specifies the form of the factorization and whether the
              vector E is the superdiagonal of the upper bidiagonal factor
              U or the subdiagonal of the lower bidiagonal factor L.
              = 1:  A = U**H *D*U, E is the superdiagonal of U
              = 0:  A = L*D*L**H, E is the subdiagonal of L

    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
              factorization A = U**H *D*U or A = L*D*L**H.

    E

              E is COMPLEX*16 array, dimension (N-1)
              If IUPLO = 1, the (n-1) superdiagonal elements of the unit
              bidiagonal factor U from the factorization A = U**H*D*U.
              If IUPLO = 0, the (n-1) subdiagonal elements of the unit
              bidiagonal factor L from the factorization A = L*D*L**H.

    B

              B is COMPLEX*16 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:
    June 2016

Definition at line 115 of file zptts2.f.

Author

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