complexPTcomputational(3)

NAME
complexPTcomputational - complex

SYNOPSIS

  Functions
subroutine cptcon (n, d, e, anorm, rcond, rwork, info)
CPTCON
subroutine cpteqr (compz, n, d, e, z, ldz, work, info)
CPTEQR
subroutine cptrfs (uplo, n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CPTRFS
subroutine cpttrf (n, d, e, info)
CPTTRF
subroutine cpttrs (uplo, n, nrhs, d, e, b, ldb, info)
CPTTRS
subroutine cptts2 (iuplo, n, nrhs, d, e, b, ldb)
CPTTS2 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 complex computational functions for PT matrices

Function Documentation

  subroutine cptcon (integer n, real, dimension( * ) d, complex, dimension( * ) e, real anorm, real rcond, real, dimension( * ) rwork, integer info)

CPTCON

Purpose:

     CPTCON 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
     CPTTRF.
     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 CPTTRF.

    E

              E is COMPLEX 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 CPTTRF.

    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.

    RWORK

              RWORK 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.

Definition at line 118 of file cptcon.f.

  subroutine cpteqr (character compz, integer n, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

CPTEQR

Purpose:

     CPTEQR computes all eigenvalues and, optionally, eigenvectors of a
     symmetric positive definite tridiagonal matrix by first factoring the
     matrix using SPTTRF and then calling CBDSQR 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 CHETRD, CHPTRD, or CHBTRD 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 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 COMPLEX 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 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.

Definition at line 144 of file cpteqr.f.

  subroutine cptrfs (character uplo, integer n, integer nrhs, real, dimension( * ) d, complex, dimension( * ) e, real, dimension( * ) df, complex, dimension( * ) ef, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CPTRFS

Purpose:

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

    E

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

    DF

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

    EF

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

    B

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

    RWORK

              RWORK is REAL 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.

Definition at line 181 of file cptrfs.f.

  subroutine cpttrf (integer n, real, dimension( * ) d, complex, dimension( * ) e, integer info)

CPTTRF

Purpose:

     CPTTRF 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 REAL 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 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.

Definition at line 91 of file cpttrf.f.

  subroutine cpttrs (character uplo, integer n, integer nrhs, real, dimension( * ) d, complex, dimension( * ) e, complex, dimension( ldb, * ) b, integer ldb, integer info)

CPTTRS

Purpose:

     CPTTRS 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 CPTTRF.
     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 REAL 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 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 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 120 of file cpttrs.f.

  subroutine cptts2 (integer iuplo, integer n, integer nrhs, real, dimension( * ) d, complex, dimension( * ) e, complex, dimension( ldb, * ) b, integer ldb)

CPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose:

     CPTTS2 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 CPTTRF.
     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 REAL 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 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 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 112 of file cptts2.f.

Author

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