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

NAME

       complexPTcomputational

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.

       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 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.

       Date:
           December 2016

   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.

       Date:
           December 2016

   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.

       Date:
           December 2016

   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.

       Date:
           June 2016

   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.

       Date:
           June 2016

Author

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