complexPTcomputational(3) complex

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:

September 2012

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:

September 2012

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:

September 2012

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:

September 2012

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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