nag_dtrexc (f08qfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dtrexc (f08qfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dtrexc (f08qfc) reorders the Schur factorization of a real general matrix.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dtrexc (Nag_OrderType order, Nag_ComputeQType compq, Integer n, double t[], Integer pdt, double q[], Integer pdq, Integer *ifst, Integer *ilst, NagError *fail)

3  Description

nag_dtrexc (f08qfc) reorders the Schur factorization of a real general matrix A=QTQT, so that the diagonal element or block of T with row index ifst is moved to row ilst.
The reordered Schur form T~ is computed by an orthogonal similarity transformation: T~=ZTTZ. Optionally the updated matrix Q~ of Schur vectors is computed as Q~=QZ, giving A=Q~T~Q~T.

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     compqNag_ComputeQTypeInput
On entry: indicates whether the matrix Q of Schur vectors is to be updated.
The matrix Q of Schur vectors is updated.
No Schur vectors are updated.
Constraint: compq=Nag_UpdateSchur or Nag_NotQ.
3:     nIntegerInput
On entry: n, the order of the matrix T.
Constraint: n0.
4:     t[dim]doubleInput/Output
Note: the dimension, dim, of the array t must be at least max1,pdt×n.
The i,jth element of the matrix T is stored in
  • t[j-1×pdt+i-1] when order=Nag_ColMajor;
  • t[i-1×pdt+j-1] when order=Nag_RowMajor.
On entry: the n by n upper quasi-triangular matrix T in canonical Schur form, as returned by nag_dhseqr (f08pec).
On exit: t is overwritten by the updated matrix T~. See also Section 9.
5:     pdtIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array t.
Constraint: pdt max1,n .
6:     q[dim]doubleInput/Output
Note: the dimension, dim, of the array q must be at least
  • max1,pdq×n when compq=Nag_UpdateSchur;
  • 1 when compq=Nag_NotQ.
The i,jth element of the matrix Q is stored in
  • q[j-1×pdq+i-1] when order=Nag_ColMajor;
  • q[i-1×pdq+j-1] when order=Nag_RowMajor.
On entry: if compq=Nag_UpdateSchur, q must contain the n by n orthogonal matrix Q of Schur vectors.
On exit: if compq=Nag_UpdateSchur, q contains the updated matrix of Schur vectors.
If compq=Nag_NotQ, q is not referenced.
7:     pdqIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
  • if compq=Nag_UpdateSchur, pdq max1,n ;
  • if compq=Nag_NotQ, pdq1.
8:     ifstInteger *Input/Output
9:     ilstInteger *Input/Output
On entry: ifst and ilst must specify the reordering of the diagonal elements or blocks of T. The element or block with row index ifst is moved to row ilst by a sequence of exchanges between adjacent elements or blocks.
On exit: if ifst pointed to the second row of a 2 by 2 block on entry, it is changed to point to the first row. ilst always points to the first row of the block in its final position (which may differ from its input value by ±1).
Constraint: 1ifstn and 1ilstn.
10:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, compq=value, pdq=value and n=value.
Constraint: if compq=Nag_UpdateSchur, pdq max1,n ;
if compq=Nag_NotQ, pdq1.
Two adjacent diagonal elements or blocks could not be successfully exchanged. This error can only occur if the exchange involves at least one 2 by 2 block; it implies that the problem is very ill-conditioned, and that the eigenvalues of the two blocks are very close. On exit, T may have been partially reordered, and ilst points to the first row of the current position of the block being moved; Q (if requested) is updated consistently with T.
On entry, n=value.
Constraint: n0.
On entry, pdq=value.
Constraint: pdq>0.
On entry, pdt=value.
Constraint: pdt>0.
On entry, pdt=value and n=value.
Constraint: pdt max1,n .
On entry, n=value, ifst=value and ilst=value.
Constraint: 1ifstn and 1ilstn.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7  Accuracy

The computed matrix T~ is exactly similar to a matrix T+E, where
E2 = Oε T2 ,
and ε is the machine precision.
Note that if a 2 by 2 diagonal block is involved in the reordering, its off-diagonal elements are in general changed; the diagonal elements and the eigenvalues of the block are unchanged unless the block is sufficiently ill-conditioned, in which case they may be noticeably altered. It is possible for a 2 by 2 block to break into two 1 by 1 blocks, i.e., for a pair of complex eigenvalues to become purely real. The values of real eigenvalues however are never changed by the reordering.

8  Parallelism and Performance

nag_dtrexc (f08qfc) is not threaded by NAG in any implementation.
nag_dtrexc (f08qfc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The total number of floating-point operations is approximately 6nr if compq=Nag_NotQ, and 12nr if compq=Nag_UpdateSchur, where r=ifst-ilst.
The input matrix T must be in canonical Schur form, as is the output matrix T~. This has the following structure.
If all the computed eigenvalues are real, T is upper triangular and its diagonal elements are the eigenvalues.
If some of the computed eigenvalues form complex conjugate pairs, then T has 2 by 2 diagonal blocks. Each diagonal block has the form
tii ti,i+1 ti+1,i ti+1,i+1 = α β γ α
where βγ<0. The corresponding eigenvalues are α±βγ.
The complex analogue of this function is nag_ztrexc (f08qtc).

10  Example

This example reorders the Schur factorization of the matrix T so that the 2 by 2 block with row index 2 is moved to row 1, where
T = 0.80 -0.11 0.01 0.03 0.00 -0.10 0.25 0.35 0.00 -0.65 -0.10 0.20 0.00 0.00 0.00 -0.10 .

10.1  Program Text

Program Text (f08qfce.c)

10.2  Program Data

Program Data (f08qfce.d)

10.3  Program Results

Program Results (f08qfce.r)

nag_dtrexc (f08qfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014