NAG Library Function Document

nag_ztrexc (f08qtc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_ztrexc (f08qtc) reorders the Schur factorization of a complex general matrix.

2
Specification

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

3
Description

nag_ztrexc (f08qtc) reorders the Schur factorization of a complex general matrix A=QTQH, so that the diagonal element of T with row index ifst is moved to row ilst.
The reordered Schur form T~ is computed by a unitary similarity transformation: T~=ZHTZ. Optionally the updated matrix Q~ of Schur vectors is computed as Q~=QZ, giving A=Q~T~Q~H.

4
References

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

5
Arguments

1:     order Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     compq Nag_ComputeQTypeInput
On entry: indicates whether the matrix Q of Schur vectors is to be updated.
compq=Nag_UpdateSchur
The matrix Q of Schur vectors is updated.
compq=Nag_NotQ
No Schur vectors are updated.
Constraint: compq=Nag_UpdateSchur or Nag_NotQ.
3:     n IntegerInput
On entry: n, the order of the matrix T.
Constraint: n0.
4:     t[dim] ComplexInput/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 triangular matrix T, as returned by nag_zhseqr (f08psc).
On exit: t is overwritten by the updated matrix T~.
5:     pdt IntegerInput
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] ComplexInput/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 unitary 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:     pdq IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
  • if compq=Nag_UpdateSchur, pdq max1,n ;
  • if compq=Nag_NotQ, pdq1.
8:     ifst IntegerInput
9:     ilst IntegerInput
On entry: ifst and ilst must specify the reordering of the diagonal elements of T. The element with row index ifst is moved to row ilst by a sequence of exchanges between adjacent elements.
Constraint: 1ifstn and 1ilstn.
10:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_ENUM_INT_2
On entry, compq=value, pdq=value and n=value.
Constraint: if compq=Nag_UpdateSchur, pdq max1,n ;
if compq=Nag_NotQ, pdq1.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pdq=value.
Constraint: pdq>0.
On entry, pdt=value.
Constraint: pdt>0.
NE_INT_2
On entry, pdt=value and n=value.
Constraint: pdt max1,n .
NE_INT_3
On entry, n=value, ifst=value and ilst=value.
Constraint: 1ifstn and 1ilstn and
1ifstn and
1ilstn.
NE_INTERNAL_ERROR
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

The computed matrix T~ is exactly similar to a matrix T+E, where
E2 = Oε T2 ,  
and ε is the machine precision.
The values of the eigenvalues are never changed by the reordering.

8
Parallelism and Performance

nag_ztrexc (f08qtc) 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 x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9
Further Comments

The total number of real floating-point operations is approximately 20nr if compq=Nag_NotQ, and 40nr if compq=Nag_UpdateSchur, where r=ifst-ilst.
The real analogue of this function is nag_dtrexc (f08qfc).

10
Example

This example reorders the Schur factorization of the matrix T so that element t11 is moved to t44, where
T = -6.00-7.00i 0.36-0.36i -0.19+0.48i 0.88-0.25i 0.00+0.00i -5.00+2.00i -0.03-0.72i -0.23+0.13i 0.00+0.00i 0.00+0.00i 8.00-1.00i 0.94+0.53i 0.00+0.00i 0.00+0.00i 0.00+0.00i 3.00-4.00i .  

10.1
Program Text

Program Text (f08qtce.c)

10.2
Program Data

Program Data (f08qtce.d)

10.3
Program Results

Program Results (f08qtce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017