nag_ztgexc (f08ytc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_ztgexc (f08ytc)

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_ztgexc (f08ytc) reorders the generalized Schur factorization of a complex matrix pair in generalized Schur form.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_ztgexc (Nag_OrderType order, Nag_Boolean wantq, Nag_Boolean wantz, Integer n, Complex a[], Integer pda, Complex b[], Integer pdb, Complex q[], Integer pdq, Complex z[], Integer pdz, Integer ifst, Integer *ilst, NagError *fail)

3  Description

nag_ztgexc (f08ytc) reorders the generalized complex n by n matrix pair S,T in generalized Schur form, so that the diagonal element of S,T with row index i1 is moved to row i2, using a unitary equivalence transformation. That is, S and T are factorized as
S = Q^ S^ Z^H ,   T= Q^ T^ Z^H ,  
where S^,T^ are also in generalized Schur form.
The pair S,T are in generalized Schur form if S and T are upper triangular as returned, for example, by nag_zgges (f08xnc), or nag_zhgeqz (f08xsc) with job=Nag_Schur.
If S and T are the result of a generalized Schur factorization of a matrix pair A,B 
A = QSZH ,   B= QTZH  
then, optionally, the matrices Q and Z can be updated as QQ^ and ZZ^.

4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

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 2.3.1.3 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:     wantq Nag_BooleanInput
On entry: if wantq=Nag_TRUE, update the left transformation matrix Q.
If wantq=Nag_FALSE, do not update Q.
3:     wantz Nag_BooleanInput
On entry: if wantz=Nag_TRUE, update the right transformation matrix Z.
If wantz=Nag_FALSE, do not update Z.
4:     n IntegerInput
On entry: n, the order of the matrices S and T.
Constraint: n0.
5:     a[dim] ComplexInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the matrix S in the pair S,T.
On exit: the updated matrix S^.
6:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
7:     b[dim] ComplexInput/Output
Note: the dimension, dim, of the array b must be at least max1,pdb×n.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the matrix T, in the pair S,T.
On exit: the updated matrix T^ 
8:     pdb IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbmax1,n.
9:     q[dim] ComplexInput/Output
Note: the dimension, dim, of the array q must be at least
  • max1,pdq×n when wantq=Nag_TRUE;
  • 1 otherwise.
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 wantq=Nag_TRUE, the unitary matrix Q.
On exit: if wantq=Nag_TRUE, the updated matrix QQ^.
If wantq=Nag_FALSE, q is not referenced.
10:   pdq IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
  • if wantq=Nag_TRUE, pdq max1,n ;
  • otherwise pdq1.
11:   z[dim] ComplexInput/Output
Note: the dimension, dim, of the array z must be at least
  • max1,pdz×n when wantz=Nag_TRUE;
  • 1 otherwise.
The i,jth element of the matrix Z is stored in
  • z[j-1×pdz+i-1] when order=Nag_ColMajor;
  • z[i-1×pdz+j-1] when order=Nag_RowMajor.
On entry: if wantz=Nag_TRUE, the unitary matrix Z.
On exit: if wantz=Nag_TRUE, the updated matrix ZZ^.
If wantz=Nag_FALSE, z is not referenced.
12:   pdz IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
  • if wantz=Nag_TRUE, pdz max1,n ;
  • otherwise pdz1.
13:   ifst IntegerInput
14:   ilst Integer *Input/Output
On entry: the indices i1 and i2 that specify the reordering of the diagonal elements of S,T. The element with row index ifst is moved to row ilst, by a sequence of swapping between adjacent diagonal elements.
On exit: ilst points to the row in its final position.
Constraint: 1ifstn and 1ilstn.
15:   fail NagError *Input/Output
The NAG error argument (see Section 2.7 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_CONSTRAINT
On entry, wantq=value, pdq=value and n=value.
Constraint: if wantq=Nag_TRUE, pdq max1,n ;
otherwise pdq1.
On entry, wantz=value, pdz=value and n=value.
Constraint: if wantz=Nag_TRUE, pdz max1,n ;
otherwise pdz1.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdq=value.
Constraint: pdq>0.
On entry, pdz=value.
Constraint: pdz>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
NE_INT_3
On entry, ifst=value, ilst=value and n=value.
Constraint: 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.
An unexpected error has been triggered by this function. Please contact NAG.
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.
NE_SCHUR
The transformed matrix pair would be too far from generalized Schur form; the problem is ill-conditioned. S,T may have been partially reordered, and ilst points to the first row of the current position of the block being moved.

7  Accuracy

The computed generalized Schur form is nearly the exact generalized Schur form for nearby matrices S+E and T+F, where
E2 = Oε S2   and   F2= Oε T2 ,  
and ε is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details of error bounds for the generalized nonsymmetric eigenproblem.

8  Parallelism and Performance

nag_ztgexc (f08ytc) is not threaded in any implementation.

9  Further Comments

The real analogue of this function is nag_dtgexc (f08yfc).

10  Example

This example exchanges rows 4 and 1 of the matrix pair S,T, where
S = 4.0+4.0i 1.0+1.0i 1.0+1.0i 2.0-1.0i 0.0i+0.0 2.0+1.0i 1.0+1.0i 1.0+1.0i 0.0i+0.0 0.0i+0.0 2.0-1.0i 1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 6.0-2.0i  
and
T = 2.0 1.0+1.0i 1.0+1.0i 3.0-1.0i 0.0 1.0i+0.0 2.0+1.0i 1.0+1.0i 0.0 0.0i+0.0 1.0i+0.0 1.0+1.0i 0.0 0.0i+0.0 0.0i+0.0 2.0i+0.0 .  

10.1  Program Text

Program Text (f08ytce.c)

10.2  Program Data

Program Data (f08ytce.d)

10.3  Program Results

Program Results (f08ytce.r)


nag_ztgexc (f08ytc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016