nag_dtgexc (f08yfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dtgexc (f08yfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dtgexc (f08yfc) reorders the generalized Schur factorization of a matrix pair in real generalized Schur form.

2  Specification

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

3  Description

nag_dtgexc (f08yfc) reorders the generalized real n by n matrix pair S,T in real generalized Schur form, so that the diagonal element or block of S,T with row index i1 is moved to row i2, using an orthogonal equivalence transformation. That is, S and T are factorized as
S = Q^ S^ Z^T ,   T= Q^ T^ Z^T ,
where S^,T^ are also in real generalized Schur form.
The pair S,T are in real generalized Schur form if S is block upper triangular with 1 by 1 and 2 by 2 diagonal blocks and T is upper triangular as returned, for example, by nag_dgges (f08xac), or nag_dhgeqz (f08xec) with job=Nag_Schur.
If S and T are the result of a generalized Schur factorization of a matrix pair A,B 
A = QSZT ,   B= QTZT
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:     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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     wantqNag_BooleanInput
On entry: if wantq=Nag_TRUE, update the left transformation matrix Q.
If wantq=Nag_FALSE, do not update Q.
3:     wantzNag_BooleanInput
On entry: if wantz=Nag_TRUE, update the right transformation matrix Z.
If wantz=Nag_FALSE, do not update Z.
4:     nIntegerInput
On entry: n, the order of the matrices S and T.
Constraint: n0.
5:     a[dim]doubleInput/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:     pdaIntegerInput
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]doubleInput/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:     pdbIntegerInput
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]doubleInput/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 orthogonal matrix Q.
On exit: if wantq=Nag_TRUE, the updated matrix QQ^.
If wantq=Nag_FALSE, q is not referenced.
10:   pdqIntegerInput
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]doubleInput/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 orthogonal matrix Z.
On exit: if wantz=Nag_TRUE, the updated matrix ZZ^.
If wantz=Nag_FALSE, z is not referenced.
12:   pdzIntegerInput
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:   ifstInteger *Input/Output
14:   ilstInteger *Input/Output
On entry: the indices i1 and i2 that specify the reordering of the diagonal blocks of S,T. The block with row index ifst is moved to row ilst, by a sequence of swapping between adjacent blocks.
On exit: if ifst pointed on entry to the second row of a 2 by 2 block, 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 or -1).
Constraint: 1ifstn and 1ilstn.
15:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.
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  Further Comments

The complex analogue of this function is nag_ztgexc (f08ytc).

9  Example

This example exchanges blocks 2 and 1 of the matrix pair S,T, where
S = 4.0 1.0 1.0 2.0 0.0 3.0 4.0 1.0 0.0 1.0 3.0 1.0 0.0 0.0 0.0 6.0   and   T= 2.0 1.0 1.0 3.0 0.0 1.0 2.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 2.0 .

9.1  Program Text

Program Text (f08yfce.c)

9.2  Program Data

Program Data (f08yfce.d)

9.3  Program Results

Program Results (f08yfce.r)


nag_dtgexc (f08yfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012