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NAG Toolbox: nag_lapack_ztgexc (f08yt)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_ztgexc (f08yt) reorders the generalized Schur factorization of a complex matrix pair in generalized Schur form.

Syntax

[a, b, q, z, ilst, info] = f08yt(wantq, wantz, a, b, q, z, ifst, ilst, 'n', n)
[a, b, q, z, ilst, info] = nag_lapack_ztgexc(wantq, wantz, a, b, q, z, ifst, ilst, 'n', n)

Description

nag_lapack_ztgexc (f08yt) 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_lapack_zgges (f08xn), or nag_lapack_zhgeqz (f08xs) with job='S'.
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^.

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

Parameters

Compulsory Input Parameters

1:     wantq – logical scalar
If wantq=true, update the left transformation matrix Q.
If wantq=false, do not update Q.
2:     wantz – logical scalar
If wantz=true, update the right transformation matrix Z.
If wantz=false, do not update Z.
3:     alda: – complex array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The matrix S in the pair S,T.
4:     bldb: – complex array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,n.
The matrix T, in the pair S,T.
5:     qldq: – complex array
The first dimension, ldq, of the array q must satisfy
  • if wantq=true, ldq max1,n ;
  • otherwise ldq1.
The second dimension of the array q must be at least max1,n if wantq=true, and at least 1 otherwise.
If wantq=true, the unitary matrix Q.
6:     zldz: – complex array
The first dimension, ldz, of the array z must satisfy
  • if wantz=true, ldz max1,n ;
  • otherwise ldz1.
The second dimension of the array z must be at least max1,n if wantz=true, and at least 1 otherwise.
If wantz=true, the unitary matrix Z.
7:     ifst int64int32nag_int scalar
8:     ilst int64int32nag_int scalar
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.
Constraint: 1ifstn and 1ilstn.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the arrays a, b and the second dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)
n, the order of the matrices S and T.
Constraint: n0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
The updated matrix S^.
2:     bldb: – complex array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,n.
The updated matrix T^
3:     qldq: – complex array
The first dimension, ldq, of the array q will be
  • if wantq=true, ldq= max1,n ;
  • otherwise ldq=1.
The second dimension of the array q will be max1,n if wantq=true and 1 otherwise.
If wantq=true, the updated matrix QQ^.
If wantq=false, q is not referenced.
4:     zldz: – complex array
The first dimension, ldz, of the array z will be
  • if wantz=true, ldz= max1,n ;
  • otherwise ldz=1.
The second dimension of the array z will be max1,n if wantz=true and 1 otherwise.
If wantz=true, the updated matrix ZZ^.
If wantz=false, z is not referenced.
5:     ilst int64int32nag_int scalar
ilst points to the row in its final position.
6:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: wantq, 2: wantz, 3: n, 4: a, 5: lda, 6: b, 7: ldb, 8: q, 9: ldq, 10: z, 11: ldz, 12: ifst, 13: ilst, 14: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
   info=1
The transformed matrix pair S^,T^ 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.

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.

Further Comments

The real analogue of this function is nag_lapack_dtgexc (f08yf).

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 .  
function f08yt_example


fprintf('f08yt example results\n\n');

% exchanges rows 4 and 1 of the matrix pair S,T, where
s = [ 4 + 4i,  1 + 1i,  1 + 1i,  2 - 1i;
      0 + 0i,  2 + 1i,  1 + 1i,  1 + 1i;
      0 + 0i,  0 + 0i,  2 - 1i,  1 + 1i;
      0 + 0i,  0 + 0i,  0 + 0i,  6 - 2i];
t = [ 2,       1 + 1i,  1 + 1i,  3 - 1i;
      0 + 0i,  1 + 0i,  2 + 1i,  1 + 1i;
      0 + 0i,  0 + 0i,  1 + 0i,  1 + 1i;
      0 + 0i,  0 + 0i,  0 + 0i,  2 + 0i];

wantq = false;
wantz = false;
q = complex(zeros(1, 4));
z = complex(zeros(1, 4));
ifst = int64(1);
ilst = int64(4);
[s, t, q, z, ilst, info] = ...
  f08yt( ...
         wantq, wantz, s, t, q, z, ifst, ilst);

disp('Reordered Schur matrix S');
disp(s);
disp('Reordered Schur matrix T');
disp(t);


f08yt example results

Reordered Schur matrix S
   3.7081 + 3.7081i  -2.0834 - 0.5688i   2.6374 + 1.0772i   0.2845 + 0.7991i
   0.0000 + 0.0000i   1.6097 + 1.5656i  -0.0634 + 1.9234i  -0.0301 + 0.9720i
   0.0000 + 0.0000i   0.0000 + 0.0000i   4.7029 - 2.1187i   1.1379 - 3.1199i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   2.3085 - 1.8289i

Reordered Schur matrix T
   2.2249 + 0.7416i  -1.1631 + 1.5347i   2.2608 + 2.0851i   1.1094 - 0.3205i
   0.0000 + 0.0000i   0.3308 + 0.9482i   0.3919 + 1.8172i  -0.6305 + 1.6053i
   0.0000 + 0.0000i   0.0000 + 0.0000i   1.6227 - 0.1653i   0.9966 - 0.9074i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.1199 - 1.0343i


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