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NAG Toolbox: nag_lapack_dtgexc (f08yf)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dtgexc (f08yf) reorders the generalized Schur factorization of a matrix pair in real generalized Schur form.

Syntax

[a, b, q, z, ifst, ilst, info] = f08yf(wantq, wantz, a, b, q, z, ifst, ilst, 'n', n)
[a, b, q, z, ifst, ilst, info] = nag_lapack_dtgexc(wantq, wantz, a, b, q, z, ifst, ilst, 'n', n)

Description

nag_lapack_dtgexc (f08yf) 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_lapack_dgges (f08xa), or nag_lapack_dhgeqz (f08xe) with job='S'.
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^.

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: – double 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: – double 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: – double 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 orthogonal matrix Q.
6:     zldz: – double 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 orthogonal matrix Z.
7:     ifst int64int32nag_int scalar
8:     ilst int64int32nag_int scalar
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.
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: – double 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: – double 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: – double 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: – double 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:     ifst int64int32nag_int scalar
6:     ilst int64int32nag_int scalar
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).
7:     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: work, 15: lwork, 16: 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 complex analogue of this function is nag_lapack_ztgexc (f08yt).

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 .  
function f08yf_example


fprintf('f08yf example results\n\n');

% Generalized Schur form matrix pair
n = 4;
S = [4, 1, 1, 2;
     0, 3, 4, 1;
     0, 1, 3, 1;
     0, 0, 0, 6];
T = [2, 1, 1, 3;
     0, 1, 2, 1;
     0, 0, 1, 1;
     0, 0, 0, 2];

% Move block with row index 2 to row 1
ifst = int64(2);
ilst = int64(1);

% Do not update Q and Z
wantq = false;
wantz = false;
q = zeros(n,1);
z = q;

[S, T, ~, ~, ifst, ilst, info] = ...
    f08yf( ...
           wantq, wantz, S, T, q, z, ifst, ilst);

% Normalize: first nonzero element of each row of S is positive

for i = 1:n
  j = max(i-1,1);
  while (S(i,j)==0 && j<n)
    j = j + 1;
  end
  if S(i,j)<0
    S(i,j:n) = -S(i,j:n);
    T(i,i:n) = -T(i,i:n);
  end
end
disp('Reordered Schur matrix S');
disp(S);
disp('Reordered Schur matrix T');
disp(T);


f08yf example results

Reordered Schur matrix S
    4.1926    1.2591    2.5578    0.4520
    0.8712   -0.8627   -2.7912   -1.1383
         0         0    4.2426    2.1213
         0         0         0    6.0000

Reordered Schur matrix T
    1.7439         0    0.7533    0.0661
         0   -0.5406   -1.8972   -1.7308
         0         0    2.1213    2.8284
         0         0         0    2.0000


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