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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dtgsyl (f08yh)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dtgsyl (f08yh) solves the generalized real quasi-triangular Sylvester equations.

Syntax

[c, f, scale, dif, info] = f08yh(trans, ijob, a, b, c, d, e, f, 'm', m, 'n', n)
[c, f, scale, dif, info] = nag_lapack_dtgsyl(trans, ijob, a, b, c, d, e, f, 'm', m, 'n', n)

Description

nag_lapack_dtgsyl (f08yh) solves either the generalized real Sylvester equations
AR-LB =αC DR-LE =αF, (1)
or the equations
ATR+DTL =αC RBT+LET =-αF, (2)
where the pair A,D are given m by m matrices in real generalized Schur form, B,E are given n by n matrices in real generalized Schur form and C,F are given m by n matrices. The pair R,L are the m by n solution matrices, and α is an output scaling factor determined by the function to avoid overflow in computing R,L.
Equations (1) are equivalent to equations of the form
Zx=αb ,  
where
Z = IA-BTI ID-ETI  
and  is the Kronecker product. Equations (2) are then equivalent to
ZTy = αb .  
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 on the diagonal and T is upper triangular as returned, for example, by nag_lapack_dgges (f08xa), or nag_lapack_dhgeqz (f08xe) with job='S'.
Optionally, the function estimates DifA,D,B,E, the separation between the matrix pairs A,D and B,E, which is the smallest singular value of Z. The estimate can be based on either the Frobenius norm, or the 1-norm. The 1-norm estimate can be three to ten times more expensive than the Frobenius norm estimate, but makes the condition estimation uniform with the nonsymmetric eigenproblem. The Frobenius norm estimate provides a low cost, but equally reliable estimate. For more information see Sections 2.4.8.3 and 4.11.1.3 of Anderson et al. (1999) and Kågström and Poromaa (1996).

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
Kågström B (1994) A perturbation analysis of the generalized Sylvester equation AR-LB,DR-LE=c,F SIAM J. Matrix Anal. Appl. 15 1045–1060
Kågström B and Poromaa P (1996) LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs ACM Trans. Math. Software 22 78–103

Parameters

Compulsory Input Parameters

1:     trans – string (length ≥ 1)
If trans='N', solve the generalized Sylvester equation (1).
If trans='T', solve the ‘transposed’ system (2).
Constraint: trans='N' or 'T'.
2:     ijob int64int32nag_int scalar
Specifies what kind of functionality is to be performed when trans='N'.
ijob=0
Solve (1) only.
ijob=1
The functionality of ijob=0 and 3.
ijob=2
The functionality of ijob=0 and 4.
ijob=3
Only an estimate of Dif A,D,B,E  is computed based on the Frobenius norm.
ijob=4
Only an estimate of Dif A,D,B,E  is computed based on the 1-norm.
If trans='T', ijob is not referenced.
Constraint: if trans='N', 0ijob4.
3:     alda: – double array
The first dimension of the array a must be at least max1,m.
The second dimension of the array a must be at least max1,m.
The upper quasi-triangular matrix A.
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 upper quasi-triangular matrix B.
5:     cldc: – double array
The first dimension of the array c must be at least max1,m.
The second dimension of the array c must be at least max1,n.
Contains the right-hand-side matrix C.
6:     dldd: – double array
The first dimension of the array d must be at least max1,m.
The second dimension of the array d must be at least max1,m.
The upper triangular matrix D.
7:     elde: – double array
The first dimension of the array e must be at least max1,n.
The second dimension of the array e must be at least max1,n.
The upper triangular matrix E.
8:     fldf: – double array
The first dimension of the array f must be at least max1,m.
The second dimension of the array f must be at least max1,n.
Contains the right-hand side matrix F.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the first dimension of the arrays a, c, d, f and the second dimension of the arrays a, d.
m, the order of the matrices A and D, and the row dimension of the matrices C, F, R and L.
Constraint: m0.
2:     n int64int32nag_int scalar
Default: the first dimension of the arrays b, e and the second dimension of the arrays b, c, e, f.
n, the order of the matrices B and E, and the column dimension of the matrices C, F, R and L.
Constraint: n0.

Output Parameters

1:     cldc: – double array
The first dimension of the array c will be max1,m.
The second dimension of the array c will be max1,n.
If ijob=0, 1 or 2, c stores the solution matrix R.
If trans='N' and ijob=3 or 4, c holds R, the solution achieved during the computation of the Dif estimate.
2:     fldf: – double array
The first dimension of the array f will be max1,m.
The second dimension of the array f will be max1,n.
If ijob=0, 1 or 2, f stores the solution matrix L.
If trans='N' and ijob=3 or 4, f holds L, the solution achieved during the computation of the Dif estimate.
3:     scale – double scalar
α, the scaling factor in (1) or (2).
If 0<scale<1, c and f hold the solutions R and L, respectively, to a slightly perturbed system but the input arrays a, b, d and e have not been changed.
If scale=0, c and f hold the solutions R and L, respectively, to the homogeneous system with C=F=0. In this case dif is not referenced.
Normally, scale=1.
4:     dif – double scalar
The estimate of Dif. If ijob=0, dif is not referenced.
5:     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: trans, 2: ijob, 3: m, 4: n, 5: a, 6: lda, 7: b, 8: ldb, 9: c, 10: ldc, 11: d, 12: ldd, 13: e, 14: lde, 15: f, 16: ldf, 17: scale, 18: dif, 19: work, 20: lwork, 21: iwork, 22: 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>0
A,D and B,E have common or close eigenvalues and so no solution could be computed.

Accuracy

See Kågström (1994) for a perturbation analysis of the generalized Sylvester equation.

Further Comments

The total number of floating-point operations needed to solve the generalized Sylvester equations is approximately 2mnn+m. The Frobenius norm estimate of Dif does not require additional significant computation, but the 1-norm estimate is typically five times more expensive.
The complex analogue of this function is nag_lapack_ztgsyl (f08yv).

Example

This example solves the generalized Sylvester equations
AR-LB =αC DR-LE =αF,  
where
A = 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 ,   B= 1.0 1.0 1.0 1.0 0.0 3.0 4.0 1.0 0.0 1.0 3.0 1.0 0.0 0.0 0.0 4.0 ,  
D = 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 ,   E= 1.0 1.0 1.0 2.0 0.0 1.0 4.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 1.0 ,  
C = -4.0 7.0 1.0 12.0 -9.0 2.0 -2.0 -2.0 -4.0 2.0 -2.0 8.0 -7.0 7.0 -6.0 19.0   and   F= -7.0 5.0 0.0 7.0 -5.0 1.0 -8.0 0.0 -1.0 2.0 -3.0 5.0 -3.0 2.0 0.0 5.0 .  
function f08yh_example


fprintf('f08yh example results\n\n');

% Given generalized Schur form pairs (A,D) and (B,E), and matrices C and F:
% solve AR - LB = alpha*C
%       DR - LE = alpha*F
% for R, L and alpha
  
% Schur form pairs
A = [4, 1, 1, 2;
     0, 3, 4, 1;
     0, 1, 3, 1;
     0, 0, 0, 6];
D = [2, 1, 1, 3;
     0, 1, 2, 1;
     0, 0, 1, 1;
     0, 0, 0, 2];
B = [1, 1, 1, 1;
     0, 3, 4, 1;
     0, 1, 3, 1;
     0, 0, 0, 4];
E = [1, 1, 1, 2;
     0, 1, 4, 1;
     0, 0, 1, 1;
     0, 0, 0, 1];
% Right hand matrices
C = [-4, 7,  1, 12;
     -9, 2, -2, -2;
     -4, 2, -2,  8;
     -7, 7, -6, 19];
F = [-7, 5,  0,  7;
     -5, 1, -8,  0;
     -1, 2, -3,  5;
     -3, 2,  0,  5];

% Solve
trans = 'No transpose';
ijob = int64(0);
[R, L, alpha, ~, info] = f08yh( ...
                                trans, ijob, A, B, C, D, E, F);


[ifail] = x04ca( ...
                 'Gen', ' ', R, 'Solution matrix R');

fprintf('\n');
[ifail] = x04ca( ...
                 'Gen', ' ', L, 'Solution matrix L');

fprintf('\nalpha = %10.2e\n', alpha);


f08yh example results

 Solution matrix R
             1          2          3          4
 1      1.0000     1.0000     1.0000     1.0000
 2     -1.0000     2.0000    -1.0000    -1.0000
 3     -1.0000     1.0000     3.0000     1.0000
 4     -1.0000     1.0000    -1.0000     4.0000

 Solution matrix L
             1          2          3          4
 1      4.0000    -1.0000     1.0000    -1.0000
 2      1.0000     3.0000    -1.0000     1.0000
 3     -1.0000     1.0000     2.0000    -1.0000
 4      1.0000    -1.0000     1.0000     1.0000

alpha =   1.00e+00

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