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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_ztgsyl (f08yv)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_ztgsyl (f08yv) solves the generalized complex triangular Sylvester equations.

Syntax

[c, f, scale, dif, info] = f08yv(trans, ijob, a, b, c, d, e, f, 'm', m, 'n', n)

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

Description

nag_lapack_ztgsyl (f08yv) solves either the generalized complex Sylvester equations

\begin{matrix} A R - L B & = α C \\ D R - L E & = α F, \end{matrix}

(1)

or the equations

\begin{matrix} A^{H} R + D^{H} L & = α C \\ R B^{H} + L E^{H} & = - α F, \end{matrix}

(2)

where the pair

(A, D)

are given

m

m

matrices in generalized Schur form,

(B, E)

are given

n

n

matrices in generalized Schur form and

(C, F)

are given

m

n

matrices. The pair

(R, L)

are the

m

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

Z x = α b,

where

Z = (\begin{matrix} I \otimes A - B^{H} \otimes I \\ I \otimes D - E^{H} \otimes I \end{matrix})

and

\otimes

is the Kronecker product. Equations (2) are then equivalent to

Z^{H} y = α b .

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'

Optionally, the function estimates

Dif [(A, 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

(A R - L B, D R - L E) = (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)

trans ='N'

, solve the generalized Sylvester equation (1).

trans ='C'

, solve the ‘conjugate transposed’ system (2).

Constraint:

trans ='N'

'C'

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.

trans ='C'

, ijob is not referenced.

Constraint: if

trans ='N'

0 \leq ijob \leq 4

3: $a (lda :)$ – complex array

The first dimension of the array a must be at least

\max (1, m)

The second dimension of the array a must be at least

\max (1, m)

The upper triangular matrix

A

4: $b (ldb :)$ – complex array

The first dimension of the array b must be at least

\max (1, n)

The second dimension of the array b must be at least

\max (1, n)

The upper triangular matrix

B

5: $c (ldc :)$ – complex array

The first dimension of the array c must be at least

\max (1, m)

The second dimension of the array c must be at least

\max (1, n)

Contains the right-hand-side matrix

C

6: $d (ldd :)$ – complex array

The first dimension of the array d must be at least

\max (1, m)

The second dimension of the array d must be at least

\max (1, m)

The upper triangular matrix

D

7: $e (lde :)$ – complex array

The first dimension of the array e must be at least

\max (1, n)

The second dimension of the array e must be at least

\max (1, n)

The upper triangular matrix

E

8: $f (ldf :)$ – complex array

The first dimension of the array f must be at least

\max (1, m)

The second dimension of the array f must be at least

\max (1, 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: $m \geq 0$ .
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: $n \geq 0$ .

Output Parameters

1: $c (ldc :)$ – complex array: The first dimension of the array c will be $\max (1, m)$ .
The second dimension of the array c will be $\max (1, n)$ .

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: $f (ldf :)$ – complex array: The first dimension of the array f will be $\max (1, m)$ .
The second dimension of the array f will be $\max (1, n)$ .

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

8 m n (n + 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 real analogue of this function is nag_lapack_dtgsyl (f08yh).

Example

This example solves the generalized Sylvester equations

\begin{matrix} A R - L B & = α C \\ D R - L E & = α F, \end{matrix}

where

A = (\begin{array}{r} 4.0 + 4.0 i & 1.0 + 1.0 i & 1.0 + 1.0 i & 2.0 - 1.0 i \\ 0 & 2.0 + 1.0 i & 1.0 + 1.0 i & 1.0 + 1.0 i \\ 0 & 0 & 2.0 - 1.0 i & 1.0 + 1.0 i \\ 0 & 0 & 0 & 6.0 - 2.0 i \end{array}),

B = (\begin{array}{r} 2.0 & 1.0 + 1.0 i & 1.0 + 1.0 i & 3.0 - 1.0 i \\ 0 & 1.0 & 2.0 + 1.0 i & 1.0 + 1.0 i \\ 0 & 0 & 1.0 & 1.0 + 1.0 i \\ 0 & 0 & 0 & 2.0 \end{array}),

D = (\begin{array}{r} 1.0 + 1.0 i & 1.0 - 1.0 i & 1.0 + 1.0 i & 1.0 - 1.0 i \\ 0 & 6.0 - 4.0 i & 1.0 - 1.0 i & 1.0 + 1.0 i \\ 0 & 0 & 2.0 + 4.0 i & 1.0 - 1.0 i \\ 0 & 0 & 0 & 2.0 + 3.0 i \end{array}),

E = (\begin{array}{r} 1.0 & 1.0 + 1.0 i & 1.0 - 1.0 i & 1.0 + 1.0 i \\ 0 & 2.0 & 1.0 + 1.0 i & 1.0 - 1.0 i \\ 0 & 0 & 2.0 & 1.0 + 1.0 i \\ 0 & 0 & 0 & 1.0 \end{array}),

C = (\begin{array}{r} - 13.0 + 9.0 i & 2.0 + 8.0 i & - 2.0 + 8.0 i & - 2.0 + 5.0 i \\ - 9.0 - 1.0 i & 0.0 + 5.0 i & - 7.0 - 3.0 i & - 6.0 - 0.0 i \\ - 1.0 + 1.0 i & 4.0 + 2.0 i & 4.0 - 5.0 i & 9.0 - 5.0 i \\ - 6.0 + 6.0 i & 9.0 + 1.0 i & - 2.0 + 4.0 i & 22.0 - 8.0 i \end{array})

and

F = (\begin{array}{r} - 6.0 + 5.0 i & 4.0 - 4.0 i & - 3.0 + 11.0 i & 3.0 - 7.0 i \\ - 5.0 + 11.0 i & 12.0 - 4.0 i & - 2.0 + 2.0 i & 0.0 + 14.0 i \\ - 5.0 - 1.0 i & 0.0 + 4.0 i & - 2.0 + 10.0 i & 3.0 - 1.0 i \\ - 6.0 - 2.0 i & 1.0 + 1.0 i & - 7.0 - 3.0 i & 4.0 + 7.0 i \end{array}) .

Open in the MATLAB editor: f08yv_example

function f08yv_example


fprintf('f08yv 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 + 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];
D = [ 1 + 1i,  1 - 1i,  1 + 1i,  1 - 1i;
      0 + 0i,  6 - 4i,  1 - 1i,  1 + 1i;
      0 + 0i,  0 + 0i,  2 + 4i,  1 - 1i;
      0 + 0i,  0 + 0i,  0 + 0i,  2 + 3i];
B = [ 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];
E = [ 1,       1 + 1i,  1 - 1i,  1 + 1i;
      0 + 0i,  2 + 0i,  1 + 1i,  1 - 1i;
      0 + 0i,  0 + 0i,  2 + 0i,  1 + 1i;
      0 + 0i,  0 + 0i,  0 + 0i,  1 + 0i];

% Right hand matrices
C = [-13 +  9i,  2 + 8i, -2 +  8i, -2 +  5i;
      -9 -  1i,  0 + 5i, -7 -  3i, -6 +  0i;
      -1 +  1i,  4 + 2i,  4 -  5i,  9 -  5i;
      -6 +  6i,  9 + 1i, -2 +  4i, 22 -  8i];
F = [ -6 +  5i,  4 - 4i, -3 + 11i,  3 -  7i;
      -5 + 11i, 12 - 4i, -2 +  2i,  0 + 14i;
      -5 -  1i,  0 + 4i, -2 + 10i,  3 -  1i;
      -6 -  2i,  1 + 1i, -7 -  3i,  4 +  7i];

% Solve
trans = 'No transpose';
ijob = int64(0);
[R, L, alpha, ~, info] = f08yv( ...
                                trans, ijob, A, B, C, D, E, F);

disp('Solution matrix R');
disp(R); 
disp('Solution matrix L');
disp(L); 
fprintf('alpha = %10.2e\n', alpha);

f08yv example results

Solution matrix R
   1.0000 + 1.0000i   1.0000 + 1.0000i   1.0000 + 1.0000i   1.0000 + 1.0000i
  -1.0000 + 1.0000i   2.0000 + 1.0000i  -1.0000 + 1.0000i  -1.0000 + 1.0000i
  -1.0000 + 1.0000i   1.0000 + 1.0000i   3.0000 + 1.0000i   1.0000 + 1.0000i
  -1.0000 + 1.0000i   1.0000 + 1.0000i  -1.0000 + 1.0000i   4.0000 + 1.0000i

Solution matrix L
   4.0000 + 1.0000i  -1.0000 + 1.0000i   1.0000 + 1.0000i  -1.0000 + 1.0000i
   1.0000 + 1.0000i   3.0000 + 1.0000i  -1.0000 + 1.0000i   1.0000 + 1.0000i
  -1.0000 + 1.0000i   1.0000 + 1.0000i   2.0000 + 1.0000i  -1.0000 + 1.0000i
   1.0000 + 1.0000i  -1.0000 + 1.0000i   1.0000 + 1.0000i   1.0000 + 1.0000i

alpha =   1.00e+00

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Chapter Contents

Chapter Introduction

NAG Toolbox