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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dtrsyl (f08qh)

▸▿ 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_dtrsyl (f08qh) solves the real quasi-triangular Sylvester matrix equation.

Syntax

[c, scale, info] = f08qh(trana, tranb, isgn, a, b, c, 'm', m, 'n', n)

[c, scale, info] = nag_lapack_dtrsyl(trana, tranb, isgn, a, b, c, 'm', m, 'n', n)

Description

nag_lapack_dtrsyl (f08qh) solves the real Sylvester matrix equation

op (A) X \pm X op (B) = α C,

where

op (A) = A

A^{T}

, and the matrices

A

and

B

are upper quasi-triangular matrices in canonical Schur form (as returned by nag_lapack_dhseqr (f08pe));

α

is a scale factor (

\leq 1

) determined by the function to avoid overflow in

X

;

A

m

m

and

B

n

n

while the right-hand side matrix

C

and the solution matrix

X

are both

m

n

. The matrix

X

is obtained by a straightforward process of back-substitution (see Golub and Van Loan (1996)).

Note that the equation has a unique solution if and only if

α_{i} \pm β_{j} \neq 0

, where

\{α_{i}\}

and

\{β_{j}\}

are the eigenvalues of

A

and

B

respectively and the sign (

+

-

) is the same as that used in the equation to be solved.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Higham N J (1992) Perturbation theory and backward error for

A X - X B = C

Numerical Analysis Report University of Manchester

Parameters

Compulsory Input Parameters

1: $trana$ – string (length ≥ 1)

Specifies the option

op (A)

$trana ='N'$: $op (A) = A$ .
$trana ='T'$ or $'C'$: $op (A) = A^{T}$ .

Constraint:

trana ='N'

'T'

'C'

2: $tranb$ – string (length ≥ 1)

Specifies the option

op (B)

$tranb ='N'$: $op (B) = B$ .
$tranb ='T'$ or $'C'$: $op (B) = B^{T}$ .

Constraint:

tranb ='N'

'T'

'C'

3: $isgn$ – int64int32nag_int scalar

Indicates the form of the Sylvester equation.

$isgn = +1$: The equation is of the form $op (A) X + X op (B) = α C$ .
$isgn = - 1$: The equation is of the form $op (A) X - X op (B) = α C$ .

Constraint:

isgn = +1

- 1

4: $a (lda :)$ – double 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

m

m

upper quasi-triangular matrix

A

in canonical Schur form, as returned by nag_lapack_dhseqr (f08pe).

5: $b (ldb :)$ – double 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

n

n

upper quasi-triangular matrix

B

in canonical Schur form, as returned by nag_lapack_dhseqr (f08pe).

6: $c (ldc :)$ – double 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)

The

m

n

right-hand side matrix

C

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the array a and the second dimension of the array a. (An error is raised if these dimensions are not equal.)
$m$ , the order of the matrix $A$ , and the number of rows in the matrices $X$ and $C$ .

Constraint: $m \geq 0$ .
2: $n$ – int64int32nag_int scalar: Default: the first dimension of the array b and the second dimension of the array b. (An error is raised if these dimensions are not equal.)
$n$ , the order of the matrix $B$ , and the number of columns in the matrices $X$ and $C$ .

Constraint: $n \geq 0$ .

Output Parameters

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

c stores the solution matrix $X$ .
2: $scale$ – double scalar: The value of the scale factor $α$ .
3: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$info = - i$: If $info = - i$ , parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: trana, 2: tranb, 3: isgn, 4: m, 5: n, 6: a, 7: lda, 8: b, 9: ldb, 10: c, 11: ldc, 12: scale, 13: 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.

W $info = 1$: $A$ and $B$ have common or close eigenvalues, perturbed values of which were used to solve the equation.

Accuracy

Consider the equation

A X - X B = C

. (To apply the remarks to the equation

A X + X B = C

, simply replace

B

- B

Let

\tilde{X}

be the computed solution and

R

the residual matrix:

R = C - (A \tilde{X} - \tilde{X} B) .

Then the residual is always small:

{‖R‖}_{F} = O (ε) ({‖A‖}_{F} + {‖B‖}_{F}) {‖\tilde{X}‖}_{F} .

However,

\tilde{X}

is not necessarily the exact solution of a slightly perturbed equation; in other words, the solution is not backwards stable.

For the forward error, the following bound holds:

{‖\tilde{X} - X‖}_{F} \leq \frac{{‖R‖}_{F}}{sep (A, B)}

but this may be a considerable over estimate. See Golub and Van Loan (1996) for a definition of

sep (A, B)

, and Higham (1992) for further details.

These remarks also apply to the solution of a general Sylvester equation, as described in Further Comments.

Further Comments

The total number of floating-point operations is approximately

m n (m + n)

To solve the general real Sylvester equation

A X \pm X B = C

where

A

and

B

are general nonsymmetric matrices,

A

and

B

must first be reduced to Schur form (by calling nag_lapack_dgees (f08pa), for example):

A = Q_{1} \tilde{A} Q_{1}^{T} and B = Q_{2} \tilde{B} Q_{2}^{T}

where

\tilde{A}

and

\tilde{B}

are upper quasi-triangular and

Q_{1}

and

Q_{2}

are orthogonal. The original equation may then be transformed to:

\tilde{A} \tilde{X} \pm \tilde{X} \tilde{B} = \tilde{C}

where

\tilde{X} = Q_{1}^{T} X Q_{2}

and

\tilde{C} = Q_{1}^{T} C Q_{2}

\tilde{C}

may be computed by matrix multiplication; nag_lapack_dtrsyl (f08qh) may be used to solve the transformed equation; and the solution to the original equation can be obtained as

X = Q_{1} \tilde{X} Q_{2}^{T}

The complex analogue of this function is nag_lapack_ztrsyl (f08qv).

Example

This example solves the Sylvester equation

A X + X B = C

, where

A = (\begin{array}{r} 0.10 & 0.50 & 0.68 & - 0.21 \\ - 0.50 & 0.10 & - 0.24 & 0.67 \\ 0.00 & 0.00 & 0.19 & - 0.35 \\ 0.00 & 0.00 & 0.00 & - 0.72 \end{array}),

B = (\begin{array}{r} - 0.99 & - 0.17 & 0.39 & 0.58 \\ 0.00 & 0.48 & - 0.84 & - 0.15 \\ 0.00 & 0.00 & 0.75 & 0.25 \\ 0.00 & 0.00 & - 0.25 & 0.75 \end{array})

and

C = (\begin{array}{r} 0.63 & - 0.56 & 0.08 & - 0.23 \\ - 0.45 & - 0.31 & 0.27 & 1.21 \\ 0.20 & - 0.35 & 0.41 & 0.84 \\ 0.49 & - 0.05 & - 0.52 & - 0.08 \end{array}) .

Open in the MATLAB editor: f08qh_example

function f08qh_example


fprintf('f08qh example results\n\n');

% Quasi-triangular (Schur form) matrices A and B, and general matrix C 
a = [ 0.10,  0.50,  0.68, -0.21;
     -0.50,  0.10, -0.24,  0.67;
      0,     0,     0.19, -0.35;
      0,     0,     0,    -0.72];
b = [-0.99, -0.17,  0.39,  0.58;
      0,     0.48, -0.84, -0.15;
      0,     0,     0.75,  0.25;
      0,     0,    -0.25,  0.75];
c = [ 0.63, -0.56,  0.08, -0.23;
     -0.45, -0.31,  0.27,  1.21;
      0.20, -0.35,  0.41,  0.84;
      0.49, -0.05, -0.52, -0.08];

% Solve Sylvester equation AX + XB = C for X
trana = 'No transpose';
tranb = 'No transpose';
isgn = int64(1);
[X, scale, info] = f08qh( ...
			  trana, tranb, isgn, a, b, c);

disp('Solution matrix');
disp(X);

f08qh example results

Solution matrix
   -0.4209    0.1764    0.2438   -0.9577
    0.5600   -0.8337   -0.7221    0.5386
   -0.1246   -0.3392    0.6221    0.8691
   -0.2865    0.4113    0.5535    0.3174

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Chapter Contents

Chapter Introduction

NAG Toolbox