Integer, Intent (In)	::	isgn, m, n, lda, ldb, ldc
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	a(lda,), b(ldb,)
Real (Kind=nag_wp), Intent (Inout)	::	c(ldc,*)
Real (Kind=nag_wp), Intent (Out)	::	scale
Character (1), Intent (In)	::	trana, tranb

C Header Interface

#include nagmk26.h

void	f08qhf_ ( const char trana, const char tranb, const Integer isgn, const Integer m, const Integer n, const double a[], const Integer lda, const double b[], const Integer ldb, double c[], const Integer ldc, double scal, Integer info, const Charlen length_trana, const Charlen length_tranb)

The routine may be called by its LAPACK name dtrsyl.

3

Description

f08qhf (dtrsyl) 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 f08pef (dhseqr));

α

is a scale factor (

\leq 1

) determined by the routine 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.

4

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

5

Arguments

1: $trana$ – Character(1)Input

On entry: specifies the option

op (A)

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

Constraint:

trana ='N'

'T'

'C'

2: $tranb$ – Character(1)Input

On entry: specifies the option

op (B)

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

Constraint:

tranb ='N'

'T'

'C'

3: $isgn$ – IntegerInput

On entry: 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: $m$ – IntegerInput

On entry:

m

, the order of the matrix

A

, and the number of rows in the matrices

X

and

C

Constraint:

m \geq 0

5: $n$ – IntegerInput

On entry:

n

, the order of the matrix

B

, and the number of columns in the matrices

X

and

C

Constraint:

n \geq 0

6: $a (lda *)$ – Real (Kind=nag_wp) arrayInput

Note: the second dimension of the array a must be at least

\max (1, m)

On entry: the

m

m

upper quasi-triangular matrix

A

in canonical Schur form, as returned by f08pef (dhseqr).

7: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08qhf (dtrsyl) is called.

Constraint:

lda \geq \max (1, m)

8: $b (ldb *)$ – Real (Kind=nag_wp) arrayInput

Note: the second dimension of the array b must be at least

\max (1, n)

On entry: the

n

n

upper quasi-triangular matrix

B

in canonical Schur form, as returned by f08pef (dhseqr).

9: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f08qhf (dtrsyl) is called.

Constraint:

ldb \geq \max (1, n)

10: $c (ldc *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array c must be at least

\max (1, n)

On entry: the

m

n

right-hand side matrix

C

On exit: c is overwritten by the solution matrix

X

11: $ldc$ – IntegerInput

On entry: the first dimension of the array c as declared in the (sub)program from which f08qhf (dtrsyl) is called.

Constraint:

ldc \geq \max (1, m)

12: $scale$ – Real (Kind=nag_wp)Output

On exit: the value of the scale factor

α

13: $info$ – IntegerOutput

On exit:

info = 0

unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

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

7

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 Section 9.

8

Parallelism and Performance

f08qhf (dtrsyl) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

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 f08paf (dgees), 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; f08qhf (dtrsyl) 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 routine is f08qvf (ztrsyl).

10

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}) .

NAG Library Routine Document

f08qhf (dtrsyl)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results