void	nag_dtrsyl (Nag_OrderType order, Nag_TransType trana, Nag_TransType tranb, Nag_SignType sign, Integer m, Integer n, const double a[], Integer pda, const double b[], Integer pdb, double c[], Integer pdc, double scale, NagError fail)

3

Description

nag_dtrsyl (f08qhc) 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_dhseqr (f08pec));

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.

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: $order$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $trana$ – Nag_TransTypeInput

On entry: specifies the option

op (A)

$trana = Nag_NoTrans$: $op (A) = A$ .
$trana = Nag_Trans$ or $Nag_ConjTrans$: $op (A) = A^{T}$ .

Constraint:

trana = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

3: $tranb$ – Nag_TransTypeInput

On entry: specifies the option

op (B)

$tranb = Nag_NoTrans$: $op (B) = B$ .
$tranb = Nag_Trans$ or $Nag_ConjTrans$: $op (B) = B^{T}$ .

Constraint:

tranb = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

4: $sign$ – Nag_SignTypeInput

On entry: indicates the form of the Sylvester equation.

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

Constraint:

sign = Nag_Plus

Nag_Minus

5: $m$ – 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

6: $n$ – 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

7: $a [\dim]$ – const doubleInput

Note: the dimension, dim, of the array a must be at least

\max (1, pda \times m)

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

m

m

upper quasi-triangular matrix

A

in canonical Schur form, as returned by nag_dhseqr (f08pec).

8: $pda$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq \max (1, m)

9: $b [\dim]$ – const doubleInput

Note: the dimension, dim, of the array b must be at least

\max (1, pdb \times n)

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

n

upper quasi-triangular matrix

B

in canonical Schur form, as returned by nag_dhseqr (f08pec).

10: $pdb$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraint:

pdb \geq \max (1, n)

11: $c [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array c must be at least

$\max (1, pdc \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pdc)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

C

is stored in

$c [(j - 1) \times pdc + i - 1]$ when $order = Nag_ColMajor$ ;
$c [(i - 1) \times pdc + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

m

n

right-hand side matrix

C

On exit: c is overwritten by the solution matrix

X

12: $pdc$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array c.

Constraints:

if $order = Nag_ColMajor$ , $pdc \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pdc \geq \max (1, n)$ .

13: $scale$ – double *Output

On exit: the value of the scale factor

α

14: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .

On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .

On entry, $pdc = 〈value〉$ .
Constraint: $pdc > 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pda \geq \max (1, m)$ .

On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdc = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdc \geq \max (1, m)$ .

On entry, $pdc = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdc \geq \max (1, n)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_PERTURBED: $A$ 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

nag_dtrsyl (f08qhc) 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 function. 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 :

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_dtrsyl (f08qhc) 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_ztrsyl (f08qvc).

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 Function Document

nag_dtrsyl (f08qhc)

▸▿ 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