NAG Library Function Document

nag_ztrsyl (f08qvc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_ztrsyl (f08qvc) solves the complex triangular Sylvester matrix equation.

2 Specification

#include <nag.h>

#include <nagf08.h>

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

3 Description

nag_ztrsyl (f08qvc) solves the complex Sylvester matrix equation

op (A) X \pm X op (B) = α C,

where

op (A) = A

A^{H}

, and the matrices

A

and

B

are upper triangular;

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: trana – Nag_TransTypeInput

On entry: specifies the option

op (A)

$trana = Nag_NoTrans$: $op (A) = A$ .
$trana = Nag_ConjTrans$: $op (A) = A^{H}$ .

Constraint:

trana = Nag_NoTrans

Nag_ConjTrans

3: tranb – Nag_TransTypeInput

On entry: specifies the option

op (B)

$tranb = Nag_NoTrans$: $op (B) = B$ .
$tranb = Nag_ConjTrans$: $op (B) = B^{H}$ .

Constraint:

tranb = Nag_NoTrans

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 ComplexInput

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 triangular matrix

A

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 ComplexInput

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 triangular matrix

B

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[ $\dim$ ] – ComplexInput/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: scal – double *Output

On exit: the value of the scale factor

α

14: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
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 8.

8 Further Comments

The total number of real floating point operations is approximately

4 m n (m + n)

To solve the general complex Sylvester equation

A X \pm X B = C

where

A

and

B

are general matrices,

A

and

B

must first be reduced to Schur form :

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

where

\tilde{A}

and

\tilde{B}

are upper triangular and

Q_{1}

and

Q_{2}

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

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

where

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

and

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

\tilde{C}

may be computed by matrix multiplication; nag_ztrsyl (f08qvc) 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}^{H}

The real analogue of this function is nag_dtrsyl (f08qhc).

9 Example

This example solves the Sylvester equation

A X + X B = C

, where

A = (\begin{array}{r} - 6.00 - 7.00 i & 0.36 - 0.36 i & - 0.19 + 0.48 i & 0.88 - 0.25 i \\ 0.00 + 0.00 i & - 5.00 + 2.00 i & - 0.03 - 0.72 i & - 0.23 + 0.13 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & 8.00 - 1.00 i & 0.94 + 0.53 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & 0.00 + 0.00 i & 3.00 - 4.00 i \end{array}),

B = (\begin{array}{r} 0.50 - 0.20 i & - 0.29 - 0.16 i & - 0.37 + 0.84 i & - 0.55 + 0.73 i \\ 0.00 + 0.00 i & - 0.40 + 0.90 i & 0.06 + 0.22 i & - 0.43 + 0.17 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & - 0.90 - 0.10 i & - 0.89 - 0.42 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & 0.00 + 0.00 i & 0.30 - 0.70 i \end{array})

and

C = (\begin{array}{r} 0.63 + 0.35 i & 0.45 - 0.56 i & 0.08 - 0.14 i & - 0.17 - 0.23 i \\ - 0.17 + 0.09 i & - 0.07 - 0.31 i & 0.27 - 0.54 i & 0.35 + 1.21 i \\ - 0.93 - 0.44 i & - 0.33 - 0.35 i & 0.41 - 0.03 i & 0.57 + 0.84 i \\ 0.54 + 0.25 i & - 0.62 - 0.05 i & - 0.52 - 0.13 i & 0.11 - 0.08 i \end{array}) .

NAG Library Function Documentnag_ztrsyl (f08qvc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_ztrsyl (f08qvc)