Integer, Intent (In)	::	ijob, m, n, lda, ldb, ldc, ldd, lde, ldf, lwork
Integer, Intent (Out)	::	iwork(max(1,m+n+6)), info
Real (Kind=nag_wp), Intent (In)	::	a(lda,), b(ldb,), d(ldd,), e(lde,)
Real (Kind=nag_wp), Intent (Inout)	::	c(ldc,), f(ldf,)
Real (Kind=nag_wp), Intent (Out)	::	scale, dif, work(max(1,lwork))
Character (1), Intent (In)	::	trans

C Header Interface

#include <nag.h>

void

f08yhf_ (const char *trans, const Integer *ijob, const Integer *m, const Integer *n, const double a[], const Integer *lda, const double b[], const Integer *ldb, double c[], const Integer *ldc, const double d[], const Integer *ldd, const double e[], const Integer *lde, double f[], const Integer *ldf, double *scal, double *dif, double work[], const Integer *lwork, Integer iwork[], Integer *info, const Charlen length_trans)

The routine may be called by the names f08yhf, nagf_lapackeig_dtgsyl or its LAPACK name dtgsyl.

3 Description

f08yhf solves either the generalized real 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^{T} R + D^{T} L & = α C \\ R B^{T} + L E^{T} & = - α F, \end{matrix}

(2)

where the pair

(A, D)

are given

m \times m

matrices in real generalized Schur form,

(B, E)

are given

n \times n

matrices in real generalized Schur form and

(C, F)

are given

m \times n

matrices. The pair

(R, L)

are the

m \times n

solution matrices, and

α

is an output scaling factor determined by the routine 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^{T} \otimes I \\ I \otimes D - E^{T} \otimes I \end{matrix})

and

\otimes

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

Z^{T} y = α b .

The pair

(S, T)

are in real generalized Schur form if

S

is block upper triangular with

1 \times 1

and

2 \times 2

diagonal blocks on the diagonal and

T

is upper triangular as returned, for example, by f08xcf, or f08xef with

job ='S'

Optionally, the routine 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).

4 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 https://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

5 Arguments

1: $trans$ – Character(1) Input

On entry: if

trans ='N'

, solve the generalized Sylvester equation (1).

trans ='T'

, solve the ‘transposed’ system (2).

Constraint:

trans ='N'

'T'

2: $ijob$ – Integer Input

On entry: 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 ='T'

, ijob is not referenced.

Constraint: if

trans ='N'

0 \leq ijob \leq 4

3: $m$ – Integer Input

On entry:

m

, the order of the matrices

A

and

D

, and the row dimension of the matrices

C

F

R

and

L

Constraint:

m > 0

4: $n$ – Integer Input

On entry:

n

, the order of the matrices

B

and

E

, and the column dimension of the matrices

C

F

R

and

L

Constraint:

n > 0

5: $a (lda, *)$ – Real (Kind=nag_wp) array Input

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

\max (1, m)

On entry: the upper quasi-triangular matrix

A

6: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08yhf is called.

Constraint:

lda \geq \max (1, m)

7: $b (ldb, *)$ – Real (Kind=nag_wp) array Input

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

\max (1, n)

On entry: the upper quasi-triangular matrix

B

8: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f08yhf is called.

Constraint:

ldb \geq \max (1, n)

9: $c (ldc, *)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n)

On entry: contains the right-hand-side matrix

C

On exit: if

ijob = 0

1

2

, c is overwritten by the solution matrix

R

trans ='N'

and

ijob = 3

4

, c holds

R

, the solution achieved during the computation of the Dif estimate.

10: $ldc$ – Integer Input

On entry: the first dimension of the array c as declared in the (sub)program from which f08yhf is called.

Constraint:

ldc \geq \max (1, m)

11: $d (ldd, *)$ – Real (Kind=nag_wp) array Input

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

\max (1, m)

On entry: the upper triangular matrix

D

12: $ldd$ – Integer Input

On entry: the first dimension of the array d as declared in the (sub)program from which f08yhf is called.

Constraint:

ldd \geq \max (1, m)

13: $e (lde, *)$ – Real (Kind=nag_wp) array Input

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

\max (1, n)

On entry: the upper triangular matrix

E

14: $lde$ – Integer Input

On entry: the first dimension of the array e as declared in the (sub)program from which f08yhf is called.

Constraint:

lde \geq \max (1, n)

15: $f (ldf, *)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n)

On entry: contains the right-hand side matrix

F

On exit: if

ijob = 0

1

2

, f is overwritten by the solution matrix

L

trans ='N'

and

ijob = 3

4

, f holds

L

, the solution achieved during the computation of the Dif estimate.

16: $ldf$ – Integer Input

On entry: the first dimension of the array f as declared in the (sub)program from which f08yhf is called.

Constraint:

ldf \geq \max (1, m)

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

On exit:

α

, the scaling factor in (1) or (2).

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.

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

18: $dif$ – Real (Kind=nag_wp) Output

On exit: the estimate of

Dif

. If

ijob = 0

, dif is not referenced.

19: $work (\max (1, lwork))$ – Real (Kind=nag_wp) array Workspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

20: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08yhf is called.

lwork = −1

, a workspace query is assumed; the routine only calculates the minimum size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Constraints:

lwork \neq −1

if $trans ='N'$ and $ijob = 1$ or $2$ , $lwork \geq \max (1, 2 \times m \times n)$ ;
otherwise $lwork \geq 1$ .

21: $iwork (\max (1, m + n + 6))$ – Integer array Workspace

22: $info$ – Integer Output

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.

$info > 0$: $(A, D)$ and $(B, E)$ have common or close eigenvalues and so no solution could be computed.

7 Accuracy

See Kågström (1994) for a perturbation analysis of the generalized Sylvester equation.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08yhf 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 needed to solve the generalized Sylvester equations is approximately

2 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 complex analogue of this routine is f08yvf.

10 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 & 1.0 & 1.0 & 2.0 \\ 0 & 3.0 & 4.0 & 1.0 \\ 0 & 1.0 & 3.0 & 1.0 \\ 0 & 0 & 0 & 6.0 \end{array}), B = (\begin{array}{r} 1.0 & 1.0 & 1.0 & 1.0 \\ 0 & 3.0 & 4.0 & 1.0 \\ 0 & 1.0 & 3.0 & 1.0 \\ 0 & 0 & 0 & 4.0 \end{array}),

D = (\begin{array}{r} 2.0 & 1.0 & 1.0 & 3.0 \\ 0 & 1.0 & 2.0 & 1.0 \\ 0 & 0 & 1.0 & 1.0 \\ 0 & 0 & 0 & 2.0 \end{array}), E = (\begin{array}{r} 1.0 & 1.0 & 1.0 & 2.0 \\ 0 & 1.0 & 4.0 & 1.0 \\ 0 & 0 & 1.0 & 1.0 \\ 0 & 0 & 0 & 1.0 \end{array}),

C = (\begin{array}{r} - 4.0 & 7.0 & 1.0 & 12.0 \\ - 9.0 & 2.0 & - 2.0 & - 2.0 \\ - 4.0 & 2.0 & - 2.0 & 8.0 \\ - 7.0 & 7.0 & - 6.0 & 19.0 \end{array}) and F = (\begin{array}{r} - 7.0 & 5.0 & 0.0 & 7.0 \\ - 5.0 & 1.0 & - 8.0 & 0.0 \\ - 1.0 & 2.0 & - 3.0 & 5.0 \\ - 3.0 & 2.0 & 0.0 & 5.0 \end{array}) .

f08yh: FL CL CPP AD PY MB

NAG FL Interfacef08yhf (dtgsyl)

▸▿ 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

NAG FL Interface
f08yhf (dtgsyl)