F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08YHF (DTGSYL)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08YHF (DTGSYL) solves the generalized real quasi-triangular Sylvester equations.

## 2  Specification

 SUBROUTINE F08YHF ( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
 INTEGER IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, IWORK(max(1,M+N+6)), INFO REAL (KIND=nag_wp) A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), E(LDE,*), F(LDF,*), SCALE, DIF, WORK(max(1,LWORK)) CHARACTER(1) TRANS
The routine may be called by its LAPACK name dtgsyl.

## 3  Description

F08YHF (DTGSYL) solves either the generalized real Sylvester equations
 $AR-LB =αC DR-LE =αF,$ (1)
or the equations
 $ATR+DTL =αC RBT+LET =-αF,$ (2)
where the pair $\left(A,D\right)$ are given $m$ by $m$ matrices in real generalized Schur form, $\left(B,E\right)$ are given $n$ by $n$ matrices in real generalized Schur form and $\left(C,F\right)$ are given $m$ by $n$ matrices. The pair $\left(R,L\right)$ are the $m$ by $n$ solution matrices, and $\alpha$ is an output scaling factor determined by the routine to avoid overflow in computing $\left(R,L\right)$.
Equations (1) are equivalent to equations of the form
 $Zx=αb ,$
where
 $Z = I⊗A-BT⊗I I⊗D-ET⊗I$
and $\otimes$ is the Kronecker product. Equations (2) are then equivalent to
 $ZTy = αb .$
The pair $\left(S,T\right)$ are in real generalized Schur form if $S$ is block upper triangular with $1$ by $1$ and $2$ by $2$ diagonal blocks on the diagonal and $T$ is upper triangular as returned, for example, by F08XAF (DGGES), or F08XEF (DHGEQZ) with ${\mathbf{JOB}}=\text{'S'}$.
Optionally, the routine estimates $\mathrm{Dif}\left[\left(A,D\right),\left(B,E\right)\right]$, the separation between the matrix pairs $\left(A,D\right)$ and $\left(B,E\right)$, 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 http://www.netlib.org/lapack/lug
Kågström B (1994) A perturbation analysis of the generalized Sylvester equation $\left(AR-LB,DR-LE\right)=\left(c,F\right)$ 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  Parameters

1:     $\mathrm{TRANS}$ – CHARACTER(1)Input
On entry: if ${\mathbf{TRANS}}=\text{'N'}$, solve the generalized Sylvester equation (1).
If ${\mathbf{TRANS}}=\text{'T'}$, solve the ‘transposed’ system (2).
Constraint: ${\mathbf{TRANS}}=\text{'N'}$ or $\text{'T'}$.
2:     $\mathrm{IJOB}$ – INTEGERInput
On entry: specifies what kind of functionality is to be performed when ${\mathbf{TRANS}}=\text{'N'}$.
${\mathbf{IJOB}}=0$
Solve (1) only.
${\mathbf{IJOB}}=1$
The functionality of ${\mathbf{IJOB}}=0$ and $3$.
${\mathbf{IJOB}}=2$
The functionality of ${\mathbf{IJOB}}=0$ and $4$.
${\mathbf{IJOB}}=3$
Only an estimate of $\mathrm{Dif}\left[\left(A,D\right),\left(B,E\right)\right]$ is computed based on the Frobenius norm.
${\mathbf{IJOB}}=4$
Only an estimate of $\mathrm{Dif}\left[\left(A,D\right),\left(B,E\right)\right]$ is computed based on the $1$-norm.
If ${\mathbf{TRANS}}=\text{'T'}$, IJOB is not referenced.
Constraint: if ${\mathbf{TRANS}}=\text{'N'}$, $0\le {\mathbf{IJOB}}\le 4$.
3:     $\mathrm{M}$ – INTEGERInput
On entry: $m$, the order of the matrices $A$ and $D$, and the row dimension of the matrices $C$, $F$, $R$ and $L$.
Constraint: ${\mathbf{M}}\ge 0$.
4:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the order of the matrices $B$ and $E$, and the column dimension of the matrices $C$, $F$, $R$ and $L$.
Constraint: ${\mathbf{N}}\ge 0$.
5:     $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
On entry: the upper quasi-triangular matrix $A$.
6:     $\mathrm{LDA}$ – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
7:     $\mathrm{B}\left({\mathbf{LDB}},*\right)$ – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the upper quasi-triangular matrix $B$.
8:     $\mathrm{LDB}$ – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
9:     $\mathrm{C}\left({\mathbf{LDC}},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array C must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: contains the right-hand-side matrix $C$.
On exit: if ${\mathbf{IJOB}}=0$, $1$ or $2$, C is overwritten by the solution matrix $R$.
If ${\mathbf{TRANS}}=\text{'N'}$ and ${\mathbf{IJOB}}=3$ or $4$, C holds $R$, the solution achieved during the computation of the Dif estimate.
10:   $\mathrm{LDC}$ – INTEGERInput
On entry: the first dimension of the array C as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint: ${\mathbf{LDC}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
11:   $\mathrm{D}\left({\mathbf{LDD}},*\right)$ – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array D must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
On entry: the upper triangular matrix $D$.
12:   $\mathrm{LDD}$ – INTEGERInput
On entry: the first dimension of the array D as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint: ${\mathbf{LDD}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
13:   $\mathrm{E}\left({\mathbf{LDE}},*\right)$ – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array E must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the upper triangular matrix $E$.
14:   $\mathrm{LDE}$ – INTEGERInput
On entry: the first dimension of the array E as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint: ${\mathbf{LDE}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
15:   $\mathrm{F}\left({\mathbf{LDF}},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array F must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: contains the right-hand side matrix $F$.
On exit: if ${\mathbf{IJOB}}=0$, $1$ or $2$, F is overwritten by the solution matrix $L$.
If ${\mathbf{TRANS}}=\text{'N'}$ and ${\mathbf{IJOB}}=3$ or $4$, F holds $L$, the solution achieved during the computation of the Dif estimate.
16:   $\mathrm{LDF}$ – INTEGERInput
On entry: the first dimension of the array F as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint: ${\mathbf{LDF}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
17:   $\mathrm{SCALE}$ – REAL (KIND=nag_wp)Output
On exit: $\alpha$, the scaling factor in (1) or (2).
If $0<{\mathbf{SCALE}}<1$, 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.
If ${\mathbf{SCALE}}=0$, 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, ${\mathbf{SCALE}}=1$.
18:   $\mathrm{DIF}$ – REAL (KIND=nag_wp)Output
On exit: the estimate of $\mathrm{Dif}$. If ${\mathbf{IJOB}}=0$, DIF is not referenced.
19:   $\mathrm{WORK}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)\right)$ – REAL (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{WORK}}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.
20:   $\mathrm{LWORK}$ – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08YHF (DTGSYL) is called.
If ${\mathbf{LWORK}}=-1$, 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:
if ${\mathbf{LWORK}}\ne -1$,
• if ${\mathbf{TRANS}}=\text{'N'}$ and ${\mathbf{IJOB}}=1$ or $2$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{M}}×{\mathbf{N}}\right)$;
• otherwise ${\mathbf{LWORK}}\ge 1$.
21:   $\mathrm{IWORK}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}+{\mathbf{N}}+6\right)\right)$ – INTEGER arrayWorkspace
22:   $\mathrm{INFO}$ – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
$\left(A,D\right)$ and $\left(B,E\right)$ 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

F08YHF (DTGSYL) is not threaded by NAG in any implementation.
F08YHF (DTGSYL) 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.

The total number of floating-point operations needed to solve the generalized Sylvester equations is approximately $2mn\left(n+m\right)$. The Frobenius norm estimate of $\mathrm{Dif}$ 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 (ZTGSYL).

## 10  Example

This example solves the generalized Sylvester equations
 $AR-LB =αC DR-LE =αF,$
where
 $A = 4.0 1.0 1.0 2.0 0.0 3.0 4.0 1.0 0.0 1.0 3.0 1.0 0.0 0.0 0.0 6.0 , B= 1.0 1.0 1.0 1.0 0.0 3.0 4.0 1.0 0.0 1.0 3.0 1.0 0.0 0.0 0.0 4.0 ,$
 $D = 2.0 1.0 1.0 3.0 0.0 1.0 2.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 2.0 , E= 1.0 1.0 1.0 2.0 0.0 1.0 4.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 1.0 ,$
 $C = -4.0 7.0 1.0 12.0 -9.0 2.0 -2.0 -2.0 -4.0 2.0 -2.0 8.0 -7.0 7.0 -6.0 19.0 and F= -7.0 5.0 0.0 7.0 -5.0 1.0 -8.0 0.0 -1.0 2.0 -3.0 5.0 -3.0 2.0 0.0 5.0 .$

### 10.1  Program Text

Program Text (f08yhfe.f90)

### 10.2  Program Data

Program Data (f08yhfe.d)

### 10.3  Program Results

Program Results (f08yhfe.r)