# NAG Library Routine Document

## 1Purpose

f08yvf (ztgsyl) solves the generalized complex triangular Sylvester equations.

## 2Specification

Fortran Interface
 Subroutine f08yvf ( ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, dif, work, info)
 Integer, Intent (In) :: ijob, m, n, lda, ldb, ldc, ldd, lde, ldf, lwork Integer, Intent (Out) :: iwork(m+n+2), info Real (Kind=nag_wp), Intent (Out) :: scale, dif Complex (Kind=nag_wp), Intent (In) :: a(lda,*), b(ldb,*), d(ldd,*), e(lde,*) Complex (Kind=nag_wp), Intent (Inout) :: c(ldc,*), f(ldf,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: trans
#include nagmk26.h
 void f08yvf_ (const char *trans, const Integer *ijob, const Integer *m, const Integer *n, const Complex a[], const Integer *lda, const Complex b[], const Integer *ldb, Complex c[], const Integer *ldc, const Complex d[], const Integer *ldd, const Complex e[], const Integer *lde, Complex f[], const Integer *ldf, double *scal, double *dif, Complex work[], const Integer *lwork, Integer iwork[], Integer *info, const Charlen length_trans)
The routine may be called by its LAPACK name ztgsyl.

## 3Description

f08yvf (ztgsyl) solves either the generalized complex Sylvester equations
 $AR-LB =αC DR-LE =αF,$ (1)
or the equations
 $AHR+DHL =αC RBH+LEH =-αF,$ (2)
where the pair $\left(A,D\right)$ are given $m$ by $m$ matrices in generalized Schur form, $\left(B,E\right)$ are given $n$ by $n$ matrices in 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
 $Z = I⊗A-BH⊗I I⊗D-EH⊗I$
and $\otimes$ is the Kronecker product. Equations (2) are then equivalent to
 $ZHy = αb .$
The pair $\left(S,T\right)$ are in generalized Schur form if $S$ and $T$ are upper triangular as returned, for example, by f08xnf (zgges), or f08xsf (zhgeqz) 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).

## 4References

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

## 5Arguments

1:     $\mathbf{trans}$ – Character(1)Input
On entry: if ${\mathbf{trans}}=\text{'N'}$, solve the generalized Sylvester equation (1).
If ${\mathbf{trans}}=\text{'C'}$, solve the ‘conjugate transposed’ system (2).
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'C'}$.
2:     $\mathbf{ijob}$ – 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 ${\mathbf{trans}}=\text{'C'}$, ijob is not referenced.
Constraint: if ${\mathbf{trans}}=\text{'N'}$, $0\le {\mathbf{ijob}}\le 4$.
3:     $\mathbf{m}$ – 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: ${\mathbf{m}}>0$.
4:     $\mathbf{n}$ – 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: ${\mathbf{n}}>0$.
5:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (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 triangular matrix $A$.
6:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08yvf (ztgsyl) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
7:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (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 triangular matrix $B$.
8:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08yvf (ztgsyl) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:     $\mathbf{c}\left({\mathbf{ldc}},*\right)$ – Complex (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:   $\mathbf{ldc}$ – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which f08yvf (ztgsyl) is called.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
11:   $\mathbf{d}\left({\mathbf{ldd}},*\right)$ – Complex (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:   $\mathbf{ldd}$ – IntegerInput
On entry: the first dimension of the array d as declared in the (sub)program from which f08yvf (ztgsyl) is called.
Constraint: ${\mathbf{ldd}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
13:   $\mathbf{e}\left({\mathbf{lde}},*\right)$ – Complex (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:   $\mathbf{lde}$ – IntegerInput
On entry: the first dimension of the array e as declared in the (sub)program from which f08yvf (ztgsyl) is called.
Constraint: ${\mathbf{lde}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
15:   $\mathbf{f}\left({\mathbf{ldf}},*\right)$ – Complex (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:   $\mathbf{ldf}$ – IntegerInput
On entry: the first dimension of the array f as declared in the (sub)program from which f08yvf (ztgsyl) is called.
Constraint: ${\mathbf{ldf}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
17:   $\mathbf{scale}$ – 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:   $\mathbf{dif}$ – Real (Kind=nag_wp)Output
On exit: the estimate of $\mathrm{Dif}$. If ${\mathbf{ijob}}=0$, dif is not referenced.
19:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Complex (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
20:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08yvf (ztgsyl) 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 $2$, ${\mathbf{lwork}}\ge 2×{\mathbf{m}}×{\mathbf{n}}$;
• otherwise ${\mathbf{lwork}}\ge 1$.
21:   $\mathbf{iwork}\left({\mathbf{m}}+{\mathbf{n}}+2\right)$ – Integer arrayWorkspace
22:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error 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.

## 7Accuracy

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

## 8Parallelism and Performance

f08yvf (ztgsyl) 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 $8mn\left(n+m\right)$. 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 real analogue of this routine is f08yhf (dtgsyl).

## 10Example

This example solves the generalized Sylvester equations
 $AR-LB =αC DR-LE =αF,$
where
 $A = 4.0+4.0i 1.0+1.0i 1.0+1.0i 2.0-1.0i 0.0i+0.0 2.0+1.0i 1.0+1.0i 1.0+1.0i 0.0i+0.0 0.0i+0.0 2.0-1.0i 1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 6.0-2.0i ,$
 $B = 2.0 1.0+1.0i 1.0+1.0i 3.0-1.0i 0.0 1.0i+0.0 2.0+1.0i 1.0+1.0i 0.0 0.0i+0.0 1.0i+0.0 1.0+1.0i 0.0 0.0i+0.0 0.0i+0.0 2.0i+0.0 ,$
 $D = 1.0+1.0i 1.0-1.0i 1.0+1.0i 1.0-1.0i 0.0i+0.0 6.0-4.0i 1.0-1.0i 1.0+1.0i 0.0i+0.0 0.0i+0.0 2.0+4.0i 1.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 2.0+3.0i ,$
 $E = 1.0 1.0+1.0i 1.0-1.0i 1.0+1.0i 0.0 2.0i+0.0 1.0+1.0i 1.0-1.0i 0.0 0.0i+0.0 2.0i+0.0 1.0+1.0i 0.0 0.0i+0.0 0.0i+0.0 1.0i+0.0 ,$
 $C = -13.0+9.0i 2.0+8.0i -2.0+8.0i -2.0+5.0i -9.0-1.0i 0.0+5.0i -7.0-3.0i -6.0-0.0i -1.0+1.0i 4.0+2.0i 4.0-5.0i 9.0-5.0i -6.0+6.0i 9.0+1.0i -2.0+4.0i 22.0-8.0i$
and
 $F = -6.0+05.0i 4.0-4.0i -3.0+11.0i 3.0-07.0i -5.0+11.0i 12.0-4.0i -2.0+02.0i 0.0+14.0i -5.0-01.0i 0.0+4.0i -2.0+10.0i 3.0-01.0i -6.0-02.0i 1.0+1.0i -7.0-03.0i 4.0+07.0i .$

### 10.1Program Text

Program Text (f08yvfe.f90)

### 10.2Program Data

Program Data (f08yvfe.d)

### 10.3Program Results

Program Results (f08yvfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017