# NAG Library Routine Document

## 1Purpose

f08qhf (dtrsyl) solves the real quasi-triangular Sylvester matrix equation.

## 2Specification

Fortran Interface
 Subroutine f08qhf ( isgn, m, n, a, lda, b, ldb, c, ldc, info)
 Integer, Intent (In) :: isgn, m, n, lda, ldb, ldc Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: a(lda,*), b(ldb,*) Real (Kind=nag_wp), Intent (Inout) :: c(ldc,*) Real (Kind=nag_wp), Intent (Out) :: scale Character (1), Intent (In) :: trana, tranb
#include <nagmk26.h>
 void f08qhf_ (const char *trana, const char *tranb, const Integer *isgn, const Integer *m, const Integer *n, const double a[], const Integer *lda, const double b[], const Integer *ldb, double c[], const Integer *ldc, double *scal, Integer *info, const Charlen length_trana, const Charlen length_tranb)
The routine may be called by its LAPACK name dtrsyl.

## 3Description

f08qhf (dtrsyl) solves the real Sylvester matrix equation
 $opAX ± XopB = αC ,$
where $\mathrm{op}\left(A\right)=A$ or ${A}^{\mathrm{T}}$, and the matrices $A$ and $B$ are upper quasi-triangular matrices in canonical Schur form (as returned by f08pef (dhseqr)); $\alpha$ is a scale factor ($\text{}\le 1$) determined by the routine to avoid overflow in $X$; $A$ is $m$ by $m$ and $B$ is $n$ by $n$ while the right-hand side matrix $C$ and the solution matrix $X$ are both $m$ by $n$. 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 ${\alpha }_{i}±{\beta }_{j}\ne 0$, where $\left\{{\alpha }_{i}\right\}$ and $\left\{{\beta }_{j}\right\}$ are the eigenvalues of $A$ and $B$ respectively and the sign ($+$ or $-$) is the same as that used in the equation to be solved.

## 4References

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 $AX-XB=C$ Numerical Analysis Report University of Manchester

## 5Arguments

1:     $\mathbf{trana}$ – Character(1)Input
On entry: specifies the option $\mathrm{op}\left(A\right)$.
${\mathbf{trana}}=\text{'N'}$
$\mathrm{op}\left(A\right)=A$.
${\mathbf{trana}}=\text{'T'}$ or $\text{'C'}$
$\mathrm{op}\left(A\right)={A}^{\mathrm{T}}$.
Constraint: ${\mathbf{trana}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
2:     $\mathbf{tranb}$ – Character(1)Input
On entry: specifies the option $\mathrm{op}\left(B\right)$.
${\mathbf{tranb}}=\text{'N'}$
$\mathrm{op}\left(B\right)=B$.
${\mathbf{tranb}}=\text{'T'}$ or $\text{'C'}$
$\mathrm{op}\left(B\right)={B}^{\mathrm{T}}$.
Constraint: ${\mathbf{tranb}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
3:     $\mathbf{isgn}$ – IntegerInput
On entry: indicates the form of the Sylvester equation.
${\mathbf{isgn}}=+1$
The equation is of the form $\mathrm{op}\left(A\right)X+X\mathrm{op}\left(B\right)=\alpha C$.
${\mathbf{isgn}}=-1$
The equation is of the form $\mathrm{op}\left(A\right)X-X\mathrm{op}\left(B\right)=\alpha C$.
Constraint: ${\mathbf{isgn}}=+1$ or $-1$.
4:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the order of the matrix $A$, and the number of rows in the matrices $X$ and $C$.
Constraint: ${\mathbf{m}}\ge 0$.
5:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $B$, and the number of columns in the matrices $X$ and $C$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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 $m$ by $m$ upper quasi-triangular matrix $A$ in canonical Schur form, as returned by f08pef (dhseqr).
7:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08qhf (dtrsyl) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
8:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – 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 $n$ by $n$ upper quasi-triangular matrix $B$ in canonical Schur form, as returned by f08pef (dhseqr).
9:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08qhf (dtrsyl) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10:   $\mathbf{c}\left({\mathbf{ldc}},*\right)$ – 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: the $m$ by $n$ right-hand side matrix $C$.
On exit: c is overwritten by the solution matrix $X$.
11:   $\mathbf{ldc}$ – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which f08qhf (dtrsyl) is called.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
12:   $\mathbf{scale}$ – Real (Kind=nag_wp)Output
On exit: the value of the scale factor $\alpha$.
13:   $\mathbf{info}$ – 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.
${\mathbf{info}}=1$
$A$ and $B$ have common or close eigenvalues, perturbed values of which were used to solve the equation.

## 7Accuracy

Consider the equation $AX-XB=C$. (To apply the remarks to the equation $AX+XB=C$, simply replace $B$ by $-B$.)
Let $\stackrel{~}{X}$ be the computed solution and $R$ the residual matrix:
 $R = C - AX~ - X~B .$
Then the residual is always small:
 $RF = Oε AF + BF X~F .$
However, $\stackrel{~}{X}$ 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:
 $X~ - X F ≤ RF sep A,B$
but this may be a considerable over estimate. See Golub and Van Loan (1996) for a definition of $\mathit{sep}\left(A,B\right)$, and Higham (1992) for further details.
These remarks also apply to the solution of a general Sylvester equation, as described in Section 9.

## 8Parallelism and Performance

f08qhf (dtrsyl) 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 is approximately $mn\left(m+n\right)$.
To solve the general real Sylvester equation
 $AX ± XB = C$
where $A$ and $B$ are general nonsymmetric matrices, $A$ and $B$ must first be reduced to Schur form (by calling f08paf (dgees), for example):
 $A = Q1 A~ Q1T and B = Q2 B~ Q2T$
where $\stackrel{~}{A}$ and $\stackrel{~}{B}$ are upper quasi-triangular and ${Q}_{1}$ and ${Q}_{2}$ are orthogonal. The original equation may then be transformed to:
 $A~ X~ ± X~ B~ = C~$
where $\stackrel{~}{X}={Q}_{1}^{\mathrm{T}}X{Q}_{2}$ and $\stackrel{~}{C}={Q}_{1}^{\mathrm{T}}C{Q}_{2}$. $\stackrel{~}{C}$ may be computed by matrix multiplication; f08qhf (dtrsyl) may be used to solve the transformed equation; and the solution to the original equation can be obtained as $X={Q}_{1}\stackrel{~}{X}{Q}_{2}^{\mathrm{T}}$.
The complex analogue of this routine is f08qvf (ztrsyl).

## 10Example

This example solves the Sylvester equation $AX+XB=C$, where
 $A = 0.10 0.50 0.68 -0.21 -0.50 0.10 -0.24 0.67 0.00 0.00 0.19 -0.35 0.00 0.00 0.00 -0.72 ,$
 $B = -0.99 -0.17 0.39 0.58 0.00 0.48 -0.84 -0.15 0.00 0.00 0.75 0.25 0.00 0.00 -0.25 0.75$
and
 $C = 0.63 -0.56 0.08 -0.23 -0.45 -0.31 0.27 1.21 0.20 -0.35 0.41 0.84 0.49 -0.05 -0.52 -0.08 .$

### 10.1Program Text

Program Text (f08qhfe.f90)

### 10.2Program Data

Program Data (f08qhfe.d)

### 10.3Program Results

Program Results (f08qhfe.r)