f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_ztrsyl (f08qvc)

## 1  Purpose

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

## 2  Specification

 #include #include
 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
 $opAX ± XopB = αC ,$
where $\mathrm{op}\left(A\right)=A$ or ${A}^{\mathrm{H}}$, and the matrices $A$ and $B$ are upper triangular; $\alpha$ is a scale factor ($\text{}\le 1$) determined by the function 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.

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

## 5  Arguments

1:    $\mathbf{order}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{trana}$Nag_TransTypeInput
On entry: specifies the option $\mathrm{op}\left(A\right)$.
${\mathbf{trana}}=\mathrm{Nag_NoTrans}$
$\mathrm{op}\left(A\right)=A$.
${\mathbf{trana}}=\mathrm{Nag_ConjTrans}$
$\mathrm{op}\left(A\right)={A}^{\mathrm{H}}$.
Constraint: ${\mathbf{trana}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_ConjTrans}$.
3:    $\mathbf{tranb}$Nag_TransTypeInput
On entry: specifies the option $\mathrm{op}\left(B\right)$.
${\mathbf{tranb}}=\mathrm{Nag_NoTrans}$
$\mathrm{op}\left(B\right)=B$.
${\mathbf{tranb}}=\mathrm{Nag_ConjTrans}$
$\mathrm{op}\left(B\right)={B}^{\mathrm{H}}$.
Constraint: ${\mathbf{tranb}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_ConjTrans}$.
4:    $\mathbf{sign}$Nag_SignTypeInput
On entry: indicates the form of the Sylvester equation.
${\mathbf{sign}}=\mathrm{Nag_Plus}$
The equation is of the form $\mathrm{op}\left(A\right)X+X\mathrm{op}\left(B\right)=\alpha C$.
${\mathbf{sign}}=\mathrm{Nag_Minus}$
The equation is of the form $\mathrm{op}\left(A\right)X-X\mathrm{op}\left(B\right)=\alpha C$.
Constraint: ${\mathbf{sign}}=\mathrm{Nag_Plus}$ or $\mathrm{Nag_Minus}$.
5:    $\mathbf{m}$IntegerInput
On entry: $m$, the order of the matrix $A$, and the number of rows in the matrices $X$ and $C$.
Constraint: ${\mathbf{m}}\ge 0$.
6:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $B$, and the number of columns in the matrices $X$ and $C$.
Constraint: ${\mathbf{n}}\ge 0$.
7:    $\mathbf{a}\left[\mathit{dim}\right]$const ComplexInput
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{m}}\right)$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $m$ by $m$ upper triangular matrix $A$.
8:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
9:    $\mathbf{b}\left[\mathit{dim}\right]$const ComplexInput
Note: the dimension, dim, of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $n$ upper triangular matrix $B$.
10:  $\mathbf{pdb}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
11:  $\mathbf{c}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array c must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdc}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pdc}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $C$ is stored in
• ${\mathbf{c}}\left[\left(j-1\right)×{\mathbf{pdc}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{c}}\left[\left(i-1\right)×{\mathbf{pdc}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $m$ by $n$ right-hand side matrix $C$.
On exit: c is overwritten by the solution matrix $X$.
12:  $\mathbf{pdc}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
13:  $\mathbf{scal}$double *Output
On exit: the value of the scale factor $\alpha$.
14:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}>0$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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 $AX-XB=C$. (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.

## 8  Parallelism and Performance

nag_ztrsyl (f08qvc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of real floating-point operations is approximately $4mn\left(m+n\right)$.
To solve the general complex Sylvester equation
 $AX ± XB = C$
where $A$ and $B$ are general matrices, $A$ and $B$ must first be reduced to Schur form :
 $A = Q1 A~ Q1H and B = Q2 B~ Q2H$
where $\stackrel{~}{A}$ and $\stackrel{~}{B}$ are upper triangular and ${Q}_{1}$ and ${Q}_{2}$ are unitary. The original equation may then be transformed to:
 $A~ X~ ± X~ B~ = C~$
where $\stackrel{~}{X}={Q}_{1}^{\mathrm{H}}X{Q}_{2}$ and $\stackrel{~}{C}={Q}_{1}^{\mathrm{H}}C{Q}_{2}$. $\stackrel{~}{C}$ 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 $X={Q}_{1}\stackrel{~}{X}{Q}_{2}^{\mathrm{H}}$.
The real analogue of this function is nag_dtrsyl (f08qhc).

## 10  Example

This example solves the Sylvester equation $AX+XB=C$, where
 $A = -6.00-7.00i 0.36-0.36i -0.19+0.48i 0.88-0.25i 0.00+0.00i -5.00+2.00i -0.03-0.72i -0.23+0.13i 0.00+0.00i 0.00+0.00i 8.00-1.00i 0.94+0.53i 0.00+0.00i 0.00+0.00i 0.00+0.00i 3.00-4.00i ,$
 $B = 0.50-0.20i -0.29-0.16i -0.37+0.84i -0.55+0.73i 0.00+0.00i -0.40+0.90i 0.06+0.22i -0.43+0.17i 0.00+0.00i 0.00+0.00i -0.90-0.10i -0.89-0.42i 0.00+0.00i 0.00+0.00i 0.00+0.00i 0.30-0.70i$
and
 $C = 0.63+0.35i 0.45-0.56i 0.08-0.14i -0.17-0.23i -0.17+0.09i -0.07-0.31i 0.27-0.54i 0.35+1.21i -0.93-0.44i -0.33-0.35i 0.41-0.03i 0.57+0.84i 0.54+0.25i -0.62-0.05i -0.52-0.13i 0.11-0.08i .$

### 10.1  Program Text

Program Text (f08qvce.c)

### 10.2  Program Data

Program Data (f08qvce.d)

### 10.3  Program Results

Program Results (f08qvce.r)