F08QHF (DTRSYL) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08QHF (DTRSYL)

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.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08QHF (DTRSYL) solves the real quasi-triangular Sylvester matrix equation.

2  Specification

SUBROUTINE F08QHF ( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO)
INTEGER  ISGN, M, N, LDA, LDB, LDC, INFO
REAL (KIND=nag_wp)  A(LDA,*), B(LDB,*), C(LDC,*), SCALE
CHARACTER(1)  TRANA, TRANB
The routine may be called by its LAPACK name dtrsyl.

3  Description

F08QHF (DTRSYL) solves the real Sylvester matrix equation
opAX ± XopB = αC ,  
where opA = A  or AT, and the matrices A and B are upper quasi-triangular matrices in canonical Schur form (as returned by F08PEF (DHSEQR)); α is a scale factor (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 αi±βj0, where αi and βj 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:     TRANA – CHARACTER(1)Input
On entry: specifies the option opA.
TRANA='N'
opA=A.
TRANA='T' or 'C'
opA=AT.
Constraint: TRANA='N', 'T' or 'C'.
2:     TRANB – CHARACTER(1)Input
On entry: specifies the option opB.
TRANB='N'
opB=B.
TRANB='T' or 'C'
opB=BT.
Constraint: TRANB='N', 'T' or 'C'.
3:     ISGN – INTEGERInput
On entry: indicates the form of the Sylvester equation.
ISGN=+1
The equation is of the form opAX+XopB=αC.
ISGN=-1
The equation is of the form opAX-XopB=αC.
Constraint: ISGN=+1 or -1.
4:     M – INTEGERInput
On entry: m, the order of the matrix A, and the number of rows in the matrices X and C.
Constraint: M0.
5:     N – INTEGERInput
On entry: n, the order of the matrix B, and the number of columns in the matrices X and C.
Constraint: N0.
6:     ALDA* – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array A must be at least max1,M.
On entry: the m by m upper quasi-triangular matrix A in canonical Schur form, as returned by F08PEF (DHSEQR).
7:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08QHF (DTRSYL) is called.
Constraint: LDAmax1,M.
8:     BLDB* – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array B must be at least max1,N.
On entry: the n by n upper quasi-triangular matrix B in canonical Schur form, as returned by F08PEF (DHSEQR).
9:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08QHF (DTRSYL) is called.
Constraint: LDBmax1,N.
10:   CLDC* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array C must be at least max1,N.
On entry: the m by n right-hand side matrix C.
On exit: C is overwritten by the solution matrix X.
11:   LDC – INTEGERInput
On entry: the first dimension of the array C as declared in the (sub)program from which F08QHF (DTRSYL) is called.
Constraint: LDCmax1,M.
12:   SCALE – REAL (KIND=nag_wp)Output
On exit: the value of the scale factor α.
13:   INFO – INTEGEROutput
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=1
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 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, 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 sepA,B, 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

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.

9  Further Comments

The total number of floating-point operations is approximately mnm+n.
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 A~ and B~ are upper quasi-triangular and Q1 and Q2 are orthogonal. The original equation may then be transformed to:
A~ X~ ± X~ B~ = C~  
where X~ = Q1T X Q2  and C~ = Q1T C Q2 . 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 = Q1 X~ Q2T .
The complex analogue of this routine is F08QVF (ZTRSYL).

10  Example

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 .  

10.1  Program Text

Program Text (f08qhfe.f90)

10.2  Program Data

Program Data (f08qhfe.d)

10.3  Program Results

Program Results (f08qhfe.r)


F08QHF (DTRSYL) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016