nag_dtrsyl (f08qhc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dtrsyl (f08qhc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dtrsyl (f08qhc) solves the real quasi-triangular Sylvester matrix equation.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dtrsyl (Nag_OrderType order, Nag_TransType trana, Nag_TransType tranb, Nag_SignType sign, Integer m, Integer n, const double a[], Integer pda, const double b[], Integer pdb, double c[], Integer pdc, double *scale, NagError *fail)

3  Description

nag_dtrsyl (f08qhc) 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 nag_dhseqr (f08pec)); α is a scale factor (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 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:     orderNag_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 order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     tranaNag_TransTypeInput
On entry: specifies the option opA.
trana=Nag_Trans or Nag_ConjTrans
Constraint: trana=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3:     tranbNag_TransTypeInput
On entry: specifies the option opB.
tranb=Nag_Trans or Nag_ConjTrans
Constraint: tranb=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
4:     signNag_SignTypeInput
On entry: indicates the form of the Sylvester equation.
The equation is of the form opAX+XopB=αC.
The equation is of the form opAX-XopB=αC.
Constraint: sign=Nag_Plus or Nag_Minus.
5:     mIntegerInput
On entry: m, the order of the matrix A, and the number of rows in the matrices X and C.
Constraint: m0.
6:     nIntegerInput
On entry: n, the order of the matrix B, and the number of columns in the matrices X and C.
Constraint: n0.
7:     a[dim]const doubleInput
Note: the dimension, dim, of the array a must be at least max1,pda×m.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the m by m upper quasi-triangular matrix A in canonical Schur form, as returned by nag_dhseqr (f08pec).
8:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,m.
9:     b[dim]const doubleInput
Note: the dimension, dim, of the array b must be at least max1,pdb×n.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the n by n upper quasi-triangular matrix B in canonical Schur form, as returned by nag_dhseqr (f08pec).
10:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbmax1,n.
11:   c[dim]doubleInput/Output
Note: the dimension, dim, of the array c must be at least
  • max1,pdc×n when order=Nag_ColMajor;
  • max1,m×pdc when order=Nag_RowMajor.
The i,jth element of the matrix C is stored in
  • c[j-1×pdc+i-1] when order=Nag_ColMajor;
  • c[i-1×pdc+j-1] when order=Nag_RowMajor.
On entry: the m by n right-hand side matrix C.
On exit: c is overwritten by the solution matrix X.
12:   pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
  • if order=Nag_ColMajor, pdcmax1,m;
  • if order=Nag_RowMajor, pdcmax1,n.
13:   scaledouble *Output
On exit: the value of the scale factor α.
14:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdc=value.
Constraint: pdc>0.
On entry, pda=value and m=value.
Constraint: pdamax1,m.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdc=value and m=value.
Constraint: pdcmax1,m.
On entry, pdc=value and n=value.
Constraint: pdcmax1,n.
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.
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

nag_dtrsyl (f08qhc) is not threaded by NAG in any implementation.
nag_dtrsyl (f08qhc) 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 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 :
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; nag_dtrsyl (f08qhc) 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 function is nag_ztrsyl (f08qvc).

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
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 (f08qhce.c)

10.2  Program Data

Program Data (f08qhce.d)

10.3  Program Results

Program Results (f08qhce.r)

nag_dtrsyl (f08qhc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014