nag_dtrsyl (f08qhc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C 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 3.2.1.3 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_NoTrans
opA=A.
trana=Nag_Trans or Nag_ConjTrans
opA=AT.
Constraint: trana=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3:     tranbNag_TransTypeInput
On entry: specifies the option opB.
tranb=Nag_NoTrans
opB=B.
tranb=Nag_Trans or Nag_ConjTrans
opB=BT.
Constraint: tranb=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
4:     signNag_SignTypeInput
On entry: indicates the form of the Sylvester equation.
sign=Nag_Plus
The equation is of the form opAX+XopB=αC.
sign=Nag_Minus
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.
Constraints:
  • 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

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
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.
NE_INT_2
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.
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.
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 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 8.

8  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).

9  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 .

9.1  Program Text

Program Text (f08qhce.c)

9.2  Program Data

Program Data (f08qhce.d)

9.3  Program Results

Program Results (f08qhce.r)


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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012