NAG CL Interface
f08qhc (dtrsyl)

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1 Purpose

f08qhc solves the real quasi-triangular Sylvester matrix equation.

2 Specification

#include <nag.h>
void  f08qhc (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)
The function may be called by the names: f08qhc, nag_lapackeig_dtrsyl or nag_dtrsyl.

3 Description

f08qhc solves the real Sylvester matrix equation
op(A)X ± Xop(B) = αC ,  
where op(A) = A or AT, and the matrices A and B are upper quasi-triangular matrices in canonical Schur form (as returned by f08pec); α is a scale factor (1) determined by the function to avoid overflow in X; A is m×m and B is n×n while the right-hand side matrix C and the solution matrix X are both m×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: order Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: trana Nag_TransType Input
On entry: specifies the option op(A).
trana=Nag_NoTrans
op(A)=A.
trana=Nag_Trans or Nag_ConjTrans
op(A)=AT.
Constraint: trana=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3: tranb Nag_TransType Input
On entry: specifies the option op(B).
tranb=Nag_NoTrans
op(B)=B.
tranb=Nag_Trans or Nag_ConjTrans
op(B)=BT.
Constraint: tranb=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
4: sign Nag_SignType Input
On entry: indicates the form of the Sylvester equation.
sign=Nag_Plus
The equation is of the form op(A)X+Xop(B)=αC.
sign=Nag_Minus
The equation is of the form op(A)X-Xop(B)=αC.
Constraint: sign=Nag_Plus or Nag_Minus.
5: m Integer Input
On entry: m, the order of the matrix A, and the number of rows in the matrices X and C.
Constraint: m0.
6: n Integer Input
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 double Input
Note: the dimension, dim, of the array a must be at least max(1,pda×m).
The (i,j)th 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×m upper quasi-triangular matrix A in canonical Schur form, as returned by f08pec.
8: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax(1,m).
9: b[dim] const double Input
Note: the dimension, dim, of the array b must be at least max(1,pdb×n).
The (i,j)th 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×n upper quasi-triangular matrix B in canonical Schur form, as returned by f08pec.
10: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbmax(1,n).
11: c[dim] double Input/Output
Note: the dimension, dim, of the array c must be at least
  • max(1,pdc×n) when order=Nag_ColMajor;
  • max(1,m×pdc) when order=Nag_RowMajor.
The (i,j)th 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×n right-hand side matrix C.
On exit: c is overwritten by the solution matrix X.
12: pdc Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraints:
  • if order=Nag_ColMajor, pdcmax(1,m);
  • if order=Nag_RowMajor, pdcmax(1,n).
13: scale double * Output
On exit: the value of the scale factor α.
14: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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: pdamax(1,m).
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdc=value and m=value.
Constraint: pdcmax(1,m).
On entry, pdc=value and n=value.
Constraint: pdcmax(1,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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface 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 X~ be the computed solution and R the residual matrix:
R = C - (AX~-X~B) .  
Then the residual is always small:
RF = O(ε) (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~-XF RF sep (A,B)  
but this may be a considerable over estimate. See Golub and Van Loan (1996) for a definition of sep(A,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

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 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.

9 Further Comments

The total number of floating-point operations is approximately mn(m+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; 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 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 )  
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 (f08qhce.c)

10.2 Program Data

Program Data (f08qhce.d)

10.3 Program Results

Program Results (f08qhce.r)