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NAG[f08qhc] NAG[nag_dtrsyl] - Solve real Sylvester matrix equation , and are upper quasi-triangular or transposes

Calling Sequence

f08qhc(trana, tranb, sign_type, a, b, c, scale, 'm'=m, 'n'=n, 'fail'=fail)

nag_dtrsyl(. . .)

Parameters

trana - String;

On entry: specifies the option .

"Nag_Trans" or "Nag_ConjTrans" .

Constraint: "Nag_NoTrans", "Nag_Trans" or "Nag_ConjTrans". .

tranb - String;

On entry: specifies the option .

"Nag_Trans" or "Nag_ConjTrans" .

Constraint: "Nag_NoTrans", "Nag_Trans" or "Nag_ConjTrans". .

sign_type - String;

On entry: indicates the form of the Sylvester equation.

The equation is of the form .

Constraint: "Nag_Plus" or "Nag_Minus". .

a - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: the by upper quasi-triangular matrix in canonical Schur form, as returned by f08pec (nag_dhseqr).

b - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: the by upper quasi-triangular matrix in canonical Schur form, as returned by f08pec (nag_dhseqr).

c - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: the by right-hand side matrix .

On exit: is overwritten by the solution matrix .

scale - assignable;

Note: On exit the variable scale will have a value of type float.

On exit: the value of the scale factor .

'm'=m - integer; (optional)

Default value: the first dimension of the array a and the second dimension of the array athe arrays a, c.

On entry: , the order of the matrix , and the number of rows in the matrices and .

Constraint: . .

'n'=n - integer; (optional)

Default value: the dimension of the array b.

On entry: , the order of the matrix , and the number of columns in the matrices and .

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

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

Description

nag_dtrsyl (f08qhc) solves the real Sylvester matrix equation

where or , and the matrices and are upper quasi-triangular matrices in canonical Schur form (as returned by f08pec (nag_dhseqr)); is a scale factor () determined by the function to avoid overflow in ; is by and is by while the right-hand side matrix and the solution matrix are both by . The matrix 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 , where and are the eigenvalues of and respectively and the sign ( or ) is the same as that used in the equation to be solved.

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument had an illegal value.

"NE_INT"

On entry, . Constraint: .

"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 consult NAG for assistance.

"NE_PERTURBED"

and have common or close eigenvalues, perturbed values of which were used to solve the equation.

Accuracy

Consider the equation . (To apply the remarks to the equation , simply replace by .)

Let be the computed solution and the residual matrix:

Then the residual is always small:

However, 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:

but this may be a considerable overestimate. See Golub and Van Loan (1996) for a definition of , and Higham (1992) for further details.

These remarks also apply to the solution of a general Sylvester equation, as described in Section [Further Comments].

Further Comments

The total number of floating-point operations is approximately .

To solve the general real Sylvester equation

where and are general non-symmetric matrices, and must first be reduced to Schur form :

and

where and are upper quasi-triangular and and are orthogonal. The original equation may then be transformed to:

where and . 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 .

The complex analogue of this function is f08qvc (nag_ztrsyl).

Examples

trana := "Nag_NoTrans":
tranb := "Nag_NoTrans":
sign_type := "Nag_Plus":
m := 4:
n := 4:
a := Matrix([[0.1, 0.5, 0.68, -0.21], [-0.5, 0.1, -0.24, 0.67], [0, 0, 0.19, -0.35], [0, 0, 0, -0.72]], datatype=float[8]):
b := Matrix([[-0.99, -0.17, 0.39, 0.58], [0, 0.48, -0.84, -0.15], [0, 0, 0.75, 0.25], [0, 0, -0.25, 0.75]], datatype=float[8]):
c := Matrix([[0.63, -0.5600000000000001, 0.08, -0.23], [-0.45, -0.31, 0.27, 1.21], [0.2, -0.35, 0.41, 0.84], [0.49, -0.05, -0.52, -0.08]], datatype=float[8]):
NAG:-f08qhc(trana, tranb, sign_type, a, b, c, scale, 'm' = m, 'n' = n):

scale;

	See Also
	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 Numerical Analysis Report University of Manchester f08 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.