NAG Library Function Document

nag_dtgsyl (f08yhc)

nag_dtgsyl (Nag_OrderType order, Nag_TransType trans, Integer ijob, Integer m, Integer n, const double a[], Integer pda, const double b[], Integer pdb, double c[], Integer pdc, const double d[], Integer pdd, const double e[], Integer pde, double f[], Integer pdf, double *scale, double *dif, NagError *fail)

3 Description

nag_dtgsyl (f08yhc) solves either the generalized real Sylvester equations

\begin{matrix} A R - L B & = α C \\ D R - L E & = α F, \end{matrix}

(1)

or the equations

\begin{matrix} A^{T} R + D^{T} L & = α C \\ R B^{T} + L E^{T} & = - α F, \end{matrix}

(2)

where the pair

(A, D)

are given

m

m

matrices in real generalized Schur form,

(B, E)

are given

n

n

matrices in real generalized Schur form and

(C, F)

are given

m

n

matrices. The pair

(R, L)

are the

m

n

solution matrices, and

α

is an output scaling factor determined by the function to avoid overflow in computing

(R, L)

Equations (1) are equivalent to equations of the form

Z x = α b,

where

Z = (\begin{matrix} I \otimes A - B^{T} \otimes I \\ I \otimes D - E^{T} \otimes I \end{matrix})

and

\otimes

is the Kronecker product. Equations (2) are then equivalent to

Z^{T} y = α b .

The pair

(S, T)

are in real generalized Schur form if

S

is block upper triangular with

1

1

and

2

2

diagonal blocks on the diagonal and

T

is upper triangular as returned, for example, by nag_dgges (f08xac), or nag_dhgeqz (f08xec) with

job = Nag_Schur

Optionally, the function estimates

Dif [(A, D), (B, E)]

, the separation between the matrix pairs

(A, D)

and

(B, E)

, which is the smallest singular value of

Z

. The estimate can be based on either the Frobenius norm, or the

1

-norm. The

1

-norm estimate can be three to ten times more expensive than the Frobenius norm estimate, but makes the condition estimation uniform with the nonsymmetric eigenproblem. The Frobenius norm estimate provides a low cost, but equally reliable estimate. For more information see Sections 2.4.8.3 and 4.11.1.3 of Anderson et al. (1999) and Kågström and Poromaa (1996).

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

Kågström B (1994) A perturbation analysis of the generalized Sylvester equation

(A R - L B, D R - L E) = (c, F)

SIAM J. Matrix Anal. Appl. 15 1045–1060

Kågström B and Poromaa P (1996) LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs ACM Trans. Math. Software 22 78–103

5 Arguments

1: order – Nag_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

Nag_ColMajor

2: trans – Nag_TransTypeInput

On entry: if

trans = Nag_NoTrans

, solve the generalized Sylvester equation (1).

trans = Nag_Trans

, solve the ‘transposed’ system (2).

Constraint:

trans = Nag_NoTrans

Nag_Trans

3: ijob – IntegerInput

On entry: specifies what kind of functionality is to be performed when

trans = Nag_NoTrans

$ijob = 0$: Solve (1) only.
$ijob = 1$: The functionality of $ijob = 0$ and $3$ .
$ijob = 2$: The functionality of $ijob = 0$ and $4$ .
$ijob = 3$: Only an estimate of $Dif [(A, D), (B, E)]$ is computed based on the Frobenius norm.
$ijob = 4$: Only an estimate of $Dif [(A, D), (B, E)]$ is computed based on the $1$ -norm.

trans = Nag_Trans

, ijob is not referenced.

Constraint: if

trans = Nag_NoTrans

0 \leq ijob \leq 4

4: m – IntegerInput

On entry:

m

, the order of the matrices

A

and

D

, and the row dimension of the matrices

C

F

R

and

L

Constraint:

m \geq 0

5: n – IntegerInput

On entry:

n

, the order of the matrices

B

and

E

, and the column dimension of the matrices

C

F

R

and

L

Constraint:

n \geq 0

6: a[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array a must be at least

\max (1, pda \times m)

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the upper quasi-triangular matrix

A

7: pda – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq \max (1, m)

8: b[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array b must be at least

\max (1, pdb \times n)

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the upper quasi-triangular matrix

B

9: pdb – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraint:

pdb \geq \max (1, n)

10: c[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array c must be at least

$\max (1, pdc \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pdc)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

C

is stored in

$c [(j - 1) \times pdc + i - 1]$ when $order = Nag_ColMajor$ ;
$c [(i - 1) \times pdc + j - 1]$ when $order = Nag_RowMajor$ .

On entry: contains the right-hand-side matrix

C

On exit: if

ijob = 0

1

2

, c is overwritten by the solution matrix

R

trans = Nag_NoTrans

and

ijob = 3

4

, c holds

R

, the solution achieved during the computation of the Dif estimate.

11: pdc – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array c.

Constraints:

if $order = Nag_ColMajor$ , $pdc \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pdc \geq \max (1, n)$ .

12: d[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array d must be at least

\max (1, pdd \times m)

The

(i, j)

th element of the matrix

D

is stored in

$d [(j - 1) \times pdd + i - 1]$ when $order = Nag_ColMajor$ ;
$d [(i - 1) \times pdd + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the upper triangular matrix

D

13: pdd – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array d.

Constraint:

pdd \geq \max (1, m)

14: e[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array e must be at least

\max (1, pde \times n)

The

(i, j)

th element of the matrix

E

is stored in

$e [(j - 1) \times pde + i - 1]$ when $order = Nag_ColMajor$ ;
$e [(i - 1) \times pde + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the upper triangular matrix

E

15: pde – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array e.

Constraint:

pde \geq \max (1, n)

16: f[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array f must be at least

$\max (1, pdf \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pdf)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

F

is stored in

$f [(j - 1) \times pdf + i - 1]$ when $order = Nag_ColMajor$ ;
$f [(i - 1) \times pdf + j - 1]$ when $order = Nag_RowMajor$ .

On entry: contains the right-hand side matrix

F

On exit: if

ijob = 0

1

2

, f is overwritten by the solution matrix

L

trans = Nag_NoTrans

and

ijob = 3

4

, f holds

L

, the solution achieved during the computation of the Dif estimate.

17: pdf – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array f.

Constraints:

if $order = Nag_ColMajor$ , $pdf \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pdf \geq \max (1, n)$ .

18: scale – double *Output

On exit:

α

, the scaling factor in (1) or (2).

0 < scale < 1

, c and f hold the solutions

R

and

L

, respectively, to a slightly perturbed system but the input arrays a, b, d and e have not been changed.

scale = 0

, c and f hold the solutions

R

and

L

, respectively, to the homogeneous system with

C = F = 0

. In this case dif is not referenced.

Normally,

scale = 1

19: dif – double *Output

On exit: the estimate of

Dif

. If

ijob = 0

, dif is not referenced.

20: fail – NagError *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_EIGENVALUES: $(A, D)$ and $(B, E)$ have common or close eigenvalues and so no solution could be computed.
NE_ENUM_INT: On entry, $ijob = ⟨value⟩$ and $trans = ⟨value⟩$ .
Constraint: if $trans = Nag_NoTrans$ , $0 \leq ijob \leq 4$ .
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pda = ⟨value⟩$ .
Constraint: $pda > 0$ .
On entry, $pdb = ⟨value⟩$ .
Constraint: $pdb > 0$ .
On entry, $pdc = ⟨value⟩$ .
Constraint: $pdc > 0$ .
On entry, $pdd = ⟨value⟩$ .
Constraint: $pdd > 0$ .
On entry, $pde = ⟨value⟩$ .
Constraint: $pde > 0$ .
On entry, $pdf = ⟨value⟩$ .
Constraint: $pdf > 0$ .
NE_INT_2: On entry, $pda = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $pda \geq \max (1, m)$ .
On entry, $pdb = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdb \geq \max (1, n)$ .
On entry, $pdc = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $pdc \geq \max (1, m)$ .
On entry, $pdc = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdc \geq \max (1, n)$ .
On entry, $pdd = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $pdd \geq \max (1, m)$ .
On entry, $pde = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pde \geq \max (1, n)$ .
On entry, $pdf = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $pdf \geq \max (1, m)$ .
On entry, $pdf = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdf \geq \max (1, n)$ .
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.

7 Accuracy

See Kågström (1994) for a perturbation analysis of the generalized Sylvester equation.

8 Parallelism and Performance

nag_dtgsyl (f08yhc) is not threaded by NAG in any implementation.

nag_dtgsyl (f08yhc) 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 needed to solve the generalized Sylvester equations is approximately

2 m n (n + m)

. The Frobenius norm estimate of

Dif

does not require additional significant computation, but the

1

-norm estimate is typically five times more expensive.

The complex analogue of this function is nag_ztgsyl (f08yvc).

10 Example

This example solves the generalized Sylvester equations

\begin{matrix} A R - L B & = α C \\ D R - L E & = α F, \end{matrix}

where

A = (\begin{array}{r} 4.0 & 1.0 & 1.0 & 2.0 \\ 0 & 3.0 & 4.0 & 1.0 \\ 0 & 1.0 & 3.0 & 1.0 \\ 0 & 0 & 0 & 6.0 \end{array}), B = (\begin{array}{r} 1.0 & 1.0 & 1.0 & 1.0 \\ 0 & 3.0 & 4.0 & 1.0 \\ 0 & 1.0 & 3.0 & 1.0 \\ 0 & 0 & 0 & 4.0 \end{array}),

D = (\begin{array}{r} 2.0 & 1.0 & 1.0 & 3.0 \\ 0 & 1.0 & 2.0 & 1.0 \\ 0 & 0 & 1.0 & 1.0 \\ 0 & 0 & 0 & 2.0 \end{array}), E = (\begin{array}{r} 1.0 & 1.0 & 1.0 & 2.0 \\ 0 & 1.0 & 4.0 & 1.0 \\ 0 & 0 & 1.0 & 1.0 \\ 0 & 0 & 0 & 1.0 \end{array}),

C = (\begin{array}{r} - 4.0 & 7.0 & 1.0 & 12.0 \\ - 9.0 & 2.0 & - 2.0 & - 2.0 \\ - 4.0 & 2.0 & - 2.0 & 8.0 \\ - 7.0 & 7.0 & - 6.0 & 19.0 \end{array}) and F = (\begin{array}{r} - 7.0 & 5.0 & 0.0 & 7.0 \\ - 5.0 & 1.0 & - 8.0 & 0.0 \\ - 1.0 & 2.0 & - 3.0 & 5.0 \\ - 3.0 & 2.0 & 0.0 & 5.0 \end{array}) .

NAG Library Function Documentnag_dtgsyl (f08yhc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_dtgsyl (f08yhc)