F08WHF (DGGBAL) : NAG Library, Mark 24

NAG Library Routine Document

F08WHF (DGGBAL)

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

3 Description

Balancing may reduce the

1

-norms of the matrices and improve the accuracy of the computed eigenvalues and eigenvectors in the real generalized eigenvalue problem

A x = λ B x .

F08WHF (DGGBAL) is usually the first step in the solution of the above generalized eigenvalue problem. Balancing is optional but it is highly recommended.

The term ‘balancing’ covers two steps, each of which involves similarity transformations on

A

and

B

. The routine can perform either or both of these steps. Both steps are optional.

The routine first attempts to permute $A$ and $B$ to block upper triangular form by a similarity transformation:
$P A P^{T} = F = (\begin{matrix} F_{11} & F_{12} & F_{13} \\ F_{22} & F_{23} \\ F_{33} \end{matrix})$
$P B P^{T} = G = (\begin{matrix} G_{11} & G_{12} & G_{13} \\ G_{22} & G_{23} \\ G_{33} \end{matrix})$
where $P$ is a permutation matrix, $F_{11}$ , $F_{33}$ , $G_{11}$ and $G_{33}$ are upper triangular. Then the diagonal elements of the matrix pairs $(F_{11}, G_{11})$ and $(F_{33}, G_{33})$ are generalized eigenvalues of $(A, B)$ . The rest of the generalized eigenvalues are given by the matrix pair $(F_{22}, G_{22})$ which are in rows and columns $i_{lo}$ to $i_{hi}$ . Subsequent operations to compute the generalized eigenvalues of $(A, B)$ need only be applied to the matrix pair $(F_{22}, G_{22})$ ; this can save a significant amount of work if $i_{lo} > 1$ and $i_{hi} < n$ . If no suitable permutation exists (as is often the case), the routine sets $i_{lo} = 1$ and $i_{hi} = n$ .
The routine applies a diagonal similarity transformation to $(F, G)$ , to make the rows and columns of $(F_{22}, G_{22})$ as close in norm as possible:
$D F \hat{D} = (\begin{matrix} I & 0 & 0 \\ 0 & D_{22} & 0 \\ 0 & 0 & I \end{matrix}) (\begin{matrix} F_{11} & F_{12} & F_{13} \\ F_{22} & F_{23} \\ F_{33} \end{matrix}) (\begin{matrix} I & 0 & 0 \\ 0 & {\hat{D}}_{22} & 0 \\ 0 & 0 & I \end{matrix})$
$D G \hat{D} = (\begin{matrix} I & 0 & 0 \\ 0 & D_{22} & 0 \\ 0 & 0 & I \end{matrix}) (\begin{matrix} G_{11} & G_{12} & G_{13} \\ G_{22} & G_{23} \\ G_{33} \end{matrix}) (\begin{matrix} I & 0 & 0 \\ 0 & {\hat{D}}_{22} & 0 \\ 0 & 0 & I \end{matrix})$
This transformation usually improves the accuracy of computed generalized eigenvalues and eigenvectors.

5 Parameters

1: JOB – CHARACTER(1)Input

On entry: specifies the operations to be performed on matrices

A

and

B

$JOB ='N'$: No balancing is done. Initialize $ILO = 1$ , $IHI = N$ , $LSCALE (i) = 1.0$ and $RSCALE (i) = 1.0$ , for $i = 1, 2, \dots, n$ .
$JOB ='P'$: Only permutations are used in balancing.
$JOB ='S'$: Only scalings are are used in balancing.
$JOB ='B'$: Both permutations and scalings are used in balancing.

Constraint:

JOB ='N'

'P'

'S'

'B'

2: N – INTEGERInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

N \geq 0

3: A(LDA,

*

) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array A must be at least

\max (1, N)

On entry: the

n

n

matrix

A

On exit: A is overwritten by the balanced matrix. If

JOB ='N'

, A is not referenced.

4: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08WHF (DGGBAL) is called.

Constraint:

LDA \geq \max (1, N)

5: B(LDB,

*

) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array B must be at least

\max (1, N)

On entry: the

n

n

matrix

B

On exit: B is overwritten by the balanced matrix. If

JOB ='N'

, B is not referenced.

6: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08WHF (DGGBAL) is called.

Constraint:

LDB \geq \max (1, N)

7: ILO – INTEGEROutput

8: IHI – INTEGEROutput

On exit:

i_{lo}

and

i_{hi}

are set such that

A (i, j) = 0

and

B (i, j) = 0

i > j

and

1 \leq j < i_{lo}

i_{hi} < i \leq n

JOB ='N'

'S'

i_{lo} = 1

and

i_{hi} = n

9: LSCALE(N) – REAL (KIND=nag_wp) arrayOutput

On exit: details of the permutations and scaling factors applied to the left side of the matrices

A

and

B

. If

P_{i}

is the index of the row interchanged with row

i

and

d_{i}

is the scaling factor applied to row

i

, then

$LSCALE (i) = P_{i}$ , for $i = 1, 2, \dots, i_{lo} - 1$ ;
$LSCALE (i) = d_{i}$ , for $i = i_{lo}, \dots, i_{hi}$ ;
$LSCALE (i) = P_{i}$ , for $i = i_{hi} + 1, \dots, n$ .

The order in which the interchanges are made is

n

i_{hi} + 1

, then

1

i_{lo} - 1

10: RSCALE(N) – REAL (KIND=nag_wp) arrayOutput

On exit: details of the permutations and scaling factors applied to the right side of the matrices

A

and

B

P_{j}

is the index of the column interchanged with column

j

and

{\hat{d}}_{j}

is the scaling factor applied to column

j

, then

$RSCALE (j) = P_{j}$ , for $j = 1, 2, \dots, i_{lo} - 1$ ;
$RSCALE (j) = {\hat{d}}_{j}$ , for $j = i_{lo}, \dots, i_{hi}$ ;
$RSCALE (j) = P_{j}$ , for $j = i_{hi} + 1, \dots, n$ .

The order in which the interchanges are made is

n

i_{hi} + 1

, then

1

i_{lo} - 1

11: WORK(

*

) – REAL (KIND=nag_wp) arrayWorkspace

Note: the dimension of the array WORK must be at least

\max (1, 6 \times N)

JOB ='S'

'B'

and at least

1

JOB ='N'

'P'

12: INFO – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

8 Further Comments

F08WHF (DGGBAL) is usually the first step in computing the real generalized eigenvalue problem but it is an optional step. The matrix

B

is reduced to the upper triangular form using the

Q R

factorization routine F08AEF (DGEQRF) and this orthogonal transformation

Q

is applied to the matrix

A

by calling F08AGF (DORMQR). This is followed by F08WEF (DGGHRD) which reduces the matrix pair into the generalized Hessenberg form.

If the matrix pair

(A, B)

is balanced by this routine, then any generalized eigenvectors computed subsequently are eigenvectors of the balanced matrix pair. In that case, to compute the generalized eigenvectors of the original matrix, F08WJF (DGGBAK) must be called.

The total number of floating point operations is approximately proportional to

n^{2}

The complex analogue of this routine is F08WVF (ZGGBAL).

NAG Library Routine Document

F08WHF (DGGBAL)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

NAG Library Routine DocumentF08WHF (DGGBAL)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

NAG Library Routine Document

F08WHF (DGGBAL)