f08wv:: Least Squares and Eigenvalue Problems (LAPACK) (NAG Toolbox)

nag_lapack_zggbal (f08wv) balances a pair of complex square matrices

(A, B)

of order

n

. Balancing usually improves the accuracy of computed generalized eigenvalues and eigenvectors.

Balancing may reduce the

1

-norm of the matrices and improve the accuracy of the computed eigenvalues and eigenvectors in the complex generalized eigenvalue problem

A x = λ B x .

nag_lapack_zggbal (f08wv) 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 function can perform either or both of these steps. Both steps are optional.

The function 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}

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 function sets

i_{lo} = 1

and

i_{hi} = n

The function 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 D^{- 1} = (\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.

Ward R C (1981) Balancing the generalized eigenvalue problem SIAM J. Sci. Stat. Comp. 2 141–152

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'

The first dimension of the array a must be at least

\max (1, n)

The second dimension of the array a must be at least

\max (1, n)

The

n

n

matrix

A

The first dimension of the array b must be at least

\max (1, n)

The second dimension of the array b must be at least

\max (1, n)

The

n

n

matrix

B

Default: the first dimension of the arrays a, b and the second dimension of the arrays a, b.

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

The first dimension of the array a will be

\max (1, n)

The second dimension of the array a will be

\max (1, n)

a stores the balanced matrix. If

job ='N'

, a is not referenced.

The first dimension of the array b will be

\max (1, n)

The second dimension of the array b will be

\max (1, n)

b stores the balanced matrix. If

job ='N'

, b is not referenced.

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

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

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

info = 0

unless the function detects an error (see Error Indicators and Warnings).

info = - i

, parameter

i

had an illegal value on entry. The parameters are numbered as follows:

1: job, 2: n, 3: a, 4: lda, 5: b, 6: ldb, 7: ilo, 8: ihi, 9: lscale, 10: rscale, 11: work, 12: info.

It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

The errors are negligible, compared to those in subsequent computations.

nag_lapack_zggbal (f08wv) is usually the first step in computing the complex generalized eigenvalue problem but it is an optional step. The matrix

B

is reduced to the triangular form using the

Q R

factorization function nag_lapack_zgeqrf (f08as) and the unitary transformation

Q

is applied to the matrix

A

by calling nag_lapack_zunmqr (f08au). This is followed by nag_lapack_zgghrd (f08ws) which reduces the matrix pair into the generalized Hessenberg form.

If the matrix pair

(A, B)

is balanced by this function, 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, nag_lapack_zggbak (f08ww) must be called.

The total number of floating-point operations is approximately proportional to

n^{2}

The real analogue of this function is nag_lapack_dggbal (f08wh).

NAG Toolbox: nag_lapack_zggbal (f08wv)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example