nag_dgebal (f08nhc) : NAG Library, Mark 23

nag_dgebal (f08nhc) balances a real general matrix in order to improve the accuracy of computed eigenvalues and/or eigenvectors.

nag_dgebal (f08nhc) balances a real general matrix

A

. The term ‘balancing’ covers two steps, each of which involves a similarity transformation of

A

. The function can perform either or both of these steps.

The function first attempts to permute $A$ to block upper triangular form by a similarity transformation:
$P A P^{T} = A^{'} = (\begin{matrix} A_{11}^{'} & A_{12}^{'} & A_{13}^{'} \\ 0 & A_{22}^{'} & A_{23}^{'} \\ 0 & 0 & A_{33}^{'} \end{matrix})$
where $P$ is a permutation matrix, and $A_{11}^{'}$ and $A_{33}^{'}$ are upper triangular. Then the diagonal elements of $A_{11}^{'}$ and $A_{33}^{'}$ are eigenvalues of $A$ . The rest of the eigenvalues of $A$ are the eigenvalues of the central diagonal block $A_{22}^{'}$ , in rows and columns $i_{lo}$ to $i_{hi}$ . Subsequent operations to compute the eigenvalues of $A$ (or its Schur factorization) need only be applied to these rows and columns; 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 $i_{hi} = n$ , and $A_{22}^{'}$ is the whole of $A$ .

The function applies a diagonal similarity transformation to

A^{'}

, to make the rows and columns of

A_{22}^{'}

as close in norm as possible:

A^{''} = D A^{'} D^{- 1} = (\begin{matrix} I & 0 & 0 \\ 0 & D_{22} & 0 \\ 0 & 0 & I \end{matrix}) (\begin{matrix} A_{11}^{'} & A_{12}^{'} & A_{13}^{'} \\ 0 & A_{22}^{'} & A_{23}^{'} \\ 0 & 0 & A_{33}^{'} \end{matrix}) (\begin{matrix} I & 0 & 0 \\ 0 & D_{22}^{- 1} & 0 \\ 0 & 0 & I \end{matrix}) .

This scaling can reduce the norm of the matrix (i.e.,

‖A_{22}^{''}‖ < ‖A_{22}^{'}‖

) and hence reduce the effect of rounding errors on the accuracy of computed eigenvalues and eigenvectors.

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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

or Nag_ColMajor.

On entry: indicates whether

A

is to be permuted and/or scaled (or neither).

$job = Nag_DoNothing$: $A$ is neither permuted nor scaled (but values are assigned to ilo, ihi and scale).
$job = Nag_Permute$: $A$ is permuted but not scaled.
$job = Nag_Scale$: $A$ is scaled but not permuted.
$job = Nag_DoBoth$: $A$ is both permuted and scaled.

Constraint:

job = Nag_DoNothing

Nag_Permute

Nag_Scale

Nag_DoBoth

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

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

\max (1, pda \times n)

Where

A (i, j)

appears in this document, it refers to the array element

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

On entry: the

n

n

matrix

A

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

job = Nag_DoNothing

, a is not referenced.

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

Constraint:

pda \geq \max (1, n)

On exit: the values

i_{lo}

and

i_{hi}

such that on exit

A (i, j)

is zero if

i > j

and

1 \leq j < i_{lo}

i_{hi} < i \leq n

job = Nag_DoNothing

Nag_Scale

i_{lo} = 1

and

i_{hi} = n

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

A

. More precisely, if

p_{j}

is the index of the row and column interchanged with row and column

j

and

d_{j}

is the scaling factor used to balance row and column

j

then

scale [j - 1] = \{\begin{cases} p_{j}, & j = 1, 2, \dots, i_{lo} - 1 \\ d_{j}, & j = i_{lo}, i_{lo} + 1, \dots, i_{hi} and \\ p_{j}, & j = i_{hi} + 1, i_{hi} + 2, \dots, n . \end{cases}

The order in which the interchanges are made is

n

i_{hi} + 1

then

1

i_{lo} - 1

The NAG error argument (see Section 3.6 in the Essential Introduction).

Dynamic memory allocation failed.

On entry, argument

〈value〉

had an illegal value.

On entry,

n = 〈value〉

.
Constraint:

n \geq 0

On entry,

pda = 〈value〉

.
Constraint:

pda > 0

On entry,

pda = 〈value〉

and

n = 〈value〉

.
Constraint:

pda \geq \max (1, n)

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.

The errors are negligible.

If the matrix

A

is balanced by nag_dgebal (f08nhc), then any eigenvectors computed subsequently are eigenvectors of the matrix

A^{''}

(see Section 3) and hence nag_dgebak (f08njc) must then be called to transform them back to eigenvectors of

A

If the Schur vectors of

A

are required, then this function must not be called with

job = Nag_Scale

Nag_DoBoth

, because then the balancing transformation is not orthogonal. If this function is called with

job = Nag_Permute

, then any Schur vectors computed subsequently are Schur vectors of the matrix

A^{''}

, and nag_dgebak (f08njc) must be called (with

side = Nag_RightSide

) to transform them back to Schur vectors of

A

The total number of floating point operations is approximately proportional to

n^{2}

The complex analogue of this function is nag_zgebal (f08nvc).

This example computes all the eigenvalues and right eigenvectors of the matrix

A

, where

A = (\begin{array}{r} 5.14 & 0.91 & 0.00 & - 32.80 \\ 0.91 & 0.20 & 0.00 & 34.50 \\ 1.90 & 0.80 & - 0.40 & - 3.00 \\ - 0.33 & 0.35 & 0.00 & 0.66 \end{array}) .

The program first calls nag_dgebal (f08nhc) to balance the matrix; it then computes the Schur factorization of the balanced matrix, by reduction to Hessenberg form and the

Q R

algorithm. Then it calls nag_dtrevc (f08qkc) to compute the right eigenvectors of the balanced matrix, and finally calls nag_dgebak (f08njc) to transform the eigenvectors back to eigenvectors of the original matrix

A

NAG Library Function Document

nag_dgebal (f08nhc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Documentnag_dgebal (f08nhc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_dgebal (f08nhc)