f08nhf (dgebal) : NAG Library, Mark 28

f08nhf balances a real general matrix in order to improve the accuracy of computed eigenvalues and/or eigenvectors.

The routine may be called by the names f08nhf, nagf_lapackeig_dgebal or its LAPACK name dgebal.

f08nhf balances a real general matrix

A

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

A

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

1.The routine 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 routine sets $i_{lo} = 1$ and $i_{hi} = n$ , and $A_{22}^{'}$ is the whole of $A$ .

2.The routine 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: indicates whether

A

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

$job ='N'$: $A$ is neither permuted nor scaled (but values are assigned to ilo, ihi and scale).
$job ='P'$: $A$ is permuted but not scaled.
$job ='S'$: $A$ is scaled but not permuted.
$job ='B'$: $A$ is both permuted and scaled.

Constraint:

job ='N'

'P'

'S'

'B'

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

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

\max (1, n)

On entry: the

n \times n

matrix

A

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

job ='N'

, a is not referenced.

On entry: the first dimension of the array a as declared in the (sub)program from which f08nhf is called.

Constraint:

lda \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 ='N'

'S'

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) = {\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

On exit:

info = 0

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

info = - i

, argument

i

had an illegal value. An explanatory message is output, and execution of the program is terminated.

The errors are negligible.

Background information to multithreading can be found in the Multithreading documentation.

f08nhf 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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

If the matrix

A

is balanced by f08nhf, then any eigenvectors computed subsequently are eigenvectors of the matrix

A^{''}

(see Section 3) and hence f08njf must then be called to transform them back to eigenvectors of

A

If the Schur vectors of

A

are required, then this routine must not be called with

job ='S'

'B'

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

job ='P'

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

A^{''}

, and f08njf must be called (with

side ='R'

) 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 routine is f08nvf.

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 f08nhf 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 f08qkf to compute the right eigenvectors of the balanced matrix, and finally calls f08njf to transform the eigenvectors back to eigenvectors of the original matrix

A

NAG FL Interface
f08nhf (dgebal)

▸▿ 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 FL Interfacef08nhf (dgebal)

▸▿ 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 FL Interface
f08nhf (dgebal)