NAG Library Routine Document

1Purpose

f08nvf (zgebal) balances a complex general matrix in order to improve the accuracy of computed eigenvalues and/or eigenvectors.

2Specification

Fortran Interface
 Subroutine f08nvf ( job, n, a, lda, ilo, ihi, info)
 Integer, Intent (In) :: n, lda Integer, Intent (Out) :: ilo, ihi, info Real (Kind=nag_wp), Intent (Out) :: scale(n) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Character (1), Intent (In) :: job
#include nagmk26.h
 void f08nvf_ (const char *job, const Integer *n, Complex a[], const Integer *lda, Integer *ilo, Integer *ihi, double scal[], Integer *info, const Charlen length_job)
The routine may be called by its LAPACK name zgebal.

3Description

f08nvf (zgebal) balances a complex 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:
 $PAPT = A′ = A11′ A12′ A13′ 0 A22′ A23′ 0 0 A33′$
where $P$ is a permutation matrix, and ${A}_{11}^{\prime }$ and ${A}_{33}^{\prime }$ are upper triangular. Then the diagonal elements of ${A}_{11}^{\prime }$ and ${A}_{33}^{\prime }$ are eigenvalues of $A$. The rest of the eigenvalues of $A$ are the eigenvalues of the central diagonal block ${A}_{22}^{\prime }$, in rows and columns ${i}_{\mathrm{lo}}$ to ${i}_{\mathrm{hi}}$. 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 ${i}_{\mathrm{lo}}>1$ and ${i}_{\mathrm{hi}}. If no suitable permutation exists (as is often the case), the routine sets ${i}_{\mathrm{lo}}=1$ and ${i}_{\mathrm{hi}}=n$, and ${A}_{22}^{\prime }$ is the whole of $A$.
2. The routine applies a diagonal similarity transformation to ${A}^{\prime }$, to make the rows and columns of ${A}_{22}^{\prime }$ as close in norm as possible:
 $A′′ = DA′D-1 = I 0 0 0 D22 0 0 0 I A11′ A12′ A13′ 0 A22′ A23′ 0 0 A33′ I 0 0 0 D22-1 0 0 0 I .$
This scaling can reduce the norm of the matrix (i.e., $‖{A}_{22}^{\prime \prime }‖<‖{A}_{22}^{\prime }‖$) and hence reduce the effect of rounding errors on the accuracy of computed eigenvalues and eigenvectors.

4References

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

5Arguments

1:     $\mathbf{job}$ – Character(1)Input
On entry: indicates whether $A$ is to be permuted and/or scaled (or neither).
${\mathbf{job}}=\text{'N'}$
$A$ is neither permuted nor scaled (but values are assigned to ilo, ihi and scale).
${\mathbf{job}}=\text{'P'}$
$A$ is permuted but not scaled.
${\mathbf{job}}=\text{'S'}$
$A$ is scaled but not permuted.
${\mathbf{job}}=\text{'B'}$
$A$ is both permuted and scaled.
Constraint: ${\mathbf{job}}=\text{'N'}$, $\text{'P'}$, $\text{'S'}$ or $\text{'B'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ matrix $A$.
On exit: a is overwritten by the balanced matrix. If ${\mathbf{job}}=\text{'N'}$, a is not referenced.
4:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08nvf (zgebal) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5:     $\mathbf{ilo}$ – IntegerOutput
6:     $\mathbf{ihi}$ – IntegerOutput
On exit: the values ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ such that on exit ${\mathbf{a}}\left(i,j\right)$ is zero if $i>j$ and $1\le j<{i}_{\mathrm{lo}}$ or ${i}_{\mathrm{hi}}.
If ${\mathbf{job}}=\text{'N'}$ or $\text{'S'}$, ${i}_{\mathrm{lo}}=1$ and ${i}_{\mathrm{hi}}=n$.
7:     $\mathbf{scale}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
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
 $scalej = pj, j=1,2,…,ilo-1 dj, j=ilo,ilo+1,…,ihi and pj, j=ihi+1,ihi+2,…,n.$
The order in which the interchanges are made is $n$ to ${i}_{\mathrm{hi}}+1$ then $1$ to ${i}_{\mathrm{lo}}-1$.
8:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7Accuracy

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

8Parallelism and Performance

f08nvf (zgebal) 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 f08nvf (zgebal), then any eigenvectors computed subsequently are eigenvectors of the matrix ${A}^{\prime \prime }$ (see Section 3) and hence f08nwf (zgebak) 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 ${\mathbf{job}}=\text{'S'}$ or $\text{'B'}$, because then the balancing transformation is not unitary. If this routine is called with ${\mathbf{job}}=\text{'P'}$, then any Schur vectors computed subsequently are Schur vectors of the matrix ${A}^{\prime \prime }$, and f08nwf (zgebak) must be called (with ${\mathbf{side}}=\text{'R'}$) to transform them back to Schur vectors of $A$.
The total number of real floating-point operations is approximately proportional to ${n}^{2}$.
The real analogue of this routine is f08nhf (dgebal).

10Example

This example computes all the eigenvalues and right eigenvectors of the matrix $A$, where
 $A = 1.50-2.75i 0.00+0.00i 0.00+0.00i 0.00+0.00i -8.06-1.24i -2.50-0.50i 0.00+0.00i -0.75+0.50i -2.09+7.56i 1.39+3.97i -1.25+0.75i -4.82-5.67i 6.18+9.79i -0.92-0.62i 0.00+0.00i -2.50-0.50i .$
The program first calls f08nvf (zgebal) to balance the matrix; it then computes the Schur factorization of the balanced matrix, by reduction to Hessenberg form and the $QR$ algorithm. Then it calls f08qxf (ztrevc) to compute the right eigenvectors of the balanced matrix, and finally calls f08nwf (zgebak) to transform the eigenvectors back to eigenvectors of the original matrix $A$.

10.1Program Text

Program Text (f08nvfe.f90)

10.2Program Data

Program Data (f08nvfe.d)

10.3Program Results

Program Results (f08nvfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017