Page 128 - Compact Numerical Methods For Computers

The algebraic eigenvalue problem 117
Example 9.2. Eigensolutions of a complex matrix
The following output from a Data General NOVA operating in 23-bit binary
arithmetic shows the computation of the eigensolutions of a complex matrix due
to Eberlein from the test set published by Gregory and Karney (1969). The
notation ( , ) is used to indicate a complex number, real part followed by
imaginary part. Note that the residuals computed are quite large by comparison
with those for a real symmetric matrix. This is due to the increased difficulty of
the problem, to the extra operations needed to take account of the complex
numbers and to the standardisation of the eigenvectors, which will introduce some
additional errors (for instance, in the first eigenvector, 5·96046E–8 for zero).
This comment must be tempered by the observation that the norm of the matrix is
quite large, so that the residuals divided by this norm are still only a reasonably
small multiple of the machine precision.

RUN
ENHCMG - COMEIG AT SEPT 3 74
ORDER? 3
ELEMENT( 1 , 1 );REAL=? 1 IMAGINARY? 2
ELEMENT( 1 , 2 );REAL=? 3 IMAGINARY? 4
ELEMENT( 1 , 3 );REAL=? 21 IMAGINARY? 22
ELEMENT( 2 , 1 );REAL=? 43 IMAGINARY? 44
ELEMENT( 2 , 2 );REAL=? 13 IMAGINARY? 14
ELEMENT( 2 , 3 );REAL=? 15 IMAGINARY? 16
ELEMENT( 3 , 1 );REAL=? 5 IMAGINARY? 6
ELEMENT( 3 , 2 );REAL=? 7 IMAGINARY? 8
ELEMENT( 3 , 3 );REAL=? 25 IMAGINARY? 26
TAU= 194 AT ITN 1
TAU= 99,7552 AT ITN 2
TAU= 64,3109 AT ITN 3
TAU= 25,0133 AT ITN 4
TAU= 7,45953 AT ITN 5
TAU= .507665 AT ITN 6
TAU= 6.23797E-4 AT ITN 7
TAU= 1.05392E-7 AT ITN 8
EIGENSOLUTIONS
RAW VECTOR 1
( .371175 ,-.114606 )
( .873341 ,-.29618 )
( .541304 ,-.178142 )
EIGENVALUE 1 =( 39,7761 , 42,9951 )
VECTOR
( .42108 , 1.15757E-2 )
( 1 , 5.96046E-8 )
( .617916 , 5.57855E-3 )
RESIDUALS
( 2.2918E-4 , 2.34604E-4 )
( 5.16415E-4 , 5.11169E-4 )
( 3.70204E-4 , 3.77655E-4 )
RAW VECTOR 2
(-9.52917E-2 ,-.491205 )
( 1.19177 , .98026 )
(-.342159 ,-9.71221E-2 )