1.2. NUMERICA L ROO T FINDIN G 3
1/2 2/ 3 1/2 1/2 1/3 1/2
+ 6 I ( 3 (108 + 12 6 9 ) + 8 6 9 + 8 (108 + 12 6 9 ) )
/ 1/2 1/3
+ 7 2 ) / (108 + 12 6 9 )
/
The numbe r 4 8 is the order o f the Galoi s group an d it s name i s "6T11". O f course,
the use r no w ha s to consul t help(galois ) i n order t o lear n more .
1.2. Numerica l Roo t Findin g
In symboli c computation , w e frequentl y conside r a polynomia l proble m a s
solved i f i t ha s bee n reduce d t o finding th e root s o f on e polynomia l i n on e vari -
able. Naturally , th e latte r proble m ca n stil l b e ver y interestin g an d challengin g
from th e perspectiv e o f numerica l analysis , especiall y i f d get s ver y larg e o r i f th e
cii are give n b y floating poin t approximations . I n th e problem s studie d i n thi s
book, however , the a * are usually exact rationa l number s with reasonably smal l nu -
merators an d denominators , an d th e degre e d rarely exceed s 100. Fo r numericall y
solving univariat e polynomial s i n thi s range , i t ha s bee n th e author' s experienc e
that mapl e doe s reasonabl y wel l an d ma t lab ha s n o difficulty whatsoever .
Digits := 6:
f := x~200 - x~157 + 8 * x~101 - 23 * x~61 + 1:
fsolve(f,x);
.950624, 1.01796
This polynomial hasonlytwo realroots.To lis t thecomplex roots, wesay:
fsolve(f,x,complex);
-1.02820-.0686972 I, -1.02820+.0686972 I, -1.01767-.0190398 I,
-1.01767+.0190398 I, -1.01745-.118366 I, -1.01745 + .118366 I,
-1.00698-.204423 I, -1.00698+.204423 I, -1.00028 - .160348 I,
-1.00028+.160348 I, -.996734-.252681 I, -.996734 + .252681 I,
-.970912-.299748 I, -.970912+.299748 I, -.964269 - .336097 I,
ETC...ETC..
Our polynomial p(x) i s represented in mat lab a s the row vector of its coefficient s
[ad cid-i ^2^1 flo]- For instance, the following two commands compute the thre e
roots o f the dens e cubi c p(x)
31x3
+ 23x
2
+ 19x + 11.
» p = [3 1 2 3 19 11];
» roots(p )
ans =
-0.0486 + 0.7402 i
-0.0486 - 0.7402 1
-0.6448
Representing the sparse polynomial p(x) = x
200

x157 +8x101
23x
61+
l considere d
above requires introducin g lot s o f zero coefficients :
» p=[ l zeros(l,42 ) - 1 zeros(l,55 ) 8 zeros(l,39 ) -2 3 zeros(l,60 ) 1]
roots(p )
ans =
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