2 1. INTRODUCTIO N
FIGURE 1.1 . Newton' s Metho d
Suppose w e are given two polynomials o f two complex variables. W e are intereste d
in thei r commo n roots .
r A(Z,W) =
I B{z,w) =
}
Then th e pai r o f polynomial s ca n b e considere d a s a ma p R :
C2H- C2,
R =
(A, B). Th e wel l known Newton' s metho d the n consist s o f guessing a root p\ clos e
to a n actua l roo t an d gettin g hopefull y a n even closer point p2 to the root fro m th e
formula
p2=Pi-(R'(pi))-1R(pi)
See figure 1.1 . Henc e w e shoul d obtai n a roo t o f R
iterations o f the ma p
0 afte r infinitel y man y
F(p)=p-[(R')-1R}(P).
We get a sequence {p n}
p
n +
i = F(p
n
)
For som e initia l value s p\ th e sequenc e {p n} doe s not converg e to an y root . S o we
come back to our initial problem again ; find th e se t o f points p\ fo r whic h {p
n
}
converges t o som e root .
As a n example , w e consider th e equations :
! » - \*
2
f A(z,w) = i
\ B(z,w) -
Obviously (0,0 ) i s a commo n root . W e wil l see ho w Newton' s metho d work s fo r
this case . T o simplify, on e ca n first replac e R' i n the formul a fo r Newton' s metho d
by the constan t matri x
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