Nonlinear Algebraic Equations - Elements of Scientific Computing - page 125

Here

ı n1 ı C ı 2 C ::: C ı mn ı m1 C ı m2 C ::: C ı n D

X

ı n1 ı k

kD1

1
1 ı ;

ı n1

( 4.118 )
D

so

ı n1
1 ı j v 1 v 0 j :

j v m v n j

This means that for any ">0, we can find an integer M such that 5

j v m v n j <";

provided that m, n M , and consequently f v k g is a Cauchy sequence. A Cauchy
sequence is convergent and the limit, as seen above, solves the equation

v D h. v /:

(4.127)

We can now conclude that if h is a contractive mapping satisfying ( 4.109 )and
( 4.110 )onpage 121 ,then( 4.127 ) has a unique solution that can be computed by the
simple iteration

v kC1 D h. v k /:

4.6 Systems of Nonlinear Equations

We have seen above that equations of the form

f.x/ D 0 (4.128)

5 It is sufficient to choose M such that

ı M 1
1

<";

ı

so

ln.".1 ı//

M C 1:

ln.ı/