Roots of Equations - Numerical Methods in Engineering with MATLAB - page 173

Forapolynomialofdegree n , the procedurecan besummarizedas

P 0 ( x ) = a 1

P i ( x ) = a n + i + xP i − 1 ( x ), i = 1 , 2 ,..., n (4.10)

leading to the algorithm

p = a(1);
fori=1:n
p=p*x+a(i+1)

end

The last algorithm involves only n multiplications, making it moreefficientfor

n Butcomputational economy is not the prime reasonwhy this algorithm should

be used. Because the result of each multiplicationis rounded off, the procedure with
the least number of multiplications invariablyaccumulates the smallest roundoff
error.
Some root-finding algorithms, including Laguerre's method, also requireeval-
uation of the first and second derivatives of P n ( x )

> 3 .

. FromEq. (4.10) weobtain by

differentiation

P 0 ( x ) 0 P i ( x ) xP

i

= = P i − 1 ( x ) +

1 ( x ),

i = 1 , 2 ,..., n (4.11a)

−

P
0

0 P
i

2 P
i

xP
i

( x ) = ( x ) =

1 ( x )

+

1 ( x ),

i = 1 , 2 ,..., n (4.11b)

− −

evalPoly

Here is the function thatevaluates apolynomial and its derivatives:

function [p,dp,ddp] = evalpoly(a,x)
% Evaluates the polynomial
%p=a(1)*xˆn+a(2)*xˆ(n-1)+...+a(n+1)
% and its first two derivatives dp and ddp.
% USAGE: [p,dp,ddp] = evalpoly(a,x)

n = length(a) - 1;
p=a(1);dp=0.0;ddp=0.0;
fori=1:n
ddp = ddp*x + 2.0*dp;
dp=dp*x+p;
p=p*x+a(i+1);

end