> restart;

HermiteHorner := proc(N,X,F,DF)

local i,j,x,HX,HD,value;

printf("Specify [f(x1)...f(xn)] and [f'(x1)...f'(xn)] (2*n conditions),\n");

printf(" find Hermite interpolating polynomial,\n");

printf(" and evaluate it at a given point.\n");

HX := array(1..2*N);

HD := array(1..2*N);

for i from 1 to N do

HX[2*i-1] := X[i];

HX[2*i] := X[i];

HD[2*i] := DF[i];

od;

HD[1]:= F[1];

# 1st Divided Differences

for i from N to 2 by -1 do

HD[2*i-1]:=(F[i]-F[i-1])/(X[i]-X[i-1]);

od;

# 2nd,..., (2*N)-th Divided Differences

for j from 3 to 2*N do

for i from 2*N to j by -1 do

HD[i]:=(HD[i]-HD[i-1])/(HX[i]-HX[i-j+1]);

od;

od;

printf("\nNewton Divided differences:\n ");

for i from 1 to 2*N do printf("%f ",HD[i]); od;

printf("\n");

# Horner's scheme:

value := HD[2*N];

printf("\nEnter x to evaluate interpolating polynomial:\n");

x := scanf("%f")[1];

for i from 2*N-1 to 1 by -1 do

value := HD[i]+(x-HX[i])*value;

od;

printf("p(%f)=%f\n",x,value);

end:

> N:=2: X:=array([1,2]): F:=array([3,4]): DF:=array([5,6]):

HermiteHorner(N,X,F,DF);

Specify [f(x1)...f(xn)] and [f'(x1)...f'(xn)] (2*n conditions),

find Hermite interpolating polynomial,

and evaluate it at a given point.

Newton Divided differences:

3.000000 5.000000 -4.000000 9.000000

Enter x to evaluate interpolating polynomial:

> 3

p(3.000000)=33.000000