Write

a MATLAB function to calculate the coefficients of Hermite interpolating polynomial,

H

(using divided difference)

a MATLAB function to evaluate the Hermite interpolating polynomial.

a MATLAB script to compare the error of using Hermite polynomial which fits the data at =0,0.5,1

x

=

0

,

0.5

,

1

to approximate the function

e

x

at =0.3,0.7,0.9

x

=

0.3

,

0.7

,

0.9

and plot a graph of the exact function

e

x

and the Hermite polynomial

H

.

First function:

Input: x, y, dy where x is a vector consisting of

x

i

,

y

is a vector consisting of ()

f

(

x

i

)

,

d

y

is a vector consisting of ()

f

(

x

i

)

,

Output: c where c is a vector consisting of the coefficients of the Hermite Polynomial, that is,

()=1+2(1)+2(1)2++2+2=1().

H

(

x

)

=

c

1

+

c

2

(

x

x

1

)

+

c

2

(

x

x

1

)

2

+

+

c

2

n

+

2

i

=

1

k

(

x

z

i

)

.

where 21=2=,

z

2

i

1

=

z

2

i

=

x

i

,

for =1,,

i

=

1

,

,

n

.

function [c] = Divided_difference_Hermite_NetID(x,y,dy)

%% Step 1: F = zeros(2n,2n);

F(1:2:end,1) = y; F(2:2:end,1) = y; F(2:2:end,2) = dy;

z = zeros(2*n,1); z(1:2:end) = x; z(2:2:end) = x;

%% Step 2: for j = 1, ..., n-1

F(2*j+1,2) = (F(2*j+1,1)-F(2*j,1))/(z(2*j+1)-z(2*j));

%% Step 3: for i = 3, ..., 2n

for j = 1, ..., i-1

(Using the Divided_differences to calculate the coefficient)

%% Step 4: set c = diag(F);

end

Second function:

Input: c, x, X, where c is a vector consisting of the coefficients of the Hermite Polynomial and x is a vector consisting of

x

i

.

Output: HX which is the function value of

H

at point X

function [HX] = Hermite_polynomial_NetID(c,x,X)

%% Step 1: PX = c(1); S = 1; z = zeros(2*n,1); z(1:2:end) = x; z(2:2:end) = x;

%% Step 2: for i = 2,3,..., 2*n

S = S*(X-z(i-1))

PX = PX + c(i)*S;

end

Main script:

%% Step 1: define the function

f

and

f

f = @(x) exp(x);

df = @(x) exp(x);

%% Step 2: define the vector x

%%% type your code here

%% Step 3: define the test points X

%%% type your code here

%% Step 4: calculate the vector y and dy

y = f(x);

dy = df(x);

%% Step 5: calculate the coefficients of the Hermite Polynomial

%%% type your code here

%% Step 6: evaluate the polynomial and the function at X

%%% type your code here

%% Step 7: compute the error

%%% type your code here

%% Step 8: plot the graph

N = 100;

HX = zeros(N+1,1);

X_plot = (0:1/N:1)';

for i = 1:N+1

HX(i)= Hermite_polynomial_NetID(c,x,X_plot(i));

end

plot(X_plot,HX,'.-')

hold on

plot(X_plot,f(X_plot),'r-.-')

Assuming the main script is saved as "main_NetID.m". We can use the following Matlab comment to check your code:

Command Output

x = [-1,0,1]';

y = [-1,0,1]';

dy = [3,0,3]';

c = Divided_difference_Hermite_NetID(x,y,dy)

X = 1/2;

HX = Hermite_polynomial_NetID(c,x,X)

c =

-1

3

-2

1

0

0

HX =

0.1250

main_NetID

error =

1.0e-05 *

0.3962

0.4195

0.3174