Solution for Ex 1. At rst write the following function.

(1)

Solution for Ex 1. At rst write the following function.

1

^function [yt,b]=NewtonInterp(x,y,xt)

2

% Newton form of the interpolating polynomial

3

n=length(x); b=y;

4

^for ^i=2:n

5

^for ^j=2:i

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add=eps*((x(i)

−

x(j

−

1))<1.e

−

5);

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^b(i)=(b(i)

−

b(j

−

1))/(x(i)

−

x(j

−

1)+add);

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^end

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^end

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% part2: Horner scheme for evaluating the polynomial

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^for i=1:length(xt)

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yt(i)=b(n);

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^for ^k=n

−

1:

−

1:1

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yt(i)=yt(i)*(xt(i)

−

x(k))+b(k);

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^end

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^end

Then, on the script Esercizio1:

1

^{clear all}

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^{close all}

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^clc

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f=@(x) x+exp(x)+20./(1+x.^2)

−

5; n=12;

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x=linspace(

−

2,2,n); % interpolation nodes

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y=f(x); xt=linspace(

−

2,2); % function values

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p=NewtonInterp(x,y,xt);

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err1=max(abs((p

−

f(xt))))

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figure(), plot(x,y,'or',xt,p,'

−−

r',xt,f(xt),'k');

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legend('Equispaced points','Newt. interp.',...

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'Original function');

1

(2)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -5

0 5 10 15 20

Equispaced points Newt. interp.

Original function

2

(3)

Solution for Ex 2. On the script named Esercizio2 write 1

^{clear all}

2

^{close all}

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^{a=0; b=1;}

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h =0.05; x=a:h:b; % equispaced points

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y=0.1+x+(10^(

−

1))*rand(size(x)); % the data

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xx=linspace(a,b); % for the evaluation

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P=polyfit(x,y,1); % construct the approximant

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Y = polyval(P,xx); % evaluate the approximant

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plot(x,y,'or',xx,Y,'

−

') % plot

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legend('Data',...

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'Approximating polynomial')

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.2 0.4 0.6 0.8 1 1.2

Data

Approximating polynomial

3

(4)

Solution for Ex 3. Modify the given script as follows.

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^{clear all}

2

^{close all}

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a=0; b=2*pi;

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h=0.01; x=a:h:b; % nodes

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y=cos(2*x)+(10^(

−

1))*rand(size(x)); % perturbed data

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^{kk = 0;}

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^while kk <= 14

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^{n = kk;}

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% computes the coefficients

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coeff=polyfit(x,y,n);

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% evaluate the coefficients

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z=polyval(coeff,x);

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% computes the residual

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err2=norm(z

−

y,2);

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fprintf('\n \t n: %2.0f norma2: %1.2e',n,err2);

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% plot the results

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ht=1/10000; u=a:ht:b;

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v=polyval(coeff,u);

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plot(x,y,'r.'); hold on;

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plot(u,v,'k

−

','LineWidth',2);

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titlestr=strcat('Minimi quadrati di grado:',...

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num2str(n));

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title(titlestr);

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legend('Dati','Approssimante Minimi quadrati');

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^{hold off;}

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^pause(.2);

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^{kk = kk+1;}

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^end

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fprintf('\n \n');

4

 Solution for Ex 1. At rst write the following function.