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# MATLAB EXAMPLES Interpolation and Integration - PowerPoint PPT Presentation

MATLAB EXAMPLES Interpolation and Integration. 58:110 Computer-Aided Engineering Spring 2005. Department of Mechanical and industrial engineering January 2005. Some useful functions. Interpolation.

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MATLAB EXAMPLESInterpolation and Integration

58:110 Computer-Aided Engineering Spring 2005

Department of Mechanical and industrial engineering

January 2005

Here is the example to get the polynomial fitting by Lagrange interpolation:

Suppose the original data set is:

• There are five sets of (x,y) above, polyfit can give the 4th order polynomial form by Lagrange interpolation. To compare, we also use interp1 to give the more smooth fitting curve by piecewise cubic Hermite interpolation.

• The M-file (L_interperlation.m) is given to plot the fitting curve in the following:

• % Lagrange interpolating polynomial fitting

• x=[-2 -1 0 1 2];

• y=[-9 -15 -5 -3 39];

• [m n]=size(x)

• p=polyfit(x,y,n-1)

• x1=linspace(-2,2,50);

• y1=polyval(p,x1);

• y2=interp1(x,y,x1,'cubic');

• plot(x,y,'o',x1,y1,'-',x1,y2,'.');

• xlabel('x'),ylabel('y=f(x)')

• title ('Lagrange and Piecewise cubic Hermite interpolation')

To run this example in MATLAB:

>> L_interpolation

p =

3.0000 2.0000 -7.0000 4.0000 -5.0000

That means, the 4th order polynomial is:

• The right figure shows the fitting curve and the original points (circle).

• The solid line is the 4th order polynomial by lagrange interpolation. The dot curve is fitted by piecewise cubic Hermite interpolation.

Consider the following integration:

The accurate integration is:

• Use the numerical integration, here only try trapezoidal method and Simpson method. Assuming the step size is h = 0.1. In MATLAB, first use the trapezoidal method:

• >>x=linspace(0,1,11);

• >> y=x.*exp(x.^2);

• >>s=trapz(x,y);

• >> num2str(s,'%20.9f')

• ans =

• 0.865089832

To try Simpson’s method, we need write some codes (I_simpson.m):

% Simpson's method of Numerical integration

x=linspace(0,1,11);

s=0.0;

f = inline('x.*exp(x.^2)');

for k=1:5

f0=f(x(k*2-1));

f1=f(x(k*2));

f2=f(x(k*2+1));

s=s+(f0+4*f1+f2)*0.1/3.;

end

num2str(s,'%20.9f')

Run this M-file in command window:

>> I_simpson

ans =

0.859193792

Use the built-in function quad, which uses more accurate adaptive Simpson

quadrature. The default error tolerance is 10e-6.

>> f=inline('x.*exp(x.^2)')

>> num2str(s,'%20.9f')

ans =

0.859140934

Compare the results of different methods (all keep 9 digits) in the following table:

To estimate error by step size h:

For Trapezoidal method, it’s O(h2)=O(0.01)

For Simpson’s method, it’s O(h4)=O(0.0001)