16.11 Problem Runge's function is written as
f ( x)
1 1 25 x 2
Generate five equidistant points in the interval [1, 1] evaluate the function there and fit it using various options. First the data: >> x=linspace(1,1,5) x = 1.0000 0.5000 0 0.5000 1.0000 >> runge=@(x) 1./(1+25*x.^2) runge = @(x)1./(1+25*x.^2) >> y=runge(x) y = 0.0385 0.1379 1.0000 0.1379 0.0385 Let us first use the crudest interpolation, which is to use the value of the nearest point >> xx=linspace(1,1); >> yy=runge(xx); >> yexact=runge(xx); >> yy=interp1(x,y,xx,'nearest'); >> plot(x,y,'o',xx,yy,xx,yexact,'')
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Next we will do linear interpolation, which is also linear spline, and the default method of interp1 >> ylin=interp1(x,y,xx); >> plot(x,y,'o',xx,ylin,xx,yexact,'')
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we Next will fit it with a cubic spline. We can do that by choosing `spline' in the method in interp1, or with the spline function (which gives additional flexibility on end conditions) >> yspl=spline(x,y,xx); >> plot(x,y,'o',xx,yspl,xx,yexact,'')
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We can reduce the oscillation by specifying zero slope end conditions >> yspl0=spline(x,[0 y 0],xx); >> plot(x,y,'o',xx,yspl0,xx,yexact,'')
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Next we do Hermite (cubic interpolation) >> yher=interp1(x,y,xx,'cubic'); >> plot(x,y,'o',xx,yher,xx,yexact,'')
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Finally, we use polyfit to do a single quartic polynomial
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yquart=polyval(p,xx); >> plot(x,y,'o',xx,yquart,xx,yexact,'')