Maximum error in not-a-knot spline of bessel function

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Rick 2014-7-25

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/143374-maximum-error-in-not-a-knot-spline-of-bessel-function

The syntax is as follows: If Xdata and Ydata are vectors with the same number of elements, then four various splines can be created as

SN = splineE7(Xdata,Ydata,'N');        % Natural
SK = splineE7(Xdata,Ydata,'K');        % not-a-knot
SP = splineE7(Xdata,Ydata,'P');        % Periodic
SE = splineE7(Xdata,Ydata,'E',v1,vN);  % end-slope

If xG is an array, then the various splines can be evaluated at the values in xG using splinevalueE7, for example

yG = splinevalueE7(S,xG)

where S is one of the created splines.

Define a function using Bessel functions of the 2nd kind, which is the solution to an important differential equation arising in acoustics and other engineering disciplines. Define a function f via an anonymous function,

f = @(x) besselj(x,2);

Consider a not-a-knot-spline fit to the function f(x), using 10 points, with {xi}10i=1 linearly spaced from 0 to 2. Associated with this spline, compute the maximum absolute value of the error, evaluated on a denser grid, with 500 points linearly spaced from 0 to 2. Which of the numbers below is approximately equal to that maximum absolute error?

Now consider a natural spline fit to the function f(x), using 10 points, with {xi}10i=1 linearly spaced from 0 to 2. Associated with this spline, compute the maximum absolute value of the error, evaluated on a denser grid, with 500 points linearly spaced from 0 to 2. Which of the numbers below is approximately equal to that maximum absolute error?

Here is my attempt

f = @(x) besselj(x,2);
x = linspace(0,2,10);
y = f(x);
S = splineE7(x,f(x),'K')
S = 
      Coeff: [4x9 double]
          x: [0 0.2222 0.4444 0.6667 0.8889 1.1111 1.3333 1.5556 1.7778 2]
          y: [1x10 double]