Question
Transcribed Text
1. Consider f(x) = xe−6x and the interpolating points {xi}
5
i=0 = {0, 0.2, 0.4, 0.6, 0.8, 1}.
You may use the built-in MATLAB function spline or myspline on the course webpage in the following calculations.
(a) Complete the MATLAB code myspline.m on the course webpage by adding the
endpoint conditions for clamped splines. Attach your code to the report.
(b) Compute the natural cubic spline S(x) that interpolates f(x) at the given points.
Find S0(x) and S4(x), and verify that they satisfy the natural boundary conditions. You need to use myslpine for this problem since the built-in spline does
not have the option of natural spline.
(c) Compute the clamped cubic spline s(x) that interpolates f(x) at the given points.
Find s0(x) and s4(x), and verify the boundary conditions at x0 and x5.
(d) Compute the not-a-knot cubic spline Sˆ(x) that interpolates f(x) at the given
points. Find Sˆ
0(x) and Sˆ
1(x), and verify that Sˆ
(k)
0
(x1) = Sˆ
(k)
1
(x1) for k =
0, 1, 2, 3.
(e) Compare f(x) and these three cubic splines on the same plot. For each method,
compute the maximum error at points x = 0 : 0.001 : 1. (You may use MATLAB
function ppval to evaluate a piecewise polynomial at different points.) Discuss
the quality of the approximations. Which one is the most accurate and which is
the least accurate? Explain the reason in your own words.
2. Consider
f(x) = xe−6x
, 0 ≤ x ≤ 1.
(a) Write a MATLAB script file using spline (with the option of clampled spline)
to interpolate this function at equidistant points xi = ih, i = 0, 1, · · · n, where
h = 1/n. Generate data to complete the following table. For each grid spacing,
obtain the actual (numerical) error enum(h) and compute the numerical order of
accuracy α. Is α consistent with your expectation? Find the error bound for
each grid spacing. Does the error bound provide a good estimation of the actual
error? Attach your script file to the report.
k nk hk |enum(hk)| αk |ebound(hk)|
1 4 1/4 —
2 8 1/8
3 16 1/16
4 32 1/32
5 64 1/64
Here, |enum(hk)| is defined as the maximum error evaluated at x = 0 : 0.001 : 1,
for the piecewise interpolating polynomial phk
(x) with grid spacing hk:
|enum(hk)| = max
x=0:0.001:1
|f(x) − phk
(x)|.
The order of accuarcy α is a number such that the error e(h) = O(h
α). The
numerical order can be obtained from
αk =
ln(|enum(hk)|/|enum(hk−1)|)
ln(hk/hk−1)
.
1
(b) Repeat (a) using piecewise linear interpolation. You may use interp1 with
method=‘linear’.
2
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function Question2()%clamped spline has endtype 2
%f(x)=x e^(-6x) f'(x)=(1-6x)e^(-6x)
%f''(x)=(-12+36x)e^(-6x)
f=@(x)(x.* exp(-6*x)); df=@(x)((1-6*x).*exp(-6*x));
endtype=2;
h=zeros(1,5);
enum=zeros(1,5);%vector of errors
alpha=zeros(1,5);%vector of orders of precision
X=0:0.001:1;
for k=1:5
n=2^k;
h(k)=1/n;
x=0:h(k):1;
Y2=[df(0),f(x),df(1)];
[pp2]=myspline(x,Y2,endtype);
[breaks2, coefs2]=unmkpp(pp2);
enum(k)=max(abs(ppval(pp2,X)-f(X)));
end
for k=2:5
alpha(k)=log(enum(k)/enum(k-1))/log(h(k)/h(k-1));
end
EnumClampedSpline=enum
alphaClampedSpline=alpha...
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