Question

5.2. This problem (a) Find the piecewise linear interpolation function g(r) that fits these data. (b) Find the global interpolation polynomial p3(x) that fits these data concerns the data in Table 5.6. -1 0 1 2 0 1 0 1 y

Answer 1

a) The piecewise linear interpolant is:

x if - 1 < 0 if 0 1 2 xif 1x2 T: y f(x) 1

Diagram of the data points (in red) and linear interpolants (in blue) is given below:

1:5 (0, 1) (1,1) 0:5 (-1,0) (2,0) -1.5 -1 -0.5 0 0.5 1 1.5 2.5 -0:5 LC LC

b) For cubic interpolant we solve for the data-set to get the set of equations:

yarbr cd

0 ab-c+ d, 1 d, 1 = a +b+c+ d, 0 8a +4b+2c + d

Solving this, we have

,c= d = 1 2 a 0, b P 2

And so our cubic interpolant is graphed below in green over the original data points (in red)

1 2 y f(x) 2 +

(1,1) (0, 1) 0.5 (-1-0) (2,0) -1.5 -0.5 O.5 1 1.5 2' 2.5 -0.5 LO

$\blacksquare$

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