Let s(x) be the spline fitted to 3 equally spaced nodes x0 < x1 < x2, with spacing

xi+1 xi = h, i = 0, 1, 2 and the output values are y0, y1, y2.

(a) (5 points.) Write down the expression of the spline function s(x), if we require s(x) to

pass through (xi, yi), i = 0, 1, 2 exactly. (Hint: use second derivative to parameterize

s(x) as we did in class, and note that s(x0) = s(x2) = 0.)

(b) (10 points.) Now we take into account noise and do NOT require the spline to pass

through (xi, yi), i = 0, 1, 2 exactly. Instead, we fit s(x) by solving the following

min

s

X2

i=0

(yi fi)2 +

Z x2

x0

[s(x)]2dx

where fi = s(xi). This optimization problem can be solved explicitly. Write down the

expression of the spline function (or a linear equation that determines the parameters),

which minimizes the above cost function for = 1.