Description
AMATH 740, CS 770, CM 750
1. (a) Derive explicit formulas for natural cubic spline interpolation, i.e. give the linear system
that needs to be solved to find spline coefficients. You can assume that points xi are
equidistributed.
(b) Implement the above algorithm. You can use the Matlab build-in linear solver or write
your own. Submit the printout of your code.
2. Construct the intepolating polynomial passing through the points (-1,-5), (0,1), (1,1),(2,1)
using the monomial, Lagrange and Newton bases.
3. (a) Consider linear interpolation of f(x) = x
3 at x0 = 0 and x1 = 1. Find the value of ξ
for the error function Enf(x) derived in class.
(b) Repeat for f(x) = (2x − 1)4
.
4. Interpolate f(x) = sin(πx), f(x) = 1/(1 + 25x
2 AMATH 740, CS 770, CM 750
), f(x) = |x| on [−1, 1] with degree 10
polynomials using the equidistant and Chebyshev points, and with cubic splines with 11
nodes. Compare results and make a meaningful conclusion. Show your work.
5. (a) Show that
Z 1
0
1
1 + x
2
= π
(b) Approximate the integral using the 6 point Gauss-Legendre quadrature and a composite
trapezoidal rule with the same number of function evaluations. Compare results and
make a meaningful conclusion. Show your work. Nodes and weights can be found in
Abramowitz and Stegun: Handbook of Mathematical Functions (available online) or
other electronic resorces.
6. Let Q(n) be the composite trapezoidal rule approximation to R b
a
f(x)dx with [a, b] divided
into n subintervals. Show that for f(x) of sufficient smoothness
Q(n) − Q(2n)
Q(2n) − Q(4n)
→ 4, n → ∞
7. Derive the composite Simpson’s Rule and find the expression for its error.
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