MATH 448-548
Midterm project assignment
1. Use Taylor formula to derive the error term for the approximate formula
f ′(x) ≈ 1
2h
(−3f(x) + 4f(x + h)− f(x + 2h)) .
2. Consider the following variation of the Newton’s method:
xn+1 = xn − f(xn)
f ′(x0)
.
Find constants C and s such that
en+1 = Ce
s
n.
3. Consider the linear system Ax = b, where
A =
 6.25 −1 0.5−1 5 2.12
0.5 2.12 3.6
 ,
and
b =
 7.5−8.68
−0.24
 .
Write a MATLAB program for LU-factorization with a unit lower triangular L (meaning that
the diagonal entries should be equal to one). Then write a program for the Cholesky factorization.
WARNING: avoid using MATLAB shortcuts. The programming should be done ”from
scratch”.
4. Do Problem 5 from Computer Problems 2.2 (p. 62) in the book. You can use the previous
Problem 4 for guidance.
5. Implement and analyze the Ridders’ method for solving the equation f(x) = 0. For guidance,
you might wish to read the Wikipedia page on the Ridders’ method. Also feel free to consult other
sources.
Brief Description of the method. Given the initial bracketing interval [x0, x2], contain-
ing one solution, and such that f(x0) and f(x2) have opposite signs, compute the midpoint
x1 = (x0 + x2)/2. Then define a new function h(x) = f(x)e
ax. Find the parameter a that ensures
h(x1) =
1
2 (h(x0) + h(x2)). Then define x3 as the x-intercept of the line passing through (x0, h(x0))
and (x2, h(x2)). Use x3 as one of the endpoints of the next interval bracketing the solution. The
other endpoint is x1 if f(x1)f(x3) < 0. Otherwise, choose either x0 or x2 based on the requirement
that the sign of f(x) at the chosen point must be opposite to the sign of f(x3). Continue until the
desired accuracy is reached.
Questions and items to implement
1. Show that a is uniquely defined and give the equation for finding it.
2. Work out a formula for x3 in detail.
3. Implement Ridders’ method in MATLAB together with the standard bisection method for
finding all solutions of
ex − x2 = 0.
4. Numerically, compare the convergence rate of both algorithms. Discuss the results. What is
the approximate order of convergence for the Ridders’ algorithm?