Homework 2
Problem 1
Consider the iteration formula (Steffensen’s method)
xn+1 = xn −
f (xn )2
f (xn + f (xn )) − f (xn )
for solving f (x) = 0. Show (analytically) that the method is quadratically convergent, under
some suitable assumptions.
Problem 2
A cylindrical object, with a uniform circular section has a temperature on one side T =
140C and ambient temperature Ta = 40C. It has thermal conductivity of k = 70watts/cmK
and a heat-transfer coefficient of h = 5watts/cm2 K. When the convection loss from the end
A is also considered, the nodal temperatures T1 , T2 and T3 are governed by the equation
T1 = 140
72.6668T2 − 23.8333T3 = 4336
−23.8333T2 + 41.3334T3 = 700.
(1)
Determine the values of the nodal temperatures using LU decomposition.
Ta=40C
140C
T1
T3
T2
2 cm
l=5cm
Figure 1: Problem 2
Problem 3
1. Prove that if U is upper triangular and invertible, then U −1 is upper diagonal.
2. Prove that if L is lower triangular and invertible, then L−1 is lower diagonal.
3. Prove that the product of two upper diagonal matrices is upper diagonal.
Problem 4
Write a program to produce the Doolittle factorization of the following matrix.
The matrix A (n × n) is a tri-diagonal matrix with 2 on the main diagonal and −1 on
sub- and super- diagonal (in matlab
A = 2 ∗ diag(ones(n, 1)) + (−1) ∗ diag(ones(n − 1, 1), 1) + (−1) ∗ diag(ones(n − 1, 1), −1)).
1