Homework 2
Problem 1
Consider the iteration formula (Steffensenโs method)
xn+1 =xnโf(xn)2
f(xn+f(xn)) โf(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=
140Cand ambient temperature Ta= 40C. It has thermal conductivity of k= 70watts/cmK
and a heat-transfer coefficient of h= 5watts/cm2K. When the convection loss from the end
Ais also considered, the nodal temperatures T1,T2and T3are governed by the equation
T1= 140
72.6668T2โ23.8333T3= 4336
โ23.8333T2+ 41.3334T3= 700.
(1)
Determine the values of the nodal temperatures using LU decomposition.
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l=5cm
140C
T1 T2 T3 2 cm
Ta=40C
Figure 1: Problem 2
Problem 3
1. Prove that if Uis upper triangular and invertible, then Uโ1is upper diagonal.
2. Prove that if Lis lower triangular and invertible, then Lโ1is 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(nรn) is a tri-diagonal matrix with 2 on the main diagonal and โ1 on
sub- and super- diagonal (in matlab
A= 2 โdiag(ones(n, 1)) + (โ1) โdiag(ones(nโ1,1),1) + (โ1) โdiag(ones(nโ1,1),โ1)).
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