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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