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(i) Consider the LU factorization 0 0 [b, C1 a2 b2 C2 0 0 a3 bz C3 as b, 14 0 0 01[1 u 0 0 01 U2 0 1 U3 1 (1.1) %3D 0 a3 l3 lo o 0 0 a, 4JL0 o 0 Work out the product on the right hand side to show that the result has the same non-zero structure as the left hand side. Hence, show that the elements ly and uu are given by the following formulae. l1 = b1, ui =7, 4+1 = b+1 - a¿+1U¡, i = 1,2,3. (1.2) %3D

(i) Consider the LU factorization 0 0 [b, C1 a2 b2 C2 0 0 a3 bz C3 as b, 14 0 0 01[1 u 0 0 01 U2 0 1 U3 1 (1.1) %3D 0 a3 l3 lo o 0 0 a, 4JL0 o 0 Work out the product on the right hand side to show that the result has the same non-zero structure as the left hand side. Hence, show that the elements ly and uu are given by the following formulae. l1 = b1, ui =7, 4+1 = b+1 - a¿+1U¡, i = 1,2,3. (1.2) %3D

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.4: The Lu Factorization
Problem 34EQ
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Question 1 [25 marks] (i) Consider the LU factorization [b C1 0 0 0 0 0 a3 [1 u1 1 01 U2 0 1 a2 b2 C2 0 az l2 (1.1) bz C3 0 0 0 az lo o Uz 1. Work out the product on the right hand side to show that the result has the same non-zero structure as the left hand side. Hence, show that the elements l, and uu are given by the following formulae. l1 = b1, uj =4, l+1 = bi+1– ai+1ui, i = 1,2,3. (1.2) (ii) Consider the differential equation y"(x) = y(x) (1.3) with domain 0 <x< 2 and boundary values y(0) = 1, y(2) = 0.25. If we use the central finite difference formula for the second derivative to discretise the differential equation and take the step size to be h = 0.4, show the discretization results in following system of equations: 图-L [2.25 -1 0 0 1 2.25 -1 0 2.25 y2 -1 0 -1 0 0 (1.4) -1 Уз -1 2.25ly4 L0.0820850I Use the formulas (1.2) to find the LU factorisation of the matrix in (1.4). Use 4 significant figures for your calculations. Then, solve the system using LU factorization and show that y1 = 0.609427,y2 = 0.371211, y3 = 0.225799, y4 = 0.136837. (1.5) (ii) Show that the exact solution to equation (1.3) which also satisfies the boundary conditions y(0) and y(2) is given by y(x) = e-x. Use this result to determine the discretization error in the numerical solution obtained in part (ii). kel
Transcribed Image Text:Question 1 [25 marks] (i) Consider the LU factorization [b C1 0 0 0 0 0 a3 [1 u1 1 01 U2 0 1 a2 b2 C2 0 az l2 (1.1) bz C3 0 0 0 az lo o Uz 1. Work out the product on the right hand side to show that the result has the same non-zero structure as the left hand side. Hence, show that the elements l, and uu are given by the following formulae. l1 = b1, uj =4, l+1 = bi+1– ai+1ui, i = 1,2,3. (1.2) (ii) Consider the differential equation y"(x) = y(x) (1.3) with domain 0 <x< 2 and boundary values y(0) = 1, y(2) = 0.25. If we use the central finite difference formula for the second derivative to discretise the differential equation and take the step size to be h = 0.4, show the discretization results in following system of equations: 图-L [2.25 -1 0 0 1 2.25 -1 0 2.25 y2 -1 0 -1 0 0 (1.4) -1 Уз -1 2.25ly4 L0.0820850I Use the formulas (1.2) to find the LU factorisation of the matrix in (1.4). Use 4 significant figures for your calculations. Then, solve the system using LU factorization and show that y1 = 0.609427,y2 = 0.371211, y3 = 0.225799, y4 = 0.136837. (1.5) (ii) Show that the exact solution to equation (1.3) which also satisfies the boundary conditions y(0) and y(2) is given by y(x) = e-x. Use this result to determine the discretization error in the numerical solution obtained in part (ii). kel
(iv) Using centred-differences, express the partial differential equation a'u x² + y?, for 0 < x < 0.75, 0 < y < 0.75 (1.6) ax2 ду? 0, u(0, y) 0, u(x, 0.75) = subject to the boundary conditions u(x,0) 0.75x and u(0.75, y) = y² in finite difference form with h = 0.25. Make sure %3D %3D your system of equations has the form: 01 ru11 U21 U12 (Each "*" represents a number and these numbers can be different.) Draw a diagram of the grid, label the grid points, and put the boundary values on the grid. Note that you do not have to solve the system.
Transcribed Image Text:(iv) Using centred-differences, express the partial differential equation a'u x² + y?, for 0 < x < 0.75, 0 < y < 0.75 (1.6) ax2 ду? 0, u(0, y) 0, u(x, 0.75) = subject to the boundary conditions u(x,0) 0.75x and u(0.75, y) = y² in finite difference form with h = 0.25. Make sure %3D %3D your system of equations has the form: 01 ru11 U21 U12 (Each "*" represents a number and these numbers can be different.) Draw a diagram of the grid, label the grid points, and put the boundary values on the grid. Note that you do not have to solve the system.
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