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8.1 Use the given LU-decomposition of the matrix B to solve the system of equations B = b. Show all the steps clearly and give your answer in vector form. 1 0 0 0 1 0 -2 1 1 1 1 0 0 1 1 -2 B = 3 1 8.2 Determine the LU-decomposition of the matrix 1 1 0 01 7 5 0 2 A = 15 1 -1 -1 0 1 8.3 Calculate the norm ||A||. .

8.1 Use the given LU-decomposition of the matrix B to solve the system of equations B = b. Show all the steps clearly and give your answer in vector form. 1 0 0 0 1 0 -2 1 1 1 1 0 0 1 1 -2 B = 3 1 8.2 Determine the LU-decomposition of the matrix 1 1 0 01 7 5 0 2 A = 15 1 -1 -1 0 1 8.3 Calculate the norm ||A||. .

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 7 Consider the initial value problem zy? z' = z+ (3t – 1)y y(0) = 1 z(0) = 2 The improved Euler algorithm is applied with h 0.2 and some of the results are tabulated below: %3D k Yk 0 1.000 2.000 1 2.624 0.868 2 3.169 1.009 Use the values in the table and find an approximation for the solution of the system of differential equations at t = 0.6. 2 Question 8 8.1 Use the given LU-decomposition of the matrix B to solve the system of equations Bã = b. Show all the steps clearly and give your answer in vector form. 1 0 0 0 1 0 1 1 0 0 1 1 0 0 1 -2 B = 3 -2 1 1 8.2 Determine the LU-decomposition of the matrix 1 1 0 0 2 7 5 0 A = 15 1 -1 -1 0 1 8.3 Calculate the norm ||A|| .
Transcribed Image Text:Question 7 Consider the initial value problem zy? z' = z+ (3t – 1)y y(0) = 1 z(0) = 2 The improved Euler algorithm is applied with h 0.2 and some of the results are tabulated below: %3D k Yk 0 1.000 2.000 1 2.624 0.868 2 3.169 1.009 Use the values in the table and find an approximation for the solution of the system of differential equations at t = 0.6. 2 Question 8 8.1 Use the given LU-decomposition of the matrix B to solve the system of equations Bã = b. Show all the steps clearly and give your answer in vector form. 1 0 0 0 1 0 1 1 0 0 1 1 0 0 1 -2 B = 3 -2 1 1 8.2 Determine the LU-decomposition of the matrix 1 1 0 0 2 7 5 0 A = 15 1 -1 -1 0 1 8.3 Calculate the norm ||A|| .
Question 5 5.1 Consider the graph of the solution of an initial value problem on the interval [0, 1]. Euler's method is used to approximate the solution using two steps. Redraw the graph on your answer sheet and illustrate the global error, E1, and the local error, e2, clearly on the sketch. 0.4 0.5 1 5.2 Consider the initial value problem dr f(t, x), #(0) = 10. dt Assume that r(1) = 77.43. An approximation r4 = method with 4 steps. Another approximation r, = 77.39 is calculated for r(1) using more steps. Determine a probable step length that was used in the second approximation by keeping the order of the global error for Runge-Kutta in mind. Explain your answer. 76.77 is calculated for r(1) using the Runge-Kutta Question 6 Write the given initial value problem as an initial value problem for a first order system. y"(t) + y(t) + cos(3t) = 5t y(1) = -1, y'(1) = 2, y"(1) = 0.
Transcribed Image Text:Question 5 5.1 Consider the graph of the solution of an initial value problem on the interval [0, 1]. Euler's method is used to approximate the solution using two steps. Redraw the graph on your answer sheet and illustrate the global error, E1, and the local error, e2, clearly on the sketch. 0.4 0.5 1 5.2 Consider the initial value problem dr f(t, x), #(0) = 10. dt Assume that r(1) = 77.43. An approximation r4 = method with 4 steps. Another approximation r, = 77.39 is calculated for r(1) using more steps. Determine a probable step length that was used in the second approximation by keeping the order of the global error for Runge-Kutta in mind. Explain your answer. 76.77 is calculated for r(1) using the Runge-Kutta Question 6 Write the given initial value problem as an initial value problem for a first order system. y"(t) + y(t) + cos(3t) = 5t y(1) = -1, y'(1) = 2, y"(1) = 0.
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