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# Homework 7: Solving system of linear equations

(due on Friday, Dec 3, 2021)

• Upload your solution as a PDF and the relevant .mlx and/or .m files to the Canvas page.

• Purpose: introduce you to computer methods for solving systems of simultaneous linear equations. You will perform hand calculations that solves the equations using Gauss Elimination (direct solver) and Gauss-Seidel (iterative solver) approaches and then and write a computer program that does the same.

Problem 1: Given the system of equations Ax = b, as defined below:

 ? 2 3 ? −1 −2 2 2
1. Use MATLAB’s backslash operator ‘\’ to solve

2. Determine the solution by hand using Gauss Elimination
3. Use Gauss-Seidel method to solve (by implementing a function that performs Gauss-Seidel iterations for a given matrix A and vector b.)

Problem 2: Given the system of equations Ax = b, as defined below:

 ? 8 2 ? −3 2 5 1
1. Use MATLAB’s backslash operator ‘\’ to solve
2. Carry out three iterations of the Gauss-Seidel method by hand, assuming an initial values of x equal to zero. After the third iteration, compute the error for each estimate with relative to the true values (you can use backslash operator to obtain the true solution.)
3. Use Gauss-Seidel method to solve (by implementing a function that performs Gauss-Seidel iterations for a given matrix A and vector b.)

1

The University of Texas at Austin

Dept. of Civil, Arch. & Env. Eng.

CE 311K, Fall 2021

Homework no. 7

Problem 3: Solve the axial forces Fi for the following truss with pin-joints and 13 members. The resulting system of 13 equations is:

F2 + 0.707F1 = 0 F3− 0.707F1− 2000 = 0

0.707F1 + F4 + 6229 = 0 −F2 + 0.659F5 + F6 = 0

F4− 0.753F5− 600 = 0

F3− 0.659F5 + F7 = 0

F8 + 0.753F5 = 0

F6 + 0.659F9 + F10 = 0

F8− 0.753F9− 800 = 0

F7− 0.659F9 + F11 = 0

F12 + 0.753F9− 2429 = 0

F10 + 0.707F13 = 0

F12− 0.7071F13− 600 = 0

1. How many unknowns and how many equations does the system of equations have?
2. Solve this system of equations using the backslash operator
3. Solve this system of equations using your Gauss-Seidel function (implemented in Problem 1) using initial values of F equal to zero.
4. Explain what is happening when you try to solve this problem using Gauss-Seidel.

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