Complete the following problems with a single script and custom functions as appropriate. Important lines of code should include descriptive comments.
Microsoft Word - AERO 300 Laboratory 4 - Iterative Methods to Solve Matrix Equations.docx AERO 300 Laboratory 4 Iterative Methods to Solve Linear Matrix Equations The pre-lab assignment (Section 4) is due at the beginning of this lab (can be hand-written or typed). The lab (Section 5) will be due before your next lab section (.zip file submitted to polylearn). 1 Objectives This lab introduces concepts and methods used to solving matrix equations iteratively. The topics covered are: • Solving linear systems of equations like !" = $ with MATLAB. • Jacobi iterative method • Gauss-Seidel iterative method • Convergence criteria At the completion of this lab you should be able to solve matrix equations using both the Jacobi and Gauss-Seidel method. 2 Introduction Linear systems of equations can be expressed in matrix form. Consider the system of equations: 3" + 5( = 3 (1) 2" + 6( = 1 (2) This system can be expressed in the form !" = $ , where !is a matrix of the coefficients, " is the solution vector we seek, and $ is the right-hand side (or known part) of the system. Thus, the above system is written in matrix form as: -3 52 6. - "(. = -31. (3) There are many methods and schemes for solving systems of equations including: algebraic substitution, algebraic addition, graphically, matrix decomposition, Gaussian elimination, and iterative schemes. Iterative schemes differ from direct methods in that they converge on a numeric approximation of the exact solution. Of course, direct and iterative schemes are always limited by machine precision. So, normally an engineer should balance the scheme with the size of the problem. For relatively small problems, direct methods are sufficient. When the size of the problem is large (or the system is sparse – most of the elements of the matrix ! are zeros), iterative schemes are preferred. Practical uses of iterative schemes include applications in circuit analysis, boundary value problems and partial differential equations. In Aerospace Engineering, iterative schemes for solving systems of equations are used in Computational Fluid Dynamics (CFD), Finite Element Analysis (FEA) for structures, and Computational Thermal Dynamics to understand heat flow. Iteration is a popular technique for finding roots of equations. A generalization of fixed-point iteration can be applied to systems of linear equations to produce accurate results. The Jacobi iteration method is attributed to Carl Jacobi (1804-1851) and Gauss-Seidel iteration is attributed to Johann Carl Friedrich Gauss (1777-1855) and Philipp Ludwig von Seidel (1821- 1896). In both schemes, the solution vector " of the system of equations is found without directly solving for the inverse of !. 2.1 Jacobi Iteration Method Consider the following set of coupled linear equations. (Note that this method can be applied to any number of equations. A set of four was chosen for this explanation.) !" = $ (4) Now, if we let ! = / + 0 + 1, we can write " = /234$ − (0 + 1)"8 (5) Now, using an iterative scheme, we have, "9:3 = /234$ − (0 + 1)"98 (6) Jacobi iteration uses all previous step information to calculate the next approximate solution. In the above equations, the left-hand side is the “next approximate solution” and the right-hand side is the “previous step information.” Equation (6) is known as Jacobi Iteration and can used to solve for the unknown vector " in equation (4). Since we do not have previous time step data when we first begin the iteration, we must make an initial guess,";. The better the guess, the faster the iteration will converge. As with some of our other methods, we are using MATLAB to “cheat” with built in matrix multiplication. To implement the algorithm using the coefficients of ! and $ and for loops, we can reformulate the problem by noticing, for example, that the first two elements of " are updated as, "39:3 = 1<33 4$3="" −=""><3="=9 −=""><3>">9 −⋯− <3@"@98 "="9:3" ==""><== 4$="−"><=3"39 −=""><=>">9 − ⋯− <=@"@98 (7)="" this="" relationship="" can="" be="" written="" in="" general="" as,="" "a9:3="">
> A(2,1) This will pull the value, 2, from matrix A. We can store this element in a variable called x by writing: >> x = A(2,1); The variable x now has the value 2 stored in it. In general, if we want to refer to the information stored in row i, and column j, we write: >> A(i, j) There are times when we will want to access an entire column of a matrix. This is done using the colon operator. If we want to store the second column of matrix A into the vector y, we write: >> y = A(:,2); This statement will store every row in column two of the A matrix in y. If we now display y we get: >> disp(y) 5 6 6 Similarly, if we want to access the entire third row of A, we write: >> A(3,:) This statement retrieves every column in row three of matrix A. If only some subset of rows or column is desired, then these can also be accessed using the colon operator. If we want the top two values in column three, we would write: >> x = A(1:2, 3); Now, if we display x, we get: >> disp(x) 1 8 Similarly, if we want the 2x2 matrix comprised of the lower right portion of matrix A we can write: >> A(2:3,2:3) ans = 6 8 6 9 In general, if we want to access rows, a to b, and columns c to d, we can write: >> A(a:b, c:d) 3 Help Potentially useful terms for this lab: • linspace • for • function • tic, toc • while • disp • plot • subplot • axis • grid • hold • legend • title • xlabel • ylabel 4 Pre-lab Assignment NOTE: This must be done before lab. If it is not completed, you will not be allowed into lab. For a general 4 × 4 matrix ! = W