Solution of Simultaneous Linear Algebraic Equations: Lecture (I) Chapter 3 Solution of Simultaneous Linear Algebraic Equations: Lecture (I) Dr. Jie Zou PHY 3320

Outline Introduction Scientific and engineering applications Simultaneous linear algebraic equations? Scientific and engineering applications Statics: Force analysis Circuit analysis: Currents and voltages Introduction to (1) matrix algebra and (2) MATLAB built-in functions* *Reference: Applied Numerical Methods with MATLAB for Engineers and Scientists, S. Chapra, Ch. 8 Dr. Jie Zou PHY 3320

Introduction What are simultaneous linear algebraic equations? The general form: a11x1 + a12x2 + + a1nxn = b1 a21x1 + a22x2 + + a2nxn = b2 an1x1 + an2x2 + + annxn = bn Dr. Jie Zou PHY 3320

Scientific and engineering applications: Statics Example 3.1: A scaffolding system, consisting of three rigid bars and six wire ropes, is used to support the loads P1, P2, and P3 as shown in Fig. (a). Find the tensions developed in the ropes A, B, C, D, E, and F where P1 = 2000 lb, P2 = 1000 lb, and P3 = 500 lb. Free-body diagram Dr. Jie Zou PHY 3320

Scientific and engineering applications: Circuit analysis Example 3.2: An electrical network consists of six resistors as shown. If the voltages at nodes 1 and 6 are specified as 200 and 0 volts, respectively, determine the voltages at the nodes 2, 3, 4, and 5. Dr. Jie Zou PHY 3320

Introduction to matrix algebra A matrix Row vectors: 1  n matrices Column vectors: m  1 matrices Dr. Jie Zou PHY 3320

Types of matrices (1) A square matrix: (2) A symmetric matrix: (3) A diagonal matrix: (4) An identity matrix [I]: A diagonal matrix where all the diagonal elements are equal to 1. (5) An upper triangular matrix: (6) A banded matrix: Principal or main diagonal Bandwidth of 3 Dr. Jie Zou PHY 3320

Matrix operating rules Addition of two matrices [C] = [A]+[B]: cij = aij + bij Multiplication of a matrix by a scalar [D] = g[A]: dij = gaij Product of two matrices [C] = [A][B]: [A]: m  n; [B]: n  l; [C]: m  l [A][B]  [B][A] Dr. Jie Zou PHY 3320

Matrix operating rules (cont.) Inverse of a matrix [A]-1: Definition: [A][A]-1 = [A]-1[A] = [I] The inverse of a 2  2 matrix: For higher-dimensional matrices, the computation is much more complicated. Transpose of a matrix [A]T: Dr. Jie Zou PHY 3320

MATLAB matrix manipulations Hands-on exercise-Type the following in the MATLAB command window: (1) A=[1 5 6;7 4 2;-3 6 7] (2) A’ (3) x=[8 6 9]; y=[-5 8 1]; z=[4 8 2]; B=[x; y; z] (4) C=A+B (5) D=C-B (6) A*B (7) A.*B (8) AI=inv(A) (9) A*AI (10) I=eye(3) Dr. Jie Zou PHY 3320

Represent linear algebraic equations in the matrix form a11x1 + a12x2 + a13x3 = b1 a21x1 + a22x2 + a23x3 = b2 a31x1 + a32x2 + a33x3 = b3 Matrix form: [A]x = b A formal way to solve [A]x = b: x=[A]-1b; Note: Involving the calculation for [A]-1; very inefficient. Dr. Jie Zou PHY 3320

Solve linear algebra equations with MATLAB MATLAB provides two direct ways: (1) Use the backslash or “left-division”: x=A\b (2) Use matrix inversion: x=inv(A)*b Method (1) is 2 or 3 times as fast. Exercise: Write an M-file and use the “left-division” and matrix inversion to solve Example 3.2 (Rao). Dr. Jie Zou PHY 3320