1 Chapter: 3a System of Linear Equations Dr. Asaf Varol

2 Introduction (I) Linear Equation: an equations where only the first exponent of the unknown is present System of Linear Equations: the set of linear equations where there are more than one unknown

3 Introduction (II)

4 Introduction Example - Vibration of Three Railroad Cars

5

6 Introduction Example - Forces Acting on a Simple Truss

7

8

9 Review of Matrix Algebra

10 Review of Matrix Algebra - Addition

11 Review of Matrix Algebra - Multiplication

12 Example E3.2.1

13 Transpose and Determinant of a Matrix

14 Example E Find the Determinant of a Matrix

15 Identity Matrix and the Inverse of a Matrix

16 Inverse of a matrix

17 Inverse of a matrix (Cont’d)

18 Rules Governing Matrix Algebra

19 Matrix Norms

20 Matrix Norms (II)

21 Example E3.2.3

22 Condition Number and Ill-Conditioned Systems The condition number of a matrix, A. is defined by Cond([A]) = where the symbol indicates any norm of the matrix. It can be shown by matrix algebra, that the condition number of a matrix is always greater than or equal to one., i.e. Cond ([A]) = If the condition number of a matrix is “large” it is said to be ill-conditioned. Systems of equations that involve ill-conditioned matrices are difficult to solve. These systems must be first preconditioned before attempting to find a numerical solution. How large should the condition number be before one can call a matrix truly ill- conditioned is a matter of order of magnitudes. For ill-conditioned systems, a small change in coefficients leads to large changes in the solution.

23 Example E3.3.1 Problem:Find the condition number of the following matrix: [A] = [A] -1 = (1/56) Solution: Using the uniform matrix norm: = Max (4, 7, 12) = 12; = Max[(1/56) (36, 24, 18)] = 36/56 = 9/14 Cond ([A]) = (12)(9/14) = 54/7 Using the Column norm = Max (6, 9, 8) = 9; = Max[(1/56) (40, 24, 14)] = 40/56 = 5/7 Cond ([A]) = (9)(5/7) = 45/7

24 Rules of Thumb for Checking the Condition of a Matrix Apply the following tests after scaling (or normalizing) the matrix, [A], such that the largest element in each row is one. (i) If the sum of the absolute value of off-diagonal elements is less than the absolute value of the diagonal element for each row separately, this matrix is most likely to be well-conditioned. Note that interchanging the rows, i.e. partial pivoting, does not improve the condition number of a matrix. (ii) If det[A]  0. This matrix is ill-conditioned. (iii) If there are elements of [A] -1 which are several order of magnitude larger than one, it is most likely that this matrix is ill-conditioned. (iv) Let [I]* = [A] [A] -1 ; if [I]* is significantly different (i.e. the diagonal elements are not close to one) than the identity matrix, [I], then the matrix [A] is likely to be ill-conditioned. (v) Let [A]* = {[A] -1 } -1 ; if [A]* is not close to the original matrix [A] the matrix is most likely to be ill-conditioned.

25 Example E3.3.2 Problem: Given the following Show that the condition number of the matrix does not change when the rows are interchanged. Solution: Now interchange the rows to obtain One can show that the inverse of the matrices are right by checking if the product [B][B] -1 = I is satisfied. Using the Froberius norm we find cond([A]) = ||[A]|| e ||[A] -1 || e = (1.4177)(1.4177) = cond([B]) = ||[B]|| e ||[B] -1 || e = (1.4177)(1.4177) = 2.001

26 Direct Methods for Solving Linear Systems Survey of several commonly used direct methods meaning that when the outlined procedure or (algorithm) is completed we get the solution, hence there is no need for iteration to improve an approximate solution. Cramer’s Method: This method is the oldest but the most cumbersome and expensive method to solve a given system of linear equations. It requires computation of (n+1) determinants of nxn matrices, where n is the number of unknowns. The rule is given by x i = det{[A]*}/det[A] for i=1,2,3,...,n (3.4.1) where [A]* is the modified matrix where the “i”th column is replaced by the right hand side of the equations, i.e. by {c} T = (c 1,c 2,c 3,...

27 Example E3.4.1 Problem: Find the solution of the system of equations given by Equation (3.1.4). Solution: First we write the equations in matrix form [A]{X}={C} and observe that x 1 = a 0 = det[A*] 1 /det[A]; x 2 = a 1 = det[A*] 2 /det[A]; x 3 = a 2 = det[A*] 3 /det[A] where It follows thendet[A] = 6 a 0 = 8/6a 1 = 3/6a 2 = -5/6

28 References Celik, Ismail, B., “Introductory Numerical Methods for Engineering Applications”, Ararat Books & Publishing, LCC., Morgantown, 2001 Fausett, Laurene, V. “Numerical Methods, Algorithms and Applications”, Prentice Hall, 2003 by Pearson Education, Inc., Upper Saddle River, NJ Rao, Singiresu, S., “Applied Numerical Methods for Engineers and Scientists, 2002 Prentice Hall, Upper Saddle River, NJ Mathews, John, H.; Fink, Kurtis, D., “Numerical Methods Using MATLAB” Fourth Edition, 2004 Prentice Hall, Upper Saddle River, NJ Varol, A., “Sayisal Analiz (Numerical Analysis), in Turkish, Course notes, Firat University,