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Chapter 3

Chapter 3. Solution of Simultaneous Linear Algebraic Equations: Lecture (III). Note: Besides the main textbook, also see Ref: Applied Numerical Methods with MATLAB for Engineers and Scientists , by S. Chapra, Ch. 9. Naïve Gauss Elimination: The general algorithm.

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Chapter 3

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  1. Chapter 3 Solution of Simultaneous Linear Algebraic Equations: Lecture (III) Note: Besides the main textbook, also see Ref: Applied Numerical Methods with MATLAB for Engineers and Scientists, by S. Chapra, Ch. 9.

  2. Naïve Gauss Elimination: The general algorithm Problem: Solve a general set of n equations: a11x1+a12x2+ +a1nxn=b1(1) a21x1+a22x2+ +a2nxn=b2(2) an1x1+an2x2+ +annxn=bn(n) (I) Forward Elimination of Unknowns Step 1: Eliminate x1 from Eq. (2) through Eq. (n). Eq. (2) - (a21/a11)  Eq. (1) Eq. (n) – (an1/a11)  Eq. (1) The modified system: a11x1+a12x2+ +a1nxn=b1(1’) a’22x2+ +a’2nxn=b’2(2’) a’n2x2+ +a’nnxn=b’n(n’) Repeated

  3. Naïve Gauss Elimination: The general algorithm (cont.) Step 2: Eliminate x2 from Eq. (3’) through Eq. (n’). Eq. (3’) - (a’32/a’22)  Eq. (2’) Eq. (n’) – (a’n2/a’22)  Eq. (2’) The modified system: a11x1+a12x2+ a13x3+ +a1nxn=b1(1”) a’22x2+a’23x3+ +a’2nxn=b’2 (2”) a”33x3+ +a”3nxn=b”3 (3”) a”n3x3+ +a”nnxn=b”n(n’) Repeat the procedure … Step n-1: Eliminate xn-1 from the nth equation. Eq. (n) – (ann-1/an-1n-1)  Eq. (n-1) The modified system: a11x1+a12x2+ a13x3+ +a1nxn=b1 a’22x2+a’23x3+ +a’2nxn=b’2 a”33x3+ +a”3nxn=b”3 a(n-1)nnxn=b(n-1)n Repeated

  4. Naïve Gauss Elimination: The general algorithm (cont.) (II) Back Substitution Step 1: Solve xn from the last equation a(n-1)nnxn=b(n-1)n . xn = bn(n-1)/ann(n-1) • Note: the superscript (n-1) indicates that the elements have been modified (n-1) times. Step 2: Back-substitute the result into the (n-1)th equation to solve for xn-1; repeat forxn-2, …, x1. For example: After xn and xn-1 have been solved, xn-2 is given by xn-2=(bn-2-an-2 n-1xn-1-an-2 nxn)/an-2 n-2, or xn-2=(bn-2-[an-2 n-1 an-2 n]*[xn-1 xn]’)/an-2 n-2 (*) Note: (*) will be useful when implementing back substitution on a computer.

  5. Summary Augmented Matrix • Two phases of Gauss Elimination: • Forward elimination • Back substitution • The end result: An upper triangular system. • Your turn: How to implement Gauss elimination on a computer?

  6. Initilization: Define the original [A] and b; return the size of matrix A:[m,n]=size(A); define the augmented matrix: Aug=[A b]; set nb=n+1. Start j=1; j is the index for the unknown, xj. T j  n-1 i  n Outer Loop F Start i=j+1 Inner Loop i=i+1 T j=j+1 F End outer loop End inner loop Eliminate xj from Row i of Aug Flowchart: Forward elimination

  7. An Exercise • Example 9.3 (Ref. by Chapra): Use Gauss elimination to solve 3x1 – 0.1x2 – 0.2x3 = 7.85 0.1x1 + 7x2 – 0.3x3 = -19.3 0.3x1 – 0.2x2 + 10x3 = 71.4 • (a) By hand. Show detailed work step by step. • (b) Write an M-file MyGaussElimination.m. A copy of the code will be handed out later.

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