# 8 - 8: Linear Algebraic Equations 8: Gauss Elimination...

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8: Linear Algebraic Equations 8: Linear Algebraic Equations Gauss Elimination Gauss Elimination

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Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 2 Objectives understand concepts of singularity and ill-condition implement forward elimination and back substitution as in Gauss elimination count flops as a measure of algorithmic efficiency implement partial pivoting solve a tridiagonal system efficiently
Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 3 Introduction Consider the following system of linear algebraic equations [ A ]{ x } = { b } MATLAB provides two direct methods for solving the above system of equations: left-division: >> x = A\b matrix-inversion: >> x = inv(A)*b An understanding of how MATLAB obtains the solution is needed Gauss elimination involves combining equations to “eliminate” unknowns

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Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 4 8.1 Solving Small Numbers of Equations Besides Gauss elimination, several methods are appropriate for solving small ( n ≤ 3 ) sets of equations: graphical method Cramer's rule elimination of unknowns
Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 5 8.1.1: The Graphical Method (1) used for solving two linear equations in two unknowns plot each equation as a straight line the solution is the intersection of the two lines Example: 3x 1 2x 2 = 18 x 1 2x 2 = 2 x 2 =− 3 2 x 1 9 x 2 = 1 2 x 1 1 Figure 8.1

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Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 6 The Graphical Method (2) problems that can be encountered: (a) singular system – no solution (b) singular system – infinite number of solutions (c) ill-conditioned system – solution exists but sensitive to round-off error Figure 8.2
Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 7 8.1.2: Determinants and Cramer's Rule (1) consider the following system of linear algebraic equations [ A ]{ x } = { b } [ A ] is the coefficient matrix the determinant of this system is represented by [ A ] = [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] D = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 a 22 a 23 a 32 a 33 a 12 a 21 a 23 a 31 a 33 a 13 a 21 a 22 a 31 a 32 = a 11 a 22 a 23 a 32 a 23 a 12 a 21 a 33 a 31 a 23 a 13 a 21 a 32 a 31 a 22

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Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 8 Determinants and Cramer's Rule (2) Example 8.1: Fig. 8.1: non-singular well-conditioned: Fig. 8.2a: singular: Fig. 8.2b: singular: Fig. 8.2c: non-singular ill-conditioned: D = 3 2 1 2 = 3 2 − 2 − 1 = 8 D = 1 / 2 1 1 / 2 1 =− 1 / 2 1 − 1 − 1 / 2 = 0 D = 1 / 2 1 1 2 =− 1 / 2 2 − 1 − 1 = 0 D = 1 / 2 1 2.3 / 5 1 1 / 2 1 − 1 − 2.3 / 5 =− 0.04
Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 9 Determinants and Cramer's Rule (3) Cramer's rule: As the size of the system of equations increases, computing the determinants becomes very time consuming and Cramer's rule becomes less practical Example: x 1 = b 1 a 12 a 13 b 2 a 22 a 23 b 3 a 32 a 33 D , x 2 = a 11 b 1 a 13 a 21 b 2 a 23 a 31 b 3 a 33 D , x 3 = a 11 a 12 b 1 a 21 a 22 b 2 a 31 a 32 b 3 D [ 0.3 0.52 1 0.5 1 1.9

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## This note was uploaded on 11/13/2011 for the course CIVE 2*** taught by Professor - during the Spring '11 term at Carleton CA.

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8 - 8: Linear Algebraic Equations 8: Gauss Elimination...

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