ch20

Advanced Engineering Mathematics

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CHAPTER 20 Numeric Linear Algebra SECTION 20.1. Linear Systems: Gauss Elimination, page 833 Purpose. To explain the Gauss elimination, which is a solution method for linear systems of equations by systematic elimination (reduction to triangular form). Main Content, Important Concepts Gauss elimination, back substitution Pivot equation, pivot, choice of pivot Operations count, order [e.g., O ( n 3 )] Comments on Content This section is independent of Chap. 7 on matrices (in particular, independent of Sec. 7.3, where the Gauss elimination is also considered). Gauss’s method and its variants (Sec. 20.2) are the most important solution methods for those systems (with matrices that do not have too many zeros). The Gauss–Jordan method (Sec. 20.2) is less practical because it requires more operations than the Gauss elimination. Cramer’s rule (Sec. 7.7) would be totally impractical in numeric work, even for systems of modest size. SOLUTIONS TO PROBLEM SET 20.1, page 839 2. x 1 5 0.65 x 2 , x 2 arbitrary. Both equations represent the same straight line. 4. x 1 5 0, x 2 52 3 6. x 1 5 (30.6 1 15.48 x 2 )/25.38, x 2 arbitrary 8. No solution; the matrix obtained at the end is YZ . 10. x 1 5 0.5, x 2 0.5, x 3 5 3.5 12. x 1 5 0.142857, x 2 5 0.692308, x 3 0.173913 14. x 1 5 1.05, x 2 5 0, x 3 0.45, x 4 5 0.5 16. Team Project. (a) (i) a Þ 1 to make D 5 a 2 1 Þ 0; (ii) a 5 1, b 5 3; (iii) a 5 1, b Þ 3. (b) x 1 5 1 _ 2 (3 x 3 2 1), x 2 5 1 _ 2 ( 2 5 x 3 1 7), x 3 arbitrary is the solution of the first system. The second system has no solution. (c) det A 5 0 can change to det A Þ 0 because of roundoff. (d) (1 2 1/ e ) x 2 5 2 2 1/ eventually becomes x 2 / < 1/ , x 2 5 1, x 1 5 (1 2 x 2 )/ < 0. The exact solution is x 1 5 1/(1 2 ), x 2 5 (1 2 2 )/(1 2 ). We obtain it if we take x 1 1 x 2 5 2 as the pivot equation. (e) The exact solution is x 1 5 1, x 2 4. The 3-digit calculation gives x 2 4.5, x 1 5 1.27 without pivoting and x 2 6, x 1 5 2.08 with pivoting. This shows that 2 2 3 5 1 8 0 3 2 4 0 5 0 0 313 im20.qxd 9/21/05 1:21 PM Page 313
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3S is simply not enough. The 4-digit calculations give x 2 52 4.095, x 1 5 1.051 without pivoting and the exact result x 2 4, x 1 5 1 with pivoting. SECTION 20.2. Linear Systems: LU-Factorization, Matrix Inversion, page 840 Purpose. To discuss Doolittle’s, Crout’s, and Cholesky’s methods, three methods for solving linear systems that are based on the idea of writing the coefficient matrix as a product of two triangular matrices (“LU-factorization”). Furthermore, we discuss matrix inversion by the Gauss–Jordan elimination. Main Content, Important Concepts Doolittle’s and Crout’s methods for arbitrary square matrices Cholesky’s method for positive definite symmetric matrices Numerical matrix inversion Short Courses. Doolittle’s method and the Gauss–Jordan elimination. Comment on Content L suggests “lower triangular” and U “upper triangular.” For Doolittle’s method, these are the same as the matrix of the multipliers and of the triangular system in the Gauss elimination.
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ch20 - im20.qxd 9/21/05 1:21 PM Page 313 CHAPTER 20 Numeric...

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