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Numerical Methods in Engineering with Python Phần 2

Python là một ngôn ngữ lập trình thông dịch do Guido van Rossum tạo ra năm 1990. Python hoàn toàn tạo kiểu động và dùng cơ chế cấp phát bộ nhớ tự động; do vậy nó tương tự như Perl, Ruby, Scheme, Smalltalk, và Tcl. Python được phát triển trong một dự án mã mở, do tổ chức phi lợi nhuận Python Software Foundation quản lý. | 33 2.2 Gauss Elimination Method 2.2 Solution We first solve the equations Ly b by forward substitution 2yi 28 -yi 2y2 -40 y1 - y2 y3 33 yi 28 2 14 y2 -40 yi 2 -40 14 2 -13 y3 33 - y1 y2 33 - 14 - 13 6 The solution x is then obtained from Ux y by back substitution 2X3 y3 4X2 - 3X3 y2 4X1 - 3X2 X3 y1 X3 y3 2 6 2 3 X2 y2 3X3 4 -13 3 3 4 -1 X1 y1 3X2 - X3 4 14 3 -1 - 3 4 2 Hence the solution is x 2 -1 3 Gauss Elimination Method Introduction Gauss elimination is the most familiar method for solving simultaneous equations. It consists of two parts the elimination phase and the solution phase. As indicated in Table 2.1 the function of the elimination phase is to transform the equations into the form Ux c. The equations are then solved by back substitution. In order to illustrate the procedure let us solve the equations 4X1 2X2 X3 11 a 2X1 4X2 2X3 16 b X1 2X2 4X3 17 c Elimination Phase The elimination phase utilizes only one of the elementary operations listed in Table 2.1 - multiplying one equation say equation j by a constant Ấ and subtracting it from another equation equation i . The symbolic representation of this operation is Eq. i Eq. i - Ấ X Eq. j 2.6 The equation being subtracted namely Eq. j is called the pivot equation. We start the elimination by taking Eq. a to be the pivot equation and choosing the multipliers Ấ so as to eliminate X1 from Eqs. b and c Eq. b Eq. b 0.5 X Eq. a Eq. c Eq. c 0.25 X Eq. a 34 Systems of Linear Algebraic Equations After this transformation the equations become 4X1 2x2 x3 11 a 3x2 1.5x3 10.5 b 1.5x2 3.75x3 14.25 c This completes the first pass. Now we pick b as the pivot equation and eliminate x2 from c Eq. c - Eq. c 0.5 X Eq. b which yields the equations 4x1 2x2 x3 11 a 3x2 1.5x3 10.5 b 3x3 9 c The elimination phase is now complete. The original equations have been replaced by equivalent equations that can be easily solved by back substitution. As pointed out before the augmented coefficient matrix is a more convenient instrument for