tailieunhanh - Quantitative Methods and Applications in GIS - Chapter 11 (end)

Giải quyết một hệ thống phương trình tuyến tính và ứng dụng Mô phỏng cấu trúc đô thị Chương này giới thiệu các phương pháp để giải quyết một hệ thống phương trình tuyến tính. Kỹ thuật này được sử dụng trong nhiều ứng dụng, bao gồm cả phân tích đầu vào-đầu ra phổ biến (ví dụ, Hewings, 1985, xem Phụ lục 11A đối với một giới thiệu ngắn gọn). Ở đây, phương pháp được minh họa trong việc giải quyết mô hình Garin-Lowry, một mô hình được sử dụng rộng rãi bởi các nhà quy hoạch đô thị và địa. | 11 Solving a System of Linear Equations and Application in Simulating Urban Structure This chapter introduces the method for solving a system of linear equations. The technique is used in many applications including the popular input-output analysis . Hewings 1985 see Appendix 11A for a brief introduction . Here the method is illustrated in solving the Garin-Lowry model a model widely used by urban planners and geographers for analyzing urban land use structure. A case study using a hypothetical city shows how the distributions of population and employment interact with each other and how the patterns can be affected by the transportation network. The GIS usage in the case study involves the computation of a travel time matrix and other data preparation tasks. The method is fundamental in numerical analysis NA and is often used as a building block in many NA tasks such as solving a system of nonlinear equations and the eigenvalue problem. Appendix 11B shows how the task of solving a system of linear equations is also imbedded in the method of solving a system of nonlinear equations. SOLVING A SYSTEM OF LINEAR EQUATIONS A system of n linear equations with n unknowns x1 x2 . xn is written as a11 X1 a12 x2 . a1nxn b1 21X1 a22 X2 . a2 nxn b2 a 1 X1 a 2 X2 . annxn bn In the matrix form it is b1 b2 bn 219 2006 by Taylor Francis Group LLC 220 Quantitative Methods and Applications in GIS or simply Ax b If matrix A has a diagonal structure Equation becomes a11 0 0 x1 b1 0 a22 0 x2 b2 0 0 an J _x _ _b _ The solution is simple xi bị ail If aii 0 and bi 0 xi can be any real number and if aii 0 and bi 0 there is no solution for the system. There are two other simple systems with easy solutions. If matrix A has a lower triangular structure . all elements above the main diagonal are 0 Equation becomes b1 b2 b Assuming aii 0 for all i the forward-substitution algorithm is used to solve the system by obtaining x1 from the first equation substituting x1 in