19.0 Introduction 831 http://www.nr.com or call 1-800-872- Permission is read able files A (including this one) granted fori 1988-1992 by Cambridge -7423 (North America to any server computer,is tusers to make one paper from NUMERICAL RECIPES IN C: e University Press. THE only),or st st Programs Xo Copyright (C) Figure 19.0.2.Finite-difference representation of a second-order elliptic equation on a two-dimensional to dir grid.The second derivatives at the point A are evaluated using the points to which A is shown connected. The second derivatives at point B are evaluated using the connected points and also using"right-hand ART OF SCIENTIFIC COMPUTING(ISBN side"boundary information,shown schematically as. rectcustser are boundary points where either u or its derivative has been specified.If we pull v@cam all this "known"information over to the right-hand side of equation (19.0.8),then 1988-1992 by Numerical Recipes 10-521 the equation takes the form 43108 A·u=b (19.0.10) where A has the form shown in Figure 19.0.3.The matrix A is called"tridiagonal (outside with fringes."A general linear second-order elliptic equation North Software. 02u du 02u a(z,y) 2+b( +c(,) 7+d红，而 (19.0.11) 02u visit website machine +e(x,y) xOy +f(x,y)u=g(x,y) will lead to a matrix of similar structure except that the nonzero entries will not be constants. As a rough classification,there are three different approaches to the solution of equation (19.0.10),not all applicable in all cases:relaxation methods,"rapid" methods (e.g.,Fourier methods),and direct matrix methods.19.0 Introduction 831 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). yL ∆ y1 y0 x0 xJ x1 ... ∆ A B Figure 19.0.2. Finite-difference representation of a second-order elliptic equation on a two-dimensional grid. The second derivatives at the point A are evaluated using the points to which A is shown connected. The second derivatives at point B are evaluated using the connected points and also using “right-hand side” boundary information, shown schematically as ⊗. are boundary points where either u or its derivative has been specified. If we pull all this “known” information over to the right-hand side of equation (19.0.8), then the equation takes the form A · u = b (19.0.10) where A has the form shown in Figure 19.0.3. The matrix A is called “tridiagonal with fringes.” A general linear second-order elliptic equation a(x, y) ∂2u ∂x2 + b(x, y) ∂u ∂x + c(x, y) ∂2u ∂y2 + d(x, y) ∂u ∂y + e(x, y) ∂2u ∂x∂y + f(x, y)u = g(x, y) (19.0.11) will lead to a matrix of similar structure except that the nonzero entries will not be constants. As a rough classification, there are three different approaches to the solution of equation (19.0.10), not all applicable in all cases: relaxation methods, “rapid” methods (e.g., Fourier methods), and direct matrix methods