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17.3 Relaxation Methods. 0 (17.3.2) The form of the FDE in (17.3.2) illustrates the idea, but not uniquely: There are many ways to turn the ODE into an FDE. When the problem involves N coupled first-order ODEs represented by FDEs on a mesh of M points, a solution consists of values for N dependent functions given at each of the...
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17.4 A Worked Example: Spheroidal Harmonics. 17.3), and then by the methods of shooting. 17.1) and shooting to a fitting point. 0 they are called “oblate.” The equation has singular points at x. We write the eigenvalues of (17.4.1) as λ mn (c) and the eigenfunctions as S mn (x. in equation (17.4.1), we find that the regular solution has...
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17.5 Automated Allocation of Mesh Points 783. 17.5 Automated Allocation of Mesh Points. dy dq = g dx. In terms of q, equation (17.5.2) as an FDE might be written y k − y k − 1 − 1 2. Its reciprocal dq/dx is proportional to the density of mesh points.. For example, we might want dq/dx to be larger...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. Written in terms of the mesh variable q, this equation is dx. Otherwise (17.5.7) can have a zero in its denominator.. To use automated mesh spacing, you add the three ODEs (17.5.5) and (17.5.7) to...
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Integral Equations and Inverse Theory. 18.0 Introduction. Many people, otherwise numerically knowledgable, imagine that the numerical solution of integral equations must be an extremely arcane topic, since, until recently, it was almost never treated in numerical analysis textbooks. Actually there is a large and growing literature on the numerical solution of integral equations. Because this correspondence lies at the heart...
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18.1 Fredholm Equations of the Second Kind 791. 18.4–18.7 we face up to the issues of inverse problems. 18.4 is an introduction to this large subject.. It should go without saying that this one short chapter can only barely touch on a few of the most basic methods involved in this complicated subject.. Delves, L.M., and Mohamed, J.L. 1985, Computational...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. However, since the weights w j are not equal for most quadrature rules, the matrix K (equation 18.1.5) is not symmetric. Then equation (18.1.7) becomes. Equation (18.1.8) is now in the form of a symmetric...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. The general consensus is that the best of the higher order methods is the block-by-block method (see [1. You should also be on the lookout for singularities in the integrand. If you find them, then...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. 18.4 Inverse Problems and the Use of A Priori Information. 0 (18.4.2) where λ 1 is a Lagrange multiplier. Notice that b is absent in the second equality, since it doesn’t depend on u.. Instead...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. minimize: A + λ B (18.4.12). along the so-called trade-off curve (see Figure 18.4.1), and then to settle on a “best” value of λ by one or another criterion, ranging from fairly objective (e.g., making...
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18.6 Backus-Gilbert Method 815. The uniqueness of the solution is also not well understood, although for two-dimensional images of reasonable complexity it is believed to be unique.. In particular, rearranging terms somewhat, we can write the iteration (18.5.21) as. (18.5.27). (18.5.28) (or, instead of P i ’s, the T i operators of equation 18.5.26), then it can be shown that...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. y = R, and then substitute y into both the numerators and denominators of 18.6.12 or 18.6.13.). Equations (18.6.12) and (18.6.13) have a completely different character from the linearly regularized solutions to (18.5.7) and (18.5.8)....
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19.0 Introduction. The prototypical example of a hyperbolic equation is the one-dimensional wave equation. where v = constant is the velocity of wave propagation. The prototypical parabolic equation is the diffusion equation. where D is the diffusion coefficient. The prototypical elliptic equation is the Poisson equation. Equations (19.0.1) and (19.0.2) both define initial value or Cauchy problems: If information on...
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19.4), but they do not work well for problems with discontinuities.. 1973, An Analysis of the Finite Element Method (Englewood Cliffs, NJ:. 19.1 Flux-Conservative Initial Value Problems. (The physicist-reader may recognize equations (19.1.3) as analogous to Maxwell’s equations for one-dimensional propagation of electromagnetic waves.). We will consider, in this section, a prototypical example of the general flux- conservative equation (19.1.1),...
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19.2 Diffusive Initial Value Problems. We will assume D ≥ 0, otherwise equation (19.2.1) has physically unstable solutions: A small disturbance evolves to become more and more concentrated instead of dispersing. (Don’t make the mistake of trying to find a stable differencing scheme for a problem whose underlying PDEs are themselves unstable!) Even though (19.2.1) is of the form already...
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1 + 1 2 iH∆t (19.2.35) In other words,. ψ n j (19.2.36). 19.3 Initial Value Problems in Multidimensions. The methods described in § 19.1 and § 19.2 for problems in 1 + 1 dimension (one space and one time dimension) can easily be generalized to N + 1 dimensions.. As an example, we show how to generalize the Lax...
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Solution of Linear Algebraic Equations. A set of linear algebraic equations looks like this:. N are known numbers, as are the right-hand side quantities b i , i = 1, 2. If N = M then there are as many equations as unknowns, and there is a good chance of solving for a unique solution set of x j ’s....
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. Coleman, T.F., and Van Loan, C. Forsythe, G.E., and Moler, C.B. Wilkinson, J.H., and Reinsch, C. Johnson, L.W., and Riess, R.D. Ralston, A., and Rabinowitz, P. 2.1 Gauss-Jordan Elimination. For inverting a matrix, Gauss-Jordan elimination...
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for (i=1;i<=n;i. for (j=i+1;j<=n;j. for (k=i;k<j;k. Wilkinson, J.H., and Reinsch, C. Gill, P.E., Murray, W., and Wright, M.H. Dahlquist, G., and Bjorck, A. Golub, G.H., and Van Loan, C.F. 2.10 QR Decomposition. We write a Householder matrix in the form 1 − u ⊗ u/c where c = 1 2 u · u. Thus we arrange for the first Householder...
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Let us reinterpret (19.3.19) to have a different meaning: Let U 1 now denote an updating method that includes algebraically all the pieces of the total operator L , but which is desirably stable only for the L 1 piece. (19.3.22). The timestep for each fractional step in (19.3.22) is now only 1/m of the full timestep, because each partial...