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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. free_vector(psum,1,ndim);. Extrapolates by a factor fac through the face of the simplex across from the high point, tries it, and replaces the high point if the new point is better.. Evaluate the function at the...
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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. (ed.) 1977, The State of the Art in Numerical Analysis (London: Academic Press), pp. 10.6 Conjugate Gradient Methods in Multidimensions. A rough counting argument will show how advantageous it is to use the gradient information:...
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free_vector(xicom,1,n);. free_vector(pcom,1,n);. free_vector(df,1,ncom);. free_vector(xt,1,ncom);. (ed.) 1977, The State of the Art in Numerical Analysis (London: Academic Press), Chapter III.1.7 (by K.W. Stoer, J., and Bulirsch, R. 10.7 Variable Metric Methods in Multidimensions. The goal of variable metric methods, which are sometimes called quasi-Newton methods, is not different from the goal of conjugate gradient methods: to accumulate information from successive line...
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10.5) and frprmn. 10.6), however, need more accurate line minimization, which is carried out by the routine linmin.. There is a simple fix for this rare problem, the same as was mentioned in § 10.4: In case of any doubt, you should restart the algorithm at the claimed minimum point, and see if it goes anywhere. Then, instead of calculating...
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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. Stoer, J., and Bulirsch, R. Wilkinson, J.H., and Reinsch, C. 10.9 Simulated Annealing Methods. For practical purposes, simulated annealing has effectively “solved” the famous traveling salesman problem of finding the shortest cyclical itinerary for a...
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11.0 Introduction. An N × N matrix A is said to have an eigenvector x and corresponding eigenvalue λ if. (The zero vector is not considered to be an eigenvector at all.) Evidently (11.0.1) can hold only if. 0 (11.0.2) which, if expanded out, is an N th degree polynomial in λ whose roots are the eigen- values. Root-searching in...
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Stoer, J., and Bulirsch, R. Wilkinson, J.H., and Reinsch, C. Golub, G.H., and Van Loan, C.F. Horn, R.A., and Johnson, C.R. 11.1 Jacobi Transformations of a Symmetric Matrix. The Jacobi method consists of a sequence of orthogonal similarity transforma- tions of the form of equation (11.0.14). Each transformation (a Jacobi rotation) is just a plane rotation designed to annihilate one...
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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. for (j=i+1;j<=n;j. for (j=1;j<=n;j. 11.2 Reduction of a Symmetric Matrix to Tridiagonal Form: Givens and Householder Reductions. The Givens reduction is a modification of the Jacobi method. This allows the procedure to be carried out...
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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. Form K, equation (11.2.11).. for (j=1;j<=l;j. for (k=1;k<=j;k. Reduce a, equation (11.2.13).. Next statement can be omitted if eigenvectors not wanted. Contents of this loop can be omitted if eigenvectors not wanted except for statement...
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11.4 Hermitian Matrices 481. 11.4 Hermitian Matrices. Note that the 2n × 2n matrix in (11.4.2) is symmetric: A T = A and B T. λ n are the eigenvalues of C, then the 2n eigenvalues of the augmented problem (11.4.2) are λ 1 , λ 1 , λ 2 , λ 2. Thus we solve the augmented problem (11.4.2),...
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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. Note that the 2n × 2n matrix in (11.4.2) is symmetric: A T = A and B T. λ n are the eigenvalues of C, then the 2n eigenvalues of the augmented problem (11.4.2) are...
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for (j=m-1;j<=n;j. for (i=m+1;i<=n;i. for (j=m;j<=n;j. for (j=1;j<=n;j. 11.6 The QR Algorithm for Real Hessenberg Matrices. The QR transformation preserves the upper Hessenberg form of the original matrix A ≡ A 1 , and the workload on such a matrix is O(n 2 ) per iteration as opposed to O(n 3 ) on a general matrix. As we pointed out...
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11.7 Eigenvalues or Eigenvectors by Inverse Iteration 493. 11.7 Improving Eigenvalues and/or Finding Eigenvectors by Inverse Iteration. The basic idea behind inverse iteration is quite simple. Then (11.7.1) gives. A · x n = λ n x n (11.7.7) so. x n = (λ n − τ k )x n (11.7.8) Since y of (11.7.6) is an improved approximation to...
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12.0 Introduction. A very large class of important computational problems falls under the general rubric of “Fourier transform methods” or “spectral methods.” For some of these problems, the Fourier transform is simply an efficient computational tool for accomplishing certain common manipulations of data. In other cases, we have problems for which the Fourier transform (or the related “power spectrum”) is...
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At those frequencies, it becomes a delta-function, i.e., a sharp spike, whose width gets narrower and narrower, but whose area converges to be the mean square amplitude of the discrete sine or cosine component at that frequency.. We have by now stated all of the analytical formalism that we will need in this chapter with one exception: In computational work,...
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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. There are also discrete analogs to the convolution and correlation theorems (equations 12.0.9 and 12.0.11), but we shall defer them to § 13.1 and § 13.2, respectively.. 1974, The Fast Fourier Transform (Englewood Cliffs, NJ:...
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1982, Fast Fourier Transform and Convolution Algorithms (New York: Springer- Verlag).. 1974, The Fast Fourier Transform (Englewood Cliffs, NJ: Prentice-Hall). 1992, Computational Frameworks for the Fast Fourier Transform (Philadelphia:. 12.3 FFT of Real Functions, Sine and Cosine Transforms. However, the use of the full complex FFT algorithm for real data is inefficient, both in execution time and in storage required....
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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. Sorensen, H.V., Jones, D.L., Heideman, M.T., and Burris, C.S. Gonzalez, R.C., and Wintz, P. Chen, W., Smith, C.H., and Fralick, S.C. 12.4 FFT in Two or More Dimensions. Given a complex function h(k 1 ,...
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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. 12.5 Fourier Transforms of Real Data in Two and Three Dimensions. Two-dimensional FFTs are particularly important in the field of image process- ing. Use of the FFT is by far the most efficient technique.. In...
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In fact, an algorithm of just this description was developed by Singleton [1] very soon after the discovery of the FFT. The algorithm requires four sequential storage devices, each capable of holding half of the input data. The first half of the input data is initially on one device, the second half on another.. Singleton’s algorithm is based on the...