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The best way to solve equations of the form including the constant coefficient problem (19.0.3), is a combination of Fourier analysis and cyclic reduction, the FACR method [3-6. If at the rth stage of CR we Fourier analyze the equations of the form (19.4.32) along y, that is, with respect to the suppressed vector index, we will have a tridiagonal...
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Given u n , one solves (19.5.36) for u n+1/2 , substitutes on the right-hand side of and then solves for u n+1 . As usual, the goal is to minimize the spectral radius of the iteration matrix.. 19.6 Multigrid Methods for Boundary Value Problems. The “rapid” direct elliptic solvers discussed in § 19.4 solve special kinds of elliptic equations...
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2.2 Gaussian Elimination with Backsubstitution 41. 2.2 Gaussian Elimination with Backsubstitution. The usefulness of Gaussian elimination with backsubstitution is primarily pedagogical. Gaussian elimination reduces a matrix not all the way to the identity matrix, but only halfway, to a matrix whose components on the diagonal and above (say) remain nontrivial. The procedure up to this point is termed Gaussian elimination.....
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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. Isaacson, E., and Keller, H.B. Johnson, L.W., and Riess, R.D. For the case of a 4 × 4 matrix A, for example, equation (2.3.1) would look like this:. (2.3.2) We can use a decomposition such...
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Golub, G.H., and Van Loan, C.F. Forsythe, G.E., Malcolm, M.A., and Moler, C.B. Forsythe, G.E., and Moler, C.B. Stoer, J., and Bulirsch, R. Ralston, A., and Rabinowitz, P. Horn, R.A., and Johnson, C.R. 2.4 Tridiagonal and Band Diagonal Systems of Equations. unsigned long n). Solves for a vector u[1..n] the tridiagonal linear set given by equation (2.4.1). unsigned long j;....
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Wilkinson, J.H., and Reinsch, C. Golub, G.H., and Van Loan, C.F. a 11 a 12 a 21 a 22. b 11 b 12 b 21 b 22. c 11 c 12 c 21 c . c 11 = a 11 × b 11 + a 12 × b 21 c 12 = a 11 × b 12 + a 12...
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Iterative improvement of the solution to A · x = b. Obviously it is not easy to obtain greater precision for the solution of a linear set than the precision of your computer’s floating-point word. Unfortunately, for large sets of linear equations, it is not always easy to obtain precision equal to, or even comparable to, the computer’s limit. If...
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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 exists a very powerful set of techniques for dealing with sets of equations or matrices that are either singular or else numerically very close to singular. In many cases where Gaussian elimination and LU...
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It is wasteful to use general methods of linear algebra on such problems, because most of the O(N 3 ) arithmetic operations devoted to solving the set of equations or inverting the matrix involve zero operands.. Direct methods for solving sparse equations, then, depend crucially on the precise pattern of sparsity of the matrix. You should also be aware of...
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In § 2.4 the case of a tridiagonal matrix was treated specially, because that particular type of linear system admits a solution in only of order N operations, rather than of order N 3 for the general linear problem. (Other than the operations count, these two types having nothing in common.) Matrices of the first type, termed Vandermonde matrices, occur...
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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 matrix is stored somewhere.. for(i=1;i<=n;i. The matrix A e is the diagonal part of A, stored in the first n elements of sa. (ed.) 1977, The State of the Art in Numerical Analysis (London:...
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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. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call North America only),or send email to [email protected] (outside North America).. Less-Numerical Algorithms. 20.0 Introduction. This final chapter is an idiosyncratic collection of “less-numerical recipes”...
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Authors of computer science texts, we’ve noticed, like to throw in a token numerical subject (usually quite a dull one — quadrature, for example). One topic, Gray codes, was already used in the construction of quasi-random sequences. Two other topics, on diagnosing a computer’s floating-point parameters, and on arbitrary precision arithmetic, give additional insight into the machinery behind the casual...
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20.2 Gray Codes. An example of a Gray code (in fact, the most commonly used one) is the sequence and 1000, for i = 0. The spelling is “Gray,” not “gray”: The codes are named after one Frank Gray, who first patented the idea for use in shaft encoders. 20.2 Gray Codes 895. Single-bit operations for calculating the Gray code...
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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. if (is >= 0) This is the easy direction!. This is the more complicated direction: In hierarchical stages, starting with a one-bit right shift, cause each bit to be XORed with all more significant bits.....
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If the desired x is in between the largest and smallest of the x i ’s, the problem is called interpolation. if x is outside that range, it is called extrapolation, which is considerably more hazardous (as many former stock-market analysts can attest).. In the case of interpolation, you are given the function f at points not of your own...
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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. 1988, Communications of the ACM , vol. 1987, Communications of the ACM , vol. 1989, Communications of the ACM , vol. 20.4 Huffman Coding and Compression of Data. References [3-6] give the flavor of some...
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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. 20.5 Arithmetic Coding. We saw in the previous section that a perfect (entropy-bounded) coding scheme would use L i. log 2 p i bits to encode character i (in the range 1 ≤ i ≤...
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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. Bell, T.C., Cleary, J.G., and Witten, I.H. Witten, I.H., Neal, R.M., and Cleary, J.G. 1987, Communications of the ACM , vol. 20.6 Arithmetic at Arbitrary Precision. In doing so, we’ll learn some things about multiple...
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unsigned long *ilob,*iupb,*ncumfq,jdif,nc,minint,nch,ncum,nrad;. unsigned long *icod,*ncod,*left,*right,nch,nodemax;. void arcmak(unsigned long nfreq. unsigned long nchh, unsigned long nradd, arithcode *acode);. void arcsum(unsigned long iin. unsigned long iout. unsigned long ja, int nwk, unsigned long nrad, unsigned long nc);. void asolve(unsigned long n, double b. void atimes(unsigned long n, double x. unsigned long n, float *ave, float *var);. void banbks(float **a, unsigned long...