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comparison toolbox/cauchy.m @ 2:c124219d7bfa draft

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Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author Antonio Pino Robles <data.script93@gmail.com>
date Thu, 07 May 2015 18:36:24 +0200
parents 8f23314345f4
children
comparison
equal deleted inserted replaced
1:e471a92d17be 2:c124219d7bfa
1 function C = cauchy(x, y)
2 %CAUCHY Cauchy matrix.
3 % C = CAUCHY(X, Y), where X, Y are N-vectors, is the N-by-N matrix
4 % with C(i,j) = 1/(X(i)+Y(j)). By default, Y = X.
5 % Special case: if X is a scalar CAUCHY(X) is the same as CAUCHY(1:X).
6 % Explicit formulas are known for DET(C) (which is nonzero if X and Y
7 % both have distinct elements) and the elements of INV(C).
8 % C is totally positive if 0 < X(1) < ... < X(N) and
9 % 0 < Y(1) < ... < Y(N).
10
11 % References:
12 % N.J. Higham, Accuracy and Stability of Numerical Algorithms,
13 % Society for Industrial and Applied Mathematics, Philadelphia, PA,
14 % USA, 1996; sec. 26.1.
15 % D.E. Knuth, The Art of Computer Programming, Volume 1,
16 % Fundamental Algorithms, second edition, Addison-Wesley, Reading,
17 % Massachusetts, 1973, p. 36.
18 % E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications,
19 % Linear Algebra and Appl., 149 (1991), pp. 1-18.
20 % O. Taussky and M. Marcus, Eigenvalues of finite matrices, in
21 % Survey of Numerical Analysis, J. Todd, ed., McGraw-Hill, New York,
22 % pp. 279-313, 1962. (States the totally positive property on p. 295.)
23
24 n = max(size(x));
25 % Handle scalar x.
26 if n == 1
27 n = x;
28 x = 1:n;
29 end
30
31 if nargin == 1, y = x; end
32
33 x = x(:); y = y(:); % Ensure x and y are column vectors.
34 if any(size(x) ~= size(y))
35 error('Parameter vectors must be of same dimension.')
36 end
37
38 C = x*ones(1,n) + ones(n,1)*y.';
39 C = ones(n) ./ C;