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author Antonio Pino Robles <data.script93@gmail.com>
date Fri, 29 May 2015 09:48:36 +0200
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8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
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1 function C = cauchy(x, y)
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2 %CAUCHY Cauchy matrix.
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3 % C = CAUCHY(X, Y), where X, Y are N-vectors, is the N-by-N matrix
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4 % with C(i,j) = 1/(X(i)+Y(j)). By default, Y = X.
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5 % Special case: if X is a scalar CAUCHY(X) is the same as CAUCHY(1:X).
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6 % Explicit formulas are known for DET(C) (which is nonzero if X and Y
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7 % both have distinct elements) and the elements of INV(C).
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8 % C is totally positive if 0 < X(1) < ... < X(N) and
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9 % 0 < Y(1) < ... < Y(N).
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10
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11 % References:
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12 % N.J. Higham, Accuracy and Stability of Numerical Algorithms,
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13 % Society for Industrial and Applied Mathematics, Philadelphia, PA,
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14 % USA, 1996; sec. 26.1.
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15 % D.E. Knuth, The Art of Computer Programming, Volume 1,
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16 % Fundamental Algorithms, second edition, Addison-Wesley, Reading,
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17 % Massachusetts, 1973, p. 36.
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18 % E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications,
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19 % Linear Algebra and Appl., 149 (1991), pp. 1-18.
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20 % O. Taussky and M. Marcus, Eigenvalues of finite matrices, in
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21 % Survey of Numerical Analysis, J. Todd, ed., McGraw-Hill, New York,
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22 % pp. 279-313, 1962. (States the totally positive property on p. 295.)
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23
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24 n = max(size(x));
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25 % Handle scalar x.
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26 if n == 1
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27 n = x;
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28 x = 1:n;
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29 end
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31 if nargin == 1, y = x; end
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32
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33 x = x(:); y = y(:); % Ensure x and y are column vectors.
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34 if any(size(x) ~= size(y))
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35 error('Parameter vectors must be of same dimension.')
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36 end
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37
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38 C = x*ones(1,n) + ones(n,1)*y.';
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39 C = ones(n) ./ C;