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annotate toolbox/fiedler.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
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8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
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1 function A = fiedler(c)
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2 %FIEDLER Fiedler matrix - symmetric.
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3 % A = FIEDLER(C), where C is an n-vector, is the n-by-n symmetric
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4 % matrix with elements ABS(C(i)-C(j)).
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5 % Special case: if C is a scalar, then A = FIEDLER(1:C)
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6 % (i.e. A(i,j) = ABS(i-j)).
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7 % Properties:
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8 % FIEDLER(N) has a dominant positive eigenvalue and all the other
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
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9 % eigenvalues are negative (Szego, 1936).
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10 % Explicit formulas for INV(A) and DET(A) are given by Todd (1977)
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11 % and attributed to Fiedler. These indicate that INV(A) is
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12 % tridiagonal except for nonzero (1,n) and (n,1) elements.
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13 % [I think these formulas are valid only if the elements of
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14 % C are in increasing or decreasing order---NJH.]
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15
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16 % References:
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17 % G. Szego, Solution to problem 3705, Amer. Math. Monthly,
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18 % 43 (1936), pp. 246-259.
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19 % J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
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20 % Birkhauser, Basel, and Academic Press, New York, 1977, p. 159.
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21
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22 n = max(size(c));
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23
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24 % Handle scalar c.
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25 if n == 1
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26 n = c;
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27 c = 1:n;
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28 end
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30 c = c(:).'; % Ensure c is a row vector.
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31
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32 A = ones(n,1)*c;
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33 A = abs(A - A.'); % NB. array transpose.