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annotate toolbox/kms.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 = kms(n, rho)
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2 %KMS Kac-Murdock-Szego Toeplitz matrix.
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Antonio Pino Robles <data.script93@gmail.com>
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3 % A = KMS(N, RHO) is the N-by-N Kac-Murdock-Szego Toeplitz matrix with
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
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4 % A(i,j) = RHO^(ABS((i-j))) (for real RHO).
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5 % If RHO is complex, then the same formula holds except that elements
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
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6 % below the diagonal are conjugated.
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7 % RHO defaults to 0.5.
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8 % Properties:
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9 % A has an LDL' factorization with
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10 % L = INV(TRIW(N,-RHO,1)'),
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11 % D(i,i) = (1-ABS(RHO)^2)*EYE(N) except D(1,1) = 1.
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12 % A is positive definite if and only if 0 < ABS(RHO) < 1.
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13 % INV(A) is tridiagonal.
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14
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15 % Reference:
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16 % W.F. Trench, Numerical solution of the eigenvalue problem
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17 % for Hermitian Toeplitz matrices, SIAM J. Matrix Analysis and Appl.,
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18 % 10 (1989), pp. 135-146 (and see the references therein).
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19
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20 if nargin < 2, rho = 0.5; end
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21
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22 A = (1:n)'*ones(1,n);
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23 A = abs(A - A');
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24 A = rho .^ A;
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25 if imag(rho)
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26 A = conj(tril(A,-1)) + triu(A);
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27 end