Mercurial > matrix-functions
annotate toolbox/cauchy.m @ 0:8f23314345f4 draft
Create local repository for matrix toolboxes. Step #0 done.
author | Antonio Pino Robles <data.script93@gmail.com> |
---|---|
date | Wed, 06 May 2015 14:56:53 +0200 |
parents | |
children |
rev | line source |
---|---|
0
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
1 function C = cauchy(x, y) |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
2 %CAUCHY Cauchy matrix. |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
3 % C = CAUCHY(X, Y), where X, Y are N-vectors, is the N-by-N matrix |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
4 % with C(i,j) = 1/(X(i)+Y(j)). By default, Y = X. |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
5 % Special case: if X is a scalar CAUCHY(X) is the same as CAUCHY(1:X). |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
6 % Explicit formulas are known for DET(C) (which is nonzero if X and Y |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
7 % both have distinct elements) and the elements of INV(C). |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
8 % C is totally positive if 0 < X(1) < ... < X(N) and |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
9 % 0 < Y(1) < ... < Y(N). |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
10 |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
11 % References: |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
12 % N.J. Higham, Accuracy and Stability of Numerical Algorithms, |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
13 % Society for Industrial and Applied Mathematics, Philadelphia, PA, |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
14 % USA, 1996; sec. 26.1. |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
15 % D.E. Knuth, The Art of Computer Programming, Volume 1, |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
16 % Fundamental Algorithms, second edition, Addison-Wesley, Reading, |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
17 % Massachusetts, 1973, p. 36. |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
18 % E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
19 % Linear Algebra and Appl., 149 (1991), pp. 1-18. |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
20 % O. Taussky and M. Marcus, Eigenvalues of finite matrices, in |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
21 % Survey of Numerical Analysis, J. Todd, ed., McGraw-Hill, New York, |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
22 % pp. 279-313, 1962. (States the totally positive property on p. 295.) |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
23 |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
24 n = max(size(x)); |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
25 % Handle scalar x. |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
26 if n == 1 |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
27 n = x; |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
28 x = 1:n; |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
29 end |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
30 |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
31 if nargin == 1, y = x; end |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
32 |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
33 x = x(:); y = y(:); % Ensure x and y are column vectors. |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
34 if any(size(x) ~= size(y)) |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
35 error('Parameter vectors must be of same dimension.') |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
36 end |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
37 |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
38 C = x*ones(1,n) + ones(n,1)*y.'; |
8f23314345f4
Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
parents:
diff
changeset
|
39 C = ones(n) ./ C; |