임의의 6x6 행렬의 고유방정식. 역행렬 구하기
케일리-해밀턴의 정리와 Bocher의 공식을 이용한 Faddev-Leverrier 알고리즘 사용.
function [p,Ainv,B]=fadlev(A)
%
% FADLEV Faddeev-Leverrier approach to generate coefficients of the
% characteristic polynomial and inverse of a given matrix
% Uasage: [p,Ainv,B]=fedlev(A)
%
% Input: A - the given matrix
% Output: p - the coefficient vector of the characteristic polynomial
% B - a cell array of the sequency of matrices generated, where
% B{1} = A p(1)=trace(B{1})
% B{2} = A*(B{1}-p(1)*I) p(2)=trace(B{2})/2
% .....
% B{n} = A*(B{n-1}-p(n-1)*I) p(n)=trace(B{n})/n
%
% Ainv - The inverse of A calculated as
% Ainv = (B{n-1}-p(n-1)*I)/p(n)
%
% Example:
%{
A=magic(5);
[p,B]=fadlev(A);
fprintf('Check inverse: norm(B-inv(A))=%g\n',norm(B-inv(A)));
fprintf('Check polynomial: norm(p-poly(A))=%g\n',norm(p-poly(A)));
%}
% By Yi Cao on 17 August 2007 at Cranfield University
%
% Reference:
% Vera Nikolaevna Faddeeva, "Computational Methods of Linear
% Algebra," (Translated From The Russian By Curtis D. Benster), Dover
% Publications Inc. N.Y., Date Published: 1959 ISBN: 0486604241.
%
[n,m]=size(A);
if n~=m
error('The given matrix is not square!');
end
[B{1:n}]=deal(A);
p=ones(1,n+1);
for k=2:n
p(k)=-trace(B{k-1})/(k-1);
B{k}=A*(B{k-1}+p(k)*eye(n));
end
p(n+1)=-trace(B{n})/n;
Ainv=-(B{n-1}+p(n)*eye(n))/p(n+1);
>> A=[1 2 3 4 5 6; 2 3 5 6 12 2; 3 23 4 6 12 23; 2 5 24 6 43 2; 3 12 54 5 1 2; 12 3 5 2 1 3]
A =
1 2 3 4 5 6
2 3 5 6 12 2
3 23 4 6 12 23
2 5 24 6 43 2
3 12 54 5 1 2
12 3 5 2 1 3
>> [p,Ainv,B]=fadlev(A)
p =
1 -18 -1405 -26815 -263490 -212314 *
Ainv =
-148/7879 -67/7061 -258/46063 31/6878 -125/15789 1831/20548
-975/4492 519/4615 241/4878 -87/4384 154/19527 -107/9168
377/9736 -78/1969 -122/13761 73/8453 51/3025 -5/80231
74/2301 689/3007 -291/14494 -191/3099 139/16468 -587/21201
-95/9764 -14/919 35/15581 99/3505 -30/2767 23/29071
319/1523 -317/2030 118/287291 93/4870 -99/15637 31/4513
B =
[6x6 double] [6x6 double] [6x6 double] [6x6 double] [6x6 double] [6x6 double]
