Solving Higher Order Matrix Polynomials

7 次查看（过去 30 天）
Jing
Jing 2012-3-15
Hi, guys:
I wonder if there is some numerical method to solve a general matrix polynomial in the form: I+A1*X+A2*X^2+...+Aq*X^q=0. where X is supposed to be a matrix with the same dimension as As. A1 to Aq are k-by-k square matrices.
Any hint or reference is highly appreciated.
  2 个评论
Walter Roberson
Walter Roberson 2012-3-15
To cross-check, q is a positive integer?
Jing
Jing 2012-3-16
Yes, q is a positive integer. q=1 is a simple question, but cases for q>=2 are much more complicated. If k=1, which is just a scalar case, we can always covert it to a matrix 1st order equation and the solution is simple. However, to stack matrix up to reduce the order does not work for the multivariate case.

请先登录，再进行评论。

请先登录，再回答此问题。

回答（1 个）

Teja Muppirala
Teja Muppirala 2012-3-22
If k and q are not too large, one idea is to try to solve it as an optimization problem.
"Which elements of X will yield the smallest norm of the residual"
(you will have to save this as a function):
function solvepoly
k = 4;
qmax = 3;
for q = 1:qmax
A(:,:,q) = randn(k); %Make some random A matrices
end
[xf,fval] = fminunc(@doCost,zeros(k))
function COST = doCost(x)
COST = eye(k);
for q = 1:qmax
COST = COST + A(:,:,q)*x^q;
end
COST = sum(COST(:).^2);
end
end
  1 个评论
Jing
Jing 2012-3-22
Teja:
Thanks for the input. This searching mechanism can give a good approximation for the solution. However, I wonder if there are more efficient ways, since it is pretty easy to calculate the eigenvalues of this polynomial. The main problem is how to use these eigenvalues efficiently. In addition, There will be multiple solutions for sure, how can way make sure the searching mechanism to converge.

请先登录，再进行评论。

类别

在 Help Center 和 File Exchange 中查找有关 Linear Algebra 的更多信息

标签

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by