Timeline for On solution of a discrete-time equation
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| May 1, 2013 at 12:35 | vote | accept | eolithr | ||
| Apr 29, 2013 at 9:01 | comment | added | eolithr | Ok, I will post it later. Maybe vecotorize method together with Kronecker product will work. Thanks a lot. | |
| Apr 29, 2013 at 8:58 | comment | added | eolithr | Let's consider a more complex situation, i.e. $$X=F_{1}XF^{1}^{T}+...+F_{p}XF^{p}^{T}+C$$ where $p$ is an positive integer. If we augment $F=[F_{1}...F_{p}]$ and $Y=diag{X...X}$, then the equation becomes $$FYF^{T}-[I...0]Y[I...0]^{T}+C=$$ seems like a generalized Lyapunov equation. However, there is a constraint on $Y$ for its diagonal form. How to compute $X$? | |
| Apr 29, 2013 at 8:55 | comment | added | Federico Poloni | For a general linear matrix equation with more than two terms, I am not aware of any method faster than the $O(n^6)$ of the naive "vectorize everything" algorithm. (caveat: I know almost nothing of LMIs, so there might be a method in that area that I do not know about). Maybe you'll have more luck posting this as a separate question, either here or on scicomp.stackexchange.com. | |
| Apr 29, 2013 at 7:41 | comment | added | eolithr | Very very nice comments, Dr. Federico. Thanks. | |
| Apr 29, 2013 at 7:10 | comment | added | Federico Poloni | it more stable, and ultimately obtain a discrete-time version of the ADI method for Lyapunov equations. In short, with some algebraic manipulations everything that works in the continuous-time case rates to work in the discrete-time case as well. | |
| Apr 29, 2013 at 7:09 | comment | added | Federico Poloni |
Yep. dlyap will be fine for a small-scale problem; it should use a variant of the same Bartels-Stewart method that is used for continous-time lyapunov eqs; essentially, take a Schur form of $F$ and solve directly via back-substitution for each entry of $X$ "in the right order". For large-scale problems, you can truncate the series $X=\sum_{i=0}^{\infty} F^{i}CF^{Ti}$, or obtain the partial sum truncated at the term $2^{k}$ directly from the one truncated at $2^{k-1}$ with some manipulations (Smith methods). You can apply some Möbius transforms to $F$ without changing the solution to make...
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| Apr 29, 2013 at 3:04 | comment | added | eolithr | No, I made the problem too complex. By using 'dlyap' in Matlab can solve this equation. | |
| Apr 29, 2013 at 1:07 | comment | added | eolithr | Thanks for your kind help, Dr. Carlo and Dr. Federico. When $F$ is singular, The Sylvester method may fail to work. So, would you please give me some links or references on the equation $X-FXF^{T}=C$. It seems we can solve it using LMI techniques for obtaining numerical solutions. | |
| Apr 28, 2013 at 18:37 | comment | added | Federico Poloni | There are numerical methods dealing directly with the equation $X-FXF^T=C$, known as discrete-time Lyapunov equation, or Stein equation. There is no need to invert $F$ and convert it to a Sylvester; sometimes algorithms even go the other way round and convert a continuous-time Lyapunov equation to a discrete-time one to solve it. | |
| Apr 28, 2013 at 17:53 | history | answered | Carlo Beenakker | CC BY-SA 3.0 |