Federico Poloni
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Registered User
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Numerical linear algebra. Matrix equations + control.
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16h |
awarded | ● Nice Answer |
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1d |
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how to solve this multivariate quadratic equation? Can you add a link to the discussion on math.stackexchange? |
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2d |
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Eigenvalues of Symmetric Tridiagonal Matrices Another argument is: it's easy to reduce any symmetric matrix to tridiagonal with similarity transforms. So if this problem were easy to solve, all symmetric eigenproblems would be. |
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2d |
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Solve $(A+B)x=y$ given Cholesky decomposition of A and B In general, no. One can make low-rank updates of factorization easily, but for anything different the usual answer is "you have to compute everything from scratch". Maybe you can use CG with $A$ (or $B$) as a preconditioner, but that will be numerically convenient only in some special cases. |
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2d |
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fixedpoint or fixed point or fixed-point I am not familiar with that definition --- I call pre-fixed point a point $x$ such that $f(x)\neq x$, but $f^k(x)=f^{k+1}(x)$ for some $k\geq 1$. |
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2d |
answered | Why don’t more mathematicians improve Wikipedia articles? |
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2d |
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fixedpoint or fixed point or fixed-point "prefixed" here most likely comes from "pre-fixed", not "prefix+ed". I guess this shows once more the importance of using hyphens. :) |
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May 22 |
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What can be said about pairs of matrices P,Q that satisfies $(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$ ? Related question: mathoverflow.net/questions/63027/…. I'd start from the book suggested in the answer, Horn and Johnson's Topics in Matrix Analysis (not to be confused with Matrix Analysis by the same authors). |
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May 20 |
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What does “Vertex Solution” mean? A bit more context would help. |
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May 17 |
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How to maximize the determinant of a matrix of the form VDV^H Not sure about that: $V^HV$ is a rank-$M$ $2M\times 2M$ matrix, so its determinant would be zero. |
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May 15 |
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spectral radius monotonicity You are right! One needs an additional hypothesis (irreducibility) to get strict inequalities in P-F. |
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May 15 |
revised |
spectral radius monotonicity added 15 characters in body |
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May 14 |
revised |
Can an accumulation point be an eigenvalue? deleted 20 characters in body |
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May 13 |
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Eigenvalues of random Hamiltonian matrices I must be missing something then. Doesn't conjecture (1) (proved in a paper) say that they should be uniformly distributed in the limit? |
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May 13 |
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Eigenvalues of random Hamiltonian matrices It is curious that there seems to be an area with less eigenvalues around the imaginary axis (for Hamiltonian) or the real axis (for Ginibre). Is it an artifact of the numerical methods? I can imagine them having trouble in identifying properly purely imaginary/real eigenvalues. |
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May 6 |
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Spectral Properties of $A(I-A)^{-1}$ T is triangular. en.wikipedia.org/wiki/Schur_form |
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May 6 |
accepted | spectral radius monotonicity |
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May 5 |
answered | spectral radius monotonicity |
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May 1 |
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Problems where Conjugate gradient works much better than GMRES From the look of it, yes, but there is probably a completely theoretical question in matrix approximation theory hidden behind this one. |
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Apr 29 |
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On solution of a discrete-time equation 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. |
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Apr 29 |
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On solution of a discrete-time equation 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. |
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Apr 29 |
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On solution of a discrete-time equation 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 28 |
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On solution of a discrete-time equation 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. |
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Apr 25 |
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naming for the map $T = x \mapsto a x b$ Since you are back here: thought that the name "discrete-time Sylvester operator" makes a lot of sense for the map $S:X\to X-AXB$, since a discrete-time Sylvester equation can be written $S(x)=C$, analogously with other similarly-named operator. The name with commas returns a whopping two Google results, but I think that a person working in matrix equation would understand immediately what you're talking about if you use it. |
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Apr 22 |
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The first eigenvalue of a branching process matrix You are right, sorry for forgetting this part! |
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Apr 22 |
answered | The first eigenvalue of a branching process matrix |
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Apr 22 |
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The first eigenvalue of a branching process matrix The second "smaller" should be "larger". |
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Apr 21 |
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On solution of a recursion with rectangular matrices What is the source of this problem? Looks a lot like a discrete-time difference Riccati equation. This, and what Robert Israel said. |
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Apr 18 |
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Products of matrices of a certain form They also happen to be semiseparable. |
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Apr 18 |
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Control a linear system to the kernal space of the output matrix. Maybe it would help if you added $(t)$ and quantifiers such as $\forall t \in \mathbb{R}$ in the right places... |
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Apr 17 |
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Control a linear system to the kernal space of the output matrix. If $Cx=0 \Rightarrow x=0$, then the two tasks are equivalent, isn't it? Is there maybe a typo in your question? |
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Apr 16 |
accepted | Efficient computation of Markov chain transition probability matrix |
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Apr 15 |
answered | Efficient computation of Markov chain transition probability matrix |
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Apr 14 |
revised |
Old books still used publication date edited in, mathscinet cites |
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Apr 8 |
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Global solution for spectral clustering Please don't cross-post to two different sites before waiting at least a few days without answers. Your question is only 3 hours old, and you already got a seemingly good answer on math.se. |
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Apr 8 |
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Numerical evaluation of a triple integral can be made ? What is your question exactly? You can always write that expression; of course it will not be equal to the exact integral, but it will converge to it (slowly) under mild hypotheses. |
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Apr 2 |
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Strassen’s algorithm If you remove it, and work in finite precision arithmetic, then you can replace an arithmetic operation with a bazillion if's and assignments. But I guess that it can be removed, if one restricts to algorithms that do not "cheat" using real number representations. |
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Apr 2 |
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Strassen’s algorithm That is the same algorithm that I had in mind. |
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Apr 1 |
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Strassen’s algorithm @Dima Pasechnik not necessarily. For instance, you can easily compute the square of a real 2x2 matrix with only 5 multiplications, but this is an algorithm that exploits commutativity, and thus cannot be recast in bilinear "tensor language". |
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Apr 1 |
answered | Strassen’s algorithm |
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Mar 31 |
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Does this matrix shape have a name? @AaronMeyerowitz not that $J$ is a standard notation for the all-ones matrix... |
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Mar 15 |
revised |
Multiple Number Partitioning / “Multiprocessor Scheduling” typo in tag |
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Mar 6 |
accepted | Nth root of a matrix as an analytic function? |
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Feb 24 |
answered | Nth root of a matrix as an analytic function? |
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Feb 20 |
accepted | Sum of elements of inverse matrix |
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Feb 19 |
revised |
Fiction books about mathematicians? expanded, incorporated suggestions |
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Feb 19 |
answered | Fiction books about mathematicians? |
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Feb 18 |
answered | Sum of elements of inverse matrix |
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Feb 1 |
answered | Compute roots of sum_i c_i/(a_i + b_i x)^p |
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Jan 24 |
asked | Kronecker-structured matrix kernel |
