Federico Poloni
Reputation
7,735
303/400 score
 5h comment Find optimal value for a regularization parameter in generalized eigenvalue problem Is this a problem with large and sparse matrices? Because otherwise you shouldn't be doing inverses to compute eigenvalues, but rather use a backward stable method such as QZ, which avoids the issue completely. Feb 1 comment Have you solved problems in your sleep? I am sure lots of people have had great ideas while sitting on the toilet, but this doesn't get mentioned as often as solving problems while sleeping because it's less sexy. Jan 31 comment How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$? @hamidkamali among the hypotheses of that theorem there is "if $A$ has no eigenvalues in $\mathbb{R}^-$" --- so it works only if $A$ has no nonpositive real eigenvalues. Jan 31 comment How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$? In the applied literature, usually one defines $\log A$ only for matrices having no nonpositive real eigenvalues, and takes the principal value with eigenvalues in the strip $-\pi < \Im \lambda < \pi$. For the definition, one can use the Jordan decomposition or Cauchy integral definition here. Reference: Higham, Functions of matrices, beginning of Chapter 11. Jan 30 comment Is formula valid for relating $\pi$ with ALL of its OEIS A002485(n)/A002486(n) convergents? OP asked something very similar also on scicomp.stackexchange: scicomp.stackexchange.com/questions/21918/… Jan 28 reviewed Approve Upper bound for the minimum number of generators of the canonical module Jan 25 comment Evaluate a Function to Full Machine Precision @user85729 Step 1, use a method to compute a Padé (or min-max) approximant with sufficiently low error on all of [-1,1]. Take as much time as needed, this is a precomputation step. Step 2, store its coefficients, and write a function that applies this rational function to $x$. Jan 25 comment Is there any way to approximate the largest root of a polynomial? Please edit your original question for clarification, don't ask a new one. Jan 25 comment Is there a limit definition for the roots of a polynomial with arbitrary degree? Is your goal numerical computation? Or do you just want to express the largest root of a polynomial as a limit for theoretical purposes (for instance, you are looking for a proof strategy)? Jan 22 reviewed Reject Plugging $1-x$ into Schur polynomials Jan 22 reviewed Reject Does a Riemannian manifold have a triangulation with quantitative bounds? Jan 22 reviewed Approve Constrained optimization with known first and second derivatives Jan 22 revised Why Householder reflection is better than Givens rotation in dense linear algebra? made clearer that the cost is for a full QR factorization Jan 20 reviewed Approve hankel-matrices tag wiki excerpt Jan 16 revised Smallest length of {0,1} vectors to satisfy some orthogonality conditions sets->vectors; removed specification of orthogonality conditions Jan 15 reviewed Approve Example of progressively measurable process that is not predictable Jan 15 revised Why is the doubling dimension of any net of a metric space at most half of that of the metric space? fixed title Jan 8 comment How to find a subset of a matrix that has minimum condition number? In my answer to the duplicate question, I have a link to a Matlab toolbox (written by me) that solves a similar problem, namely, it finds a $m\times m$ submatrix $B$ of a $m\times n$ matrix $A$ such that $B^{-1}A$ has all entries bounded by 1. In my application at least, this property is sufficient to get good numerical properties. It should also possible to bound explicitly $\kappa(B)/\kappa(A)$. Jan 8 revised Optimizing the condition number explained relevance of the determinant Jan 7 reviewed Approve Solving Non-Linear Equations over a Finite Field of a Large Prime Order