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1500 questions
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votes
4 answers

fastest linear system solve for small square matrices (10x10)

I am very interested in optimizing the hell out of linear system solving for small matrices (10x10), sometimes called tiny matrices. Is there a ready solution for this? The matrix can be assumed nonsingular. This solver is to be executed in excess…
rfabbri
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How to intellligently attempt to rule out convexity?

I want to minimize a complicated objective function, and I'm not sure if it is convex. Is there a nice algorithm that attempts to prove that it is not convex? Of course the algorithm could fail to prove this, in which case I would not know if it…
MLE
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2 answers

Numerical stability of higher order Zernike polynomials

I'm trying to calculate higher order (e.g., m=0, n=46) Zernike moments for some image. However, I'm running into a problem regarding the radial polynomial (see wikipedia). This is a polynomial defined on the interval [0 1]. See the MATLAB code…
Sanchises
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2 answers

Initially Bracketing Minimum for Line Search

Leafing through a few textbooks, I've noticed that the problem of initially bracketing a minimum during a line search tends be an afterthought (at least in my undergraduate texts). Are there well-established techniques or best practices for this…
Christopher Johnson
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3 answers

How does the QR algorithm applied to a real matrix returns complex eigenvalues?

I'm a noob into eigenvalues algorithms, but something call my attention. QR algorithm works with real/complex matrices producing real/complex eigenvalues. However, it can not produce complex eigenvalues from a real matrix. Here a simplistic example…
Noel Araujo
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Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system

Problem Solving a non-linear system of equations. The number of variables is the same as the number of equations. When I fix a set of variables (say $\vec{y}$) and keep another set free (say $\vec{x}$), the system becomes an under-determined, dense,…
R zu
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Review of modern homotopy methods and practical techniques

I'm hoping someone can recommend recent literature concerning homotopy methods for solving systems of nonlinear equations. Already by the time of Layne Watson's 1986 paper there were a lot of methods, and I'm sure there have been developments since…
OskarM
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3 answers

Basin of attraction for Newton's method

Newton's method for solving nonlinear equations is known to converge quadratically when the starting guess is "sufficiently close" to the solution. What is "sufficiently close"? Is there literature about the structure of this basin of attraction?
David Ketcheson
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3 answers

What norm to choose when?

Recently, I saw this question: how to measure the error of a finite difference method I am student of simulation sciences and unfortunately, for me, it's totally unclear, what norm to use in what context. Quite often, we use the Euclidian norm or…
vanCompute
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3 answers

Galerkin method: Test functions vs. Basis functions

I'm a novice to finite element and I'm finding quite hard to find the actual difference between Test function(s) and Basis function(s). I would be glad if somone could explain me that and point out how can they differ from one another and in which…
arocha
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Easily understandable argument that normal Runge–Kutta methods cannot be generalised to SDEs?

A naïve approach to solving stochastic differential equations (SDEs) would be: take a regular multi-step Runge–Kutta method, use a sufficiently fine discretisation of the underlying Wiener process, make each step of the Runge–Kutta method analogous…
Wrzlprmft
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Updatable SVD implementation in Python, C, or Fortran?

I would like to do evolving factor analysis using the SVD: Given $m \times n$ data matrix $\mathcal{A}$, and for each $i$ from 1 to $m$, I want to calculate the singular values of: $$\mathcal{A}\left[1:i,\ 1:n\right]$$ This would be much faster if…
tillsten
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2 answers

Eigenvectors of a small norm adjustment

I have a dataset that is slowly changing, and I need to keep track of eigenvectors/eigenvalues of its covariance matrix. I've been using scipy.linalg.eigh, but it's too expensive, and it doesn't use the fact that I already have a decomposition…
Yaroslav Bulatov
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1 answer

CFL condition in polar coordinates

In this question, I suggested that the Couran-Friedrichs-Lewy (CFL) condition for the wave equation in polar coordinates reads $$C = 2c\frac{\Delta t}{\Delta r \Delta \phi} \leq C_\max \enspace ,$$ where $c$ is the phase speed. I suggested this from…
nicoguaro
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Convergence rate vs convergence order

I'm a bit confused about the concepts of convergence rate and convergence order. Let me first give you the definitions we use: [sorry for the English, it's all self translated] Let $x^{*}$ be our solution. Definition 1: The sequence $x^{(k)}$ is…
xotix
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