Computational Science Stack Exchange
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Polynomials that are orthogonal over curves in the complex plane

Various important sets of polynomials (Legendre, Chebyshev, etc.) are orthogonal over some real interval with some weighting. Are there known families of polynomials that are orthogonal over other curves in the complex plane? For instance, I would…
David Ketcheson
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Which C++ linear algebra library is probably the fastest on solving huge sparse [square matrix] linear system?

I am developing a 2D CFD solver for fluid-particle interaction. To solve Navier-Stokes equations on a grid of size $10000\times 10000$ cells (or >1 million cells), a large linear system $Ax=b$ with $A$ being the $10000\times10000$ sparse coefficient…
KOF
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How to find the nearest/a near positive definite from a given matrix?

I'm given a matrix. How do I find the nearest (or a near) positive definite from it? The matrix can have complex eigenvalues, not be symmetric, etc. However, all its entries are real valued. The resulting matrix from the algorithm must be positive…
An old man in the sea.
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Integer operations vs floating point operations

I have been working with an algorithm, which uses additions of floating point vectors, (sparse matrix of floats)x(dense vector of floats) dot products I recently found out that I can get the same functionality form the algorithm if I set…
Dionysios Georgiadis
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Are there shortcuts for numerically approximating systems of ordinary differential equations when autonomous?

Existing algorithms for solving ODEs handle functions $\frac{dy}{dt} = f(y, t)$, where $y \in \mathbb R^n$. But in many physical systems, the differential equation is autonomous, so $\frac{dy}{dt} = f(y)$, $y \in \mathbb R^n$, with the $t$ left out.…
procrastinating on actual work
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Benchmark problems for eigenvalue reordering algorithms sought

Every real matrix $A$ can be reduce to real Schur form $T = U^T A U$ using an orthogonal similiary transform $U$. Here the matrix $T$ is quasi-triangular form with 1 by 1 or 2 by 2 blocks on the main diagonal. Each 1 by 1 block corresponds to a…
Carl Christian
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Priorities for learning computational methods, when should I write my own code vs. using libraries as a beginning graduate student?

I am beginning my graduate studies in engineering and will be working on computational science projects. I noticed that there has been some discussion about the advantages and disadvantages of implementing your own algorithms here. Is the…
Godric Seer
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PDE discretization with the method of rothe and the method of lines (Modular implementation)

The Heat equation is discretized in space with FV (or FEM), and a semi-discrete equation is obtained (system of ODEs). This approach, known as the method of lines, allows to easily switch from one temporal discretization to another, without code…
gnzlbg
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Is there an algorithm to find an almost-convex hull given a tolerance angle?

I'd like to know if there is an algorithm that given a set o points and an angle computes the convex-hull if the angle is $\alpha = 0$ and given an $\alpha > 0$ computes an envelope that follows more closely the "perimeter". And if there is a…
naufraghi
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Explicit Euler method too slow for reaction-diffusion problem

I am solving Turing's reaction-diffusion system with following C++ code. It is too slow: for 128x128 pixel texture, acceptable number of iterations is 200 – which results in 2.5 seconds of delay. I need 400 iterations to obtain interesting image –…
AllCoder
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Central differencing scheme for second derivative leads to ill-conditioning

The central difference scheme: $$\frac{d^2u}{dx^2}=\frac{u_{n+1}-2u_i + u_{n-1}}{\Delta x^2}$$ yields a tridiagonal coefficient matrix [1 -2 1]; As the number of points gets larger, this matrix becomes ill-conditioned. Yet this is a popular…
ben142
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Resources on mesh generation for finite element methods

I know that this is not really apart of the rules as this is a recommendation question and these don't really have an answer per say. But, like this forum posting: https://stackoverflow.com/questions/388242/the-definitive-c-book-guide-and-list. To…
philm
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Diagonalization of Dense Ill Conditioned Matrices

I am trying to diagonalize some dense, ill-conditioned matrices. In machine precision, results are inaccurate (returning negative eigenvalues, eigenvectors do not have the expected symmetries). I switched over to Mathematica's Eigensystem[] function…
Leigh
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Commonly-used metrics to quantify the irregularity of a triangular mesh

Say you have a triangular mesh on a flat plane. This has been drawn to eventually solve some problem in mechanics, for example. A mesh of equilateral triangles is the best inasmuch as the distances between the vertices and between the centroids are…
XavierStuvw
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How much regularization to add to make SVD stable?

I've been using Intel MKL's SVD (dgesvd through SciPy) and noticed that results are are significantly different when I change precision between float32 and float64 when my matrix is badly conditioned/not full rank. Is there a guide on the minimum…
Yaroslav Bulatov
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