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Heuristic check of numerical stability

Assume I have a real valued function $f(x_1,\ldots ,x_N)$ of some variables $x_i$ which I want to evaluate numerically. In general the formula for $f$ can contain products, rationals, trancendental functions etc. and will be to long to investigate…
highsciguy
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Algorithms for linear system of ODEs

I wonder: what is the best algorithm to solve \begin{equation} \frac{du}{dt} = Au \end{equation} Where $A$ is a real $n\times n$ matrix. A is not explicitly time-dependent, usually sparse but not necessarily banded. Its eigenvalues have…
Gabriel Landi
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Absolute Value in Linear Constraints

I have the following optimization problem where I have absolute value in my constraints: Let $\mathbf{x} \in \mathbb{R}^n$ and $\mathbf{f}_0, \mathbf{f}_1, \ldots, \mathbf{f}_m$ be column vectors of size $n$ each. We would like to solve the…
Mohammad Fawaz
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Higher precision floating-point arithmetic in numerical PDE

I have the impression, from very different resources and talks with researches, that there is a growing demand for high precision computations in numerical partial differential equations. Here, high precision means more precision than just the…
shuhalo
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Numeric integration of multi-dimensional integral with known boundaries

I have a (2-dimensional) improper integral $$I=\int_A \frac{W(x,y)}{F(x,y)}\,\mbox{d}x\mbox{d}y$$ where the domain of integration $A$ is smaller than $x=[-1,1]$, $y=[-1,1]$ but further restricted by $F(x,y)>0$ . Since $F$ and $W$ are smooth and $W…
highsciguy
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good (free) software for producing publishable images?

I am producing 1d and 2d images using Matlab right now for comparison of accuracy against a given model. I need to compare my methods with the standard Gaussian .wfn model and I am going to do that by analyzing the density of a molecule as well as…
drjrm3
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For noisy or fine-structured data, are there better quadratures than the midpoint rule?

Only the first two sections of this long question are essential. The others are just for illustration. Background Advanced quadratures such as higher-degree composite Newton–Cotes, Gauß–Legendre, and Romberg seem to be mainly intended for cases…
Wrzlprmft
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For software submitted to ACM TOMS, how does the ACM software license agreement interact with other licenses?

The journal Association for Computing Machinery Transactions on Mathematical Software (ACM TOMS) publishes many articles on numerical algorithms that include software implementations. According to their editorial policy, submission of an algorithm…
Geoff Oxberry
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Inverse problem in linear ODE

I have a linear ordinary differential equation (ODE) with a system matrix with constant coefficients: $$\dot{y}(t) = \mathcal{A}\; y(t), \quad y(0) = y_0$$ with $y(t) \in \mathbb{R}^{n \times 1}$ and $\mathcal{A} \in \mathbb{R}^{n \times n}$ where…
GertVdE
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Rapidly determining whether or not a dense matrix is of low rank

In a software project that I'm working on, certain computations are vastly easier for dense low-rank matrices. Some problem instances involve dense low-rank matrices, but they're given to me in full, rather than as factors, so I'll have to check…
Brian Borchers
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DG local equation, how to interpret mean-averaged test function

In the paper http://www.sciencedirect.com/science/article/pii/S0045782509003521, an HDG element-local equation is described on page 584 equation (4), with one of the equations taking the following form $$-(u_h,\nabla q)_K = -\left\langle\hat{u}_h…
user3482876
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Libraries for solving Lyapunov's equation

The following matrix equation $$B\Sigma + \Sigma B^T + C = 0$$ in $\Sigma$ $-$ for given $B$ and $C$ matrices $-$ appears in my work as a characterization of a covariance matrix. I have learned that this equation is known, in particularly in…
NRH
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Understanding the Courant–Friedrichs–Lewy condition

I understand these equations in particular can be solved easily without use of computational methods. Although right now I am concerned with trying to solve these equations using numerical integration strictly. $$ \frac {\partial ^2 E}{\partial t…
Mathews24
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Costs of lookups versus calculations

I am interested in setting up calculations to check if a distance criterion is satisfied: that is, the distance between a vector ${\bf x}_i$ and anther vector ${\bf x}_j$ should be less than some value $r_{\rm max}$. My data is partitioned according…
aeismail
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Newton-based methods in optimization vs. solving systems of nonlinear equations

I asked for clarification about a recent question about minpack, and got the following comment: Any system of equations is equivalent to an optimization problem, which is why Newton-based methods in optimization look a lot like Newton-based methods…
Thomas Klimpel
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