# Questions tagged [numerics]

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### Scientific standards for numerical errors

In my field of research the specification of experimental errors is commonly accepted and publications which fail to provide them are highly criticized. At the same time I often find that results of ...
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### What's the state of the art in parallel ODE methods?

I'm currently looking into parallel methods for ODE integration. There is a lot of new and old literature out there describing a wide range of approaches, but I haven't found any recent surveys or ...
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### What's the state-of-the-art in highly oscillatory integral computation?

What's the state-of-the-art in the approximation of highly oscillatory integrals in both one dimension and higher dimensions to arbitrary precision?
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### Catastrophic cancellation in logsum

I'm trying to implement the following function in double-precision floating point with low relative error: $$\mathrm{logsum}(x,y) = \log(\exp(x) + \exp(y))$$ This is used extensively in statistical ...
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### What are differences between 'a priori' and 'posteriori' error estimate in numerical analysis?

I have learnt about Finite Element Method (also a little on other numerical methods) but I don't know what are exactly definition of these two errors and differences between them?
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### Efficient computation of the matrix square root inverse

A common problem in statistics is computing the square root inverse of a symmetric positive definite matrix. What would be the most efficient way of computing this? I came across some literature (...
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### How do you debug numerical code, what could be source of this oscillatory error?

Quiet a lot of insight can be gained form experience, I was just wondering if anybody has seen something similar to this before. The plot shows the initial condition (green) for the advection-...
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### How should boundary conditions be applied when using finite-volume method?

Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh, I would like to apply a Robin type boundary condition to the l.h.s. of the domain (...
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### Puzzling remark about stability region of fifth-order Runge-Kutta method

I came across a puzzling remark in the paper P. J. van der Houwen, The development of Runge-Kutta methods for partial differential equations, Appl. Num. Math. 20:261, 1996 On lines 8ff on page 264, ...
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Let $\vec{x} = (x_1, x_2, \dots, x_n) \in [0,1]^n$ and $f(\vec{x}): [0,1]^n \to \mathbb{C}$ be a function in these variables. Is there a recursive scheme for this iterated integral? $$\int_{[0,1]^n}... 2answers 359 views ### Oscillations in singularly perturbed reaction-diffusion problems with finite elements When FEM-discretizing and solving a reaction-diffusion problem, e.g.,$$ - \varepsilon \Delta u + u = 1 \text{ on } \Omega\\ u = 0 \text{ on } \partial\Omega $$with 0 < \varepsilon \ll 1 (... 1answer 7k views ### Applying the Runge-Kutta method to second order ODEs How can I replace the Euler method by Runge-Kutta 4th order to determine the free fall motion in not constant gravitional magnitude (eg. free fall from 10 000 km above ground)? So far I wrote simple ... 1answer 891 views ### Numerical methods for inverting integral transforms? I'm trying to numerically invert the following integral transform:$$F(y) = \int_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_0\left(xy\right)f(x)\;\mathrm{d}x$$So for a given F(y) ... 3answers 1k views ### How should non-constant coefficients be treated with finite-volume first order upwind scheme? Starting with the advection equation in conservation form.$$ u_t = (a(x)u)_x $$where a(x) is a velocity which depend on space, and u is a concentration of a species which is conserved. ... 1answer 449 views ### scale invariance for line-search and trust region algorithms In Nocedal & Wright's book on Numerical Optimization, there is a statement in section 2.2 (page 27), "Generally speaking, it is easier to preserve scale invariance for line search algorithms than ... 2answers 614 views ### About faster approximation of log(x) I had written a code a while ago which attempted to calculate log(x) without using library functions. Yesterday, I was reviewing the old code, and I tried to make it as fast as possible, (and ... 5answers 4k views ### Integral in log-log space I'm working with functions which, in general, are much smoother and better behaved in log-log space --- so that's where I perform interpolation/extrapolation, etc, and that works very well. Is there ... 2answers 528 views ### How much regularization to add to make SVD stable? I've been using Intel MKL's SVD (dgesvd through SciPy) and noticed that results are are significantly different when I change precision between ... 4answers 627 views ### Relevance of fixed-point and arbitrary precision computations I see very few non-floating point computing libraries/packages around. Given the various inaccuracies of floating point representation, the question arises why there aren't at least some fields where ... 3answers 494 views ### Regression testing of chaotic numerical models When we have a numerical model that represents a real physical system, and that exhibits chaos (e.g. fluid dynamics models, climate models), how can we know that the model is performing as it should? ... 2answers 477 views ### Numerical method for equation solving that works on stochastically computed functions There are many well known numerical methods for solving equations of the type$$ f(x) = 0, \quad x \in \mathbb{R}^n,$$e.g. bisection method, Newton's method, etc. In my application f(x) is ... 1answer 142 views ### Are there shortcuts for numerically approximating systems of ordinary differential equations when autonomous? Existing algorithms for solving ODEs handle functions \frac{dy}{dt} = f(y, t), where y \in \mathbb R^n. But in many physical systems, the differential equation is autonomous, so \frac{dy}{dt} = f(... 1answer 250 views ### Solving a difficult system of equations numerically I have a system of n non-linear equations that I want to solve numerically:$$\mathbf{f}(\mathbf{x})=\mathbf{a}\mathbf{f}=(f_1,\dots,f_n)\quad\mathbf{x}=(x_1,\dots,x_n)$$This system has a ... 1answer 117 views ### 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 ... 2answers 4k views ### Which Runge-Kutta method is more accurate: Dormand-Prince or Cash-Karp? I simply want to know whether the Dormand-Prince Numerical Method or the Cash-Karp Numerical Method is more accurate. 2answers 7k views ### Use of machine learning in computational fluid dynamics Background: I have only built one working numeric solution to 2d Navier-Stokes, for a course. It was a solution for lid-driven cavity flow. The course, however, discussed a number of schemas for ... 3answers 4k views ### Scientific computing vs numerical analysis I'm a double major in computer science and mathematics. I love both subjects. I'm thinking in taking a graduate career, perhaps in scientific computing. What's the real difference between scientific ... 3answers 3k views ### Finite Element Method vs Extended Finite Element Method (FEM vs XFEM) What are main differences between FEM and XFEM? When should we (not) use XFEM intead of FEM and vice versa? In other words, when I meet a new problem, how I can know to use which one of them? 2answers 2k views ### Help deciding between cubic and quadratic interpolation in line search I'm performing a line search as part of a quasi-Newton BFGS algorithm. In one step of the line search I use a cubic interpolation to move closer to the local minimizer. Let f : R \rightarrow R, f \... 2answers 1k views ### What does the Von Neumann's stability analysis tell us about non-linear finite difference equations? I am reading a paper  where they solve the following non-linear equation \begin{equation} u_t + u_x + uu_x - u_{xxt} = 0 \end{equation} using finite difference methods. They also analyse the ... 1answer 420 views ### Algorithm to calculate the exponential of an Hessenberg matrix I am interested in computing the solution of a lage system of ODEs using a krylov method as in . Such method involve functions related to the exponential (the so-called \varphi-functions). It ... 1answer 257 views ### Iterative “solver” for x^t \Sigma^{-1} x I can't imagine I'm the first to think about the following problem, so I'll be satisfied with a reference (but a complete, detailed answer is always appreciated): Say you have a symmetric positive ... 5answers 763 views ### Choice of basis in FEM Briefly, what are the different types of basis used in FEM and why is the nodal basis so popular and advantageous in the finite element context? 2answers 1k views ### Does the matrix condition number affect accuracy of iterative linear solvers? I have a rather specific question regarding the condition number. I run FEM simulations which have multiple length scales to them which results in a huge disparity between the largest entries and the ... 2answers 263 views ### Astoundingly large difference when evaulating trigonometric identity with NumPy According to Wolfram Alpha and the Sage computer algebra system, the following identity holds:$$ \cos\left(\arctan\left(\frac{l_1-l_2}{d}\right)\right) = \frac{1}{\sqrt{1 + \frac{(l_1-l_2)^2}{d^2}}} $... 5answers 929 views ### Is Discrete Exterior Calculus currently a focusing point in numerial computing world or simulation in industry, I am just wondering if Discrete Exterior Calculus, as a new numerical method , is good at numericall solving problems in elasticity, fluids or other physical/real area. 2answers 128 views ### How can I numerically solve an ODE to$N$provably correct digits? Suppose we have an initial value problem of the form $$\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t} = f(\mathbf{x}) \qquad \mathbf{x}(0) = \mathbf{x}_0$$ where$\mathbf{x}_0 \in \mathbb{R}^n$is known ... 2answers 311 views ### Lagrange multipliers space is too rich in a mathematical view Background: Lagrange multiplier method has been employed in numerous fields, such as contact problems, material interfaces, phase transformation, stiff constraints or sliding along interfaces. It is ... 1answer 2k views ### Fortran 90/95: Deallocating variables I understand the crucial importance of freeing memory when certain variables or arrays need to be reused later in the program, or may not be in use for a while. However, in my experience with ... 1answer 191 views ### Solving two inverse problems with same solution I've got two inverse problems, $$A_1 ~ x = b_1 \qquad A_2 ~ x = b_2$$ So far I've been solving them independently using Tikhonov Regularization and getting two estimates for$x$. However in my case$...
The equations of motion for an elastic solid are given by \begin{align} &\nabla \cdot \boldsymbol{\sigma} + \mathbf{f} = \rho \ddot{\mathbf{u}}\\ &\boldsymbol{\sigma} = \mathbb{C}\...
It's a fairly well known trick to avoid division in calculating square-roots to apply Newton's method to finding $1/\sqrt{x}$, and probably better known, using Newton's method to find reciprocals ...