Widely used as a synonym for numerical-analysis, in particular in the German speaking community.

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Function similar to erf that is fast at scale and allows for changing the slope at 0?

I'm interested in a function that would allow me to weight my system similar to using the error function; however computing the error function at scale would be a bottleneck. Is there something like ...
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3answers
108 views

Accurate computation of the current time in time integrator

I implement Runge--Kutta method for time integration of the system of nonlinear conservation laws $$ u_t + f(u)_x = 0. $$ As the system is nonlinear, we have to recompute time step ...
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32 views

solving tridiagonal system with multiple right hand sides

I need to solve a tridiagonal system (positive definite, diagonally dominant) $Ax = b$ in a time stepping loop. $A \in \mathbb{R}^{N \times N}$ remains constant but $b$ changes during each time ...
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1answer
37 views

Conjugate Gradient, initial direction set to initial residual

In the (iterative) Conjugate Gradient (CG) algorithm: https://en.wikipedia.org/wiki/Conjugate_gradient_method The initial search direction $p_{0}$ is set to the initial residual $r_{0}$. But I can't ...
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0answers
55 views

Preconditioned Steepest Descent

For my program assignment I need to write a preconditioned steepest descent algorithm. I have psedo-code from my professor which is here: From reading in my text on this method they say that $P$ is ...
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3answers
91 views

combination of field and particle methods for fluid dynamics

In numerical fluid dynamics there are field methods like finite-volume, finite-element, etc. and particle methods like Smoothed-Particle-Hydrodynamics – SPH and others. Both approaches have advantages ...
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1answer
53 views

Problem with Richardson extrapolation method for weak convergence in SDE

I have implemented the Richardson extrapolation of the Euler-Maruyama method to 4th order, to estimate the moments of SDE. The Euler-Maruyama works, and I would expect the Richardson extrapolation to ...
7
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120 views

Finite difference recursion and higher order

This may be a trivial question, but I've always wondered... For the classical, central finite difference schemes, if I'm interested in determining the second derivative, does applying the first ...
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1answer
58 views

$LU$ factorization routine to solve system $Lv = f$ and $Uv = f$

I implemented an $LU$ factorization algorithm on c++ for a $n$ by $n$ matrix $A$, and I wanted to now write a routine that solves the systems $Lv = f$ and $Uv = f$. How would you recommend writing ...
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2answers
187 views

$LU$ factorization

Our task is to implement a factorization routine that given A in a suitably efficient data structure returns the factors $L$ and $U$ where $L$ is unit lower triangular and $U$ is upper triangular. In ...
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32 views

Unitary matrix representing Discrete Fourier Transform

Let $F_n\in\mathbb{C}^{n\times n}$ be the unitary matrix representing the discrete Fourier transform of length $n$ and so $F_n^{H}\in\mathbb{C}^{n\times n}$ is the inverse DFT of length $n$. For ...
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84 views

Are self-convergence tests reliable?

I'm developing a solver for solving linear hyperbolic equations of first order with respect to time and spatial derivatives. The formal order of accuracy of the solver must be 5 because I use ...
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70 views

Numerical inversion involved confluent hypergeometric (1F1) (or Kummer function)

Edit: the question is solved ! Thank you for your time ! This problem arises when I tried to compute the valua of Asian call otions using Inverse Laplace transform method. Let $r=\mu = 0.15; \sigma ...
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2answers
85 views

Sum over very small exponentials: Underflow

I am trying to compute (in C) a sum like $S = \sum_i \exp( - a_i )$, where $10^{4} < a_i < 10^{5}$ are approximately normal distributed. So even if I do the Log-Sum-Exp trick $S = ...
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0answers
46 views

convergence of one step methods, order $p$

This question is not homework it is recommended exercises to prepare for the final exam. Consider the family of linear one-step methods defined by $$y_n = y_{n-1} + h(\theta f_n + (1 - ...
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104 views

Family of linear one-step method convergence question

This question is not homework it is recommended exercises to prepare for the final exam. Consider the family of linear one-step methods defined by $$y_n = y_{n-1} + h(\theta f_n + (1 - ...
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1answer
115 views

How do I develop numerical routines for the evaluation of my own special functions?

This question was previously posted to Math.SE here and had received no answers at the time of this posting. When performing computational work, I often come across a univariate function, defined ...
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2answers
179 views

Does artifical dissipation term makes scheme inconsistent?

Central schemes like JST uses artificial dissipation for the stabilization. This modification is an artificial one. Does this additional term makes system inconsistent? Can we expect this term to be ...
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1answer
57 views

How do I add some floating point numbers, keeping numerical accuracy in mind?

I am solving a problem involving the line with the set of points $(x_3,y_3)$ that are equidistant to two given points $(x_1,y_1)$ and $(x_2,y_2)$. The equation for this line is $$(x_3 - x_1)^2 + (y_3 ...
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171 views

C++ library for numerical intergration (quadrature)

I have my own little subroutine for numerical integration (quadrature), which is a C++ adaptation of an ALGOL program published by Bulirsch & Stoer in 1967 (Numerische Mathematik, 9, 271-278). I ...
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1answer
55 views

Big errors while calculating Complex Cholesky Factorization

I am using my own Routine to calculate the Cholesky-Factorization of a complex, positive definite symmetric Matrix. My Code Looks like this: ...
3
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1answer
80 views

Numerical integration with singularities

I need to compute some integrals numerically. The integrand is this: $$f(x,y) = \left ( \sum_{mn=-j}^{j}A(m,n)\dfrac{\tan^{2j+m+n}(x/2)}{(1+\tan^2(x/2))^{2j}}e^{iy(n-m)} \right )^{N}$$ Note: sums ...
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1answer
96 views

Improving my QZ-Algorithm (Include Shifts)

I Need to to solve an generalized Eigenvalue Problem and to compare two Methods (QR and QZ) concerning their convergence rate and execution time. I started with the Basic QR-Algorithm, implemented in ...
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0answers
69 views

Difference between fast and normal Givens Rotations?

would someone be so kind as to explain me the difference between the ordinary givens-rotation and the fast givens-rotation? I know that the fast givens Rotation reduces the Count of operations to ...
3
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1answer
74 views

How close observed order of accuracy should be to theoretical order of accuracy?

I try to write tests to the implementations of numerical methods. The best way to do this is to study the observed order of accuracy and check that it matches with the theoretical order of accuracy. ...
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0answers
49 views

Jacobi method converging then diverging

I am working to solve Poisson's equation in 2D axisymmetric cylindrical coordinates using the Jacobi method. The $L^2$ norm decreases from $\sim 10^3$ on the first iteration (I have a really bad ...
3
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41 views

Broadening spectral data by using FFT's

I obtain numerical discrete data of the form $$ S_{raw}(\omega) = \sum_{j}w_{j} \delta(\omega-\omega_{j}) $$ to compare the result with experimental data the delta peaks need to be broadened ...
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1answer
58 views

Reference request: theory regarding time evolution of closed loop 2D elastic shapes?

I am interested in approximating the time evolution of 2D curves. Here's an illustration: An issue that arises when naively making this approximation as illustrated above, is that as one increases ...
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170 views
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1answer
41 views

Resources exploring the problem of “volume exclusion”?

Consider the following situation: There are two boundaries -- one is denoted using grey lines, and the other is denoted using black lines. The boundaries are numerically represented using ...
2
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46 views

Need a smart way to numerically take residues in a multidimensional integral

I'm trying to do an integral of the form $\int_C f(u,v) $, where $C$ is a set of contours in $u$ and $v$. In particular, each variable's contour starts at $-\infty+i \epsilon$, goes around a branch ...
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3answers
362 views

What is the fastest opensource implementation of Bessel functions computation?

I'm looking for an open-source (to use and learn from) software which computes Bessel functions of integer order of real argument to double precision the fastest among all such implementations. ...
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1answer
81 views

Numerical quadrature when locations of singularities are approximate

I'd like to numerically integrate $\frac{1}{\sqrt{f(x)}}$ on an interval between two consecutive zeros of the function $f(x)$, which makes the integrand singular at two endpoints. Standard practice ...
2
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1answer
58 views

Resource recommendations for numerical methods involved in dynamical systems analysis

I am interested in learning numerical methods that specifically have to do with analyzing dynamical systems. In particular: drawing phase plane diagrams drawing phase portraits analyzing ...
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51 views

Quick scheme for separable first-order ODE

I'm trying to integrate an incredibly simple ODE: $$ y'(x) = -f(y),\quad y(0) = y_0 \ , $$ from $x=0$ to $x=1$. This is a decay type of equation, $f$ is the (variable) decay rate and $y$ is the ...
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0answers
111 views

Tips on improving stability in numerical scheme for non-linear PDE

I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then ...
0
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1answer
140 views

Numerical integration of sharp peaked function (position of peak known)?

What methods are available to integrate a sharply peaked function (position of peak known) on a finite interval (the interval includes the peak)? Currently I am getting underflows using some of GSL's ...
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1answer
139 views

Shallow Water Equations Boundary Conditions

I am trying to solve shallow water equations using DG methods. Flow over a bump is a common problem that comes up in this context. For example ...
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1answer
241 views

When and why is `r./sum(r)` not a good way to renormalize a vector in PageRank computation?

I experimented with the PageRank algorithm. When the number of pages is large, I encountered a situation when one formula for re-normalizing a vector (so that sum of its components is equal to 1; ...
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1answer
110 views

What does “strongly conservative” mean in the context of numerical methods?

I have a homework problem that asks me to show that 1st order unwinding or central differencing can give a strongly conservative, consistent scheme for the 1-D Burger's Equation using a finite volume ...
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1answer
253 views

How to solve Energy Balance equation by numerical method

Good Day I am new to heat transfer technique please give me some suggestion on solving energy balance equation $$a \frac{\partial T_p}{\partial t}=\frac{\partial}{\partial x}\left(b\frac{\partial ...
3
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1answer
150 views

How to integrate numerically a function with error bars?

Typically, the function that one wants to integrate numerically, $f$, is given, i.e. its values for various points $\{x_i\}$ are known precisely. The resulting error is due to the fact that we chose a ...
2
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2answers
104 views

Expected computational time for DNS computation of fluid flow

Using an established criterion involving capturing eddies down to the Kolmogorov length scale it can be reasoned that the order of grid points in the computational mesh needs to be $N^3 \ge Re^{9/4}$ ...
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152 views

Plotting renormalization group flow diagram from recursion relation

How do I plot a flow diagram (given certain initial conditions, the trajectory in y-x space) for a RG (renormalization group) flow given by the following recursion relations? \begin{eqnarray} ...
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1answer
2k views

Runge-Kutta 4th order for 4 coupled first order differential equation [closed]

I have to solve 4 coupled first order differential equations for $f(t)$ ,$g(t)$, $h(t)$ and $w(t)$ witch are only functions of $t$ , but for every reference link a function of 3 variables is assumed ...
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1answer
42 views

Modifying finite difference solution to Schrodinger eqn to account for fermion/boson effects

I have been playing with an implementation of Visscher's explicit method for solving the time dependent Schrodinger equation (Are there simple ways to numerically solve the time-dependent ...
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1answer
508 views

a circular plot from a vector which represents the temperature along the radius surface, which is the same for every radius

I have calculated the temperature of the section of a cylinder, which is subjected to a heat flow on its upper surface. Getting the temperature distribution in the 2D section. As shown in the ...
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1answer
59 views

Integer simplification of irrational inequality

I'm doing work in computational geometry where the robustness of the algorithm is important. On two separate occasions now have I come across a scenario where I compare the numerical size of two ...
5
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1answer
113 views

What spline functions are used in Section 13.9 of “Numerical Recipes in C”?

I asked a similar question on MathSE but with more added fluff, but didn't really get any straight answers, so I figured I'd ask here. Computing Fourier coefficients of a function using the FFT is ...
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1answer
68 views

Finding roots without knowing much about the function

Consider solving numerically for roots: $( x_0, y_0): f(x_0, y_0) = 0, g(x_0, y_0) = 0$ where you only know that f, g continuously differentiable but the theoretical differentiation is not a ...