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Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems.

Definitions: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral).

Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear equations for a thousand unknowns for finding forces in a truss).

Applications: With the advent of the modern high speed electronic digital computers, the numerical methods are successfully applied to study problems in mathematics, engineering, computer science and physical sciences such as biophysics, physics, atmospheric sciences and geo-sciences.

Possible topics include but are not limited to:

  1. Approximation theory, interpolations.
  2. Numerical ODE/PDE.
  3. Root finding algorithm.
  4. Numerical linear algebra, matrix computations.
  5. Discrete integral transform, FFT, etc.
  6. Linear/Non-linear programming, integer optimization.

For questions concerning matrices, please consider adding the tag.

For questions concerning optimization, please consider adding the tag.

For questions concerning Numerical ODE/PDE, please consider adding the // tag.

References:

https://en.wikipedia.org/wiki/Numerical_method

"Numerical Methods for Scientific and Engineering Computation" by M. K. Jain, S.R.K. Iyengar, R. K. Jain

14158 questions
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Reaction-diffusion equation terms

I have been given the reaction diffusion equation: $\frac{\partial u}{\partial t} = \kappa\frac{\partial^2 u}{\partial x^2} + u - u^3$ with neumann boundaries and in initial equation for the starting time. I have only worked with the diffusion…
Jeff Bezos
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divided differences polynomial

For n + 1 supporting points and values (x$_{0}$,y$_{0}$),...,(x$_{n}$,y$_{n}$) and 0 ≤ j ≤ n as well as 0 ≤ i ≤ n-j let p$_{i,j}\in$ P$_{j}$ defined by p$_{i,j}$,(x$_{k}$)=y$_k$, k = i, i + 1,...,i+j. Let the numbers y$_{i,j}$, be defined by…
Melila
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Numerical problem - finding a unique polynomial of 2nd degree

I was given the following problem to solve and I am not confident with my solution. I have tried finding a variable alpha that would help me create $p(x)$ as a Hermite polynomial, which would be of $2nd$ degree at most and unique by definition. The…
RandyMarsh
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Difference between newton forward and backward

Newton forward difference method is better for points which are closer to x0, while Newton backward difference method is better for points which are closer to xn. I dont want any example. I want formal logic.
Soumya
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How to demonstrate weak instability of Runge-Kutta 4th order?

I have to prepare a task for my Numerical Analysis class in which I compare the performance of RK4 (not A-stable) and BDF2 (A-stable) for the solution of the stiff Van der Poll equation. I've managed to implement both methods in Matlab, however, I…
Othman El Hammouchi
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Implementing BDF2 for ODEs in Matlab

I'm trying to implement the Backward Differentiation Formula of order 2 in Matlab and I've been stuck for more than a day. The goal is to solve the Van der Pol equation. I can't find a functioning example for this algorithm on the internet. My…
Othman El Hammouchi
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Randomized Kaczmarz is very slow?

import math import numpy as np #given ordered set, e.g. [A,B,C], and some value D, return 0 if D ≤ A, #return 1 if A ≤ D < B, return 2 if B ≤ D < C, and 3 if C < D #O(log(n)) time complexity def findID(partition,val): ub = len(partition) …
nabu1227
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Piecewise Polynomials in FEM

In FEM, why is it required to have the piecewise polynomials, $\phi_i(x_j) = \delta_{ij}$ i.e. what is the significance of Kronecker Delta?
Art of Juking
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How do we know the Newton-Raphson method produces a better approximation than the initial point.

I know this method works by finding the root of the tangent line to the initial point, which is the best linear approximation of the function at that point. And I also know the closer the initial point to the root, the better the approximation,…
TFR
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Leapfrog method

How do I use leapfrog method for a non-separable force ? I know how to use it for the force of the form $f(q)+g(p)$. But I don’t know how to use it for the force like $f(q)\cdot g(p) $.
Sachin Vaidya
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Mixed finite element methods and time-dependence

I've looked all over stack exchange and can see the solution to this so here's hopping somebody knows. I'm Trying to solve an time-dependent parabolic equation using mixed finite element method and backwards Euler. So I have two coupled…
user2350366
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If $E=X-x$ find $\sqrt{X}-\sqrt{x}$

In the footnotes of my old (numerical methods and types of errors) lecture notes there is a question to find $\sqrt{X}-\sqrt{x}$ given that $E=X-x$. I am assuming that $E$ represents the error difference. The notes are old and not very clear but it…
gbd
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How to address the numerical instability of a function involving erf and an exponential

We need to compute this function: $f(x) = x + \sqrt {\frac 2 \pi} \cdot \sigma_x \cdot \frac {e^{- ( \frac {x} {\sqrt 2 \cdot \sigma_x} )^2}} {1 + erf(\frac {x} {\sqrt 2 \cdot \sigma_x})}$ We noticed that, for negative values of $\frac {x} {\sqrt 2…
user6376297
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Chebyshev-Gauss quadrature with $\tan(x)$

With Chebyshev-Gauss quadrature, solve $\int_0^{\pi/4}x\cdot tan^2(x)$, for $n=3$. Needs first to determine the change in the integral, to change the limits of integrals and then reduce in form integrate…
user79847
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