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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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Selection of the numerical method

Given the transcendental equation: $$ \frac{\tan x}{x} + c = 0, $$ where $c$ is any real number. I tried Newton's method, but it is very bad fit. Which numerical method will be the smartest solution in this case?
Tim Gates
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How many problems $Ax = b_1, … , Ax = b_k$ can be solved in the same time?

Assume that it takes one second to factorize $500 \times 500$ matrix $A$ into $A = LU$. How many problems $Ax = b_1, … , Ax = b_k$ can be solved in the same time? Here are my workings, this is the first time doing this sort of calculation so I am…
mt12345
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Interpolation by Lagrange for equidistant points

For given equidistant values $u_{-1}, u_{0}, u_{1},$ and $u_{2}$, a value is interpolated by Lagrange's formula. Show that it may be written in the form $$u_{x} = yu_{0} + xu_{1} + \frac{y\left (y^{2} -1 \right )}{3!}\Delta ^{2}u_{-1} +…
AJ_
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Relative error in numerical analysis

Searching for some help with the following questions: Given the $3$ numbers $0.1329, 1.543, 23.21$, $1$ ) Add the $3$ numbers in both ascending and descending order rounding all calculations to $4$ digits. My workings: I am a little confused with…
jh123
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How to compute Soft Absolute Function in a numerically stable way?

I want to evaluate $$f(x) = x \frac{\exp(ax)+\exp(-ax)}{\exp(ax)-\exp(-ax)}$$ This is a function approximate absolute value of $x$. However,as $a$ goes to large, the better approximation it gets. But this leads to numerical overflow as it will be an…
ElleryL
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How do I equidistantly discretize a continuous function?

Let's say I have a piecewise function that's continuous in an interval [1,7]: $f(x) = \left\{ \begin{array}{lr} -5x + 6 & : x \in [1,2)\\ -4x + 14 & : x \in [2,3)\\ -0.25x + 2.75 & : x \in [3,7] \end{array} …
user8696
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Find the solution of a complex equation

Find the solution $x$ of the equation $$ \frac{{Ae^{\frac{{Cx}} {{{\left( {1 - x}\right)}^{D} }}} }}{{1 + Bx}} = x $$ where $A$,$B$,$C$ and $D$ are the constant coefficients. If the solution of above equation can't be obtained, I want to ask to…
user581447
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PDE numeric solution

I would like to know, if its possible to plot the solution of the PDE: Let $\lambda,p_1,p_2>0$ and $c_1, c_2 < 0$ I am looking now for some $f:(R^+_0)^2\rightarrow R^+_0$ such that \begin{align*} \lambda f(x,y)-\frac{1}{2}\triangle…
luna999
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discrete differences, doubt about sign

Let $G\in C(\mathbb R)$ then $\lim_{N\to +\infty}\frac{G(x-\frac{1}{N})-G(x)}{-\frac{1}{N}}=G'(x)$? I have a doubt about the sign in front of $G'$, it is a $+$ or a $-$? Thanks to everyone
user495333
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Banach's fixed how to find q

I am looking at Banach's fixed point theorem in $\Bbb R$ and I am wondering what is $q$? I have seen that it is $$\max_{ x \in [a,b] }|f'(x)|, $$ can anyone confirm this?
user420309
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How to solve with Python a non-linear system of differential equation at order 2?

I need to solve the following equation with Python : \begin{align} r''&=f_{1}(r,r',\theta, \theta',\phi,\phi') \\ \theta''&=f_{2}(r,r',\theta,\theta',\phi,\phi') \\ \phi''&=f_{3}(r,r',\theta,\theta',\phi,\phi') \end{align} $f_{1},f_{2},f_{3}$ are…
Luther Ollier
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'Numerical Differentiation' how to compute for the round off error for a 4 point formula

from Brian Bradie -Friendly introduction to numerical analysis---- numerical differentiation exercise $$ f'(x_0)=\frac{-2f(x-0-3h)+9f(x_0-2h)-18f(x_0-h)+11f(x_0)}{6h}+\frac14h^3f^{(4)}(\xi) $$ The image shows an example problem that asks to…
userTron
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How to find the optimal approximation

We consider the space of continuous functions in $[-1,1]$ that has the usual inner product $(f,g)=\int_{-1}^1 f(x) g(x) dx$. I want to characterize the optimal approximation of a function $f \in C[-1,1]$ from the space of polynomials of degree $\leq…
Evinda
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Checking the contraction of a given function

Let$$\Phi:R^2\to R^2$$ be defined as following $$\Phi(x,y)=\frac{1}{2}(1+\frac{\sin(x)}{4}+y , 1+\sin(y)+x)^T$$ And we need to check if it is contraction with respect to Norm infinity and norm 2! I have got solution but i actually it is not…
F.O
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Fixed Point Iteration - Divergence to Convergence

Fixed Point Iteration - Divergence to Convergence Please refer to the question in the image. My attempts are as follows: 1) Sub in g(x)=x for a final result of h(x)=x, then x=alpha obtains the result, but not sure if there is more to it. 2) I am…
user565684
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