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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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How to compute a number in base 16

I have to compare $ (2)_{16} $ and $ (1.\overline{F})_{16} $. $ (2)_{16} $ is very simple to compute, e.g. $ (2)_{16} = 2 \cdot 16^0 = 2 $. On the other hand, I dont know how to find out what $ (1.\overline{F})_{16} $ is because I don't understand…
Prime Cuts
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Finite Element Method for a Two-Point Problem

Given the following two-point problem: $$-y''(x)+(by)'(x)=f(x), \forall x \in [0,1]$$ $$y(0)=0, y'(1)=my(1)$$ where $ b \in C^1([0,1];\Bbb{R}), f \in C([0,1];\Bbb{R})$ and $ m \in \Bbb{R}$ a constant. What could be a finite element method for the…
Mary Star
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Numerical computation stability issue

We have a problem to find $x$ such that $$ \sum_{i}\left(\frac{a_i}{x}\right)^\lambda= \sum_{j}\left(\frac{b_j}{x}\right)^{-\lambda} $$ $a_i < x < b_j$ $a_i \approx x \approx b_j$ We get analytic solution: $$ x =…
Serg
  • 277
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Approximation of pi

Given that $\frac{\pi^2}{6}=\sum_{n=1}^{\infty}\left(\frac{1}{n^2}\right)$, I have to write a program in C that finds an approximation of $\pi$ using the formula $S_n=\sum_{i=1}^{n}\left(\frac{1}{i^2}\right)$. Then the approximation is:…
Mary Star
  • 13,956
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Eigenfunctions of Laplacian on sphere - numerical approach

Consider a Laplacian: $$\hat L=\frac1{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac1{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2},$$ where $\theta\in[0,\pi], \varphi\in[0,2\pi)$. I know…
Ruslan
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Floating Point Calculation with mantissa

i have a function f(x)=$\sqrt{x+1/x}$ - $\sqrt{x-1/x}$ (x $\ge{1}$) and want to calculate $f(2.0004 * 10^{18})$ in floating point arithmetic with a mantissa t = 4. First, I rounded the input to 2 * $10^{18}$ (since the 4 at the end would require a…
Philipp
  • 155
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How prove this "Trigonometric interpolation"?

let $f(x)$ show that $$f(x)\approx\sum_{i=0}^{2n-1}f(t_{j})L_{j}(t)$$ where $$t_{j}=\dfrac{\pi}{n}j$$ $$L_{j}(t)=\dfrac{1}{2n}\left[1+2\displaystyle\sum_{m=1}^{n-1}\cos{m(t-t_{j})}+\cos{n(t-t_{j})}\right], j=0,1,2,\cdots,2n-1$$ It is say that…
math110
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Creating a degree $n$ Taylor polynomial for $\sqrt{1+x}$

I have been asked to produce a general formula for the degree $n$ Taylor polynomial for $\sqrt{1+x}$ using a=0 as the point of approximation. Given that…
Brooks
  • 125
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Proving that Gauss-Jacobi iterations always converge when a system is strictly diagonally dominant.

Consider a system $A\mathbf x=\mathbf b$ where $A=[a_{ij}]_{n\times n}$ is a coefficient matrix, $\mathbf b$ is the constant vector $(b_1,b_2,\ldots,b_n)^T$ and $\mathbf x$ is the solution vector $(x_1,x_2,\ldots,x_n)^T$. Recall that we start off…
Nothing special
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Local truncation error of a given multi-step method

This is admittedly a homework problem, so I've also attached my best attempt at a solution. Question (abbreviated, from Numerical Analysis, Burden and Faires): Given the multistep method $w_{i+1} = -\frac{3}{2}w_{i} + 3w_{i-1} - \frac{1}{2}w_{i-2}…
Srihari P
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Is it possible to reuse computation while dividing?

Suppose I wanted to divide $1$ by $x$, and I get the answer as $\frac{1}{x}$. Is there any way to go from this to the answer to $\frac{1}{x+1}$ without having to restart the division? I know the answer's going to be pretty close, but is it possible…
avighnac
  • 355
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Can we conclude that this summation is positive?

The summation is: $$S_n=n\sum_{i=0}^n x_i^2- \left ( \sum_{i=0}^nx_i\right)^2$$ where $n>1$ and $x_1,x_2,\ldots,x_n\in \mathbb{R}$. I'm trying to prove that if $x_i\neq x_j$ for $i\neq j$ then $S>0$. If $n=2$, it's easy: $$ S_2=(x_1-x_2)^2>0$$ But…
Pedro
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Will a function with a double root always have a (local of absolute) maximum or minimum there?

I was doing some coding and wanted to code the bisection method, but since this applies to the case where $f(a)f(b)<0$, I started to think how I could generalize this to $f(a)f(b)>0$ and I thought that this would be 'codeable' because I would only…
B.K.
  • 75
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Deciding which number is bigger

Suppose that $a$ and $b$ are two algebraic numbers with $0<|a-b|\approx 10^{-50}.$ Suppose further that a calculator can evaluate $a$ and $b$ up to say 12 digits. Are there some general transformation I can do such that my calculator will output if…
finnmath
  • 55
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Any explicit formula of the smoothness indicator used by Shu for WENO schemes?

Am reading Shu's WENO notes to build a 1-D WENO reconstruction and stumbled on Eq. 2.61: $$ \beta_r = \int_{x_{i - 0.5}}^{x_{i + 0.5}} (\Delta x)^1 \cdot \left[\frac{dp_r(x)}{dx}\right]^2 + (\Delta x)^3 \cdot \left[\frac{d^2p_r(x)}{dx^2}\right]^2 +…
Yuki.F
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