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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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Nonlinear equation / iteration method

$$ 1 x^5-9.067 x^4+24.726 x^3-13.998 x^2-15.278 x+1.014 = 0 $$ Wolfram Result but i need solve roots using "iteration method" (i don't know how it correctly called in English, but Newton's method is not what I need) method, what I need, starts from…
roman
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Brent's algorithm

Use Brent's algorithm to find all real roots of the equation $$9-\sqrt{99+2x-x^2}=\cos(2x),\\ x\in[-8,10]$$ I am having difficulty understanding Brent's algorithm. I looked at an example in wikipedia and in my book but the examples given isn't…
Tom
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Step size for trapezoidal?

I feel a bit lost with this problem. I've looked into Romberg integration a bit and tried to apply it here, but haven't been able to conclude anything. I also read somewhere that there is a correlation between the difference, $T(2h) - T(h)$, and the…
Moo
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How to evaluate $\int_{0}^{\infty}e^{-x}f\left(x\right)dx$ using $f\left(0\right)$ and $f\left(1\right)$?

Let $f\in{\cal C}^{\infty}\left(\mathbb{R}\right)$ be such that $\left|f^{\left(k\right)}\left(x\right)\right|\leq M_{k}$ for every $k$. Find a way to evaluate $\int_{0}^{\infty}e^{-x}f\left(x\right)dx$ using $f\left(0\right)$ and $f\left(1\right)$,…
Jon
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Convergence order slowdown

I'm using a Runge-Kutta second-order solver (more specifically, this with $\alpha = 1$) and I'm trying to approximate the solution of the nonlinear system $$\begin{cases} T'(r)=S(r) \\ S'(r)=-\frac{S(r)}{r}-\beta e^{T(r)} \end{cases}$$ with initial…
user401936
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Loss of significance - why does it gives more accurate result here?

Having $2$ formats of same function - $f(x) = \displaystyle x(\sqrt{x+1}-\sqrt{x}) $ and edit: $f(x) \displaystyle = \frac{x}{\sqrt{x + 1} + \sqrt{x}} $ (evaluated as @vonbrand computed ..) Assume we use floating point with $6$ figures , why…
URL87
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How I can solve this : $-2^x+3^{x/2}+1=0$ without using numerical ways?

This equation : $-2^x+3^{x/2}+1=0$ is confusing me , However it has only one solution which it is an integer $ x=2$ , But i can't resolve it using clear way , I have used the varibale change $y= x/2$ in order to transforme it in equation of degree…
zeraoulia rafik
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Adams Bashforth method pi product indices

For a certain programming assignment, I have to calculate the b coefficients for the Adams Bashforth method, and to evaluate these involves this expression. I understand that the pi notation refers to the product of the terms specified by the…
Rabble
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Minimum Iterations In Bisection Method

I am using the Bisection Method to find a root for: $$\frac{1.52}{(1+x)^2}-0.5\tan^{-1}\left(\frac{1}{x}+\frac{0.65x}{1+x^2}\right)$$ At $[0.1,2]$ and for $\varepsilon=0.01$ Using $\log_2(\frac{b-a}{\varepsilon})\leq n$ I get that $7.56\leq n$ But…
newhere
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I don't know stiff equation definition

My text book is proving very difficult. The definition of a 'stiff system' in the text book is vague. Could someone explain the definition of a 'stiff system'? Also: why $y'=\lambda y, y(0)=\alpha,$ where $\lambda < 0$ is a test equation? Moreover…
seyunkim
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I do not understand a statement about the local truncation error of Runge-Kutta methods

\begin{align} w_0 &= \alpha \\ w_{i+1}&=w_i +a_1 f(t_i,w_i) + a_2 f(t_i+\alpha_2,w_i+\delta_2 f(t_i,w_i)) \end{align} for each $i =0,1,2,...,N-1$, cannot have local truncation error $O(h^3)$ for any choice of constants $\ a_1,a_2,\alpha_2 $ and…
seyunkim
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Approximating the derivative at $~x=1~$ given $~f(1), ~f(0.8)~$

Given that $f(0.8)=2.2255, f(1.0)=2.7183, \int_{0.8}^{1.0}f(x)e^{-x}dx=0.2$, I need to find an approximation of $f'(1.0)$ using a method with error of degree $~2~$. I'm studying for a test and I've seen many questions like this one. My guessing is…
Tamir Shalev
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Newton's divided difference interpolation

I'm studying numerical analysis and I'm having difficulty in solving this problem: The question is : $$f(1)=2, \quad f'(1)=3 \quad f''(1)=1$$ $$f(2)=6,\quad f'(2)=7,\quad f''(2)=8$$ and I need to find f in lowest order possible with newton's…
정세희
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Jensen's Inequality, Landau-Notation

I would love to know how to show the following: $n!n^{s} = o(n^n)$ fo $n \rightarrow \infty, s \geq 0$ As for a hint, I know that $\sum_{k=1}^n \log k \leq n* \log (\frac{n+1}{2})^{n}$ can be used, but haven't really come far.
user62487
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Approximate a function with polynomial degree $n$ instead of $n+1$

If I have a $n+1$ degree polynomial $f(x)$ that approximate the function $g(x)$ on the interval $[-1,1]$. After that, if I lose $g(x)$, just know $f(x)$, how to find a $n$ degree polynomial that is closest to $g$ in maximum norm?
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