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.


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

12913 questions
7 answers

"The Egg:" Bizarre behavior of the roots of a family of polynomials.

In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ counted the number of subspaces of an…
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Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger.

Conor McBride asks for a fast proof that $$x = {\left(\pi^\pi\right)}^{\pi^\pi}$$ is not an integer. It would be sufficient to calculate a very rough approximation, to a precision of less than $1,$ and show that $n < x < n+1$ for some integer $n$. …
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Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one. Numerical computations, to my understanding, never deal with irrational numbers, but only rational…
3 answers

What algorithm is used by computers to calculate logarithms?

I would like to know how logarithms are calculated by computers. The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that logarithms are calculated directly from the hardware. So the question is: what…
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How much does symbolic integration mean to mathematics?

(Before reading, I apologize for my poor English ability.) I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast world of mathematics. For instance, the integral…
Sangchul Lee
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What’s the difference between analytical and numerical approaches to problems?

I don't have much (good) math education beyond some basic university-level calculus. What do "analytical" and "numerical" mean? How are they different?
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What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal places of accuracy when $n\geq180$. The third…
5 answers

Recommendations for Numerical Analysis texts?

I'm in a numerical analysis course right now and it's pretty rigorous but I'm enjoying it a lot. I took a lower level course before that was more oriented towards implementation of numerical methods, so it's not my first time with the…
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Finding $1/x^2 + 1/x^3 + 1/x^5 + \dots $

The following function came up in my work: $$ f(x)=\sum_{p\text{ prime}}\frac{1}{x^p}=\frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+\cdots. $$ Naturally, this converges for $x>1$ since the geometric series does. Does this…
4 answers

Gradient descent with constraints

In order to find the local minima of a scalar function $p(x), x\in \mathbb{R}^3$, I know we can use the gradient descent method: $$x_{k+1}=x_k-\alpha_k \nabla_xp(x)$$ where $\alpha_k$ is the step size and $\nabla_xp(x)$ is the gradient of $p(x)$. My…
3 answers

Why does the Mandelbrot set appear when I use Newton's method to find the inverse of $\tan(z)$?

Why does the Mandelbrot set appear when I use Newton's method to find the inverse of $\tan(z)$ Specifically for the equation $y = \tan(z)$ I use Newton's method ($20$ iterations) to solve $0 = \tan(y) - z$ for $y$, where $y_0 = \tan(z)$ ($y_0$…
4 answers

Why does Monte-Carlo integration work better than naive numerical integration in high dimensions?

Can anyone explain simply why Monte-Carlo works better than naive Riemann integration in high dimensions? I do not understand how chosing randomly the points on which you evaluate the function can yield a more precise result than distributing these…
4 answers

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order Runge-Kutta method, so I have the…
2 answers

How is the Taylor expansion for $f(x + h)$ derived?

According to this Wikipedia article, the expansion for $f(x\pm h)$ is: $$f(x \pm h) = f(x) \pm hf'(x) + \frac{h^2}{2}f''(x) \pm \frac{h^3}{6}f^{(3)}(x) + O(h^4)$$ I'm not understanding how you are left with $f(x)$ terms on the right hand side. I…
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Why does the Newton-Raphson method not converge for some functions?

$f(x)=2x^2-x^3-2$. This is a cubic type graph as shown. The real root of this graph is $(-0.839,0)$. So, the question is to use Newton's approximation method twice to approximate a solution to this $f(x)$. I use an initial starting value of…
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