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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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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?
jbrennan
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The Convergence of Jacobi and Gauss-Seidel methods

Help me please with this question: The following system of equations is given: \begin{align} x+2y+3z&=5\\ 2x-y+2z&=1\\ 3x+y-2z&=-1 \end{align} Check if the Jacoby method or Gauss-Seidel method converges? If the methods or one of the methods…
Tina
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Convergence rate of Newton's method

Let $f(x)$ be a polynomial in one variable $x$ and let $\alpha$ be its $\delta$-multiple root ($\delta\ge2$). Show that in the Newton's $x_{k+1}=x_k-f(x_k)/f'(x_k)$, the rate of convergence to $\alpha$ is not quadratic. My solution: Suppose that…
Jimmy Wang
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How do you mathematically round a number?

How does someone mathematically round a number to its nearest integer? For example 1.2 would round down to 1 and 1.7 would round up to 2
ParoX
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Prove that $\phi(x) = 2x^2$ : $\phi[-1,1] \to [0,1]$ has two fixed points in its domain of definition

Have can one prove that there are two? I know how to prove when the image set $[a,b]$ is in the domain of definition $[a,b]$, but applying the method to this problem doesn't seem to work. If we take the derivative of $\phi'(x) = 2x^2$ = $4x$, then…
PlatinumMaths
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How to compute the pade approximation?

Like $\log(1+x)$? Is there any algorithms? I have read many materials but doesn't have an idea
Lookout
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How to derive a function to approximate $\sqrt{3}$?

The problem is to used fixed-point iteration method to find an approximation to $\sqrt{3}$. The equation from the book is $f(x) = \dfrac{1}{2}\left(x + \dfrac{3}{x}\right)$. It make senses to me however I realize there are other equations can be…
roxrook
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Explanation of Lagrange Interpolating Polynomial

Can anybody explain to me what Lagrange Interpolating Polynomial is with examples? I know the formula but it doesn't seem intuitive to me.
user87274
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Numerical solution to x = tan (x)

I needed to find, using the bisection method, the first positive value that satisfy $x = \tan(x)$. So I went to Scilab, I wrote the bisection method and I got $1.5707903$. But after some reasoning I came to the conclusion that this value is…
David Robert Jones
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Numerical Analysis-Proof That Sum of Lagrange Interpolating Polynomials is One

I am having trouble proving that given interpolation nodes $x_0, x_1, \ldots, x_n$ and function values $f_0, f_1, \ldots, f_n$ that $\sum\limits_{i=0}^n l_i(x) = 1$. Our Lagrange interpolation formula is given by: $p_n(x) = \sum\limits_{i=0}^n…
French Toast Crunch
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Newton's method for roots of multiplicity $> 1$

I'm studying numerical analysis and I read in Wikipedia that "If the root being sought has multiplicity greater than one, the convergence rate is merely linear". The article in Wikipedia also mentions that "if the multiplicity $m$ of the root is…
Belgi
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When is 2d sparse numerical integration by Voronoi regions better than using triangular mesh elements

I've inherited some numerical analysis code that integrates a 2D function that is only known at a large set of unstructured points. The way it does this is by Delauney triangulating the domain using the sample points, and then calculating the area…
Michael Anderson
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Backward stability vs Stability

I'm reading a numerical analysis book, which gave me very confusing descriptions about stability and backward stability. It says that an algorithm $f$ is stable, if for any $x$,…
Jack2019
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Largest floating-point number?

I'm trying to understand the Double-precision floating-point format: As I understand so far, a floating-point number is of the form $$ (-1)^s2^{c-1023}(1+f) $$ where $s=0,1$ is the sign indicator, $c$ is the 11-bit exponent, $f$ is the 52-bit…
user9464
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Chebyshev Polynomials

I am trying to prove a something regarding Chebyshev polynomials. Given the polynomials $T_n(x), n = 0, 1, \ldots$ which are recursively defined by $$\begin{cases} T_0(x) = 1\\ T_1(x) = x \\T_n(x) = 2x T_{n−1}(x) − T_{n−2}(x), & \text{for } n \geq…
Samuel Gregory
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