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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

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Understanding what to do for relative error when p = 0 (bisection method)

I noticed this was mentioned in class, but the detail wasn't really given as to how to deal with it (outside of using another error method such as absolute error). Given the relative error of the bisection method: $$ \frac{|P_n - P_{n-1}|}{|P_n|} $$…
user3389436
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root of the function $f(x)=\sqrt{2}-x$ by using fixed point iteration

How can I find the approximate value of $\sqrt{2}$ by using the fixed point iteration? I have tried $x-\sqrt{2}=0$, $x^2=2$, $x^2-2+x=x$, $g(x)=x^2-2+x$, $g\prime(x)=2x+1$ And i choose $x_0=-\frac{1}{2}$. But i cant find the approximate value…
Raio
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Functional Iterations (numerical analysis)

(I'm opening a new post because the claim has been found in another book and now I have an exact description) I wish to prove this claim I found in a numerical analysis book : Assume that $\varphi(x)$ is $p$ times continuously differentiable. Then…
Belgi
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Functional Iterations and order of convergence

Possible Duplicate: Functional Iterations (numerical analysis) Let F be a function $F:\mathbb R\to\mathbb R$ and Let $x_0$ be (any) real number. we define : $x_{n+1}=F(x_n)\, (\text{ for }n\ge 0)$ The book I'm reading on numerical analysis claims…
Belgi
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Approximation of values in [0,1] by sums of unit fractions

Let $U = \{(-1)^k\cdot\frac{1}{n}: n\in\mathbb{N}, k\in\{-1,1\}\}$ be the set of positive and negative unit fractions. For a positive integer $m\in\mathbb{N}$ and a real $x\in [0,1]$ we set $d(m,x) = \inf\{|x-\sum_{i=1}^m u_i|: u_i \in U \textrm{…
Dominic van der Zypen
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Fixed Point theory question (Numerical methods)

I have an exam in a previous exam paper which i have no solutions too. I am stuck on the last 2 parts of the question and have been for several days now! Any help much appreciated. Here is the question: Let $D:=[1/4,3/4]$ and consider the iteration…
Bernard.Mathews
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The 'order of error'

If I do a trapezoidal rule estimate and get 0.6386 and the true value of the integral is 0.636294, then the error is 0.002306. If I was asked to find the order of error, does it just mean the error or has it a different meaning? Also, if the…
user137090
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Why do we get oscillations in Euler's method of integration and what is the period?

When using Euler's method of integration, applied on a stochastic differential eq. : For example - given $$\dot v = -\gamma v \Delta t + \sqrt{\epsilon \cdot \Delta t }\Gamma (t) $$ we loop over $$ v[n+1] = v[n] -\gamma v[n] \Delta t +…
user1611107
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Fastest way of finding eigenvectors from eigenvalues

Given the eigenvalue of a matrix of large dimensions, I want to know if there is a fast way of finding the corresponded eigenvectors?
user133834
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Inverse Iteration to Find Eigenvalues - Question about Method

So I'm doing Inverse Iteration in Excel to find the dominant eigenvalue and eigevector of a matrix. This particular method involves estimating an eigenvalue, multiplying the identity matrix by it, then subtracting from the original matrix. From…
NMH
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Fixed-Point & Root Relation

Lets say I have found out the fixed point for a given function. I'm only given a point that is basically mapped to itself, but how do I find the root of the function and thus any solutions using this fixed point I just found?
Adrian MK
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Non-convergence of Bairstow's method

I am writing a program to compute the roots of a polynomial with real coefficients. I am using Newton's method to get the real roots, and trying to use Bairstow's method for the complex ones. I am looking for the roots of x^2 + x + 1. (Yes, I…
Jim
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Do elliptic allow for direct solvers of roots of quintic polynomials?

Galois Theory tells us that we cannot directly solve for the roots of a quintic polynomial using elementary operations and radicals. I have seen sources that use this to reason that any computer algorithm that solves for the roots of a quintic must…
Jean Valjean
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Show that $\vert\int_{-1}^1 \omega(t) dt \vert \leq 2^n \int_{-1}^1\vert \omega^{(n)}(t)\vert dt$

I am stuck with the following problem: With $\omega: [-1,1]\rightarrow \mathbb{R}$, $\omega\in C^n(-1,1)$. Suppose that $\omega$ has a finite number of zeroes $t_1
Luc M
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Find the real roots

I have the following equation: \begin{equation} (y-1)^a - C~~ y~~ \exp(b x)=0 \end{equation} where $a, b$ are real constants, $C$ may be a complex number. I need to find the real solution of the above equation for each choice of $x$, with $x \in…
Tapan
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