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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 is the difference between Chebyshev and Halley root-finding methods?

Chebyshev method $$ (1)\space\space p_{n+1}=p_n-\frac{f(p_n)}{f'(p_n)}-\frac{1}{2}\frac{[f(p_n)]^2f''(p_n)}{[f'(p_n)]^3} $$ Halley method $$ (2)\space\space p_{n+1}=p_n-\frac{f(p_n)}{f'(p_n)-\frac{f(p_n)f''(p_n)}{2f'(p_n)}} $$ First we know that…
Dante
  • 1,908
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Why does this relative error work?

Assuming $p$ is the exact solution and $p_n$ is a numerical approximation. My question is that why most of the numerical analysis books using $$ \frac{|p_n-p_{n-1}|}{|p_n|} $$ to approximate the following relative error $$ \frac{|p_n-p|}{|p|} $$ ?
Dante
  • 1,908
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Numeric Condition for non differentiable function

suppose I have a function f(x)=$\sqrt{x+1/x}$ - $\sqrt{x-1/x}$ where x $\ge{1}$ and want to find out its relative numeric condition for a) $ x \rightarrow 1 $ b) $ x \rightarrow \infty $ Now the relative condition according to my materials is…
Philipp
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Lagrange finite elements

We consider in $\mathbb{R}^2$ the set of points $$\{M_1(-1,1),M_2(0,1), M_3(2,1),M_4(-1,0),M_5(1,0),M_6(2,0)\}$$ Let $\Omega$ a rectangular structure consisting of the heads $\{M_4(-1,0),M_6(2,0), M_3(2,1),M_1(-1,1)\}$ Let the two polygones $\sum_1…
jijii
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Determining an upper bound for the error function approximation

The error function is defined by, $$\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int^{x}_{0}e^{-t^2} dt$$ I started by determining the Taylor polynomial of degree $2n$ around $x=0$ of $e^{-x^2}$. For this I used the Taylor expansion of $e^t$, which…
gordon
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Using the Taylor Series of $\ln\left(\frac{1-x}{1+x}\right)$

This is Section 1.6 of the book "An introduction to Numerical Methods and Analysis" by James F. Epperson. The questions that we will be working on: I have done question 1. I'm having trouble with question 6, and after question 6 is done, I have…
Sonny Jordan
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Numerical inversion of characteristic functions

I have a need to use the FFT in my work and am trying to learn how to use it. I am beginning by attempting to use the FFT to numerically invert the characteristic function of a normal distribution. So I have discretised the integral using the…
Freakalien
  • 229
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Show that the averaged vector field one step method is well-defined

Let $\dot{\mathbf{y}} = \mathbf{f}(\mathbf{y}), \;\mathbf{f}: D \subset \mathbb{R}^d \to \mathbb{R}^d$ be an autonomous differential equation with $\mathbf{f}$ smooth. We define the averaged vector field one step method implicitely by $$\mathbf{y}_1…
user88109
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Why is (asymptotic) stability inherited by A-stable Runge-Kutta-methods?

I wonder how A-stability of a Runge-Kutta-Method implies that (asymptotic) stability is inherited from the solution of a linear initial value problem. For a Runge-Kutta-Method $\psi^{\tau}$ there is a step size $\tau^{*}$ such that application to…
ilius123
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Local matrices - unstructured mesh

I'm facing an "assembly issue" concerning a finite element method, that is a DG method, I was working on. I'll sum up my problem. If we consider a $\mathbb{P}^1$ Lagrangian piece-wise discretization $\mathcal{T}_h$ (with finite element denoted…
Amzocks
  • 243
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Find the value $p,q,r$ that make the sequence converge with highest order as possible

When using iteration $x_{k+1}=g(x_k),\displaystyle{g(x)=px+\frac{qa}{x^2}+\frac{ra^2}{x^5}}$ in order to find the value of $a^\frac{1}{3}$ What is value of $p,q,r$ that make the sequence converge with highest order as possible?
user76608
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What tutorial videos is best for numerical methods 1?

I'm doing this course and we have only been told to work through the prescribed textbook. No study guide was given and I'm not sure how/where to start with the assignment questions. Please advise.
Dee
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Reduce error when $x$ is much smaller than 1

I have the following function $$\frac{1}{1+2x}-\frac{1-x}{1+x} $$ How to find equivalent way to compute it but when $x$ is much smaller than 1? I assume the problem here is with $1+x$ since it probably would be equal to 1. I don't know if…
Awerde
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Question about matrix discretisation numerical methods

Tomorrow I have an exam about Numerical Methods, and I came up with the following question. Let $$-\frac{d}{dr} \left ( \frac{1}{r} \frac{dy}{dr} \right ) = 1 $$with $r\in [1,2], y(1) = 1 \mbox{ and }y(2)=10$. Take $h = \frac{1}{n+1}$ and $r_i = ih…
user54297
  • 1,036
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Using Newtons Method to Approximate a solution

I am solving the following problem: Use Newton's method to find solutions accurate to within $10^{-5}$ for the following problem. $e^x + 2^{-x} + 2 \cos x - 6 = 0$ for $1 < x < 2$ I am familiar with claiming a $p_0$ and using the newtons formula,…
lo leafs
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