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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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numerical analysis : Fixed point iteration

Let $f(x)=x^3-2x+1$. To solve $f(x)=0$, the following fixed-pint problems are proposed. Derive each fixed point method and compute $p_1,p_2,p_3,p_4$. Which method seem to be appropriate? a) $x=\dfrac{1}{2}(x^3+1), \quad p_0=\dfrac{1}{2}$ b)…
Stu
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Taylor series in numerical differentiation

I'm studying some numerical methods and especially the numerical differentiation. I have some troubles with the central differentiation. How do you become from the Taylor series the formula for $f(x+h)$, $f(x-h)$, $f(x+2h)$ and $f(x-2h)$?
WinstonCherf
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About the error bound of Simpson's rule

To find the error bound formula of Simpson's rule with three equally spaced points $x_{0},x_{1},x_{2}$, the error bound of the newton forward polynomial of degree two should be integrated. As you can see in the attached photo, the integration of the…
soso sos
  • 349
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Problem with the convergence of a Nystrom algorithm

I programmed a Nystrom Algorithm specifically for my problem: This is the exact equation i want to solve: $y′′=(w2−e∗cos(t))∗sin(t)−b∗y′$ And this is my algorithm function [T U] = Nystrom(a,b,u0,h,w,v,e) % a is the initial value % b is the final…
Federico
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How should I do this? Euler's Method?

I am approximating a solution to a first order LODE using Euler's method. I made two tables, one using a step size of .01 and another using .05 ( I had to start at x=0 and end at x=1). I am not understanding the directions for the second part of my…
A A
  • 559
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Ill-conditioned conversion from roots to polynomial coefficients

I need to numerically compute polynomial coefficients from its roots. However, this seems to be numerically unstable even for a moderate number of roots. For example, computing the polynomial $z^N - 1$ from the roots of unity $e^{2j\pi k/N}$ fails…
Jiro
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Explain why this formula is inferior to $x_{n+1}=x_n-f(x_n)\left[\dfrac{x_n-x_{n-1}}{f(x_n)-f(x_{n-1})}\right]$ in practice.

Prove that the formula for the secant method can be written in the form $$x_{n+1}=\dfrac{f(x_n)x_{n-1}-x_nf(x_{n-1})}{f(x_n)-f(x_{n-1})}$$ Explain why this formula is inferior…
Username Unknown
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The forward Euler always under-estimates the exact solution

How can one show that the forward Euler always under-estimates the exact solution provided that $αΔt<1$ and $αΔt≠0$. When $$\frac{dy}{dt} =-αy$$ My solution I found the exact solution to be as $$y(t)=e^{-αt}$$ and then showed $(1-αΔt)$ $\le$…
etet112
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Truncation error in approximation of $\frac{e^x - 1}{x}$

I have a function $\frac{e^x - 1}{x}$. In order to avoid loss of significance when calculating values of this function near $x = 0$, I represent $e^x$ as Taylor series. The truncation error of $e^x$ is $\frac{x^5}{5!}*e^x$ (If i use 4 members of…
John
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Is the ratio of sequences not being bounded a necessary condition for a sequence to not be Big-Oh?

This is more for clarification than anything else, in my text we define: Definition: We say $x_n = \mathcal{O}(\alpha_n)$ if there are constants $C$ and $n_0$ such that $|x_n| \leq C|\alpha_n|$ when $n \geq n_0$. Am I right in saying that the…
Dragonite
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Show that recursion

Hi I have the following problem: Let $$c_{v,\mu}=\int_a^bw_v(t)(b-t)^{\mu}dt\:\:\:\:\:\:\:,w_v(t)=\prod_{l=0}^{v-1}(t-t_{j-l})$$ with $v=0,...$ and $\mu=0,...$ Show that $$c_{v+1,\mu}=(b-t_{j-v})c_{v,\mu}-c_{v,\mu+1}$$ I already tried to…
Tobi92sr
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improving estimation of integral

Using this set of points of the form $(x,f(x))$: $$(1.0,1.54),(1.2,1.81),(1.4,2.15),(1.6,2.58),(1.8,3.11)$$ I estimated the integral using the extended trapezoid rule: $\int\limits_1^{1.8}f(x)dx\approx 1.79$ for $h=0.4$ and…
Don Fanucci
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How to surpass Matlab precision limit for numerical calculations

In Matlab, I am trying to solve an equation that has single integral. For certain parameters the solution goes very close one even beyond the 64 bit double precision of the matlab. I am not able use "vpa" kind of symbolic math as I need to use this…
sar1729
  • 21
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What is minimum number of iterations required in the bisection method to reach at the desired accuracy?

At each iteration in the bisection method the absolute error becomes half of the previous iteration. Thus if we approximate the root $x$ of some equation in some interval $[a,b]$ (say) by means of a sequence $\{x_n\}$ converging to $x$ in the…
Arnab Chattopadhyay.
  • 2,570
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2 answers

Number of iterations for Newton Method

If I wanted to find the least number of iterations it would take for applying Newton's method on a function, is there a formula that I can use to obtain it? If so, what is that formula?
user975
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