Mathematics Stack Exchange
Mathematics Stack Exchange

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
0
votes
1 answer

Multistep method Numerical Analysis : IVP

Given an explicit multistep method , we need to find the constants of the terms such as f(xi) , f(xi-1)..... and method order ..etc. A general approach is to write the corresponding Taylor series and compare the coefficients of terms .But I don't…
Rafed Nole
  • 117
  • 3
0
votes
1 answer

What's the difference?

In my work we fit a parabola to some data in order to determine three parameters. I recently talked to someone who pointed out that the ISO standard related to the fit equation had changed. The claim is that there is some subtlety with the fit…
user141401
  • 1
0
votes
1 answer

Interpolation (Numerical Analysis)

Suppose you have a table of the logarithm function $\ln x$ for positive integer values of $x$, and you compute $\ln 11.1$ by quadratic interpolation at $x_0 = 10, x_1 = 11, x_2 = 12$. Estimate the relative error incurred. Is this problem is about…
user139071
  • 1
0
votes
1 answer

Basic definition of numerical analysis

Can someone help me who can define numerical analysis that can easily understand by a student? I googled it and still it's hard for me to understand its meaning.
jeo
  • 11
0
votes
1 answer

Why does the error in my Fourier series increase as x approaches the limits of integration?

I've been given the task of computing the first 30 coefficients for the Fourier series of a Gaussian wavepacket given by: $$ f(x)=exp(\frac{-x^2}{2\sigma^2})cos(kx) $$ for $ -L/2
Confused Physicist
  • 1
0
votes
1 answer

Numerical integration of function where the input data is not sampled uniformly in time

My goal is to integrate a function $\dot{x}=f(x,u)$ (a known function) forward in time, but the inputs are not evenly spaced. The ad-hoc solution is to just resample the input so that it is uniformly spaced, which is what I have done in Matlab's…
Mr. Fegur
  • 862
0
votes
1 answer

Error Term Trouble

"Consider the approximate formulate: $f'(x) \approx 3/(2h^3) \int_{-h}^{h} tf(x+t) dt$. Determine its error term." I was thinking taking the taylor series of $f(x+t)$ to help me actually integrate the integral, but when would I stop the taylor…
asdfnomll
  • 3
0
votes
2 answers

An exercise on polynomial interpolation

Verify that the polynomials $p(x) = 5x^3 - 27x^2 + 45x - 21$ $q(x) = x^4 - 5x^3 +8x^2 - 5x + 3$ Interpolate the data: x| 1|2|3|4 y| 2|1|6|27
asdfnomll
  • 3
0
votes
1 answer

Using Explicit Runge-Kutta to solve system of ODEs

I am trying to understand Explicit Runge-Kutta method to solve system of ODEs. First I tried Euler's as below, $ \frac{dy_1}{dt} = f_1(y_1,y_2, ..y_n) \\ \frac{dy_2}{dt} = f_2(y_1,y_2, ..y_n) $ Euler's method, the value y at step i+1 $…
Emmet B
  • 125
0
votes
1 answer

Newtons Method, Multiplicity, and Rates of Convergence

Disclaimer: I'm aware that this topic is very well documented, and I've done considerable research before asking the MST community, but most of the documentation on this topic is highly formal, and I do not have a very good grasp of formal…
user132761
  • 1
0
votes
1 answer

Finding the ROC and multiplicity of a function

Consider Newton's method for finding the root of f(x) = x-sin(x). Run it on Matlab to find what is the rate of convergence. What is the value $\lambda$ such that $|x_{n+1} - \alpha| = \lambda|x_n - \alpha|$ where $\alpha$ is the root. Find the…
BuddyD
  • 29
0
votes
1 answer

How many numbers are in this System?

Consider the System $±(.a_1a_2a_3...a_t)_N.N^{e}$ with $0\le a_i\le N-1$, $a_1≠0$ $e\in[e_{min},e_{max}]$. Let $N=8$, length of the mantissa $t=9$ and $e_{min}=-15$, $e_{max}=15$ How many numbers are in this System ? I mean, we have to find out…
OBDA
  • 1,715
0
votes
1 answer

Estimating the error of a Taylor polynomial

The information I'm given is that we are approximating $f(8)$ with a second degree Taylor polynomial for f centered around $a = 10$. Assume that the $abs(f'''(x)) \lt 3$ for $x$ on the interval $[6, 11]$. This is all the information I have. The only…
jtht
  • 217
0
votes
1 answer

bisection method, Numerical Analysis

Consider the method of bisection applied to $f(x) = \arctan(x)$, with initial interval $x = [-4.9, 5.1].$ Are the hypotheses under which the method of bisection converges valid? If so, then how many iterations would it take to obtain the solution…
James
  • 1
0
votes
1 answer

What are the problems that may arise by the application of Euler method?

What are the problems that may arise by the application of Euler method? 2 problems are described. One of them: 1.It is written that from from the convergence order it can be concluded that halving the error means halving the step-size and thus…
learner
  • 973
1 2 3
…
99 100