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
3
votes
1 answer

When is trapezoidal rule better than Gaussian quadrature?

I have used the trapezoidal rule to evaluate the integral $$\int_{-1}^1 |x| \, dx$$ and it gives an aproximation without any error (the exact solution: $1$). However, when I use Gauss–Chebyshev or Gauss–Legendre quadrature, I have an error in the…
John Keeper
  • 1,281
3
votes
1 answer

Gauss–Seidel and Jacobi methods convergence

Can you please give me criteria for convergence of these 2 iterative methods(Gauss–Seidel and Jacobi methods)? What is necessary and sufficient condition and what just a sufficient from these 3? 1.The standard convergence condition (for any…
Tina
  • 517
3
votes
1 answer

Euler's Method question

I have this differential equation $y'=\frac{y(\sin t)}{t}$, $y(0)=2$, and $h=\frac{1}{4}$. The first set of values, the inital, are $(0,2)$. For the next iteration would it be $y_1=2+(\frac{1}{4})\left(\frac{2\cdot\sin(0)}{0}\right)$? I know you…
Alti
  • 2,458
3
votes
0 answers

How should c be chosen to ensure rapid convergence of $x_{ n+1}= x_ n+c(f( x_ n))$ to $\alpha$?

Consider the rootfinding problem $f(x)=0$ with root $α$, with $f´(x)≠0$. Convert it to the fixed-point problem $x=x+cf(x)≡g(x)$ with $c$ a nonzero constant. How should c be chosen to ensure rapid convergence of $x_{n+1}=x_{n}+cf(x_{n})$ to α…
Daniela garcia Garcia
  • 31
3
votes
2 answers

Numerical approximation of an integral

I read a problem to determine the integral $\int_1^{100}x^xdx$ with error at most 5% from the book "Which way did the bicycle go". I was a bit disappointed to read the solution which used computer or calculator. I was wondering whether there is a…
user7537
3
votes
5 answers

Iterative method to solve f(x) = x

I need to solve a non linear equation of this kind: $$f(x) = x$$ where $f(x)$ is an injective function I can't isolate $x$ so I want to use an iterative method in order to find the value of $x$. In a naive way I used this strategy: $x_0 = 1$ random…
obchardon
  • 257
3
votes
2 answers

3rd order Runga Kutta method agrees with Taylor Series up to terms of order $h^3$?

Consider the initial value problem: $$y(0) = 1, y ′ (t) = λy(t)$$ Using that the solution is $y(t) = e^{λt}$, write out a Taylor series for $y(t_{i+1})$ about $y(t_i)$ up to terms of order $h^4$ (note: use that $h = t_{i+1} − t_i$). Write out what…
sonicboom
  • 9,921
  • 13
  • 50
  • 84
3
votes
2 answers

Procedure for adaptive step size for Runge Kutta 4?

I am writing a Runga Kutta 4 algorithm in MATLAB. I would like to add adaptive step sizing to this algorithm. From what I've read it seems you calculate the value of the function for two step sizes on each iteration and then from the size of the…
sonicboom
  • 9,921
  • 13
  • 50
  • 84
3
votes
1 answer

Computing a slowly-converging limit

Let $$ f(x)=-\log\log x+\sum_{2\le n\le x}\frac{1}{n\log n}. $$ How can I efficiently compute $$ f(\infty)=:\lim_{x\to\infty}f(x)? $$ Brute force suffices to find 0.7946786454... but I would like several hundred digits. It seems that I should be…
Charles
  • 32,122
3
votes
0 answers

"Leading order error" vs "order of the error"

This may be a daft question but I wanted to be sure. If I were asked to find the leading order error when using the mid-point rule to approximate a function $f(x)$ would it be the same as being asked to find the order of the error in the midpoint…
Nicky
  • 671
3
votes
1 answer

How to show that this numerical differentation formula for the 4th derivative converges?

I have to show that the numerical differentation formula $$ f^\left(IV\right) (x) = \frac{1}{h^4} \left(f(x-2h)-4f(x-h)+6f(x)-4f(x+h)+f(x+2h)\right) + R_h(f) $$ converges. How do I do that? Do I have to show something related to the truncation…
peeter pats
  • 31
3
votes
2 answers

sincos implementation

Many computational libraries contain a routine called sincos that allows one to compute sin and cos simultaneously. The existence of this routine suggests that calling this routine is more efficient than making a call to sin followed by a call to…
Arin Chaudhuri
  • 6,349
3
votes
1 answer

Using the Euler-Maclaurin integration formula to estimate the Euler-Macheroni constant, why do the terms -1/2f(1) and $-f^{2p-1}$ disappear

Background: This is from question 12.3.2 from Arfken, Weber, Harris Mathematical methods. The definition of the The Euler-Maclaurin integration formula in the text is: The formula is taken from n=1(instead of 0) to $\infty$ and rearranged to solve…
user5389726598465
  • 1,171
3
votes
1 answer

Numerical Analysis Gauss-Lobatto

I am trying to find the expression of the weights and nodes for the Gauss-Lobatto quadratures with 4 nodes. I am guessing this is a sum of weights? Does anyone here have experience working with this type of math?
Ian Yutz
  • 31
  • 1
3
votes
2 answers

Difference Between Simpsons Rule and 3/8 rule

So, I was studying Numerical Integration from Hildebrand where I came across Simpson's Rule and 3/8 rule. Now, Simpson's rule has an error of order 5 and degree of precision 3, which is same as 3/8 rule. The difference lies in the Lagrange…
Abhinav Jha
  • 862
1 2 3
…
99 100