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
4
votes
2 answers

minimum number of iteration in Bisection method

One root of the equation $e^{x}-3x^{2}=0$ lies in the interval $(3,4)$, the least number of iterations of the bisection method, so that $|\text{Error}|<10^{-3}$ is (a) $10$ (b) $6$ (c) $8$ (d) $4$
Empty
  • 13,012
3
votes
2 answers

Is there a problem for which it is known that the only solution is "iterative"?

By "iterative solution" I mean specifically the following type of iteration: given a problem whose solution is $x$, first you compute some approximate solution $x_n$, and then make use of $x_n$ to find $x_{n+1}$; and furthermore the sequence…
Matt Calhoun
  • 4,404
3
votes
2 answers

How to numerically solve for a contour curve?

Supple you have a 2D surface z = f(x,y). Given a value z, how to numerically find all the values of x and y that satisfies z = f(x,y)? I know that my surface is well behaving to the extend that the set of x,y that satisfy the requirement are all on…
user133586
  • 135
3
votes
1 answer

Failing to check if a number is a perfect square

To check if a natural number is a perfect square, programming in Python, I check if int(math.sqrt(n))==math.sqrt(n), id est if the decimal part of the square root is zero. Here my question: are there cases such that this test could fail due to…
Immanuel Weihnachten
  • 1,876
3
votes
2 answers

Iterative integration algorthm

By iterative algorithm, I mean a numerical algorithm that works by improving on a previous approximation to obtain a more accurate approximation. An example is Newton's Method. The numerical integration algorithms I know (e.g. Simpson's rule,…
michaelkovarik
  • 431
3
votes
2 answers

Approximation of $\pi$, with an error of less than $\frac{1}{2}\times 10^{-8} $

This is what I've achieved so far: $$\tan^{-1}1 = \frac{\pi}{4} \Rightarrow \pi = 4\tan^{-1}1$$ $$\tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5}+ \cdots + (-1)^k\frac{x^{2k+1}}{2k+1}$$ $$\pi = 4\tan^{-1}1 = 4\left(1-\frac{1}{3}+\frac{1}{5} +\cdots…
Sajad Rastegar
  • 275
3
votes
1 answer

What method are there for "numerically" computing arclengths!

I know the originals formula for arc-length is: $$\int_{a}^b \sqrt{1+{f'(x)}^2}$$ However most of the formulas don't have closed formed solutions, and are unsolvable in terms of this equation. So far, I tried taking…
Arbuja
  • 1
3
votes
1 answer

Euler's method vs midpoint method

Are the following methods equally accurate and if not, why? Using Euler's method with a step size of $h$. Using the midpoint method with a step size of $2h$. Even though Euler's method has a global error of $\mathcal{O}(h)$ and the midpoint…
Simeon
  • 937
3
votes
1 answer

Where comes the +1 from in this formula?

I'm working on two papers ([1] equation 8, [2] equation 2.3) and I can't figure out why there is an identic formula I can't explain on both. $p(z) \in \mathbb{C}[z]$ is a monic polynomial with simple roots, degree $n$ and complex coefficients. I fix…
Rakk
  • 31
3
votes
3 answers

ill-conditioned matrix 'Matrix is close to singular or badly scaled'

in the equation of A*q=b, A is a NxN matrix in which the numbers can be up to 10^56 and the minimum is 1. the condition number of the matrix can be as large as 3.16e+064. The SVD, QR and LUP have been used to deal with the matrix. however, when…
wwsui
  • 41
3
votes
1 answer

Fibonacci Search Algorithm

Can someone show me an example of using this method for 'find the minimum of $$F(x) = x^2 - 6x + 2 \; \text{ on } [0,10] $$' ? I'm trying to follow the algorithm detailed above, but I don't understand it. How do they know at what $k$ to stop at?…
Ozera
  • 2,050
3
votes
2 answers

Does Newton's Method converge for f(x)

Does Newton's Method converge for: $$f(x)=(x-5)^2e^{x-5}$$ So Newton's Method is: $$x_{n+1}=x_n−\frac{f(x_n)}{f′(x_n)}$$ $$x_{n}=x_{n-1}-\frac{x_{n-1}-5}{x_{n-1}-3}$$ Error: $$e_n = x_{n} -…
ltc985
  • 43
3
votes
2 answers

How can I use Numerical-Methods to solve that equation ...

Use the Newton-Raphson iteration method (NR) to find one positive root ( Xr > 0 ) of the equation $ X^3 +X -1 =0 $ ??
Ahmed Osama
  • 174
3
votes
1 answer

Interval arithmetic for finite difference error bounds

It seems that interval arithmetic can be used to quantify floating point truncation error in computational calculations. Does anyone know if it is possible to use interval arithmetic in a finite difference scheme, to automatically quantify the…
David H
  • 133
3
votes
3 answers

Fixed point iteration contractive interval

Consider the function $F(x) = x^2-2x+2$. Find an interval in which the function is contractive and find the fixed point in this interval. What is the convergence rate of the fixed point iteration: $x_{n+1} = F(x_n)$ in that interval? I'm lost on…
Neurax
  • 1,005
1 2 3
…
99 100