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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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Newton's Method for a system of nonlinear equations for a ${\it path}$ in $\mathbf{R}^n$?

I understand that Newton's Method is used to solve for the $(x_1, \ldots, x_n ) \in {\mathbf R}^n$ which ensures $f_1 = 0, \ldots, f_n = 0$ where $f_i$ is a nonlinear function of $(x_1, \ldots, x_n )$. My problem is as follows: Find the path $(x_1…
Evan
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What is the meaning of implicit and explicit numerical solution method for elliptic equation?

The usual way to define the implicit and explicit numerical (Finite difference) solution/discretization is by using a parabolic equation like the transient heat equation. How to define the implicit and explicit solution for a elliptic equation like…
GRANZER
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Numerical Integration with weights

For every $x_i$ where i=1,..,5 there exists a corresponding weight $w_i$. I am trying to calculate numerically the continuous integral $$\int x^2w \, dx$$ How would the numerical integral look like? I have used the trapezoidal rule but I cannot…
thalia nouni
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product of orders of magnitude

I have been given quite a theoretical question. "What are the problems you may encounter when taking the product of 10 orders of differing orders of magnitude. What approach might you take to help ensure the numerical stability and accuracy of the…
UpDaDubs14
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distribution of two floating point multiplication

given any two floating numbers on a PC. what is the distribution of the roundoff error of their multiplication? Using math let $x_1$ and $x_2$ be two floating numbers, the true value of their multiplications is $x=x_1 x_2$, the number represented on…
aban
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Possibility for two distinct polynomials to interpolate the same m data points

Is it ever possible for two distinct polynomials to interpolate the same m data points? If so, under what conditions, and if not, why?
efgdh
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Why is uniqueness a requirement for well-posed problems?

I understand that, by definition, a well-posed problem must have a unique solution, but why is it important? For example, the polynomial $$x^4-x^2(2a-1)+a(a-1)$$ has real roots $x=\sqrt{a},-\sqrt{a},\sqrt{a-1},-\sqrt{a-1}$ for $a>1$, and there…
elcocodrilotito
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Implicit Midpoint Rule for a Double Pendulum

I am dealing with a Double Pendulum with two masses of size $1$ linked with inelastic, weightless rods. There is assumed to be no air resistance too. The Hamiltonian $H$ satisfies the following system: $$ \dot{\theta}_1 = \frac{\partial H}{\partial…
Naji
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What is the convergence order of the numerical formula?

$\int_0^2 e^x \cos (x) \sin (x) dx$, What is the convergence order of the numerical formula? I am confusing the question. Do I have to use Trapezoidal rule or simpson rule or approximation rule? If so then how to use approximation and finding order…
MAS
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How do I show that a function has a unique root?

My question is: 1a) Show that the function $f(x)=x^3+\sin({\pi x \over 2})−1$,for $x \in [0,1]$,has a unique root $x^* \in [0,1]$. How would I show this using numerical methods? b) Write down the function g for the Newton–Raphson method to compute…
Mai Muop Nguyen
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Thomas algorithm (Numerical methods)

enter image description here Please help i've been trying to solve it since 2 days.I have solved it using the differential equation but answer is coming different from thomas algo. and DE.
NIKHIL SHARMA
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Rate of convergence of Gauss-Seidel iteration method.

Help me: $2x-y=7\\ -x+2y-z=1\\ -y-2z=1$ Show Gauss-Seidel iteration scheme converges and find the rate of convergence. My Attempt: The iteration matrix for this system of equation is $$ H = \left( \begin{array}{ccc} 0 & 1/2 & 0 \\ 0 & 1/4 & 1/2 \\…
Faisal
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Applying one step of Richardson's Extrapolation

Suppose $N(h)$ is an approximation for every $h>0$ and that $M=N(h)+Ah^{4/3}+Bh^{7/3}$. Apply one step of Richardson's Extrapolation with Value $N(3h)=2.7172$ and $N(h)=2.7182$ and compute the impoved value for $M$. If needed take $h=.0001$. I'm not…
MANONMARS45
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Use Richardson Extrapolation to compute the function

A regular n sided polygon inscribed in a circle of diameter one has the circumference as cn = 2 n Sin( Π / n ) put h=1/n How can i show whether c(h) = c1/h satisfies the assumption for repeated Richardson extrapolation Also i need to calculate…
Roushan Singh
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Conversión entre bases (Conversion between bases)

¿Es cierto que todo número que tiene una representación finita en base 3 tambien tiene una representacion finita en base $3^k$ con $k\in\mathbb N$? ¿ Por qué? English version: Is it true that every number that has a finite representation in base 3…
sukeliny
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