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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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If I need to approximate $f'(x_i), i =0..2$ with precision of $O(h^2)$ should I only use one step calculation using central difference methid?

Points given: $x = (\frac{1}{2}, 1, \frac{3}{2})^T, \ y = (-8, -5, 0)^T$ My calculation: $$f'(1) = \frac{f(x+h)-f(x-h)}{2h} = \frac{0 + 8}{1} = 8$$ Is this correct or should there be more steps to approximate $f'(x_i)$ with precision $O(h^2)$?
use1883
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finding a fixed-point ˆ x∈R for the function

I don't know how to verify this function if I can't use the Banach-Caccioppoli Contraction Principle. Please help
Amy12
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$\sqrt {144} = 12$ How can I find it without calculator?

$$12\times 12 = 144$$ If there's a question that asks me to find square roots manually, how will I find that? I know a way that's really long and boring. I actually know about the "Tree" method. Is there any other interesting and effective way to…
Himel
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Approximate Derivative, 3 Midpoint

I used 3 midpoint formula is there any better method?
Mehmet Usta
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Find the region of absolute statibility for the Forward Euler method.

One step of Forward Euler method: $u_{j+1} = u_j + hf_j$ Find the region of absolute statibility for the Forward Euler method? I'm struggling to solve this problem, could I please get a hint?
user486957
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Numerical Methods: 4th order Runga Kutta

How can I implement the 4th order Runga-Kutta method on $dy/dx$ since it is independent on time? Thanks Sum: $$ dy/dx = 0.1y +2, ~~ y(0) = 1. $$ solve for $y(t)$ from $t=0$ till $t=10$.
aid846
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How does one make a fit of $A\exp(-B|x|)$ correctly?

Suppose one has some data that looks exponentially distributed. Most, if not all, numerical subroutines can make the fit using least squares fitting of the function \begin{equation} A\exp(-Bx), \end{equation} where $A,B$ are the parameters of the…
user2820579
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Numerical solution's evolution's dependance on timestep and the absolute time passed.

After solving a heat equation (1 space dimension and the time dimension) with a scheme, if I get different results at the same absolute points of time for different durations of the timestep, does it mean that I have made a mistake? With my solution…
Nick The Dick
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Derivation of Muller's method?

I am trying to derive Muller's method. I see that if I have three points, I can come up the parabola to interpolate these three points using the Lagrange formula. After I get this very scary looking formula, I try to find the roots by simplifying…
user709452
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Derivation of the interpolatory quadrature formula $I_2$

Say we have $a = 0$, $b = 1$ and we want to compute the weights of the interpolatory quadrature formula $I_2$ with the nodes $x_0 = 0$, $x_1 = \frac{2}{5}$ and $x_2 = 1$. I want to show that $I_2(p)$ is equal to \begin{equation}\tag{1} \int_{a}^{b}…
user486957
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How do I derive the condition for the function $A(x)$ so that this Newton type method converges

Say I modify Newtons method by introducing the function $A(x)$ s.t. $$p_{ n + 1} = p_n - A(p_n) \frac{f(p_n)}{f'(p_n)}$$ If $x = a$ is a simple root of $f$ and $f$ is sufficiently differentiable, what conditions do I have put on $A(x)$ so that this…
user486957
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solution of $\frac{dy}{dx}=0.3y-10$ using euler numerical method

Finding solution of differential equation using euler numerical method $\displaystyle \frac{dy}{dx}=0.3y-10$ and $0\leq x\leq 3$ and $x=0,y=40.$ What i have tried $\displaystyle y_{n}=y_{n-1}+h\cdot f(x_{n-1},y_{n-1})$. Where $f(x,y)=0.3y-10.$ I…
jacky
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Modified Newton-Raphson Method

What is the order of convergence of modified Newton's? $$x_{n+1}=x_n-m\frac{f(x_n)}{f'(x_n)},$$ where $m$ is the multiplicity. I know it is $2$, but can someone explain it briefly?
7237372728
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Numerical Methods for Discontinuous or Indifferentiable Functions

The title says it all; by now, I have learned quite a few numerical methods for finding the roots of a function that is differentiable. However, I haven't heard of any strategy or method analogous for functions that are either discontinuous or…
Venkataram Sivaram
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Find order of consistency of IVP with Butcher Tableau

We consider this scalar IVP $\dot{y}(t)=f(t, y(t)), y(0)=y_0$ with Butcher Tableau \begin{array} {c|cccc} 0 & 0 & 0 & 0\\ \frac{1}{3} & \frac{1}{3} & 0 & 0\\ \frac{2}{3} &\frac{2}{3} & 0 & 0 \\ \hline & 0 &\frac{1}{2} &\frac{1}{2} …
user446410
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