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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

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Numerical Integration: The degree of accuracy of a quadrature

I am in a first semester numerical analysis course and we are going over numerical integration and more specifically quadrature forms. So far we have gone over standard quadrature as well as Gaussian quadrature. The problem is: Show that the…
trmpt08
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Distance between points

I am wondering how can I solve following problem. Arrange randomly n points inside a square of the side length a under the condition that the distance between any two points may not be smaller than 1. I would like to see how can it be solved.
Beginner in fort
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Numerical Integration by Undetermined coefficients

The important part of my question is after the bold "Now" The method of undetermined coefficients is defined so that the error of a function $f(x)$ to be integrated is zero. I.e. $E=\int_{a}^{b} f(x) dx - \sum_{i=0}^{n} A_i f(x_i) =0$ Where $f(x)$…
DLV
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How can we achieve absolute stability?

The following Runge-Kutta method is given. $$ \begin{array}{c|ccccc} \tau_1 =0 & a_{11}=0 & a_{12} = 0\\ \tau_2 =\frac{5}{2} & a_{21} = \frac{5}{2} & a_{22} = 0\\ \hline & b_1 = \frac{4}{5} & b_2 = \frac{1}{5} & \ \end{array} $$ I have to…
evinda
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Deferred Corrections vs Multigrid

I've been looking at the method of Deferred Corrections (see page 9 of this presentation) to numerically solve ODE IVPs. To me, the process looks identical to a V-cycle in a multigrid method if the solution was solved directly on each grid. Am I…
user245571
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turn $L\{u(t-3)(t^2)\}$ to $L\{u(t-3)[(t-3)^2+6(t-3)+9]\}$?

I was given looking at one of the examples in my textbook and it took this laplace transform $L\{u(t-3)(t^2)\}$ and turned it into this $L\{u(t-3)[(t-3)^2+6(t-3)+9]\}$ in the next step. I'm wondering how did $t^2$ become $(t-3)^2+6(t-3)+9$ I get…
a wee little worker
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Question regarding the approximant of a function

This is given as a sample exercise for our final exam in our Numerical Analysis class. However, nowhere in the course has the notion of approximant been defined or how to approach this kind of problem . What I found on the web doesn't really seem…
andyschwarz
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Gauss rule Derivation

I am working through some exam prep questions, and need a little guidance on this one: The 2 point Gauss for weight $e^{-x}$ on the interval $[0, > \infty]$ has the form: $$\int_{0}^{\infty}e^{-x}f(x)dx \approx w_1f(x_1) + w_2f(x_2)$$ (a) Use…
JackReacher
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IVP using TS and Euler

I am just working through some exam practice problems, and I am a bit stuck with this one: Consider the IVP: $$ \frac{dy}{dt} = f(t,y), \space y(0)=y_{0} $$ (a) Expand solution $y = Y(t)$ of the IVP in a two term ts with r about $t= t_{n+1}$, to…
JJJ
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Iterative methods monotonically decreasing of the residual

For a question on Iterative Methods I have to show that the 2-norm of the residual is monotonically decreasing. We are given the following formula: $r^{(k+1)} = r^{(k)} - \alpha^{(k)} A z^{(k)}$ where $r^{(k)}$ is the residual of the iterative…
penguin1311
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Summation error of n-segment Trapezoidal rule

Help me please with this question. Let $f\in C^{\infty }$ function defined as $\forall x, f(x)=f(x+2\pi )$. Let $e_{n}$ be a summation error of n-segment trapezoidal rule. Prove that $\forall \alpha \geq 0$, $\exists C>0$ so that: $\left | e_{n}…
Lilly
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Romberg Integration: accuracy

I'm applying the Romberg method to numerically integrate a data set of equally space, numerically determined values. I would like some estimate of the uncertainty (or accuracy or error) in my answer. I have found two different approaches in the…
user26718
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Gauss-Legendre Quadrature, computation of the abscissas and weights

I would like to write a program to calculate abscissas and weights of Gauss-Legendre Quadrature. I found the following source http://rosettacode.org/wiki/Numerical_integration/Gauss-Legendre_Quadrature. I don't understand why the first guess $x_0$…
LiN
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Reaction diffusion numerical scheme: discrepancy between devised scheme and scheme that works

I've been writing a PDE solver for a reaction diffusion equation in one dimension. To test it I'm using the "method of manufactured solutions" (that's the term my supervisor gave it - I'm not sure if that's a common term). The equation…
Bamboo
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Using Lagrange interpolation to determine canonical form of polynomial

I have some difficulty of determining the canonical form of a polynomial. Here is the problem: Suppose $P$ is a polynomial with degree $5$, and value of $P$ at $-2, 0, 1, 4, 5, -3$ are $2, 4, 5, -1, -2, 1$, correspondingly. Determine the canonical…
le duc quang
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