Questions tagged [numerical-calculus]

This tag is for various question on numerical calculus / numerical analysis which concerned with all aspects of the numerical solution of a problem, from the theoretical development and understanding of numerical methods to their practical implementation as reliable and efficient computer programs.

Numerical calculus / Numerical analysis provides the foundations for a major paradigm shift in what we understand as an acceptable “answer” to a scientific or technical question. In classical calculus we look for answers like $\sqrt{\sin x}$, that is,answers composed of combinations of names of functions that are familiar. This presumes we can evaluate such an expression as needed, and indeed numerical analysis has enabled the development of pocket calculators and computer software to make this routine. But numerical analysis has done much more than this. Most numerical analysts specialize in small subfields, but they share some common concerns, perspectives, and mathematical methods of analysis.

Here is some issues that numerical analysis is used in:

$1.\quad$ Solving linear/non-linear equations and finding the real roots, many methods exist like: Bisection, Newton-Raphson ... etc.

$2.\quad$Fit some points to curve, good approximation and simple solution.

$3.\quad$Interpolation, great to get any value in between a table of values. It can solve the equally spaced readings for unequally spaced methods, Newton general method is implied.

$4.\quad$Solve definite integration, simple methods is used to compute an integration based on idea that the definite integration is the bounded area by the given curve, these methods approximate the area with great approximation. Many methods there, like Simpson’s rule.

$5.\quad$Solving initial value 1st and 2nd order differential equations, good approximation and simpler than normal analysis.

$6.\quad$Solving partial differential equations like Laplace equation for wave equation, very fast solution.

Applications:

Numerical analysis / Numerical calculus is the area of mathematics and computer science that creates, analyzes, and implements algorithms for solving numerically the problems of continuous mathematics. Such problems originate generally from real-world applications of algebra, geometry, and calculus, and they involve variables which vary continuously. These problems occur throughout the natural sciences, social sciences, medicine, engineering, and business. Beginning in the 1940's, the growth in power and availability of digital computers has led to an increasing use of realistic mathematical models in science, medicine, engineering, and business; and numerical analysis of increasing sophistication has been needed to solve these more accurate and complex mathematical models of the world. The formal academic area of numerical analysis varies from highly theoretical mathematical studies to computer science issues involving the effects of computer hardware and software on the implementation of specific algorithms.

References:

https://en.wikipedia.org/wiki/Numerical_analysis

"Numerical Methods for Scientific and Engineering Computation" by M. K. Jain, S.R.K. Iyengar, R. K. Jain.

" Introduction to Numerical Analysis" by F. B. Hildebrand

"Numerical Mathematical Analysis" by James B. Scarborough

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Discretization formula for system of differential equations. "Solution to one of these is the initial condition of the other". In which sense?

Consider the following stochastic differential equation \begin{equation} dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1} \end{equation} where $A$, $B$ and $C$ are parameters and $dW$ is a Wiener increment. Equation $(1)$…
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What are the difference between some basic numerical root finding methods?

I understand the algorithms and the formulae associated with numerical methods of finding roots of functions in the real domain, such as Newton's Method, the Bisection Method, and the Secant Method. Because their formulae are constructed…
Alvin Nunez
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Finite differences second derivative as successive application of the first derivative

The finite difference expressions for the first, second and higher derivatives in the first, second or higher order of accuracy can be easily derived from Taylor's expansions. But, numerically, the successive application of the first derivative, in…
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Solving a nonlinear system of equations involving only products of unknowns

I would like to find a numerical solution of a system of $N$ equations of the form: $A^i = w_1 F(x^i_k) + w_2 F(x^i_l) + w_3 F(x^i_m) +...$ $A^j = \,\,\,\,\,\,0 \,\,\,\,\ \,\,\ + w_2 F(x^j_l) + w_3 F(x^j_m) +...$ $A^k = w_1 F(x^k_k) + w_2…
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Approximating a derivative: How to complete this proof of $f'(x_2) = \frac{f_0 - 8f_1 + 8f_3 - f_4}{12 h} + \frac{h^4}{30}f^\mathrm{V}(\xi)$?

Fix five equally spaced nodes as $x_i = x_0 + ih$ where $h > 0$, $x_0\in\mathbb{R}$, and $i = 0, 1, 2, 3, 4$. Let us also denote $f_i := f(x_i)$. Exercise. Assume that $f\in \operatorname{C^5}[x_0, x_4].$ Show that there exists some…
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(Why) can we treat a function of a variable as another independent variable?

I'm currently reading my numerical analysis textbook and something's bugging me. To get into it, let's take a look at the following differential equation; $$u'(x) = f(x, u(x))$$ In order to determine the stability of the equation, one may calculate…
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Equation of motion through the Lagrangian with Lagrange multipliers

I ask for advice, cause I'm a little confused. We have such a Lagrangian: $L=\frac{1}{2}m(\dot{x}^2+\dot{y}^2)-\lambda(x+xy+y-1)$ Here $\lambda(x+xy+y-1)$ is the constraint on the phase variables. I need to derive the equation of motion given the…
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How many positive roots can $\sum_{i}\frac{a_i}{x+b_i}$ have where $b_i$'s are all positive?

What is the maximum number of positive roots $\sum_{i}^N\frac{a_i}{x+b_i}$ can have where $b_i$'s are all positive? (everything here is a real number. To provide context, I encountered this problem while doing theoretical neuroscience research where…
CWC
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Spectral radius greater than 1

The eigenvalues of the matrix $$A=\begin{pmatrix}0 & 1/2 & -1 \\ -1/2 & 0 & 1 \\ -1 & -1 & 0\end{pmatrix}$$ are relatively complicated (roots of the polynomial $x\mapsto x^3+x/4+1$). Is there an easy way to see that the spectral radius of this…
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Smoothly connecting PDEs with finite differences

A PDE with non-smooth inhomogeneity Let $\mathcal{L}$ be a second-order, linear, elliptic differential operator acting on $\mathcal{C}^2([0,2]^2)$. I'm numerically solving the inhomogeneous…
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Approximating $\int_1^\infty x^a (x-1)^b e^{cx}dx$

After lengthy calculations, I arrived at \begin{align*} \int_1^\infty x^a (x-1)^b e^{cx}dx, \end{align*} which cannot be solved in closed-form. I thus seek to approximate the integral, potentially including special functions. The constants $a,b,c$…
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Three-Body Problem - how to find the Figure-eight solution?

Suppose the coordinates of the Earth and the Moon are fixed and let $(u,v)$ be the coordinates of the satellite. I'm looking for the numerical solution of the three-body problem: $$u'' = 2v + u - \frac{c_1(u+c_2)}{((u+c_2)^2 + v^2)^\frac{1}{2}} -…
mathbbandstuff
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Closed form of $\int_{-\infty}^{\infty}\frac{\exp(-x \tanh(x))}{1+x^2}$?

I want to approximate the integral of a function over the whole real line very accurately. The integrand should be analytic in a strip but also have exponential decay. A prototypical example is $$\int_{-\infty}^{\infty}\frac{\exp(-x…
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Proof that the roots of $\mathrm e^{-πx}=\sin πx$ approach integers as $x\to \infty$

This question is inspired by @gt6989b’s comment here. Numerical analysis suggests that the roots of the equation $\newcommand{\e}{\mathrm{e}} \e^{-πx} = \sin πx$ rapidly and closely approach integers as $x\to\infty$. Here’s a quick list of the first…
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Extrema of Chebyshev polynomials (of the first kind)

I can hardly find a proof why the extrema of the Chebyshev polynomials are $$ x_k=\cos(\frac{k}{n}\pi), k=1,...n $$ and also why there are $n+1$ of them. The Chebyshev polynomials are here defined as $$T_0=1, T_1=x, T_{n+1}=2xT_n(x)-T_{n-1}(x)$$…
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