# What is Numerical: Definition and 772 Discussions

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. The growth in computing power has revolutionized the use of realistic mathematical models in science and engineering, and subtle numerical analysis is required to implement these detailed models of the world. For example, ordinary differential equations appear in celestial mechanics (predicting the motions of planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.
Before the advent of modern computers, numerical methods often depended on hand interpolation formulas applied to data from large printed tables. Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas nevertheless continue to be used as part of the software algorithms.The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection (YBC 7289), gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square.
Numerical analysis continues this long tradition: rather than exact symbolic answers, which can only be applied to real-world measurements by translation into digits, it gives approximate solutions within specified error bounds.

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1. ### I Numerically solving a non-local PDE

I have a PDE to solve numerically on the region ##x \in [0,1]## and ##t \in (0, \infty)##. It is of the form:$$\frac{\partial f(x,t)}{\partial t} = g(x,t) + \int_0^1 h(x, x') f(x', t) dx'$$The second term is the tricky part. The change in ##f(x,t)## at ##x## depends on the value ##f(x',t)## of...
2. ### A Numerical method to Lippman-Schwinger equation

There is a question that puzzle me when I apply numerical method to principal value integral. Let me descibe it. We make use of the fact that the integral ##\int_0^\infty \frac{dk}{k^2-k_0^2}## vanishes, namely, $$\int_0^\infty \frac{dk}{k^2-k_0^2} = 0 .$$ We use this formula to express a...
3. ### What does a numerical rating mean? Mechanical equipment or device

A certain mechanical appliance or device is given a rating that says, a certain operation or a certain movement can be performed some specified thousand times. What does the rating really mean? Was that movement or operation performed until the piece failed? Does it mean some advanced test...
4. ### I Categorical vs Numerical Variables

Hello, I am generally clear on the distinction between numerical and nonnumerical (also called categorical or qualitative) variables but I still have some doubts in some regards. A numerical variable (continuous or discrete) has a value that derives from a measurement procedure (using a tool)...
5. ### Understanding Numerical Contexts: Exploring the Meaning Behind Numbers

Does anybody please know what the 'number means in these contexts? Many thanks!
6. ### Question about approximate numerical integration methods

This isn't a homework question per se but I can post more details like the data points & my work after. Suppose we are given a set of arbitrary points for which we cannot find an equation and we need to find the area under the curve without an analytical method - we can use either of the three...
7. ### Comp Sci Orbit of the Earth - numerical methods leapfrog

I am attempting this homework exercise part b). I've modified my code but I get error overflow message. My goal is to modify my code so it returns kinetic and potential energy of Earth's orbit. I made a new f(z,t) and modified the rows 99 and 100 with dz[2]=-G*M*m/r, and dz[3]=0.5*m*y**2 but...
8. ### Comp Sci Network design -- numerical problem asked in our exams

Provide IT infrastructure details for a government school with 4 departments including ISP department which are around 500 meters apart from each other. Three departments have 5 labs each with around 24 computers in each room. ISP contains server farm with server like DNS, DHCP, E-mail, FTP and...
9. ### I Numerical Calculation of Hamiltonian Overlaps

Hi all, I am currently reading through this paper: https://iopscience.iop.org/article/10.1088/1367-2630/10/4/045030 and would like to reproduce their results for N=5. My roadblock is with (9), which models the classical motion of the system. Now symbolically finding the eigenstates of the matrix...
10. ### Numerical Analysis - Richardson Extrapolation on Riemann Sum

I got something like this, but I'm not sure it is correct, because if it has the same order of convergence as trapezoidal rule which is 2, it should yield the same result as trapezoidal rule but mine doesn't (?). For example sin(x) for [0,1], n with trapezoidal rule = 0.420735... With my own...
11. ### Numerical Methods for Learning Orbital Mechanics

I'm interested in learning orbital mechanics but I haven't taken a class in numerical methods yet. Do I really need to take a whole class in numerical methods before learning orbital mechanics, or can I get by if I self-learn a smaller portion of the syllabus of a numerical methods class? If so...
12. ### COMSOL simulation of a 3D Ferrite Bar Numerical Model For Magnetic Flux

Hi, I am willing to simulate a 3D ferrite bar transmitter and reciever where coupling coefficient k and Bt magnetic flux density on the each side uses the finite element method for solving partial differential equations. The Magnetic Fields module has equation (jωσ − ω2ε0εr)A + ∇ × H = Je...
13. M

### Mathematica Numerical integration over a Green's function

Hi PF! I'm numerically integrating over a Green's function along with a few very odd functions. What I have looks like this NIntegrate[-(1/((-1. + x)^2 (1. + x)^2 (1. + y)^2)) 3.9787262092516675*^14 (3.9999999999999907 + x (-14.99999999999903 + x (20.00000000000097 -...
14. ### A Residual of PDEs as convergence criteria of numerical solution

Use a numerical method to solve a PDE f[u(x),u'(x),...]=0, where f is an operator, e.g. u'(x)+u(x)=0, and obtain a numerical solution v(x). Define f[v(x),v'(x),...] as the residual of the original PDE. Is this residual of the PDE widely used as the convergence criteria of the numerical solution...
15. ### Comp Sci I need a solved numerical example on genetic algorithm for 1 iteration

Really need this. Tried googling but not many. 1 or 2 are there. I want this algorithm solved by hand to some problem. IDK what kinds of problems exists. but one is knapsack problem. there is analytics vidya's tutorial but I want something else, more direct, more clear...Any resource you can...
16. ### Requesting suggestions for languages, libraries, and architectures for parallel (and sometimes non parallel) numerical and scientific computations

A lot of the work I am interested to do will be mostly built from scratch by myself, provided there is fair support for numerical types (like complex numbers) and high precision numerical operations (if not, I'll be happy to write those routines as well). Many of my areas of interest are...
17. ### How High Does Cheese Rise on a Released Spring?

Question is in file please these questions and send it to me Q1: A 2 kg piece of cheese is placed on a vertical spring of negligible mass and force constant k= 2000 N/m that is compressed 16 cm. when the spring is released, (I) How high does the cheese rise from this initial position? (II)...
18. ### How to determine the minimum grid length | Numerical Plasma Physics

Summary:: I am learning particle-in-cell (PIC) python 3X code. PIC currently represents one of the most important plasma simulation tools. It is particularly suited to the study of kinetic or non-Maxwellian effects. I am learning particle-in-cell (PIC) python code. PIC currently represents one...
19. ### Python Numerical integration over a disk with polar coordinates

In my job, I was given the task of calculating a force that operates an ultrasound transmitter on a receiver. The calculation is made by assuming that each point on the transmitter is a small transmitter and integration should be made on the surface of the transmitter. Since the transmitter is...
20. ### Numerical computation of the derivative

I'm not sure if this is the correct forum to post this question, or should I post it in a math forum. But I was looking at some code when I found a 'strange' implementation to compute the derivative of a function, and I wanted to know if any of you has an idea of why such an implementation is...
21. ### A Define spin operators for numerical groundstate obtained by ED

Hi, I want to measure spin components of a ground state of some models. These ground states are obtained by ED. The states for constructing the Hamiltonian are integers representing spins in binary. As the ground state (and the other eigenvectors) are now not anymore in a suitable representation...
22. ### A How to solve simple 2D space-time PDE numerically

I have a 2D space-time PDE and I want to solve it numerically over the time axis. The time initial field is already known with respect to space, i.e., the spatial distribution is already known at time t = 0`. I solved the same PDF in Mathematica and got a solution. I tried to solve it...
23. ### The convergence of a numerical series

Greetings here is the exercice My solution was as n^2+n+1/(n+1) tends asymptotically to n then the entire stuffs inside the sinus function tends to npi which make it asymptotically equal to sin(npi) which is equal to 0 and consequently the sequence is Absolutely convergent Here is the...
24. ### Help With Negative Phases in a Numerical Term

I think I've got the numerator part figured out, but I'm really stuck on what to do with those negative phases in the last term and how to get this to all come together in the end. I feel like I must have made a mistake somewhere, but can't find it. Thanks in advance for the help!
25. ### Purcell EM Problem 1.2: Theory Behind Numerical Solution?

The author start of with $\frac{1}{(y+\sqrt{3})^2} - 2 \cdot \frac{1}{1 + y^2} \left( \frac{y}{\sqrt{1+y^2}} \right) = 0$ and arrives at the equation $y = \frac{(1+y^2)^{3/2}}{2(y+\sqrt{3})^2}$ The solution is merely by iterating (use an initial guess value of y, calculate the RHS, then use this...
26. ### B Find Numerical Relativity Source Code

Where can I find numerical relativity source code?
27. ### Calculating Numerical Uncertainties in an Equation

Background Information: I am working on a pulsed NMR lab project that involves graphing out a semi-log graph of free induction decay amplitude as a function of time. After graphing out the semi-log graph, I am to determine the apparent spin-spin relaxation time (##{T_2}^*##) through the...

29. ### Laplace Equation Numerical Solution

I wonder how to incorporate point charge?
30. ### Numerical Differentiation: Comparing Central & Other Formulas

Hello there, I have found a different central differentiation formula for a first derivate from what I am used to seeing and I was wondering if they were the same one. I am struggling to find the Numerical Differentiation formulas (forward, backward and central) in scholarly articles and I have...
31. ### Unexpected result in the numerical calculation - (invalid value encountered)

from numpy import log as ln z = 3 k = 2 x = 1 - ln(1 + z) / ln(1 + k) y = 1/5 print("The x=", x) Q = x**y print(Q) The result is The x= -0.26185950714291484 c:\Users\-\Desktop\... RuntimeWarning: invalid value encountered in double_scalars Q =...
32. ### A Numerical Hartree Fock with Finite Difference Matrices for Helium

Here is the paper again: https://www.mdpi.com/2218-2004/6/2/22?type=check_update&version=2#related_content For a class project I need to calculate the energy levels of atoms using the Hartree Fock method as presented in this paper which essentially brute forces the calculation using finite...
33. ### I Bose-Einstein numerical integration

Want to integrate the total energy density over all photon energies between two temperature values from 500K to 5800K, but not sure how to proceed. Here is some examples to help:
34. ### Numerical approximation of the 2nd derivative across a diffuse interface

Imagine you create a diffuse interface in space and determine which side of the interface you are on by a local scalar value that can be between 0 and 1. We could create a circle, centered in a rectangular ynum-by-xnum grid, with such a diffuse interface with the following MATLAB code: xnum =...
35. ### B Weird stuff on infinite numerical sequences in a Soviet book

The book is Calculus: Basic Concepts for High School on the first page you are given the following sequence: 1, -1, 1/3, -1/3, 1/5, -1/5, 1/7, -1/7, ... several pages later the rule is given: in the second rule, for the first term in the sequence, the coefficient of one of the terms is 1/0...
36. ### Numerical Evaluation of the Kirchhoff Integral (Flux Pattern) (Units?)

Naturally there are vector equivalents of the Kirchhoff Integral. Taken from Jackson (10.113) ##\vec{E} \left( \vec{r} \right) = \frac{ie^{ikr}}{r} a^2 E_0 \cos \alpha \left( \vec{k} \times \vec{\epsilon}_2 \right) \frac{J_1 \left( \sin \theta \right)}{\sin \theta}##Where I just let ##\alpha =...
37. ### MHB Numerical Methods for system of integral equations

Are there any standart ways to solve such systems? $\begin{cases} m(t, x) - f(t, x)= \int_{0}^{t} q(\tau,x) \, d\tau \\ u(t,x) = \int_{-\infty}^{+\infty} \frac{1}{2 \sqrt{\pi s t}} e^{-\frac{(x-\xi)^2}{4st}} f(t,x-\xi) \, d\xi \end{cases}$ Unknown functions are $$f(t,x)$$ and \( q(t,x)...

Hey! I have been thinking about different topics on mathematics, and for some reason I feel, that applied mathematics and doing numerical computations is often overlooked. When I was studying in university, more than 95 % of all examples which would require numerical methods were either omitted...
39. ### A Numerical Solution to Random Linear Non-Homogeneous ODE

Hi I am trying to learn optimal estimation by reading Gelbs Applied Optimal Estimation, and I am having hard time with finding \Gamma defined as the following: $$\Gamma_k w_k = \int_{t_k}^{t_{k+1}} e^{F(t_{k+1} - \sigma)} G(\sigma) w(\sigma) d\sigma$$ Here F is a known matrix. So is G, and w...
40. ### MHB Numerical Methods: Second Order Runge-Kutta Scheme

I'm unsure how to begin and solve this question. Any help would be appreciated, thanks.
41. ### A Partial differential equation containing the Inverse Laplacian Operator

I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...
42. ### Engineering Numerical solution to a 3rd order D.E. - "use computer to solve"

Hello, This problem comes from boundary layer theory in fluid mechanics, but we are studying it in heat transfer. note: Since we are solving this numerically is has been suggested to replace the third boundary condition with f" = constant and then guess a constant. Then we are to check that...
43. ### Python Numerical modeling of a glacier's length -- Coding error(s)

Hey Physics Forum I am currently doing my bachelor project in geophysics, with focus on the evolution of glaciers in Greenland. My project consists partly of programming, because I want to get better at it. I have, however, hit a wall. I can't seem to figure out what is wrong with my code and I...

I've tried both c1v1 and unitary method but couldn't solve
45. ### Checking the convergence of this numerical series using the ratio test

## \sum_{n=0}^\infty \frac {(2n)!}{(n!)^2} ## ##\lim_{n \rightarrow +\infty} {\frac {a_{n+1}} {a_n}}## that becomes ##\lim_{n \rightarrow +\infty} {\frac { \frac {(2(n+1))!}{((n+1)!)^2}} { \frac {(2n)!}{(n!)^2}}}## ##\lim_{n \rightarrow +\infty} \frac {(2(n+1))!(n!)^2}{((n+1)!)^2(2n)!}##...
46. ### Numerical methods for PV modeling

Input parameter $$I_{pv}= 8.2 A$$ $$I_{o=}9.8 e -8 A$$ $$R_s=0.221 \Omega$$ $$R_p=415.405 \Omega$$ $$a=1.3$$ $$V_t=21177.82 V$$ source (Villalva, Marcelo G. 2009. Comprehensive Approach to Modeling and Simulation of Photovoltaic Arrays. IEEE Transaction On Power Electronics, Vol. 24, No. 5)...
47. ### I Numerical Solution to the Rayleigh Plesset Equation

I have been trying to numerically solve the Rayleigh Plesset equation: $R\ddot{R} + \frac{3}{2}(\dot{R})^2=\frac{p_g-p_0-p(t)}{\rho_l}-4\mu\frac{\dot{R}}{R}-\frac{2\gamma}{\rho_lR}$ using the odeint python function. The code is given below: import numpy as np from matplotlib import pyplot as...
48. ### How to judge the singularity of a matrix in numerical method?

Summary: different methods give different results. They are not consistent. Summary: different methods give different results. They are not consistent. I use two different methods to detect whether a matrix is singular. The result of calculating the determinant of a 9-order square matrix is...
49. M

### I ODE Fail: Numerical Solution Oscillations - Possible Solutions | Hi PF

Hi PF The following ODE $$\ddot x + x - x^3 = 0\\ x(0)=0,\,\,\,\dot x(0) = \frac {1}{ \sqrt 2}$$ is solve exactly with ##\tanh (t/\sqrt 2)##. However, when I try to solve this with MATLAB ode45 (ode23t looks identical) or Mathematica NDSolve I get an oscillatory numerical solution (see...
50. ### Calculating drag and pressure forces from numerical simulations

So as stated, I am calculating the pressure and drag forces on an obstacle, but have trouble with which velocities to take. This is my geometry: http://shrani.si/f/3l/13P/2Tihb3iM/projekt2.png . I am guessing that I have to take pressure just before the obstacle and just after the obstacle and...