Numerical Definition and 775 Threads

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. F

    MATLAB Finite difference numerical integration or ode45?

    I'm trying to numerically solve the time dependent Schrödinger equation and I've been told that the best approach is to numerically integrate using a finite difference method, however I don't understand why I couldn't just use ode45 to solve it?! Is the finite difference (interpolation) method...
  2. ShayanJ

    A Numerical solution of two coupled nonlinear PDEs

    One of my friends needs to numerically solve this two dimensional boundary value problem but has now idea where to begin. Could anybody help? ## [(K H )(f g_x-gf_x)]_x+[(K H )(f g_y-gf_y)]_y=0 #### K H G^2 (f^2+g^2)+\frac 1 2 [KH (f^2+g^2)_x]_x+\frac 1 2 [K H (f^2+g^2)_y]_y-K...
  3. N

    A Numerical solution to SE - variational method, many electrons

    Hi everyone, I am trying to find electron wavefunction of a system I am working in. Numerical method I choose is the Variational method (VM). This method is convenient to find the ground state of the system. More details are available here. Problem I have can be explained on a very simple...
  4. C

    MATLAB Numerical Integration with Matlab - Solve K33, K11 Integral

    Hi all, i need help solving the following integral using Matlab: * tetam is a parameter and the integration is by alpha. the answer should be function of tetam * K33, K11 are constantsThanks , Chen
  5. K

    I Two numerical questions about inflation and the flatness problem

    Good day to you all I just started studying cosmology recently using several book and especially using TASI Lectures on Inflation(https://arxiv.org/abs/0907.5424) On page 25 the fine tuning problem of the flatness is introduced with conditions as |Ω(BBN)-1|≤ ο(10-16) |Ω(GUT)-1|≤ ο(10-55)...
  6. E

    Mathematica Numerical vs. Monte-Carlo Simulations

    I have an integration that doesn't have a solution in the table of integrals. So, I evaluated it using Mathematica using the command NIntegrate. However, when I compare the result with Monte-Carlo simulations, there is a very small constant gap between the two curves. Is it because of the...
  7. astrof

    A Numerical simulation the n-body problem under SR model

    The problem is a simple n-body problem: there is n charged particles - electrons, protons, or other particles; we have the initial parameters: a mass, charge, position, velocity, at a time t = 0. In a standard (classical) numerical computation of the n-body problem I can use any numerical...
  8. R

    MATLAB Bisection method and numerical integration

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  9. Korbid

    How to Numerically Solve a Multiple Integral in Mathematica or Sage?

    I'm trying to solve numerically this multiple integral. But i don't know how to calculate it with Mathamtica or Sage software. $$\int{e^{-(\vec{v}^2_1+\vec{v}^2_2)}e^{-E(\tau)}}d\vec{r}_1d\vec{r}_2d\vec{v}_1d\vec{v}_2$$ $$E(\tau)=\frac{k}{\tau^2}e^{-\tau/\tau_0}$$...
  10. R

    MATLAB How Accurate is Numerical Differentiation of sin(x) at 0.4?

    I am trying to estimate the second derivative of ##sin(x)## at ##0.4## for ##h=10^{-k}, \ \ k=1, 2, ..., 20## using: $$\frac{f(x+h)-2f(x)+f(x-h)}{h^2}, \ \ (1)$$ I know the exact value has to be ##f''(0.4)=-sin(0.4)= -0.389418342308651.## I also want to plot the error as a function ##h## in...
  11. N

    I Finding multiple roots of polynomial using numerical methods

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  12. RJLiberator

    Fourier Series and deriving formulas for sums of numerical

    Homework Statement Homework EquationsThe Attempt at a Solution So I am tasked with answer #3 and #4. I have supplied the indicated parenthesis of 8 also with the image. Here is my thinking: Take the Fourier series for |sin(θ)|. Let θ = 0 and we see a perfect relationship. sin(0) = 0 and...
  13. awholenumber

    Few noob questions and numerical methods help....

    i was trying to work with few problems for my computer oriented numerical method ...which i had in college ... i am not in college anymore ... but i failed that subject ... i am trying to improve this maths field as much as i possibly can , because its a difficult subject for me ... let met...
  14. M

    Numerical solution of non-linear diffusion equation

    Hello. I have some questions regarding the equation: k\frac{\partial}{\partial x}\left( u\frac{\partial (u-r(x,t))}{\partial x} \right) = \kappa \frac{\partial u}{\partial t}. u is positive. r(x,t) is given as an input. I have implemented this non-linear diffusion equation using backward Euler...
  15. O

    Does Integration Scheme Choice Matter for Initial Demo of Crane Control?

    Good Day Let's say I have developed a new method to extract, more efficiently (yes, "more efficiently" is ill-defined; but bear with me) the differential equations that describe a specific phenomena (please just assume it). So now I have a system of coupled second order differential...
  16. D

    How to Integrate a Function with Steep Rise for Small x?

    I want to integrate a function numerically from 0 to infinity where for small x ##f(x)\sim x^{-5/2} \exp(-a/x)## and for large x ##f(x) \sim \exp(-bx)##. How do I best treat the steep rise for small x?
  17. mollwollfumble

    I Numerical sign problem software?

    I'm an applied mathematician with lots of experience in classical physics but none in quantum physics. What's the simplest piece of code (eg C++ on Cygwin) that illustrates the https://en.wikipedia.org/wiki/Numerical_sign_problem in quantum physics?
  18. Farah AbdlGhany

    How Are Numerical Methods Applied in Geophysics?

    I am studying numerical and programming course in this semester, I was asked to know the applications of this course (numerical methods like Bisection, Newton Raphson, Secant, Simpson, Euler and Lagrange) in geophysics and how can I exploit these methods in seismic or in geophysics in general...
  19. Y

    Finding the maximum value of a function

    Homework Statement Find the extremizing (maximum) value of the function f(x) = sin x / x using Newton's 1D method. Homework Equations [/B]The Attempt at a Solution I know the maximum point in this equation is (0, 1). When I differentiated the equation twice and used the formula above, I...
  20. jk22

    Can Numerical Simulations Reveal Violations of CHSH Inequalities?

    In the hidden variable model for CHSH, I computed the probabilities for the CHSH operators to have results -4,-2,0,2,4, then the average S. Using a computer program, I could find the minimum of that function, obtaining S>2 : #include<stdio.h> #include<math.h> double PI=4.0*atan(1.0)...
  21. Omega0

    How to Calculate Gauss Curvature from a Mesh Surface?

    Hi! I would like to calculate the curvature for a surface S:R^2->R'3 numerically. The problem: I simply have the surface as a mesh like you see in the image attached. I calculated the linearly interpolated face normals in the nodes, too. You see the vectors. Question: How would you calculate...
  22. A

    M. Kyncl's Numerical Solution of 3D Compressible Flow: Master's Thesis 2011

    Hi! Does anyone know where can I find the following thesis? M. Kyncl. Numerical solution of the three-dimensional compressible flow. Master’s thesis, Prague, 2011. Doctoral Thesis Thank you
  23. 22990atinesh

    Numerical Problem on Processor Pool Model

    Consider the case of a distributed computing system based on the processor-pool model that has ##P## processors in the pool. In this system, suppose a user start a computing job that involves compilation of a program consisting of ##F## source file ##(F < P)##. Assume that at this time the user...
  24. H

    How to check a numerical simulation is energy conservative?

    I am modelling a 1D fluid wave propagation problem and I needed to know how I can check that my results are energy conservative. please reply, urgent. thank you.
  25. H

    Relation between energy conservation and numerical stability

    Hi, Consider the conservation laws for an isothermal linear incompressible flow governed by the mass and momentum equation. The kinetic energy equation is then solved to see if energy conserved. Can anyone tell me if once it is shown energy is conserved, it implies that convergence is obtained...
  26. W

    Numerical methods of integration

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  27. Feodalherren

    Numerical Methods, need a 3D Jacobian

    Homework Statement I'm trying to write a program to solve a system of 3 non-linear equations using the Newton-Raphson method. I'm stuck on trying to figure out what the formula for a system of 3 unknowns is. I can't remember the derivation at all and after endless hours of googling and looking...
  28. E

    When is a Numerical Aperture considered High?

    When will you consider that the numerical aperture is high? I was told that the numerical Angular Spectrum Method is only valid low numerical aperture values, I want to know the boundary of this value. Thank you in advance.
  29. A

    Numerical solution of SE for cylindrical well

    I am simulating electrons inside a cylindrical well like the one shown on the first figure. My current work has been on solving the Schrodinger equation numerically for the above potential and then finding corrections to the solution such that it is consistent with Poissons equation. To do so...
  30. A

    Numerical solution of Schrodinger equation script

    I am right now working on a script that solves the Schrodinger equation numerically for arbitrary potentials using the finite difference method. The idea is that I diagonalize the Hamiltonian with elements: H(i,i+1)=1/dx^2 * constants H(i,i-1)=1/dx^2 * constants H(i,i) = -2/dx^2 * constants and...
  31. C

    Can You Help With Finite Element Analysis in Cylindrical Coordinates?

    I am trying to numerically calculate the electric potential inside a truncated cone using the finite element method (FEM). The cone is embedded in cylindrical coordinates (r,phi,z). I am assuming phi-independence on the potential, therefore the problem is essentially 2D; I am working only with...
  32. A

    Schrodinger equation numerical solution

    I have written a program that solves the Schrödinger equation numerically using the finite difference method based on the attached article. The end goal is to make a program that solves the Schrödinger and Poisson equations self-consistently for the conduction band in different heterostructures...
  33. Pyrus

    Calculating Thrust on Curved Surface of Cylindrical Vessel

    Homework Statement A large cylindrical vessel contains water to a height of 10 m. It is found that the thrust acting on the curved surface is equal to that at the bottom. If atmospheric pressure can support a water column of 10 m, the radius of the vessel is:- (a) 10 m (b) 15m (c) 5 m (d) 25 m...
  34. N

    Numerical methods that need a guess/approximate solutions

    Hello everyone! I am currently playing with an old analog computer, which could solve time-dependent ODE/PDEs pretty fast, without time-stepping. But the problem with analog computer's solutions is that they are not very accurate. I am very curious that is there any numerical method/solver which...
  35. L

    Lyapunov exponent -- Numerical calculations

    In computational physics is very often to calculate largest Lyapunov exponent. If largest Lyapunov exponent ##LE## is positive there is chaos in the system, if it is negative or zero there is no chaos in the system. But what can we say about some certain value of ##LE##. For example...
  36. T

    Cont. numerical solution to gravity

    I believe I am misunderstanding the equations I must use for the Runge Kutta of an orbit in polar coordinates. What differential equations can i use which will allow me to numerically determine both an object's distance and angle in an orbit, dependent on time?
  37. DEvens

    What books for numerical solutions to PDEs

    What are some good books (or other resources) on numerical methods of solving PDEs in 3 space and 1 time variable? I am interested both in finite element and finite volume methods. I could be interested in other methods but I don't know about them. I am interested in being able to take...
  38. M

    MHB Numerical Analysis: Project 01 | CPP

    http://www.cpp.edu/~jlcannons/mat_201_2015_spring/Project_01.pdf I don't know if this counts as cheating, but I hope someone could look this over and see what my professor is ask.
  39. PinkGeologist

    What Factors Influence the Cooling of Magma Sills in Numerical Models?

    Ok, I've built a numerical model to show the cooling of hot magma sills entered into the crust over time. The results show that the volume of the "hot" zone when the emplacement of a constant volume of hot sills is all done will vary as a matter of two things: the overall rate at which the magma...
  40. J

    Imposing normalization in numerical solution of of ODE

    Hello all, I would like to know how to impose a normalization condition to numerically solving an ODE. For simplicity let's consider the example \frac{dy}{dx}=y You could use different methods using an initial value, but if you consider the interval [x_0,x_1] and \int_{x_0}^{x_1} y(x)dx=1...
  41. M

    Mathematica Working with numerical solutions in mathematica

    Hi PF! I have just solved a non-linear ODE numerically in mathematica using the "NDSolve" command over a small interval ##[0,1.5]##. I'd like to be able to fit a polynomial to this solution, perhaps using the "Fit" command, although I've had no luck on my own. Any help is greatly appreciated!
  42. manifold

    A numerical solution of a second order ODE

    Hello everyone; i'd like some help in this problem : i want to solve num this differential equation { y"(t)+t*cos(y)=y } by the Taylor method second order expansion. i first have to make this a first order differential equation by taking this vector Z=[y' y] then we have Z'=[y" y'] which equal...
  43. evinda

    MHB Implementation of numerical method

    Hello! (Smile) Consider the initial value problem $$\left\{\begin{matrix} y'(t)=f(t,y(t)) &, a \leq t \leq b \\ y(a)=y_0& \end{matrix}\right. (1)$$ I want to write a program that implements the following numerical method to solve $(1)$ $\left\{\begin{matrix} y^{n+1}=y^n+h[\rho...
  44. Rapier

    Numerical Integration for Magnetic Field of a Loop of Wire

    Homework Statement Calculate the magnetic field of a current loop. Compare your numerical results with exact solution above the center of the loop. Investigate the effect of the grid size based on this comparison. Homework Equations dB = u0*I/4pi * (dL * R) / (R^2 + Z^2)^3/2 Bz = u0*I*R^2/ (2...
  45. C

    Numerical solution to Schrödinger equation - eigenvalues

    Not sure whether to post this here or in QM: I trying to numerically solve the Schrödinger equation for the Woods-Saxon Potential and find the energy eigenvalues and eigenfucnctions but I am confused about how exactly the eigenvalues come about. I've solved some differential equations in the...
  46. O

    Numerical Analysis Programming Language

    In two semesters or so, i will be taking a numerical analysis course as part of an Applied Mathematics major. In the course description it says that you should have knowledge of at least one programming language, but i cannot find any more information about this. What do you all think would be...
  47. R

    Applied Does anyone own Numerical Methods by Laurene Fausett?

    Hi, I'm asking this because I ordered this book today, but there is a 4-day shipping minimum. I have to do a few problems from this book, and if anyone owns this, is it possible to tell me the problem statements? Also, if I'm not allowed to ask for this, please let me know. The problems I...
  48. B

    Finite square well potential numerical solution

    hi guys i need some help with the iteration made in a numerical solution for the eigenstates in a finite square well potential using the effective length approximation First, i find the eigenstate for a infinite square well, then i define the related alpha and i use it to define an effective...
  49. E

    Heat conduction numerical model

    Hi Guys, I'm new on this forum, currently studying Aerospace Engineering and am trying to produce the model of a radioisotope thermoelectric generator using numerical methods to solve the heat conduction equation as part of my research. The way it works is that I have a radioisotope source in...
  50. G

    Algorithm for Numerical approximation to add data points

    Hi, I am working on TDR (Time Domain Reflectometry). I send a 7GHz bandwidth fast rising edge (14ns) square wave into a coax. I get a return Signal. I have an ADC with 10Msamples/sec. I am using MPLAB IDE for coding the microcontroller. Now I would like to increase the Points on the...
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