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. G

    Numerical Approximation and addition of new data points

    Hi I am new member and I am new to the Signal processing so I hope I could get some help from the members to able to understand the concepts. I have a Signal. I have a 10Msaples/sec ADC. I view the Signal on an Oscilloscope which has 20Gsamples/sec sampling rate. The Point where I am...
  2. M

    C/C++ Getting xodeint from numerical recipes to work in dev-c++

    I am taking a computational physics course, and currently I have very little knowledge of programming. I have downloaded the numerical recipes machine-readable code, and I am trying to get one of their examples to work in dev-c++. Unfortunately, I am having difficulty. I have copied xodeint.c...
  3. K

    How to improve stabilities of numerical solutions of PDEs

    This is a quite general question, but I am working with a system of partial differential equations in two variables. There is one time direction t and one spatial direction z and the numerical method is formulated by stepping forward in time. The problem is that I obtain instabilities, either at...
  4. ognik

    Tricky Intregral for numerical quadrature

    Hi - I have just started 'Computational Physics' by Koonin & Meredith, - through distance learning. Exercise 1.3 needs a program to evaluate an integral - I'm finding myself kinda rusty on integrals. The hint says - split range of integration into parts, use different change of variable in each...
  5. M

    Numerical Approximation of a Rocket's motion

    So, I ve been trying to add orientation to my model of the flight dynamics of a rocket but I ve been running into a lot of problems. I didn't bother actually doing the math for the moments of inertia and everything because I guess it really doesn't have that much of an effect on the general...
  6. QPingy

    Numerical integration - verlet algorithm - accuracy

    In my computational physics textbook, three different velocity estimators are derived for a problem with equation of motion: \ddot x = F(x) where the positions are found by using the Verlet algorithm: x(t+h) = 2 x(t) - x(t-h) + h^2 F[x(t)] The three velocity estimators are: v(t) = \frac{x(t+h)...
  7. N

    Solve 3x3 Numerical Puzzle: Can it Be Done?

    1. In a 3 x 3 square, place the numbers 2,2,2,3,3,3,4,4,4 in it so that when any line of three numbers is added up in any direction (including diagonally) the total is always 9. 2. I have tried for hours, can anyone tell me if this problem is actually possible? The best I get is when I do 234...
  8. D

    News Basel II: Numerical instabilities to stabilize banks?

    Oops, I was to quick here. At least, there is no instability. Nevertheless I would write ##\mathrm{Corr}(R)=0.12(1+\exp(-50 \mathrm{PD}))## instead of Correlation (R) = 0.12 × (1 – EXP (-50 × PD)) / (1 – EXP (-50)) + 0.24 × [1 - (1 - EXP(-50 × PD))/(1 - EXP(-50))]
  9. DivergentSpectrum

    Can the Simpsons 3/8 Rule be Extended to Calculate Double Integrals?

    how do i numerically calculate a double integral? as i understand simpsons 3/8 rule is the optimal method for a single integral, is it still true for double integrals? if so, how do i extend the 3/8s rule to do a double integral?
  10. V

    Numerical Solution for ODE with Critical Point: A General Procedure Guide

    Hello, can someone advise me how to solve numerically ODE which consist of function with "critical point" (Im not sure if it is good definition)? I mean for example this one: y'(x)=\frac{\sin{x}}{x}, where in x=0 has function a "problem". I know that limit ->1 but in numerical solutions it...
  11. S

    Numerical method for wave equation

    Hi, I am trying to plot a function subjected to a nonlinear wave equation. One of the method I found for solving the nonlinear schrodinger equation is the split step Fourier method. However I noticed that this method only works for a specific form of PDE where the equation has an analytic...
  12. V

    Solving Differential Eqs with Critical Point: Initial Conditions & Solutions

    Hello, I have two different discrepancies to this system: a) How and when is possible to have more solution of differential eq. or their system for same initial problem? For example this is happening in following system. It is written about this system: "Different value of constant \dot{M}=4\pi...
  13. C

    Creating a Numerical Model for a falling Ball

    Homework Statement A ball was dropped through a known vertical distance with the time of drop also recorded. The aim of the task is to create a numerical model in excel in order to approximate the drag coefficient. Homework Equations This is my method. 1. FBD for the ball falling. (assume...
  14. H

    Is the Numerical Dissolution Time Calculation for a Sphere Accurate?

    The problem statement Based on an analytical solution for the concentration profile of a dissolving sphere, I am supposed to use a numerical method to find the time at which the sphere has fully dissolved. This is not so much a question about specific values - but about the technique by which I...
  15. P

    Can You Solve These Tricky 2D Integrals on a Unit Circle?

    I can't compute the integral: \int \frac{\arccos(\sqrt{x^2+y^2})}{\sqrt{x^2+y^2}}\frac{x-a}/{(\sqrt{(x-1)^2+y^2})^3 dxdy on an unit circle: r < 1. for const: a = 0.01, 0.02, ect. up to 1 or 2. I used a polar coordinates, but the values jump dramatically in some places (around the 'a' values)...
  16. U

    Runge Kutta for 4 coupled differential equations

    Hi, I'm not a bright programmer , but I have to solve the fallowing equations: ##\frac{df}{dt} = \alpha f -\beta f + \theta g - (f+h)f## ##\frac{dg}{dt} = \psi f- \phi g## ##\frac{dh}{dt} = \xi f+ \mu h -\tau h + \epsilon w- (f+h)h## ##\frac{dw}{dt} = \nu h - \chi h## Where ##f(t)## ...
  17. M

    Calculating Numerical Jacobian for Matrices A and B: Step-by-Step Guide

    Hi, I have two numerical matrices, A is 150*1 matrix (A=rand(150,1)) and B is a 1*5 matrix (B=rand(1,7)), and I need to have the jacobian of A with respect to B, that should be a 150*7 matrix, anyone help is appreciated. Mathias
  18. C

    Numerical solution of a differential equation with time dependent terms

    I am would like to solve this differential equation: Where http://ieeexplore.ieee.org.ezproxy.uniandes.edu.co:8080/ielx5/8/6493417/6409989/html/img/6409989-eqdisp-3-small.png Could you give me some practical ideas about the required software and methodology? Thank you very much
  19. P

    Numerical integration of a function specified numerically

    Dear All, Can someone suggest me an appropriate routine (in Fortran) or command (in mathematica) to perform numerical integration of a function, which is specified numerically on a one dimensional grid with equal spacing (and we cannot generate additional data on other grid points)? There are...
  20. K

    Integrate Polynomial f(x^2): Reduce Sampling Points to N/2?

    Let f(x^2) be a polynomial. I would like to carry out the integration \int_a^b f(x^2) dx using quadrature rule. Suppose a and b > 0 and are arbitrary and the degree of f(x^2) is 2N. I would like to know if there is a possibility to reduce the sampling points down to N/2?
  21. L

    Diode numerical (calculation of current and voltage across diode)

    The Diode in the circuit shown below has the non linear terminal characteristics as shown in the figure.Let the voltage be "coswt" V . Question 3.1.33 and my attempt is shown in the figure.
  22. DivergentSpectrum

    Numerical second order pde solver

    Edit:whoops wrong forum mods please move 2nd edit: I just had dinner then got back on the computer, input some points and saw a beautiful elipse.(complete with a fascinating flower petal design due to inaccuracies) Weird lol! No idea why it wasnt working before Now to implement RK4 bwahahaha...
  23. F

    What is the relationship between numerical aperture and resolution in lenses?

    Hello Forum, The numerical aperture NA of a lens is usually a parameter (adimensional) that indicates the lens ability to collect light. The concept of NA is usually introduced in fiber optics where we are concern with launching light into the fiber. But NA is also discussed in microscopy...
  24. T

    Three properties that all numerical answers must have

    Homework Statement This question being for my physics summer work states "List three properties that all numerical answers must have to be correct" Homework Equations None The Attempt at a Solution 1. must be a numerical value? 2. must show accuracy. 3. must show precision.
  25. A

    Unifying Numerical Math and Symbolic Logic: Has Progress Been Made?

    In the late 1980's I asked my logic professor if there was some kind of logical and/or mathematical process which unified numerically based mathematics with true-false based symbolic logic.He told me that someone had written a lengthy book which apparently proved that it was totally impossible...
  26. T

    Help solving Simulink numerical oscillation

    Hi! I posted a new thread about a week ago, but so far no one was able to help me: https://www.physicsforums.com/showthread.php?t=761892 I wasn't sure if I posted in the right forum, so I'm posting it here hoping to reach a wider audience :) Any help is greatly appreciated! -Tusike
  27. T

    Simulink help solving numerical oscillation

    Hi! I'm trying to compute air flow between chambers for a project. I'm having a numerical oscillation problem, and I've created an example Simulink program that highlights the cause. Please see the attached Simulink model for a clear understanding; in the following I will do my best to give...
  28. C

    Numerical variational problem

    Homework Statement Here I am, trying to get the trajectory $(x(t),y(t))$ that will minimize the following Lagrangian (i.e. the integrand of the functional) between $(-1,-1)$ and $(-1,y_{eff})$ where $x_{eff}$ is defined as the point where $V(x_{eff},-1)=V(-1,-1),x<0$. So here's the...
  29. S

    Numerical methods for differential equations

    Hi, Are there any numerical techniques I can use to solve differential equations with periodic boundary conditions? I know of several techniques for other kinds of boundary conditions (such as Runge-Kutta method, Euler method etc.), but I am interested in knowing how to numerially solve...
  30. Z

    Why incompressible fluid flow is advantageous in numerical computation

    Hi, When we want to solve the Navier-Stokes equations coupled with the conservation of mass for incompressible fluids using the primitive-variable approach, we have to face to the problem that the equation for the continuity equation does not contain the pressure which leads to spurious...
  31. P

    Numerical integration using Weber force

    I need to compute numericaly n-body sys. interacting acording to the Weber force: http://en.wikipedia.org/wiki/Weber_electrodynamics and I have a problem with the acceleration on rhs: r'', because the acceleration is unknown, due to the Newton law: F = ma, and we need just 'a' to do next...
  32. majormuss

    Numerical Facts about the Milky way's halo

    I am trying to find the density, scale radius and normal radius of the milky way's halo. But these things are pretty hard to find. I want to find them so I can input them into the NFW profile:
  33. S

    How to Ensure Bracketing in Numerical Methods

    I had tried to solve but the problem is here initial guess is given as 1 and 1000 but both has positive value of function , so wat to do now how to determine x3 which is x1+x2, wat to take x1 and x2 in second iteration, just guide me how to solve Thanks a lot for ur replies
  34. A

    Area of a polygon- using numerical integration

    Hi, I need to calculate area of an irregular polygon which can be of any complex shape numerically i.e. using numerical integration techniques. Please can anyone suggest any reference material / best way of going about this efficiently? Akash
  35. E

    Proof involving numerical equivalence of sets

    Homework Statement Show that for a set A\subsetN, which is numerically equivalent to N=Z+, and the set B = A \cup{0}, it holds that A and B are numerically equivalent, i.e., that A \approxB Hint: Recall the definition of A≈B and use the fact that A is numerically equivalent to N. Note...
  36. T

    Numerical integration methods applicable to a type of definite integral

    Numerical integration methods applicable to a type of definite integrl Hey, so I've been working on a program to numerically integrate an integral of the form ∫xnf(x) dx, LIM(0 to INF.) Here n can go to negative non integral values, say -3.7 etc. and f(x) is a function of sin, cos and...
  37. Q

    Best Numerical Analysis Software for Physics

    Two criteria 1. Freely Available 2. Large support for additional scientific libraries/packages/modules for reuse http://en.wikipedia.org/wiki/Comparison_of_numerical_analysis_software Which one would you recommend and why?
  38. N

    Attenuation numerical instability via numerical diss. nonlinear ODE

    Hi, I'm solving a problem numerically that takes the form Q_{ij} \ddot{y}_j +S_{ijk}\dot{y}_j\dot{y}_k +V_i=0, where (Q_{ij},S_{ijk},V_i) are all functions of the dependent variables y_i. The dependent variables are all functions of the variable t. The resolution of this spectral...
  39. M

    Numerical solution of one dimensional Schrodinger equation

    Hi, I want to solve one dimensional Schrodinger equation for a scattering problem. The potential function is 1/ ( 1+exp(-x) ). So at -∞ it goes to 0 and at ∞ it's 1. The energy level is more than 1. I used Numerov's method and integrated it from +∞ (far enough) backwards with an initial value...
  40. W

    Numerical Solution of Differential equation

    The nonlinear oscillator y'' + f(y)=0 is equivalent to the Simple harmonic motion: y'= -z , z'= f(y) the modified Symplectic Euler equation are y'=-z+\frac {1}{2} hf(y) y'=f(y)+\frac {1}{2} hf_y z and deduce that the coresponding approximate solution lie on the family of curves...
  41. R

    Advanced numerical solution of differential equation

    Show that the explicit Runge-Kutta scheme \begin{equation} \frac {y_{n+1} -y_{n}}{h}= \frac{1}{2} [f(t,y_{n} + f(t+h, y_{n}+hk_{1})] \end{equation} where $k_{1} = f(t,y_{n})$ applied to the equation $y'= y(1-y)$ has two spurious fixed points if $h>2$. Briefy describe how you would...
  42. A

    Which numerical procedure to use

    I have a system of linear differential equations with known boundary conditions. First of all what is the general solution to such a system? I know it is exponentials with the eigenvalues, but I couldn't find any place where the exact full solution was stated. Second of all, I want to write a...
  43. R

    MHB Advanced Numerical solution of differential equations

    i) IF $\frac{dy}{dt} = - \frac{∂H}{∂z}, \frac{dz}{dt}= \frac{∂H}{∂y}$ where H is a function of $y$ and $z$, show that $H(y,z)$ is constant in time. ii) Take a $H(y,z) =Ay^2 + 2Hyz + Bz^2$ where $A,B,H$ are constants and show that solutions of the system lie on ellipses. iii) Apply the...
  44. R

    MHB Advanced topic in numerical solution of differential equation

    Investigate the stability of the PECE method where P=Predictor : y_(n+1) = y_n + hf(y_n) C=Corrector: y_(n+1) = y_n + h [(1-θ) f(y_n) + θ f(y_(n+1))], (0<θ<1) and E is the evaluation step. => substituting the predictor into corrector gives:y_(n+1) = y_n + h [(1-θ) f(y_n) + θf( y_n+ h f y_n...
  45. A

    Numerical method to solve a spring system

    Hello everyone, this is my first thread in this comunity. I explain my problem: I have a triangle mesh http://en.wikipedia.org/wiki/Triangle_mesh where at every vertex coincide a mass element and on every edge I add a spring. I would like simulate a deformation model. My idea derive...
  46. B

    Language for fast numerical integration

    I have thousands of triple integrals with very ugly integrands to run. Each of those computations takes about an hour in R on a mediocre machine. For uninteresting reasons, when I was initially coding this up, that's the language and machine I had to work with. But now my options are wide...
  47. R

    There are effectively computable numerical functions which aren’t prim

    This is from Peter Smith's Gödel without tears. I don't agree with this. If it appears in the list of p.r. functions then it is p.r. I don't see why he thinks that if x is in the list of p.r. functions then it is not p.r.
  48. C

    Numerical solution to a partial differential equation

    Homework Statement Solve the inhomogenous partial differential equation \frac{∂^{2}u}{∂t^{2}}-\frac{∂^{2}u}{∂x^{2}}=-6u^{5}+(8+4ε)u^3-(2+4ε)u by using the NDSolve function in Mathematica for the interval [0,10] x [-5,5]. Homework Equations Initial conditions: u(0,x)= tanh(x)...
  49. P

    Basic question about numerical hessian

    Hello all, Suppose I have a simple 1-D signal and I want to compute the hessian. In that case, it should generalise for second derivative for normal scalar functions. So, I observe the signal as v = [x_1, x_2, x_3, x_4...]. Then, numerically the hessian is given as (assuming I am only...
  50. M

    Taylor Polynomials and Numerical Analysis

    Homework Statement Use a Taylor Polynomial about pi/4 to approximate cos(42){degrees} to an accuracy of 10^-6. *To get an accuracy of 10^-6, use the error term to determine an nth Taylor Polynomial to use. Homework Equations x = 45 or pi/4, x0 = 42 or 7pi/30 cos(x) = Pn(x) + Rn(x)...
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