What is Numerical methods: Definition and 96 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.

View More On Wikipedia.org
  1. J

    Relativity Suggestions for graduate studies in General Relativity

    Hello everyone, I am a couple of months away of starting my PhD in Numerical Relativity and I am seeking recommendations for references to get a batter understanding on the mathematical foundations of GR and topics of Numerical Relativity. So far I have had an undergraduate introductory course...
  2. Graham87

    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...
  3. Graham87

    Comp Sci Orbit of Earth: Plotting Trajectory

    I am attempting this homework exercise but my plot does not show the whole trajectory. I don't know if it is something wrong with my equations or if it is a plotting matter. Cheers!This is my code:import matplotlib.pyplot as plt import numpy as np import scipy.integrate as spi G =...
  4. C

    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...
  5. P

    Calculate tilt angle of a bar lifted via two ropes on fixed points

    Summary: How to calculate the equilibrium angle of a bar that is lifted on its two ends with ropes attached to fixed lifting points? Hello and good day all, First of all I would like to apologize in advance for my english, I am not a native speaker so some grammar errors may be left. My...
  6. U

    Why heat PDE solution does not fully satisfy initial conditions?

    Hi, I am solving heat equation with internal heat sources both numerically and analytically. My graphs are nearly identical but! analytical one have problem at the beginning and at the end for my domain. Many people have used the same technique to solve it analytically and they got good answers...
  7. E

    Dynamical systems and cutting edge numerical methods

    Background in teaching (mathematics and physics) Signed up here as an alternative to physics.stackexchange. Pursuing inspiration for complicated dynamical systems. Actively working on cutting edge numerical methods and educational (free) software Glad to answer questions in my field.
  8. K

    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)...
  9. M

    MHB Numerical Methods: Second Order Runge-Kutta Scheme

    I'm unsure how to begin and solve this question. Any help would be appreciated, thanks.
  10. nughii

    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)...
  11. BWV

    I Neural Networks vs Traditional Numerical Methods

    As neural networks are 'universal approximators' for nonlinear functions, in general how do they perform in comparison to traditional numerical methods for nonlinear PDEs? Just googling, I can find papers on applications to Navier Stokes and other problems, but I don't really have the...
  12. S

    Error of integration in numerical methods

    Homework Statement So this is a problem for my numerical methods class. We are doing integration and derivation. The part that I am stuck on is why does the midpoint method and trapezoidal method both have error of order ##h^2##, when the first one is one point integration method while the...
  13. C

    Second order(?) ODE + Runge-Kutta method question

    Homework Statement When a rocket launches, it burns fuel at a constant rate of (kg/s) as it accelerates, maintaining a constant thrust of T. The weight of the rocket, including fuel is 1200 kg (including 900 kg of fuel). So, the mass of the rocket changes as it accelerates: m(t) = 1200 - m_ft...
  14. Telemachus

    Numerical methods for a system of coupled ODE

    Hi there. I have to solve a system of coupled ordinary differential equations. I have some initial values, but in different points of the domain. The equations are all first order. Let's suppose the system looks like this: ##\displaystyle\frac{dy_1}{dz}=y_1+y_2+0.01##...
  15. D

    Stability function for a basic RK method

    Homework Statement Hi, can someone explain how they get this stability function. Homework Equations http://blob:http://imgur.com/840ee1fe-f09c-49ab-bdbb-7f953777f900 The Attempt at a Solution I have attempted to do it two ways, and I'm not sure if I'm just having an awfully bad day at maths...
  16. awholenumber

    I The algorithms involved in finding a solution with numerical methods

    what are all the algorithms involved in finding a solution with numerical methods ?
  17. Leonardo Machado

    Problems with Runge-Kutta method

    I'm writing a program to compute an ODE solution of the Kepler's problem based on Runge-Kutta 4th order method and today I've past the whole day trying to made it work, but I've failed, maybe you could help me to kill the problem ? The solutions is cartesian. int main(){ int n...
  18. awholenumber

    I Few more numerical methods question....?

    i have few more doubts about these two numerical methods type questions ... An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation a question usually starts like this ... find the...
  19. awholenumber

    I Differential equations and numerical methods questions

    i have few doubts about differential equations and numerical methods ... in a differential equation question ... you are given an instantaneous rate of change... and you are supposed to find the function that gives , this instantaneous rate of change is this same as ... f(0)=1...
  20. Johnny_Sparx

    A Numerical Solution for Complex PDE (Ginzburg-Landau Eqn.)

    I am looking to numerically solve the (complex) Time Domain Ginzburg Landau Equation. I wish to write a python simulator to observe the nucleation of fluxons over a square 2D superconductor domain (eventually 3D, cubic domain). I am using a fourth order Runge Kutta solver for this which I made...
  21. Themis

    Numerical methods for system of ODES

    Hi there. I want to evolve a system of non linear coupled ODEs \frac{dx}{dt} = \frac{-k}{x^5(56-y^8)^{9/2}}(85+y^{5} + y^{6}) \frac{dy}{dt} = \frac{-k}{x^4(56-y^5)^{7/2}}(44+y^2) Let's say I have the initial conditions. What numerical method someone could use to solve this? adaptive step...
  22. ramzerimar

    Numerically find temperature in function of concentration....

    Homework Statement I don't even know if this is the correct forum for this question, but here we go. This exercise is from my numerical methods for engineers class, and it says the following: The coefficient of saturation of oxygen dissolved in fresh water is given by the equation: $$...
  23. N

    I Finding multiple roots of polynomial using numerical methods

    Hey all, I seek to find where the derivative of a nth order polynomial is at a 0, so far I have used secant method to find it, which works, but issue is is that that returns only one root, sliding the interval could work, but then itd always point to the edge of the interval, any help...
  24. 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...
  25. Farah AbdlGhany

    Numerical course 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...
  26. W

    Numerical methods of integration

    Mod note: Removed the color formatting, and surrounded the code with [ code ] tags. @whatisreality, please use a [code=java] tag at the beginning of your code, and a [/code] tag at the end. Inserting colors by hand is distracting and makes your code difficult to read. I have to write a piece of...
  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. T

    Mathematical modeling in particle physics

    So my university offers two programs focused on particle physics. One is simply masters in nuclear/particle physics, the other is masters in mathematical modelling with focus on particle physics. I want to go into mathematical modelling and I'm choosing what I will focus on. I'm not really...
  29. 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...
  30. D

    How to solve a very large overdetermined system numerically?

    I am doing a project on image processing and I need to solve the following set of equations: nx+nz*( z(x+1,y)-z(x,y) )=0 ny+nz*( z(x+1,y)-z(x,y) )=0 and equations of the boundary (bottom and right side of the image): nx+nz*( z(x,y)-z(x-1,y) )=0 ny+nz*( z(x,y)-z(x,y-1) )=0 nx,ny,nz is the...
  31. D

    Richardson Extrapolation with 3 steps?

    Homework Statement [/B] use richardson extrapolation to estimate the first derivative y=ln(x), x=5 using steps of 2, 1, 0.5. Four decimal points. obtain true relative error for the last estimate and comment on its value. Homework Equations [/B] deriv ln(x)=1/xThe Attempt at a Solution I know...
  32. 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...
  33. 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...
  34. ognik

    Help with Adams-Bashforth program

    Hi - trying to write a Fortran95 program to solve a trivial 1st order DE (dy/dx=-xy), the idea being that once it works it can be used for different forms of f(x,y). I use Euler to get the starting point, then A-B's 2 step formula for the next 2 starting points. It is when I apply the A-B...
  35. SalfordPhysics

    Comp Sci Fortran90: DO loop for sequence of numbers

    Homework Statement A program finds the area under the Gaussian Distribution Curve between ±σ using Simpsons Rule. Modify the program to investigate the effect of the number of strips. Do this by using a DO loop in the main program for the following sequence of number of strips (n); n-2, n-4...
  36. T

    Am I rewriting this differential correctly?

    Homework Statement I have a differential equation that I need to solve numerically by writing a program. x0, y0, x_dot0, y_dot0, α are all given Hello, I have the following differential equation: http://puu.sh/d78KC/107bd6c71f.png I want to rewrite it so I can solve it numerically by writing...
  37. A

    How to do Linearization for Non-linear least squares?

    Hy I want to know how to make linearization for some function,...what should by in Non-linear least squares problems. In my book I have only this example how to do: http://i.imgur.com/MUFiHkr.pngSomeone could me help how to do, some receipt of method what I need to do? Non-linear least...
  38. 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...
  39. 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
  40. T

    MHB Numerical Methods: Solving Exam Pickle with Factorization & Stability

    Okay, I'm in a bit of a pickle here. Got the exam on thursday and (surprise) I am utterly clueless. I cannot grasp a lot of concepts, but here's some I'd like to at least get an idea of: Factorization method. I only scrapped that it is a special case of Gauss' Exclusion method, that you take a...
  41. kini.Amith

    Are Numerical methods of any use for a physicist

    I am currently doing my bachelors in Mechanical engineering engineering and planning to pursue physics after completion. I have to choose an elective the coming semester. One of the electives offered is 'Numerical Methods for Engineers' and the modules covered include Error in numerical...
  42. C

    Period of Harmonic Oscillator using Numerical Methods

    Homework Statement Numerically determine the period of oscillations for a harmonic oscillator using the Euler-Richardson algorithm. The equation of motion of the harmonic oscillator is described by the following: \frac{d^{2}}{dt^{2}} = - \omega^{2}_{0}x The initial conditions are x(t=0)=1...
  43. M

    Numerical Methods - which one for which problem?

    Can anyone tell me for which problems you should use these numerical methods: - finite difference method - finite element method - boundary element method - method of moments For example I read that finite element method is often used for car crash simulations, and that it gives very...
  44. F

    Physics nonlinear ODE example for numerical methods project?

    I am doing a little research project into numerical methods of solving ODEs where I do 1 half of learning about the basics of numerical methods and then look at a particular method (Linear multistep) and then the second half is looking at a particular example, applying what I've learned and...
  45. C

    Any recommendations for a book on advanced numerical methods?

    I'm looking for a good book on numerical methods. I know most of the basic algorithms but one that delves more into things like adaptive quadrature and that type of thing is what I'm looking for. Any texts people could recommend?
  46. P

    Numerical methods for nonlinear PDEs in large domains

    Hi all, first post :) I have a system of z-propagated nonlinear PDEs that I solve numerically via a pseudo-spectral method which incorporates adaptive step size control using a Runge-Kutta-Fehlberg technique. This approach is fine over short propagation lengths but computation times don't...
  47. maistral

    Numerical Methods for PDEs, basic algorithm?

    This is actually a request, I don't know if these are the correct forums for me to post these kinds of things, but yeah. Alright. I intended to study and learn numerical methods with PDEs on my own. And sadly the only thing I can comprehend is the Liebmann method. :cry: And I got so little...
  48. E

    MATLAB Is a Basic Understanding of Matlab Enough for Applied Numerical Methods?

    So I'm taking an applied numerical methods course this year, and we are not putting any emphasis on Matlab. Well, we do use it on homeworks and assignments, but he does not teach us more than the very basics (for, while loops, if iterations). The weird thing is that all the math we learn in the...
  49. N

    Applications of Calculus and Numerical Methods

    Hi, I'm a new user of this forum and i wanted to know how i can use calculus and interpolation in my projects. I'm a student in engineering and i usually do home made electronics and computer programming projects, i would like to include calculus to my projects and understand the practical side...
  50. M

    Courses Preparation Numerical Methods Course

    Hi, Next year I will be a numerical methods course as part of my Mathematics minor (CS major). I want to prepare for this course in the summer. How can I best prepare for this class considering I have never used Matlab before?