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. H

    Numerical Estimation problem

    Homework Statement "Estimate the mass of water in all the World’s oceans." Homework Equations I know the following: Two-thirds of the Earth is sea. The density of seawater is 1025 kg/m3. Radius of the Earth = 6.3*106m. The Attempt at a Solution Let's assume that the...
  2. H

    Numerical Estimation of the momentum of free electrons

    Homework Statement This is a problem in numerical estimation. Estimate a typical maximum momentum pmax for the free electrons in a metal where pmax = \left(\frac{3nh^{3}}{8\pi}\right)^{1/3} (In estimating n, the number density of free electrons, assume there is one such electron...
  3. Z

    Numerical method to solve ODE boundary problem

    can anyone provide a Numerical algorithm to solve -y'' (x) +f(x)y(x) = \lambda _{n} y(x) with the boundary condition y(0)=y(a)=0 here 'a' is a parameter introduced at hand inside the program and f(x) is also introduced by hand in the program i am more interested in getting...
  4. S

    Mathematica Mathematica:How to solve the summation of numerical integration

    From the attachment. i would like to know how to find (t_1 and t_2)minimum if given t_0=0 and t_3=5.It seem like when using excel solver to find the minimum.anyone know how to do it with mathematica?
  5. S

    Mathematica Mathematica:Please help me to solve numerical integration

    i have problem regarding to the numerical method. my problem is from da attachment,if i replace x with value 1,it will get da answer by using NIntegrate.I want to know if i can get the value of t_minimum from the answer.thanks a lots.
  6. J

    Finding Vf when you dont have distance and only height without numerical value

    Homework Statement 13. A box of mass m is released from rest at point A, the top of a long, frictionless slide. Point A is at height H above the level of points B and C. Although the slide is frictionless, the horizontal surface from point B to point C is not. The coefficient of kinetic...
  7. A

    Efficient Numerical Methods for Solving Equations: MCQs and Solutions

    some one help me to solve these questions. 1. The order of convergence of Newton-Raphson Method is a) at least one b) at most two c) at least there d) four e) none of these. 2. Newton-Raphson Method fails to find the root of f(x)=0 if: a) f'(x) > 0 b) f(x)< 0 c) f'(x) = 0 d) f'(x) <0...
  8. P

    Learning Numerical Analysis for Physics Problems

    Good afternoon, I'm willing to study (teach myself) http://en.wikipedia.org/wiki/Numerical_analysis" , which I find very interesting, and which seems to be very useful for solving Physics problems (specifically, the motion of celestial bodies). I have a background in Calculus (basically...
  9. S

    Mathematica MATHEMATICA[solve numerical integration and find min value]

    i want to find the minimum sum of this equation and find the t like using solver in Excel,but since the integration cannot be integrate directly,i didnt know how to use the numerical integration to apply for this problem. i want to find the value :let say when n=2, t_0=0 and t_n=1, so i need...
  10. S

    Numerical integration simpson

    Hi, I've using numerical integration method (Simpson rule) to evaluate a definite integral in the interval [a,b]. I was wondering what is the ideal way to approximate the integral in the boundary [a,b) or (a,b] or (a,b) when for example, the function inside the integral does not exist at that...
  11. R

    MATLAB Numerical Methods (Java vs Matlab)

    Hello to all I've been wondering about the skills needed for doing a research degree. I've been looking at the Matlab language recently and it seems somewhat similar to Java. Is Java just as easy to use as Matlab for doing numerical methods? I'd be interested in hearing your views. Thank you...
  12. N

    Problem in understandind sultion in numerical analisys

    A) calculate I=\int_{0}^{1}\cos(\frac{\pi}{2}x^2) by trapeze methos in those pannels m=1,2,4,8 ? B) what is the best aproximation we can get from part A of the question? solution: the solution for part A is http://i47.tinypic.com/2vmh4b9.jpg my problem start with the solution of...
  13. V

    MATLAB MATLAB - Numerical integration

    hello, I have a problem with the use of quad function in Matlab for numerical integration. Let me try to explain on an example. When I want to find the integral of expression exp(-x^2/2) from 0 to infinity, where for the higher bound I use some very large number: Q =...
  14. A

    HP50g help with numerical solver

    Hi, can anyone tell me how to add a text line in the numerical solver calculation screen that describes a variable in an equation? for example spl = 20 log (D2/Dr) Dr is a reference distance from a source D2 is a prediction distance I want some form of description of the variable...
  15. N

    Numerical analisys question

    http://i48.tinypic.com/2yzd213.jpg how they got g1 g2 g3 ?
  16. M

    Implicit numerical differentiation

    Homework Statement I am using Crank–Nicolson to solve a logistic function, modeling population growth. To get the next time step, I have to solve a quadratic equation. The problem is that i get two solutions for y(i+1). Does it mean that I am doing it wrong? If not, can I just pick the...
  17. S

    Numerical solution of an ODE be Singular at endpoint

    Hi all, I have an ODE with the form \frac{\mathrm{d}^{2}\phi}{\mathrm{d}x^{2}}+a\exp(b\phi)=0 B.C. \frac{\mathrm{d}\phi}{\mathrm{d}{x}}\right|_{x=0}=-c,\frac{\mathrm{d}\phi}{\mathrm{d}{x}}\right|_{x=L}=0 where a,b,c,L are all positive. I solved the problem with Maple using the dsolve...
  18. M

    Adding a threshold in numerical codes

    Hi, I noticed such a strange behavior in my code every time I add a threshold: little oscillation around the threshold value itself. For example f(x) = g(x), x > x_c f(x) = 0, x <= x_c \frac{dx}{dt}=f(x) I solve the last equation and I find this oscillation. Can they be removed?
  19. A

    MATLAB Numerical integration in matlab

    i am very new in matlab. and i need to calculate the arc lengh. My equation is arc length = integration of sqrt(d.*(k.^2-2).*sin(k.*t./2).^2 - d.*(k.^2-1).*sin(k.*t./2).^4 + 1); from 0 to 2.pi where, d=(0:.1:1) and k=(0:1:10) can anybody tell me whether it is possible to calculate this arc...
  20. M

    Numerical Analysis Euler's Method

    Please help me with this problem! I have no clue. I know how to deal with only dy/dx. But this includes dy/dx and dz/dx...:cry: Please see the attachment :) Step size h = 0.2 range x = 0 to 1 y(0) = 2 z(0) = 4 Thank you, Maya
  21. K

    Numerical Approximation to Roots

    Homework Statement I'm trying to find a root-finding method for a function [tex] f: \mathbb R^n \to \mathbb R [/itex] Homework Equations x is a root of f(x) if f(x) = 0 The Attempt at a Solution There is lots of work done for this problem when n=1, and also lots of work done when...
  22. G

    Guidance sought with numerical ODE solution

    The following image is from an old heat transfer book - my problem is quite similar. Note that for this discussion, only equation 4-6 need be considered. [PLAIN]http://www.pairofdocs.net/images/eq.jpg Consider the case when the liquid is being added to the tank at a constant rate, and without...
  23. N

    Proving Numerical Equivalence of Real Number Intervals with S-B Theorem

    Homework Statement Using the Schroeder-Bernstein Theorem, prove that any two intervals of real numbers are numerically equivalent. Homework Equations Schroeder-Bernstein Theorem: Let A and B be sets, and suppose that there are injections from A into B and B into A. Then, there exists a...
  24. M

    Numerical integration - optimisation of code

    Hello I have a function which is very similar in shape to a Gaussian, except it is not a distribution and it is not analytic, so I can at best calculate a single point on the curve at a time. (In general it is a convolution of different distributions but this is not important). I need to find...
  25. S

    Numerical Optimization ( norm minim)

    Homework Statement Consider the half space defined by H = {x ∈ IRn | aT x +alpha ≥ 0} where a ∈ IRn and alpha ∈ IR are given. Formulate and solve the optimization problem for finding the point x in H that has the smallest Euclidean norm. Homework Equations The Attempt at a...
  26. R

    How to Create a Bisection Method Program for Arbitrary Functions?

    help me please "numerical method" please help me finish college assignment, I was given the task of creating programs numerical calculation method, the bisection method, the program c + + or Visual Basic, or any program language. last collection time Friday, 23 April 2010. please help me...
  27. J

    How to Use Euler's Method to Approximate Velocity in a Quarter Circle Motion

    I have an assigned project in Physics with numerical modeling, to which I am supposed to use Euler's method to approximate the velocity of a "mass sliding down a frictionless in the shape of a quarter circle of radius 2.00 m." The assignment also states: "Do three trials: t = 0.2s, t = 0.02s...
  28. N

    What is the Pattern for Finding s in Iterative Calculations?

    in this page http://i40.tinypic.com/seyrys.jpg we can't see what is the pattern of finding s? what s represents? in iteration 0 s is found by sum of an and bn in the 1st 2nd and 3rd iteration its found by their subtraction in the 4th its neither ??
  29. S

    Numerical Optimization ( steepest descent method)

    Homework Statement Consider the steepest descent method with exact line searches applied to the convex quadratic function f(x) = 1/2 xT Qx − bT x, ( T stands for transpose). show that if the initial point is such that x0 − x* ( x* is the exact solution of Qx = b) is parallel to an...
  30. N

    Derivative laws question numerical analysis

    i know the law (fg)'=f'g+fg' but here there is epsilon and it turns to someother sign i don't know what it says. i can't understand the transition ?
  31. N

    Numerical analysis mathlab question

    how to do those tables of calculation in math lab i got sum really long formullas in other question and if ill do the itterations i will spend half a day just on that
  32. edpell

    Numerical Simulations of General Relativity: 1000 Blobs

    When can we expect to see numerical simulations of GR for cases that are not highly symmetric? Say 10^3 blobs of matter in an arbitrary initial configuration.
  33. N

    Numerical analisys close numbers question

    y=\sqrt{x^2+1}-1 how do we know that there is a subtraction of two close numbers thus making loss of significance? there is no close numbers there is variable X it could give use a close result or otherwise and why multiplying and dividing by y=\sqrt{x^2+1}+1 makes it go away?
  34. N

    Numerical integration along constant and collapsing spectrum

    Hello--- I am reading a paper which describes a somewhat-unfamiliar mathematical procedure. The paper asks for a 2D spectrum U(t, \omega) of a signal s(t) which is calculated using the short-time Fourier transform (Gabor transform). This is reasonably straight-forward. However, the paper...
  35. J

    How to Terminate a While Loop in C Programming with a Printed Message or Value?

    Im using the c programming language and just wanted to ask a quick question. In a while loop how do you make the program terminate by printing a value or a message here's my code while ((fabs(func(x))>epsilon)) { if(deriv(x)==0) { print the last value of x and stop the whole program} else {...
  36. N

    Numerical integration of improper integral

    Hello-- I need to generate synthetic data to test an algorithm used to process data from an experiment. A synthetic wavelet is constructed using the following equations, but I am uncertain how to numerically evaluate the improper integral shown below. \[ u(t) = {\mathop{\rm...
  37. Simfish

    How useful is numerical analysis? (compared to other topics)

    I know that Matlab is essentially a giant numerical analysis package, so numerical analysis is helpful for anything that involves Matlab or FORTRAN or non-symbolic programming languages (or processes that use approximations of differential equations and stuff). So pretty much anything with giant...
  38. S

    Finding the Terms and Proving Induction for a Numerical Sequence

    Homework Statement If x is a real number, we de fine [x] as being the largest integer <= x. For example, [1.2] = 1, [-1.1] = -2, [1] = 1, [11/3]3 = 3, . . . Let {an}n>=1 be the numerical sequence de fined by: a1 = 3; and an = a[n/2], for n>=2 (a) Give the terms a1; a2; ... ; a8 of this...
  39. P

    Numerical Methods: Taylor Series for Diff Equation

    Homework Statement Solve the differential equation \frac{dy^2}{dx^2}=xy^2-2yy'+x^3+4 where y(1)=1 y'(1)=2 by means of the Taylor-series expansion to get the value of y at x=1.1. Use terms up to x^6 and \Delta x=0.1The Attempt at a Solution I'm unsure as to how I should go about...
  40. C

    Finding fluid flow with Numerical Method

    Homework Statement For fluid flow in pipes, friction is described by a dimensionless number, the Fanning friction factor (f). The Fanning friction factor is dependent on a number of parameters related to the size of the pipe and the fluid, which all can be represented by another dimensionless...
  41. S

    Numerical approximation of the solution

    u''(x)=f(x), boundary conditions u(a)=0,u(b)=0. (u(x+h)-2u(x)+u(x-h))/h^2=f(x); maltab code: clear all a=0; b=1; n=10; h=(b-a)/(n+1); x_with_boundary=linspace(a,b,n+2)'; x=x_with_boundary(2:n+1); A=h^(-2).*(diag(ones(1,n-1),-1)+diag(-2.*ones(1,n),0)+diag(ones(1,n-1),1))...
  42. P

    Numerical integration and errors

    Hi, I was wondering if anyone can point me to a general treatment of errors when doing numerical integration of measured variables? My problem is that I am integrating force with respect to displacement (of a piston) in an attempt to calculate work...and getting some impossible numbers...
  43. P

    Exploring Numerical Equivalence of Sets: A Brief Definition

    1. Define numerical equivalence of sets 2. I'm not sure how in depth the definition needs to be, how is my current def? 3. X is numerically equivalent to Y if \existsF:X\rightarrowY that is bijective or there are two injective functions f:X\rightarrowY and g:Y\rightarrowX
  44. A

    Excel Numerical Integration for Piston Movement Calculation

    Hi, I have an excel spreadsheet that calculates the pressure inside a piston chamber, that is a function of time. I want to calculate the distance the piston moves, this is my situation. P=F/A, therefore force on piston = PA F = M(piston).a so ma = PA, a = P(t)A/m its all numerical in...
  45. H

    Decompose Matrix A with PLU-Decomposition - Numerical Analysis

    I am supposed to write a program to decompose matrix A with PLU-decomposition. I have this text and I am trying to understand some parts of it. The text with % in front of it is in Icelandic and doesn´t matter. function [P,L,U,r] = PUL_thattun(X) n=length(X); L = zeros(n,n); U = X...
  46. C

    Numerical analysis, floating-point arithmetic

    Hi all, this (probably easy) problem from numerical analysis is giving me trouble. I can't seem to get started and need some poking in the right direction. Homework Statement Consider the following claim: if two floating point numbers x and y with the same sign differ by a factor of at...
  47. P

    Numerical differentiation using forward, backward and central finite difference

    ive been given this question for a uni assignment: given the function: f (x) = 5(x^1.3) +1.5(7x − 3)+ 3(e^− x) + ln(2.5(x^3)) find the first derivative at all possible points within the interval [0, 6], with step length h = 1 for: forward difference aproximation, backward difference...
  48. M

    Mathematica Numerical Derivatives in Mathematica

    I have some data in a text file. I want to import this data into Mathematica, and then I want to calculate the numerical derivatives of this data. In particular, I need to find the y value where the first derivative is equal to zero. I can import data. I can use the ND function for numerical...
  49. B

    Converting a problem solved with numerical analysis to a simple exponential

    I'm just a neuroscientist, so forgive me if the answer to this question is either obvious, or the answer is that it is impossible, obviously. Basically, this image should outline the question clearly http://img713.imageshack.us/img713/2569/mathsissuefixedyxs.gif (And Y0 is always 1)Also, I...
  50. J

    Numerical solution to equations with log?

    Homework Statement Merry Xmas~ I hope there's still someone online during the holiday season. I'm trying to solve a set of static laser rate equations but with log in the expression. I think, there's no analytic solution but I don't really know how solve the problem numerically with Matlab...
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