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. W

    Numerical analysis (composite numerical integration)

    using composite trapezoidal rule with n=4 how can i get a bound for the error of I=integration tan(x) from x=0 to x=pi/2 i know that the term of error in comp trapezoidal rule is (b-a)/12 h^2 f''(eita) i got the second derevative of tanx to be 2sec^2 x tanx then i don't know with what value...
  2. K

    MATLAB Euler's Method of Numerical Approximations

    Hello, For Euler's Method of Numerical Approximations, my book (Boyce&DiPrima) gives this algorithm: Step 1: define f(t,y) Step 2: input initial values t0 and y0 Step 3: input step size h and number of steps n Step 4: output t0 and y0 Step 5: for j from 1 to n do Step 6: k1 = f(t,y)...
  3. Z

    Finding Inverse of 6x6 Matrix: A Numerical Solution

    Hey all :smile: I have a house mate is doing a computer physics course and he's never before dealt with matrices bigger than 3 x 3. He was wanted to know out how to work out the inverse of 6 x 6, now it's been a long time since I've done this myself. I remembered the Gauss - Seidel Method...
  4. A

    How to Determine the Best Interpolation in Newton Forward Difference Method?

    in Newton forward differece method. how can i know that i reached the best interpolation? for example in a function like sqrt(x) for Xi=1,1.05,1.10,1.15,1.20,1.25,1.3 the best interpolation is at P3(x) why?how can i know? this really makes me conused:confused: :confused: if anyone...
  5. K

    Numerical Problem based on Newton's laws of Motion

    A 5kg block is resting on the top right hand corner of a 10kg block. The length of the top of the 10kg block is 10m. Find the time taken by the 5kg block on top to slide of the 10kg block completely if the 10kg block is accelerating at 5m/s^2. My work: aB = accleration of the 5kg block aA...
  6. L

    Numerical Solutions to GR: Exploring the Computational Physics Field

    Hello could someone give some info about the "Numerical solution" to GR...is this a field of "Computational Physics"?.. - What i know is that you take the Hyper-surface, and you " split " it into triangles..and use the ¿angles? of every triangle as finite-coordinates..then you get a problem...
  7. T

    I need tips on Calculus: Graphical Numerical Algebraic Workbook?

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  8. R

    Numerical Analysis - Finding the Rate of Convergence

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  9. M

    Which Numerical Methods Resource is Best for This Course?

    Can anyone point me to a website or set of notes that I could read up on the topics of this course? course includes: solving for the roots of algebraic and transcendental equations, solution of simultaneous linear equations, least-squares curve fits to data, interpolation, numerical...
  10. D

    What Can Computational Physics do besides Numerical Integration?

    Are there many interesting computational physics problems out there? Are there any comet trajectories that will deviate from a standard ellipse? For some reason plotting the path of a baseball just doesn't spark my interest.
  11. Astronuc

    Can Numerical Simulation Revolutionize Nuclear Reactor Design?

    This is a developing area in the nuclear industry involving large scale multiphysics computation. For the past several years Argonne National Laboratory and Purdue University have been supported by the Department of Energy (DOE) and the Electric Power Research Institute (EPRI) to develop a...
  12. P

    Numerical Analysis - Construction of a Poincare surface of section

    (I am not sure whether I'm posting in the right forum. I apologize if I do) Does anyone have an alrorithm or a code (in any language) that constructs a Poincare surface of section? I want to do so for a Hamiltonian model: A mass under the influense of the Henon-Heiles potential. It has to...
  13. A

    Numerical methods for systems of nonlinear ODEs

    I have a quick question. For a project that I'm doing, I need to numerically solve systems of nonlinear differential equations. Can anyone suggest a numerical method which I could code as a short C program? Thanks.
  14. J

    Nonlinear Parabolic BVP: Possible Numerical Methods

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  15. L

    Numerical Method Solutions ()

    I have been given the following problem as assignment: Find a numerical solution for the 1-D heat conduction (using the Explicit Method): \left\{\begin{array}U_{xx} = U_{t},\\ U(x,0) = \sin \pi x, \\ U(0,t) = U(1,t) = 0 Use h = 1, k = 0.005125 and M = 200. Can anyone help by giving...
  16. Z

    Numerical Methods Book for Groundwater Modeling & Environmental Engineering

    I am a geology major interested in groundwater modeling. Currently taking ordinary differential equations, and I'll be in a computational numerical methods course for engineers next fall. Here is the course description.. "Introduction to numerical methods for environmental engineering...
  17. H

    Looking for good book on Numerical Methods and/or Optimization

    Any recommendations? The books I have are very outdated. Extremely important to me are: - worked examples #1 criteria. Need that bridge between theory and implementation. - not overly heavy on theory (don't want to hire a PhD to explain it). I have an MS Engineering level education...
  18. M

    Problem with numerical integration

    Hi all, I am having trouble numerically integrating a function using Maple 10. Here is a bit of background on the problem: This problem is asking for two plots, one of the velocity of a sounding rocket with respect to time, and the other being the height of the sounding rocket with...
  19. B

    Question: What number should you write to win the numerical methods math riddle?

    On the piece of paper handed out in class please write a number between 0 and 100. The winner is the one who comes closest to 2/3rds of the average of numbers written by the students of the class. Question: What number should you write?
  20. G

    Numerical Analysis: Taylor Polynomials, Error, Bounds

    (a) I found the answer to be: 1/(1-x) = 1 + x + x^2 + x^3 + ... + [x^(n+1)]/(1-x) for x != 1 *Note: "^" precedes a superscript, "!=" means "does not equal" (b) Use part (a) to find a Taylor polynomial of a general (3n)th degree for: f(x) = (1/x)*Integral[(1/(1 + t^3), t, 0, x] *Note...
  21. G

    Numerical Analysis: Fixed Point Iteration

    Consider the fixed point iteration formula: *x_(n+1) = (2/3)[(x_n)^3 - 1] - 3(x_n)^2 + 4x_n = g(x) *Note: "_" precedes a subscript and "^" precedes a superscript (a) Find an interval in which every starting point x_0 will definitely converge to alpha = 1. (b) Show that the order of the...
  22. M

    Numerical Solution to ODE System - IVP or BVP?

    I have a system of spatial ODEs to solve... Actually a DAE system, but here's the issue: The equations are vaild over a specific domain, x = 0..L The equations are only bound at one point (thier "initial point") but at either 0 or L f1(0)=0 f2(0)=100 f3(L)=0 f4(L)=100 (also an...
  23. J

    C++ Cubic Spline Interpolation Source Code | Numerical Methods

    Has anybody got the source code in C++ for cubic spline interpolation?Need for my lab..
  24. S

    Deriving O(h^4) Five Point Formula to Approximate f'(x0)

    given five points of a function one can approximate the derivate of the function at some point. The standard five point formula is Derive an O(h^4) five point formula to approximate f'(x0) that uses f(x_{0}-h), f(x_{0}), f(x_{0} +h),f(x_{0} +2h),f(x_{0} +3h) . (Hint:Consider the...
  25. S

    Solving f(x)=e^(0.1x^2) with H5, H3 and Error Bounds

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  26. siddharth

    Evaluation of Numerical series by Fourier series

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  27. Clausius2

    On numerical calculation of Lift Force via Potential Flow

    Imagine: I want to compute numerically a POTENTIAL STEADY and INCOMPRESSIBLE flow over an airfoil. The set up of the problem is: \nabla^2\phi=0 \nabla\phi \cdot \overline{n}\big)_{x=surface}=0 no normal velocity component on the airfoil surface. \nabla \phi=\overline{U_\infty} as...
  28. Z

    Numerical solution of 2nd order ODE

    Got this problem and we've been given a program which can solve for x, for the equation: Ax = b Where A = \left( \begin{array}{rrrrrr} b & c & 0 & 0 & \cdots & 0 \\ a & b & c & 0 & \cdots & 0\\ 0 & \ddots & \ddots & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \ddots & 0 \\ \vdots & & &...
  29. H

    Numerical Methods - Newton Raphson

    The following 2 page example illustrates the use of the Newton-Raphson technique for solving for roots of functions. Examples included: 1. Function in a single variable 2. System of non-linear equations
  30. S

    Numerical problem about circuits

    PROBLEM:the current at the terminals of a certain current source is measured with an ammeter having an internal resistance R_i=10 ohms and is found to be 11.988 mA ;adding a 1.2 kilo -ohms resistance between the source terminals causes the ammeter reading to drop to 11.889 mA.Find i_s and R_s...
  31. S

    Solve Numerical Problems: 4x^2 - e^x = 0

    Find intervals containing solutions to 4x^2 - e^x = 0 well someone suggested i sketch the graphs of 4x^2 and 2^x but I am not sure on how to go past that point... All i have to do is find the intervals so do the intersection point(s) of these two functions indicate the intervals where the...
  32. S

    How Do You Solve 4x^2 - e^x = 0 Using Numerical Methods?

    Find the intervals containing solutions to the following equation 4x^2 - e^x = 0 I haven o clue on where to start really? WOuldi expand e^x using taylor series? i mean one could do this x - 2ln x = ln 4 so then would i do log expansion by taylor series? Or would i use bisection...
  33. S

    Numerical Methods: Calculate 4/5 + 1/3

    Not a hard question really.. Using 3 - digit arithmetic calculcate 4/5 + 1/3 and compute the relative error 4/5 + 1/3 = 17/15 chopping 4/5 = 0.800 and 1/3 = 0.333 0.800 + 0.333 = 1.133 (chop) 1.13 im assuming that after this point there is no chopping relative error aboslute value of...
  34. P

    A proof in Numerical Analysis

    Hello there! yet another proof, that i need help on I am supposed to prove that the following statement holds for the secant method dk+1/ek -> -1 for k->Infinity where dk+1 is the next change and ek is the error. I have this idea, but i want to hear whether its a valid proof. i use...
  35. happyg1

    Numerically Stable Formulas for x >= 0 and x < 0

    Hi, here's my question...We have to determine whether or not the formulas down there are numerically stable for the cases where x >=0 and x<0. I say that the formulas are stable for x>=0 but not for x<0 because you are subracting numbers that could be close to each other. My problem is that I...
  36. S

    Numerical Problems: The U.S. National Debt

    Like any loan, the government accrues interest that compounds over time on the amount it owes. If the annual interest rate is r, then the number of years it takes for the amount of money owed to increase by a factor of x is Y=1/r ln(x) where ln is the natural logarithm. The average...
  37. happyg1

    Understanding Numerical Analysis Error Bounds

    Here are my questions: "Evaluating the summation as i goes from 1 to n of a sub i in floating point arithemetic may lead to an arbirarily large error. If however, all summands a sub i are of the same sign, then this relative error is bounded. Derive a crude bound for this error, disregarding...
  38. A

    Numerical solutions of system of nonlinear algebraic equations nonlinear algebraic eq

    Could somebody who knows well the method of numerical solutions of system of nonlinear algebraic equations nonlinear algebraic equations recommand a global convergence methods? thank you very much!
  39. N

    Numerical Analysis/Methods Packages

    Any body use or know of any good numerical analysis packages in C/C++. something like LAPACK. and are they easy to use.
  40. S

    Understand Adam Moulton & Bashforth Methods for Numerical Analysis

    hello all I have been researching into numerical analysis, differential equations in particular, I underdstand how the Runge kutta methods work geometrically but I don't quit understand what is the idea behind Adam moultons method And Adam Bashforth method, Is there a graphical way of...
  41. U

    How to Solve an Improper Integral with High Precision Using Simpson's Rule?

    Im supposed to solve integral 10 to +infinity ((sin(1/x)/(1+x^3))dx with error precision of e=0.5*10^-4. Can someone please give me detailed explenation of solving this. (Supposedly by Simpson but i get lost in the way. P.S. sorry for bad spelling and lack of proper formula notions.
  42. U

    Need help with numerical integration

    Im supposed to solve integral 10 to +infinity ((sin(1/x)/(1+x^3))dx with error precision of e=0.5*10^-4. Can someone please give me detailed explenation of solving this. (Supposedly by Simpson but i get lost in the way. P.S. sorry for bad spelling and lack of proper formula notions.
  43. B

    Integral evaluation - analytical vs. numerical

    Hi, Does anyone know a reason why \int_{-\infty}^{\infty}\cosh(x)^{-n}dx (n>0) can be evaluated analytically when n = 1,2,3,..., but only numerically when n is non-integer. I don't know if there is a "reason", but I'm using this result in a Quantum Mechanics project and it would be cool if I...
  44. M

    Modern trends in the numerical solution of differential equations

    :grumpy: :cry: I have an essay to write on modern trends in the numerical solution of differential equations. Most of the journals I've been reading are quite hectic and higher grade for me. ^^, Neway, if anybody knows of any good articlkes that i could read that would be great. Do ppl still...
  45. N

    So, is there any explanation for this?Why can't python do a numerical sort?

    No numerical sort in python ? I was really surprised to see that python can't do such an elementary operation! I am trying to do a numerical sort in python on an array. I am using python for windows (DOS). In perl you can simply write sort{$a<=>$b}@a and this will sort it according to the...
  46. E

    Numerical DE: Theta method

    Hello, I am given the method: y_(n+1) = y_n + h f(t_n + w h, (1-w)y_n + w y_(n+1). I am to determine the region of absolute stability; I am also to determine for which w in [0, 1] is the method A(a) stable, i.e., the region of absolute stability contains a sector about the negative...
  47. E

    How can I combine the solutions for u(t) and y(t) to find the solution for y(t)?

    Suppose we have y'' = f(t, y); y(a) = y0; y'(a) = y0' Note all derivatives are with respect to t. Let u = y', then u' = y'' 1. u' = f(t, y), u(a) = y'(a) 2. y' = u, y(a) = y0 Question 1: For y' = u, should I think of this as dy/du = u? Otherwise, I don't see how to solve 2 because...
  48. L

    Numerical riddle 2+2=44+4=8 142+468=621 3762+8271=?

    2+2=4 4+4=8 142+468=621 3762+8271=? driving me crazy
  49. D

    Help with numerical integration

    Hello there, I've not been here in a while, but I'm stuck doing this integration and wondered if some of you kind people would help :smile: \int_0^\infty \frac{1} {(1+x)\sqrt{x}} dx (appologies for the lack of spacing in there...) anyways, I know that when x tends to infinity, the...
  50. P

    Numerical Methods Help

    Numerical Methods Help! I have been trying to understand the differences between Finite-Difference Time-Domain (FDTD), Finite Volume, and the Finite Elements methods of solving Maxwell's equations numerically. I have used the FDTD method for solving Maxwells Equations. I did this without...
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