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. N

    [Numerical] Euler Backward method

    Hi, I need to solve the following simple ODE with both the Euler Forward and Euler Backward numerical methods. I also need to answer for which values of T this can still be calculated: y'(t) = \frac{-1}{2y(t)} y(0) = 2 t \in (0, T] Obviously the analytical solution is y(t) = \sqrt{4 - t} So it...
  2. C

    Evaluating Integral with Partial Fractions: A Numerical Approach

    Homework Statement I am supposed to evaluate the integral using partial fractions. \int \frac{1}{(x+5)^2(x-1)} dx 2. The attempt at a solution So after doing all the work, I get (-1/36)ln|x+5| - (13/6)ln|x+5| + (1/36)ln|x-1| But the answer in the book appears as (-1/36)ln|x+5| -...
  3. S

    Quantitative Methods: Numerical Solution of DEs

    Homework Statement Show that if we fit f(x) over [xn,xn+1], with a first-degree polynomial that interpolates f at xn and xn+1, then f(x)=f(xn)+[f(xn+1)-f(xn)](x-xn)/h. Putting that approximation into the relevant equation 1 which is given at relevant equations part derive the approximation...
  4. P

    Numerical Analysis: Solving y0 = Ly; y(0) = 1 w/ Methods I & II

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  5. N

    Numerical Integration of Functions with Poles

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  6. U

    Numerical Integration of sin(1/x)

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  7. P

    Finding v0 in a 5-point grid using Laplace numerical solution

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  8. A

    Unravel the meaning of the variables to find numerical answers

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  9. F

    Numerical Integration: Understanding Error, Big Oh and Taylor Expansion

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  10. D

    C/C++ Integrating Numerical Methods in VC++: Library Recommendations and Compatibility

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  11. G

    Step Function: Numerical Precision Matters

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  12. T

    Numerical method needed for the following system of ODEs

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  13. T

    Is numerical analysis interesting?

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  14. P

    Numerical model validation

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  15. C

    Numerical integration - Fourier transform or brute force?

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  16. R

    Numerical method to solve a fourth order ODE

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  17. G

    Numerical integration of angular velocity

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  18. N

    MATLAB Need Help with Numerical Integration in MATLAB

    Here is an integral (similar in form to the one) I want to evaluate: $$ \int_{r_1}^{r_2} R \left\{ \int_{x_1}^{x_2} Y(x,R)\, dx \right\}^2\,dR $$ This is my approach - please correct me because I don't think it's right: I treat R as a constant and evaluate the inner integral over some vector of...
  19. B

    Third order differential equation numerical approximation

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  20. J

    Improving Numerical Approximations of Limits: A Sample Exam Question

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  21. J

    Numerical Solutions to Coupled ODEs with Boundry Values at Opposite Ends

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  22. M

    Linear stationary system of 2 partial DEs, numerical implementation

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  23. D

    Numerical Differentiation: Difference approximation on numerical data

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  24. T

    Is .999999... actually equal to 1?

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  25. H_man

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  26. M

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  27. A

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  28. T

    Problems with numerical illiteracy

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  29. H

    How to Solve a Higher Dimensional Numerical Integration Using Maple Notation?

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  30. N

    Accuracy of Numerical Integration Methods

    Hi, does anyone know what sort of methods I could use to test the accuracy of the numerical solution of the integral equation? Many thanks
  31. P

    Solve Large Numerical Values Without a Calculator

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  32. I

    Good books with numerical answers in the back

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  33. G

    Finding derivatives of vector function for numerical integration

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  34. X

    Reaction-Diffusion - numerical simulation

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  35. K

    Numerical Integration and Energy Loss

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  36. X

    Trapezoidal Rule for numerical integration

    The trapezoidal rule for numerical integration is based on the idea that when we partition our larger interval into subintervals, we can approximate the area over each subinterval by calculating the area of the trapezoid formed by connecting the value of the function at the left and right...
  37. F

    Solve the Positive Integer N Puzzle: 6N = defabc

    Homework Statement Im having problem of even starting this question: "Determine the positive integer N = abcdef such that 6N = defabc" any ideas? Homework Equations The Attempt at a Solution
  38. A

    Is the numerical value of action functional of any importance?

    While working out a problem I got a result which gave rise to this doubt regarding value of action functional. Suppose I start from an action, obtain the equation of motion and when I try to check if that solution gives a finite value of action, I get, surprisingly, vanishing value. The actual...
  39. L

    Numerical solving of elastomechanic problems

    I want to make a simple program for numerical solving of elastomechanic problems. The basic task it should perform is the calculation of deformation of a two dimensional homogeneous, isotropic body in the field of gravity and/or deformation under external force. The basic idea for each time...
  40. S

    Gravitational wave physics and numerical relativity ?

    What do you think about Gravitational wave physics and numerical relativity ? Will it expand much in future? Now few universities are giving research facilities on these two. Why ? But i really like these two.
  41. L

    Are There Integrals That Are Easier to Solve Numerically Than Analytically?

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  42. P

    How much numerical analysis do we need?

    for each of these careers: engineer, physicist, mathematician, how much numerical analysis is necessary? is just learning mathematica sufficient? or do you need at least a full class in numerical analysis?
  43. MathematicalPhysicist

    Numerical Differential Equation.

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  44. B

    Advanced Numerical Methods for Wave Eqn?

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  45. qspeechc

    Numerical Solution to 2nd Order Eqn?

    Is there a numerical method for finding solutions to 2nd order non-homogeneous differential equations? Thanks.
  46. N

    Numerical PDE's II - Circular Domain

    Homework Statement Approximate the solution of: \frac{\partial^{2}u }{{\partial r^{2} }} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^{2}}\frac{\partial^{2}u}{\partial \theta^{2}} = 0, 1<r<3, 0<\theta<\pi, u(1,\theta) = 1-\cos(\theta),u(2,\theta)=u(r,0)=0 u(r,\pi)=4-2r...
  47. D

    Numerical Evaluation of Principal Value Integrals

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  48. U

    Titius-Bode law, any analytical or numerical derivation?

    They say this is just by chance. But maybe some bright spark has come up with a mathematical derivation for it? Alternatively, what about simulating the cloud of the early solar system on a computer. Would the Titius-Bode pattern of the planet distances arise? Here's the law...
  49. P

    Motion of pointed bodies - numerical methods

    Hi, I have to solve (using numerical methods) fallowing problem: there are for example three pointed bodies in 2D. I have to find trajectories of their motions in their own gravitational field. In cartesian coordinates for each body and direction I write equation of motion (that gives me system...
  50. S

    Numerical and Non=Numerical applications of computers.

    Pliz Help me and enlighten me about the numerical and non-numerical applications of computers.
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