Numerical Definition and 775 Threads

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. T

    Engineering Concrete Slab Deflection Using Partial Differential Equations

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  2. I

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  3. J

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  4. ergospherical

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  5. HadronPhysics

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  6. S

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  7. F

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  8. M

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  9. S

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  10. Graham87

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  11. shivajikobardan

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  12. T

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  13. Graham87

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  14. C

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  15. Hasan2022

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  16. M

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  17. F

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  18. shivajikobardan

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  19. M

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  20. Saif Ali Khan

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  21. JD_PM

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  22. uzi kiko

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  23. G

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  24. W

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  26. A

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  27. B

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  29. S

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  30. Athenian

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  31. Paul Colby

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  32. jawad hussain

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  33. T

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  34. Arman777

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  35. D

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  36. T

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  37. D

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  38. I

    B Weird stuff on infinite numerical sequences in a Soviet book

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  39. P

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    Naturally there are vector equivalents of the Kirchhoff Integral. Taken from Jackson (10.113) ##\vec{E} \left( \vec{r} \right) = \frac{ie^{ikr}}{r} a^2 E_0 \cos \alpha \left( \vec{k} \times \vec{\epsilon}_2 \right) \frac{J_1 \left( \sin \theta \right)}{\sin \theta}##Where I just let ##\alpha =...
  40. 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)...
  41. Theia

    MHB Would you read a blog about numerical analysis?

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  42. Avatrin

    A Numerical Solution to Random Linear Non-Homogeneous ODE

    Hi I am trying to learn optimal estimation by reading Gelbs Applied Optimal Estimation, and I am having hard time with finding \Gamma defined as the following: $$ \Gamma_k w_k = \int_{t_k}^{t_{k+1}} e^{F(t_{k+1} - \sigma)} G(\sigma) w(\sigma) d\sigma$$ Here F is a known matrix. So is G, and w...
  43. 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.
  44. C

    A Partial differential equation containing the Inverse Laplacian Operator

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  45. R

    Engineering Numerical solution to a 3rd order D.E. - "use computer to solve"

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  46. Mikkel

    Python Numerical modeling of a glacier's length -- Coding error(s)

    Hey Physics Forum I am currently doing my bachelor project in geophysics, with focus on the evolution of glaciers in Greenland. My project consists partly of programming, because I want to get better at it. I have, however, hit a wall. I can't seem to figure out what is wrong with my code and I...
  47. Ifrah tahir

    Please help me solve this % numerical chemistry problem

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  48. D

    Checking the convergence of this numerical series using the ratio test

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  49. nughii

    Where Is the Error in My PV Modeling Code?

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  50. K

    I Numerical Solution to the Rayleigh Plesset Equation

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