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. Alexanddros81

    Simple Pendulum motion - Numerical Integration

    Homework Statement The differential equation of motion for the simple pendulum can be shown to be ##\ddot {θ} = -(g/L)sinθ##. Given that L=9.81 m and that the pendulum is released from rest at θ=60deg, determine the time required for the pendulum to reach the position θ=0deg. Use Δt=0.10s, and...
  2. F

    I Derivative and Numerical approach

    Hello, I am dealing with a function f(x) involving various complicated trig functions. I took the first derivative of the function f(x), which is doable, and set it to zero to find the at which value x the function f(x) would be maximum. However, once I have the derivative of f(x), it does not...
  3. F

    A Numerical solution of Hamiltonian systems

    The question is very general and could belong to another topic, but here it is. Suppose one wants to solve the set of differential equations $$ \frac{\partial x}{\partial t}=\frac{\partial H(x,p)}{\partial p},$$ $$\frac{\partial p}{\partial t}=-\frac{\partial H(x,p)}{\partial x},$$ with some...
  4. 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...
  5. Pushoam

    Length contraction -- numerical problem

    Homework Statement Homework EquationsThe Attempt at a Solution a) t' = l0/c b) ct1 = l0 /ϒ - vt1 t1 = l0 /ϒ(c + v) c) t2 = l0 / (ϒ v) Is this correct?
  6. Pushoam

    Length contraction numerical problem

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  7. M

    Numerical Nonlinear Lifting Line Theory in MA

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  8. awholenumber

    I Some questions from a numerical analysis book

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  9. C

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  10. Jamison Lahman

    Aperature, Numerical Aperature and Speed of Lenses

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  11. S

    Numerical integration in Python (throwing a ball)

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  12. E

    B Numerically Solving GRT Problems - Textbook/Article Recommendation

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  13. hilbert2

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  14. Ian Baughman

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  15. M

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  16. Conclaude

    Numerical Plot of Particle Orbits in Kerr Equatorial Plane

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  17. T

    Solid State Book focusing on numerical simulation of quantum systems

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  18. DeathbyGreen

    I How can I find the unitary matrix for diagonalizing a Hamiltonian numerically?

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  19. A

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  20. S

    MATLAB Is MATLAB better for numerical simulation

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  21. D

    I Numerical Derivative Formula: X's Not Same Distance

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  22. I

    Mathematica Numerical solution of integral equation with parameters

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  23. T

    Courses Computational physics course or numerical analysis course

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  24. S

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  25. A

    I Numerical solution of Schrödinger equation

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  26. chastiell

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  27. S

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  28. Telemachus

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  29. P

    I Numerical Calculus of Variations

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  30. vmr101

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  31. N

    Compute streamfunction from numerical velocity field

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  32. M

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  33. Elvis 123456789

    How can I be sure of my numerical result?

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  34. E

    I Decomposing a Function for Numerical Integration

    Is their a tutorial or a reference on how to decompose a function, specifically Fourier and Legendre decomposition, for numerical integration? The method I am going to use for the numerical integration is the Gauss Quadrature, and I suppose I need to decompose my function for the rule to work...
  35. D

    I Applications needing fast numerical conformal maps

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  36. N

    Other Numerical programming/modeling jobs with no credentials

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  37. F

    A Numerical integration of motion

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  38. MidgetDwarf

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  39. user123897

    Java Numerical integration of an harmonic oscillator using java

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  40. awholenumber

    I The algorithms involved in finding a solution with numerical methods

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  41. awholenumber

    I Few more numerical methods question....?

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  42. awholenumber

    I Differential equations and numerical methods questions

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  43. Samia qureshi

    Solve Numerical Problem: Proton Mass, Charge, Speed & Electric Field Strength

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  44. Johnny_Sparx

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

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  45. T

    Python Difference in numerical approach for PDE vs ODE

    I think I am missing something painfully obvious, but what exactly is the difference in algorithms used to solve PDEs vs ODEs? For example, I've been looking at finite difference methods and the general steps (from what I've seen, although particular approaches may vary) used to numerically...
  46. Themis

    Numerical methods for system of ODES

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  47. davidbenari

    I Numerical integration of PDEs: How do you satisfy boundary conditions

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  48. F

    MATLAB Finite difference numerical integration or ode45?

    I'm trying to numerically solve the time dependent Schrödinger equation and I've been told that the best approach is to numerically integrate using a finite difference method, however I don't understand why I couldn't just use ode45 to solve it?! Is the finite difference (interpolation) method...
  49. ShayanJ

    A Numerical solution of two coupled nonlinear PDEs

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  50. N

    A Numerical solution to SE - variational method, many electrons

    Hi everyone, I am trying to find electron wavefunction of a system I am working in. Numerical method I choose is the Variational method (VM). This method is convenient to find the ground state of the system. More details are available here. Problem I have can be explained on a very simple...
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