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In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, [5] as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.
Numerical methods for solving first-order IVPs often fall into one of two large categories: [5] linear multistep methods, or Runge–Kutta methods.A further division can be realized by dividing methods into those that are explicit and those that are implicit.
K.W. Morton and D.F. Mayers, Numerical Solution of Partial Differential Equations, An Introduction. Cambridge University Press, 2005. Autar Kaw and E. Eric Kalu, Numerical Methods with Applications, (2008) . Contains a brief, engineering-oriented introduction to FDM (for ODEs) in Chapter 08.07. John Strikwerda (2004).
The name is in analogy with quadrature, meaning numerical integration, where weighted sums are used in methods such as Simpson's rule or the trapezoidal rule. There are various methods for determining the weight coefficients, for example, the Savitzky–Golay filter. Differential quadrature is used to solve partial differential equations. There ...
Finite difference methods for heat equation and related PDEs: FTCS scheme (forward-time central-space) — first-order explicit; Crank–Nicolson method — second-order implicit; Finite difference methods for hyperbolic PDEs like the wave equation: Lax–Friedrichs method — first-order explicit; Lax–Wendroff method — second-order explicit
Adaptive quadrature is a numerical integration method in which the integral of a function is approximated using static quadrature rules on adaptively refined subintervals of the region of integration. Generally, adaptive algorithms are just as efficient and effective as traditional algorithms for "well behaved" integrands, but are also ...
The method of Gaussian elimination appears – albeit without proof – in the Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations.
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