Numerical analysis: Difference between revisions

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Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation <math>2x+5=3</math> is linear while <math>2x^2+5=3</math> is not.

Much effort has been put in the development of methods for solving [[systems of linear equations]]. Standard direct methods, i.e.that is, methods that use some [[matrix decomposition]] are [[Gaussian elimination]], [[LU decomposition]], [[Cholesky decomposition]] for [[symmetric matrix|symmetric]] (or [[hermitian matrix|hermitian]]) and [[positive-definite matrix]], and [[QR decomposition]] for non-square matrices. Iterative methods such as the [[Jacobi method]], [[Gauss–Seidel method]], [[successive over-relaxation]] and [[conjugate gradient method]]<ref>{{cite journal |last1=Hestenes |first1=Magnus R. |last2=Stiefel |first2=Eduard |title=Methods of Conjugate Gradients for Solving Linear Systems |journal=Journal of Research of the National Bureau of Standards |volume=49 |issue=6 |pages=409– |date=December 1952 |doi=10.6028/jres.049.044 |url= https://nvlpubs.nist.gov/nistpubs/jres/049/jresv49n6p409_A1b.pdf}}</ref> are usually preferred for large systems. General iterative methods can be developed using a [[matrix splitting]].

[[Root-finding algorithm]]s are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is [[derivative|differentiable]] and the derivative is known, then Newton's method is a popular choice.<ref>{{cite book |last1=Ezquerro Fernández |first1=J.A. |last2=Hernández Verón |first2=M.Á. |title=Newton's method: An updated approach of Kantorovich's theory |publisher=Birkhäuser |date=2017 |isbn=978-3-319-55976-6 |url={{GBurl|A3orDwAAQBAJ|pg=PR11}}}}</ref><ref>{{cite book |first=Peter |last=Deuflhard |title=Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms |publisher=Springer |edition=2nd |series=Computational Mathematics |volume=35 |date=2006 |isbn=978-3-540-21099-3 |url={{GBurl|l20xK__HG_kC|pg=PP1}} }}</ref> [[Linearization]] is another technique for solving nonlinear equations.