View source for Iterative method

{{Short description|Algorithm in which each approximation of the solution is derived from prior approximations}}

In [[computational mathematics]], an '''iterative method''' is a [[Algorithm|mathematical procedure]] that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") is derived from the previous ones. 

A specific implementation with [[Algorithm#Termination|termination]] criteria for a given iterative method like [[gradient descent]], [[hill climbing]], [[Newton's method]], or [[Quasi-Newton method|quasi-Newton methods]] like [[Broyden–Fletcher–Goldfarb–Shanno algorithm|BFGS]], is an [[algorithm]] of an iterative method or a '''method of successive approximation'''. An iterative method is called ''[[Convergent series|convergent]]'' if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, [[heuristic]]-based iterative methods are also common.

In contrast, '''direct methods''' attempt to solve the problem by a finite sequence of operations. In the absence of [[rounding error]]s, direct methods would deliver an exact solution (for example, solving a linear system of equations <math>A\mathbf{x}=\mathbf{b}</math> by [[Gaussian elimination]]). Iterative methods are often the only choice for [[nonlinear equation]]s. However, iterative methods are often useful even for linear problems involving many variables (sometimes on the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power.<ref>{{Cite journal|doi=10.1016/j.jcp.2015.09.040|title=Recycling Krylov subspaces for CFD applications and a new hybrid recycling solver|year=2015|last1=Amritkar|first1=Amit|last2=de Sturler|first2=Eric|last3=Świrydowicz|first3=Katarzyna|last4=Tafti|first4=Danesh|last5=Ahuja|first5=Kapil|journal=Journal of Computational Physics|volume=303|page=222|arxiv=1501.03358|bibcode=2015JCoPh.303..222A}}</ref>

==Attractive fixed points==
If an equation can be put into the form ''f''(''x'') = ''x'', and a solution '''x''' is an attractive [[fixed point (mathematics)|fixed point]] of the function ''f'', then one may begin with a point ''x''<sub>1</sub> in the [[basin of attraction]] of '''x''', and let ''x''<sub>''n''+1</sub> = ''f''(''x''<sub>''n''</sub>) for ''n''&nbsp;≥&nbsp;1, and the sequence {''x''<sub>''n''</sub>}<sub>''n''&nbsp;≥&nbsp;1</sub> will converge to the solution '''x'''. Here ''x''<sub>''n''</sub> is the ''n''th approximation or iteration of ''x'' and ''x''<sub>''n''+1</sub> is the next or ''n'' + 1 iteration of ''x''.  Alternately, superscripts in parentheses are often used in numerical methods, so as not to interfere with subscripts with other meanings.  (For example, ''x''<sup>(''n''+1)</sup> = ''f''(''x''<sup>(''n'')</sup>).) If the function ''f'' is [[continuously differentiable]], a sufficient condition for convergence is that the [[spectral radius]] of the derivative is strictly bounded by one in a neighborhood of the fixed point.  If this condition holds at the fixed point, then a sufficiently small neighborhood (basin of attraction) must exist.

==Linear systems==
In the case of a [[system of linear equations]], the two main classes of iterative methods are the '''stationary iterative methods''', and the more general [[Krylov subspace]] methods.

===Stationary iterative methods===
==== Introduction ====
Stationary iterative methods solve a linear system with an [[Operator (mathematics)|operator]] approximating the original one; and based on a measurement of the error in the result ([[Residual (numerical analysis)|the residual]]), form a "correction equation" for which this process is repeated. While these methods are simple to derive, implement, and analyze, convergence is only guaranteed for a limited class of matrices.

====Definition====
An ''iterative method'' is defined by
:<math>
  \mathbf{x}^{k+1} := \Psi (  \mathbf{x}^k ), \quad k \geq 0
</math>
and for a given linear system <math> A\mathbf x= \mathbf b </math> with exact solution <math> \mathbf{x}^* </math> the ''error'' by
:<math>
  \mathbf{e}^k := \mathbf{x}^k - \mathbf{x}^*, \quad k \geq 0.
</math>
An iterative method is called ''linear'' if there exists a matrix <math> C \in \R^{n\times n} </math> such that
:<math>
  \mathbf{e}^{k+1} = C  \mathbf{e}^k   \quad \forall k \geq 0
</math>
and this matrix is called the ''iteration matrix''.
An iterative method with a given iteration matrix <math> C </math> is called ''convergent'' if the following holds
:<math>
  \lim_{k\rightarrow \infty} C^k=0.
</math>

An important theorem states that for a given iterative method and its iteration matrix <math> C </math> it is convergent if and only if its [[spectral radius]] <math> \rho(C) </math>  is smaller than unity, that is,
:<math>
  \rho(C) < 1.
</math>

The basic iterative methods work by [[Matrix splitting|splitting]] the matrix <math> A </math> into
:<math>
  A = M - N
</math>
and here the matrix <math> M </math> should be easily [[Invertible matrix|invertible]].
The iterative methods are now defined as
:<math>
  M \mathbf{x}^{k+1} = N \mathbf{x}^k + b, \quad k \geq 0,
</math>
or, equivalently,
:<math>
  \mathbf{x}^{k+1} = \mathbf{x}^k + M^{-1} (b - A \mathbf{x}^k), \quad k \geq 0.
</math>
From this follows that the iteration matrix is given by
:<math>
  C = I - M^{-1}A = M^{-1}N.
</math>

====Examples====

Basic examples of stationary iterative methods use a splitting of the matrix <math> A </math> such as
:<math>
  A = D+L+U\,,\quad D := \text{diag}( (a_{ii})_i)
</math>
where <math> D </math> is only the diagonal part of <math> A </math>, and <math> L </math> is the strict lower [[Triangular matrix|triangular part]] of <math> A </math>.
Respectively, <math> U </math> is the strict upper triangular part of <math> A </math>.
* [[Modified Richardson iteration|Richardson method]]: <math> M:=\frac{1}{\omega} I \quad (\omega \neq 0) </math>
* [[Jacobi method]]: <math> M:=D </math>
* [[Jacobi method#Weighted Jacobi method|Damped Jacobi method]]: <math> M:=\frac{1}{\omega}D \quad (\omega \neq 0) </math>
* [[Gauss–Seidel method]]: <math> M:=D+L </math>
* [[Successive over-relaxation|Successive over-relaxation method]] (SOR): <math> M:=\frac{1}{\omega}D+L \quad (\omega \neq 0) </math>
* [[Symmetric successive over-relaxation]] (SSOR): <math> M := \frac{1}{\omega (2-\omega)} (D+\omega L) D^{-1} (D+\omega U) 
\quad (\omega \not 
\in \{0,2\}) </math>
Linear stationary iterative methods are also called [[Relaxation (iterative method)|relaxation methods]].

===Krylov subspace methods===
Krylov subspace methods<ref>Charles George Broyden and Maria Terasa Vespucci: ''Krylov Solvers for Linear Algebraic Systems: Krylov Solvers'', Elsevier, ISBN 0-444-51474-0, (2004).</ref> work by forming a [[basis (linear algebra)|basis]] of the sequence of successive matrix powers times the initial residual (the '''Krylov sequence'''). 
The approximations to the solution are then formed by minimizing the residual over the subspace formed. 
The prototypical method in this class is the [[conjugate gradient method]] (CG) which assumes that the system matrix <math> A </math> is [[Symmetric matrix|symmetric]] [[Positive-definite matrix|positive-definite]].
For symmetric (and possibly indefinite) <math> A </math> one works with the [[minimal residual method]] (MINRES).
In the case of non-symmetric matrices, methods such as the [[generalized minimal residual method]] (GMRES) and the [[biconjugate gradient method]] (BiCG) have been derived.

====Convergence of Krylov subspace methods====
Since these methods form a basis, it is evident that the method converges in ''N'' iterations, where ''N'' is the system size. However, in the presence of rounding errors this statement does not hold; moreover, in practice ''N'' can be very large, and the iterative process reaches sufficient accuracy already far earlier. The analysis of these methods is hard, depending on a complicated function of the [[spectrum of an operator|spectrum]] of the operator.

===Preconditioners===
The approximating operator that appears in stationary iterative methods can also be incorporated in Krylov subspace methods such as [[GMRES]] (alternatively, [[preconditioning|preconditioned]] Krylov methods can be considered as accelerations of stationary iterative methods), where they become transformations of the original operator to a presumably better conditioned one. The construction of preconditioners is a large research area.

==Methods of successive approximation==
Mathematical methods relating to successive approximation include:
* [[Babylonian method]], for finding square roots of numbers<ref>{{Cite web |date=December 1, 2000 |title=Babylonian mathematics |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_mathematics/ |access-date= |website=Babylonian mathematics}}</ref>
* [[Fixed-point iteration]]<ref>{{Cite book |last=day |first=Mahlon |title=Fixed-point theorems for compact convex sets |date=November 2, 1960 |publisher=Mahlon M day |language=en}}</ref>
*  Means of finding zeros of functions:
** [[Halley's method]]
** [[Newton's method]]
* Differential-equation matters:
** [[Picard–Lindelöf theorem]], on existence of solutions of differential equations
** [[Runge–Kutta methods]], for numerical solution of differential equations

===History===
[[Jamshīd al-Kāshī]] used iterative methods to calculate the sine of  1° and {{pi}} in ''The Treatise of Chord and Sine'' to high precision. 
An early iterative method for [[Gauss–Seidel method|solving a linear system]] appeared in a letter of [[Carl Friedrich Gauss|Gauss]] to a student of his.  He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest {{citation needed|date=December 2019}}.

The theory of stationary iterative methods was solidly established with the work of [[D.M. Young]] starting in the 1950s. The conjugate gradient method was also invented in the 1950s, with independent developments by [[Cornelius Lanczos]], [[Magnus Hestenes]] and [[Eduard Stiefel]], but its nature and applicability were misunderstood at the time. Only in the 1970s was it realized that conjugacy based methods work very well for [[partial differential equation]]s, especially the elliptic type.

== See also ==
{{Portal|Mathematics}}
* [[Closed-form expression]]
* [[Iterative refinement]]
* [[Kaczmarz method]]
* [[Non-linear least squares]]
* [[Numerical analysis]]
* [[Root-finding algorithm]]

==References==
{{Reflist}}

==External links==
{{Commons category|Iterative methods}}
*[http://www.netlib.org/linalg/html_templates/Templates.html Templates for the Solution of Linear Systems]
*[http://www-users.cs.umn.edu/~saad/books.html Y. Saad: ''Iterative Methods for Sparse Linear Systems'', 1st edition, PWS 1996]

{{Optimization algorithms}}
{{Authority control}}

[[Category:Iterative methods| ]]
[[Category:Numerical analysis]]