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 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 termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called 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 errors, direct methods would deliver an exact solution (for example, solving a linear system of equations A𝐱=𝐛 by Gaussian elimination). Iterative methods are often the only choice for nonlinear equations. 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.[1]

## Contents

* 1 Attractive fixed points
* 2 Linear systems
* 2.1 Stationary iterative methods
* 2.1.1 Introduction
* 2.1.2 Definition
* 2.1.3 Examples
* 2.2 Krylov subspace methods
* 2.2.1 Convergence of Krylov subspace methods
* 2.3 Preconditioners
* 3 Methods of successive approximation
* 3.1 History
* 4 See also
* 5 References
* 6 External links

## Attractive fixed points

If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x1 in the basin of attraction of x, and let xn+1 = f(xn) for n ≥ 1, and the sequence {xn}n ≥ 1 will converge to the solution x. Here xn is the nth approximation or iteration of x and xn+1 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(n+1) = f(x(n)).) 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 approximating the original one; and based on a measurement of the error in the result (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 𝐱k+1:=Ψ(𝐱k),k≥0 and for a given linear system A𝐱=𝐛 with exact solution 𝐱* the error by 𝐞k:=𝐱k−𝐱*,k≥0. An iterative method is called linear if there exists a matrix C∈ℝn×n such that 𝐞k+1=C𝐞k∀k≥0 and this matrix is called the iteration matrix. An iterative method with a given iteration matrix C is called convergent if the following holds limk->∞Ck=0.

An important theorem states that for a given iterative method and its iteration matrix C it is convergent if and only if its spectral radius ρ(C) is smaller than unity, that is, ρ(C)<1.

The basic iterative methods work by splitting the matrix A into A=M−N and here the matrix M should be easily invertible. The iterative methods are now defined as M𝐱k+1=N𝐱k+𝐛,k≥0, or, equivalently, 𝐱k+1=𝐱k+M−1(𝐛−A𝐱k),k≥0. From this follows that the iteration matrix is given by C=I−M−1A=M−1N.

#### Examples

Basic examples of stationary iterative methods use a splitting of the matrix A such as A=D+L+U,D:=diag⁡((aii)i) where D is only the diagonal part of A, and L is the strict lower triangular part of A. Respectively, U is the strict upper triangular part of A.

* Richardson method: M:=1ωI(ω≠0)
* Jacobi method: M:=D
* Damped Jacobi method: M:=1ωD(ω≠0)
* Gauss–Seidel method: M:=D+L
* Successive over-relaxation method (SOR): M:=1ωD+L(ω≠0)
* Symmetric successive over-relaxation (SSOR): M:=1ω(2−ω)(D+ωL)D−1(D+ωU)(ω∉{0,2})

Linear stationary iterative methods are also called relaxation methods.

### Krylov subspace methods

Krylov subspace methods[2] work by forming a 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 A is symmetric positive-definite. For symmetric (and possibly indefinite) A 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 the operator.

### Preconditioners

The approximating operator that appears in stationary iterative methods can also be incorporated in Krylov subspace methods such as GMRES (alternatively, 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[3]
* Fixed-point iteration[4]
* 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 π in The Treatise of Chord and Sine to high precision. An early iterative method for solving a linear system appeared in a letter of 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 .

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 equations, especially the elliptic type.

## See also

* Closed-form expression
* Iterative refinement
* Kaczmarz method
* Non-linear least squares
* Numerical analysis
* Root-finding algorithm

## References

1. ↑ Amritkar, Amit; de Sturler, Eric; Świrydowicz, Katarzyna; Tafti, Danesh; Ahuja, Kapil (2015). "Recycling Krylov subspaces for CFD applications and a new hybrid recycling solver". Journal of Computational Physics 303: 222. doi:10.1016/j.jcp.2015.09.040. Bibcode: 2015JCoPh.303..222A.
2. ↑ Charles George Broyden and Maria Terasa Vespucci: Krylov Solvers for Linear Algebraic Systems: Krylov Solvers, Elsevier, ISBN 0-444-51474-0, (2004).
3. ↑ "Babylonian mathematics". December 1, 2000. https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_mathematics/.
4. ↑ day, Mahlon (November 2, 1960) (in en). Fixed-point theorems for compact convex sets. Mahlon M day.

## External links

* Templates for the Solution of Linear Systems
* Y. Saad: Iterative Methods for Sparse Linear Systems, 1st edition, PWS 1996

* v
* t
* e

Optimization: Algorithms, methods, and heuristics

Unconstrained nonlinear

| … functions|

* Golden-section search
* Interpolation methods
* Line search
* Nelder–Mead method
* Successive parabolic interpolation

|

… and gradients|

| Convergence|

* Trust region
* Wolfe conditions

|

Quasi–Newton|

* Berndt–Hall–Hall–Hausman
* Broyden–Fletcher–Goldfarb–Shanno and L-BFGS
* Davidon–Fletcher–Powell
* Symmetric rank-one (SR1)

Other methods|

* Gauss–Newton
* Gradient
* Levenberg–Marquardt
* Conjugate gradient
* Truncated Newton

… and Hessians|

* Newton's method

Optimization computes maxima and minima.

Constrained nonlinear

| General|

* Barrier methods
* Penalty methods

|

Differentiable|

* Augmented Lagrangian methods
* Sequential quadratic programming
* Successive linear programming

Convex optimization

| Convex
minimization|

* Cutting-plane method
* Reduced gradient (Frank–Wolfe)
* Subgradient method

|

Linear and
quadratic|

| Interior point|

* Affine scaling
* Ellipsoid algorithm of Khachiyan
* Projective algorithm of Karmarkar

|

Basis-exchange|

* Simplex algorithm of Dantzig
* Revised simplex algorithm
* Criss-cross algorithm
* Principal pivoting algorithm of Lemke

Combinatorial

| Paradigms|

* Approximation algorithm
* Dynamic programming
* Greedy algorithm
* Integer programming
* Branch and bound/cut

|

Graph
algorithms|

| Minimum
spanning tree|

* Borůvka
* Prim
* Kruskal

|

* Bellman–Ford
* Dijkstra
* Floyd–Warshall

Network flows|

* Dinic
* Edmonds–Karp
* Ford–Fulkerson
* Push–relabel maximum flow

Metaheuristics

* Evolutionary algorithm
* Hill climbing
* Local search
* Simulated annealing
* Tabu search

* Software

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