Iterative refinement is a technique in which a solution is approached through a sequence of approximations instead of being computed in one closed step. A method starts from an initial approximation and applies its update rule again and again, so that each pass produces a new estimate nearer to the answer. The Digital Library of Mathematical Functions of the National Institute of Standards and Technology says that equations of this kind are usually solved by iterative methods, in which a sequence of approximations to a root converges to the solution, and it classes the speed of that convergence as linear, quadratic or cubic. In this atlas the term marks algorithms such as root finders, relaxation solvers for linear systems and eigenvalue iterations, whose accuracy improves pass by pass.
Connections
Technique Used By
Source Netlib, Templates for the Solution of Linear Systems: Implementation (BiConjugate Gradient)
Source Biconjugate gradient stabilized method (Wikipedia)
Source Wolfram MathWorld: Bisection
Source Data-flow analysis (Wikipedia)
Source Wolfram MathWorld: Gauss-Seidel Method
Source Householder's method (Wikipedia)
Source Iterative proportional fitting (Wikipedia)
Source Wolfram MathWorld: Jacobi Method
Source Kaczmarz method (Wikipedia)
Source Wolfram MathWorld: Newton's Method
Source Wolfram MathWorld: Power Method
Source Square root algorithms (Wikipedia)
Source Wolfram MathWorld: Successive Overrelaxation Method
Sources
DLMF Section 3.8 Nonlinear Equations
Square root algorithms (Wikipedia)
Technique Used By: Square root algorithms, Lead section [design-technique-used]Quote, Technique Used By: Square root algorithms, Lead section [design-technique-used]
an iterative refinement is performed until some termination criterion is met
Data-flow analysis (Wikipedia)
Technique Used By: Data-flow analysis, Lead sectionQuote, Technique Used By: Data-flow analysis, Lead section
solve them by repeatedly calculating the output from the input locally at each node until the whole system stabilizes, i.e., it reaches a fixpoint
Kaczmarz method (Wikipedia)
Technique Used By: Kaczmarz method, Lead paragraphQuote, Technique Used By: Kaczmarz method, Lead paragraph
is an iterative algorithm for solving linear equation systems
Netlib, Templates for the Solution of Linear Systems: Implementation (BiConjugate Gradient)
Netlib, Templates for the Solution of Linear SystemsTechnique Used By: Biconjugate gradient method, ImplementationQuote, Technique Used By: Biconjugate gradient method, Implementation
vector products per iteration step.
Householder's method (Wikipedia)
Technique Used By: Householder's method, Lead [design-technique-used]Quote, Technique Used By: Householder's method, Lead [design-technique-used]
The algorithm is iterative and has an order of convergence of d + 1.
Iterative proportional fitting (Wikipedia)
Technique Used By: Iterative proportional fitting, Lead [design-technique-used]Quote, Technique Used By: Iterative proportional fitting, Lead [design-technique-used]
so these steps are repeated in cycles, re-adjusting the rows and columns in turn
Biconjugate gradient stabilized method (Wikipedia)
Technique Used By: Biconjugate gradient stabilized method, Lead paragraph [design-technique-used]Quote, Technique Used By: Biconjugate gradient stabilized method, Lead paragraph [design-technique-used]
is an iterative method developed by H. A. van der Vorst
Wolfram MathWorld: Newton's Method
Wolfram MathWorldTechnique Used By: Newton-Raphson Method, MathWorld, article text (technique relation)Quote, Technique Used By: Newton-Raphson Method, MathWorld, article text (technique relation)
the algorithm can be applied iteratively to obtain
Wolfram MathWorld: Jacobi Method
Wolfram MathWorldTechnique Used By: Jacobi method, MathWorld, article text (technique relation)Quote, Technique Used By: Jacobi method, MathWorld, article text (technique relation)
The process is then iterated until it converges.
Wolfram MathWorld: Gauss-Seidel Method
Wolfram MathWorldTechnique Used By: Gauss-Seidel Method, MathWorld, article text (technique relation)Quote, Technique Used By: Gauss-Seidel Method, MathWorld, article text (technique relation)
Since each component of the new iterate depends upon all previously computed components, the updates cannot be done simultaneously as in the Jacobi method
Wolfram MathWorld: Successive Overrelaxation Method
Wolfram MathWorldTechnique Used By: Successive Over-Relaxation Algorithm, MathWorld, article text (technique relation)Quote, Technique Used By: Successive Over-Relaxation Algorithm, MathWorld, article text (technique relation)
choose a value for that will accelerate the rate of convergence of the iterates to the solution.
Wolfram MathWorld: Power Method
Wolfram MathWorldTechnique Used By: Power iteration, MathWorld, article text (technique relation)Quote, Technique Used By: Power iteration, MathWorld, article text (technique relation)
is an iterative procedure for approximating a dominant eigenvalue and corresponding eigenvector of a square real matrix
Wolfram MathWorld: Bisection
Wolfram MathWorldTechnique Used By: Bisection Method Algorithm, MathWorld, article text (technique relation)Quote, Technique Used By: Bisection Method Algorithm, MathWorld, article text (technique relation)
Bisection is a root bracketing procedure for iteratively converging on a solution which is known to lie inside some interval
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