ITERATIVE METHODS Meaning and
Definition
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Iterative methods refer to a class of mathematical techniques or algorithms used to approximate or solve complex problems by repeatedly refining a solution until it reaches a desired level of accuracy or convergence. These methods iteratively improve an initial guess or estimate, approaching the optimal solution through an incremental process. They are particularly useful when analytical or exact solutions are difficult or impossible to obtain.
In the context of numerical computing, iterative methods are commonly employed to solve linear systems of equations, optimization problems, or to find eigenvalues and eigenvectors. Rather than solving these problems directly, which can be computationally expensive or memory-intensive, iterative methods divide the problem into smaller subproblems and solve them step by step. In each iteration, an improved estimate of the solution is generated based on the previous approximation, and the process is repeated until a satisfactory result is achieved.
Iterative methods often involve iterative sequences, iterative algorithms, or iterative procedures, and can be implemented using various mathematical techniques such as fixed-point iteration, Newton's method, Jacobi method, Gauss-Seidel method, conjugate gradient method, and many others. Each of these methods has its own strengths and limitations, and the choice of which method to use depends on the nature of the problem and the desired level of accuracy.
Overall, iterative methods are powerful tools in mathematical and computational sciences, allowing researchers and engineers to efficiently solve complex problems by iteratively refining their initial estimates.
Etymology of ITERATIVE METHODS
The word "iterative" comes from the Latin word "iterare", which means "to repeat". It is formed from the root "iter", meaning "journey" or "way". An iterative process involves repeating a series of steps or actions in order to achieve a desired outcome.
The term "iterative methods" in mathematics and computer science refers to techniques or algorithms that solve problems through repeated iterations or refinements. These methods aim to approximate a solution by successively improving an initial estimate until a desired level of accuracy or convergence is reached.
