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wasi-uz-zaman
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- TL;DR Summary
- picard iteration is also known as fixed point iteration
hi, i want to figure out why it is named as fixed point iteration.
Fixed point iteration is a method used in mathematics to find the solution to a problem by repeatedly applying a function to an initial guess until a fixed point is reached. The fixed point is a value that does not change when the function is applied, and it represents the solution to the problem.
Picard iteration is a specific type of fixed point iteration, where the function used is the Picard operator. This operator is defined as the composition of a given function and a constant function. In other words, Picard iteration is a special case of fixed point iteration that uses a specific type of function.
Charles Émile Picard was a French mathematician who made significant contributions to the field of analysis, particularly in the study of differential equations. He developed the concept of Picard iteration in the late 19th century, and it was named after him as a tribute to his work.
Picard iteration is commonly used to solve differential equations, particularly those that cannot be solved analytically. It is also used in numerical analysis and optimization problems. Additionally, it has applications in physics, engineering, and economics.
Like any other numerical method, Picard iteration has its limitations. It may not always converge to the correct solution, and the convergence rate can be slow for certain problems. Additionally, it may not work for functions that do not satisfy certain conditions, such as being continuously differentiable.