Difference between revisions of "Newton's method"
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In the [[Mathematics|mathematical field]] of [[numerical analysis]], '''Newton's method''' (also known as the '''Newton–Raphson method'''), named after [[Isaac Newton]] and [[Joseph Raphson]], is a method for finding successively better approximations to the [[root of a function|roots]] (or zeroes) of a [[Real number|real]]-valued [[function (mathematics)|function]]. | In the [[Mathematics|mathematical field]] of [[numerical analysis]], '''Newton's method''' (also known as the '''Newton–Raphson method'''), named after [[Isaac Newton]] and [[Joseph Raphson]], is a method for finding successively better approximations to the [[root of a function|roots]] (or zeroes) of a [[Real number|real]]-valued [[function (mathematics)|function]]. | ||
| + | |||
| + | == Description == | ||
| + | |||
| + | The idea of the method is as follows: | ||
| + | |||
| + | One starts with an initial guess which is reasonably close to the true root | ||
| + | |||
| + | Then the function is approximated by its [[Tangent|tangent line]] (which can be computed using the tools of [[calculus]]), and one computes the x-intercept of this tangent line (which is easily done with elementary algebra). | ||
| + | |||
| + | This x-intercept will typically be a better approximation to the function's root than the original guess, and the method can be [[Iterative method|iterated]]. | ||
== See also == | == See also == | ||
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* [[Gradient descent]] | * [[Gradient descent]] | ||
* [[Integer square root]] | * [[Integer square root]] | ||
| + | * [[Iterative method]] | ||
* [[Laguerre's method]] | * [[Laguerre's method]] | ||
* [[Leonid Kantorovich]], who initiated the convergence analysis of Newton's method in Banach spaces. | * [[Leonid Kantorovich]], who initiated the convergence analysis of Newton's method in Banach spaces. | ||
Latest revision as of 20:44, 24 August 2016
In the mathematical field of numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
Description
The idea of the method is as follows:
One starts with an initial guess which is reasonably close to the true root
Then the function is approximated by its tangent line (which can be computed using the tools of calculus), and one computes the x-intercept of this tangent line (which is easily done with elementary algebra).
This x-intercept will typically be a better approximation to the function's root than the original guess, and the method can be iterated.
See also
- Aitken's delta-squared process
- Approximation
- Bisection method
- Euler method
- Fast inverse square root
- Fisher scoring
- Function (mathematics)
- Gradient descent
- Integer square root
- Iterative method
- Laguerre's method
- Leonid Kantorovich, who initiated the convergence analysis of Newton's method in Banach spaces.
- Methods of computing square roots
- Newton, Isaac
- Newton's method in optimization
- Numerical analysis
- Raphson, Joseph
- Real number
- Richardson extrapolation
- Root of a function
- Root-finding algorithm
- Secant method
- Steffensen's method
- Subgradient method
External links
- Newton's method @ Wikipedia
