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Proof of quadratic convergence for Newton's iterative method [ edit ] According to Taylor's theorem , any function f ( x ) which has a continuous second derivative can be represented by an expansion about a point that is close to a root of f ( x ) .
Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus , Newton's method (also called Newton–Raphson ) is an iterative method for finding the roots of a differentiable function f {\displaystyle f} , which are solutions to the equation f ( x ) = 0 {\displaystyle f(x)=0} .
The Kantorovich theorem, or Newton–Kantorovich theorem, is a mathematical statement on the semi-local convergence of Newton's method. It was first stated by Leonid Kantorovich in 1948. [1] [2] It is similar to the form of the Banach fixed-point theorem, although it states existence and uniqueness of a zero rather than a fixed point. [3]
In what follows, the Gauss–Newton algorithm will be derived from Newton's method for function optimization via an approximation. As a consequence, the rate of convergence of the Gauss–Newton algorithm can be quadratic under certain regularity conditions. In general (under weaker conditions), the convergence rate is linear. [9]
Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity. There are many types of sequences and modes of convergence , and different proof techniques may be more appropriate than others for proving each type of convergence of each type ...
Newton's method, also known as Newton's iteration, is an iterative method which uses the tangent lines of a single-variable function to iteratively define a sequence of numbers. Although there are exceptional cases, often this sequence is infinitely iterable and converges to a zero of the function .
In asymptotic analysis in general, one sequence () that converges to a limit is said to asymptotically converge to with a faster order of convergence than another sequence () that converges to in a shared metric space with distance metric | |, such as the real numbers or complex numbers with the ordinary absolute difference metrics, if
Scoring algorithm, also known as Fisher's scoring, [1] is a form of Newton's method used in statistics to solve maximum likelihood equations numerically, named after Ronald Fisher. Sketch of derivation