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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} .
An illustration of Newton's method. In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.
The backward Euler method is an implicit method, meaning that we have to solve an equation to find y n+1. One often uses fixed-point iteration or (some modification of) the Newton–Raphson method to achieve this.
Newton's method assumes the function f to have a continuous derivative. Newton's method may not converge if started too far away from a root. However, when it does converge, it is faster than the bisection method; its order of convergence is usually quadratic whereas the bisection method's is linear. Newton's method is also important because it ...
Newton–Raphson uses Newton's method to find the reciprocal of and multiply that reciprocal by to find the final quotient . The steps of Newton–Raphson division are: Calculate an estimate X 0 {\displaystyle X_{0}} for the reciprocal 1 / D {\displaystyle 1/D} of the divisor D {\displaystyle D} .
The line-search method first finds a descent direction along which the objective function will be reduced, and then computes a step size that determines how far should move along that direction. The descent direction can be computed by various methods, such as gradient descent or quasi-Newton method. The step size can be determined either ...
Fast-decoupled-load-flow method is a variation on Newton–Raphson that exploits the approximate decoupling of active and reactive flows in well-behaved power networks, and additionally fixes the value of the Jacobian during the iteration in order to avoid costly matrix decompositions. Also referred to as "fixed-slope, decoupled NR".
Algorithms based on the Newton–Raphson method are able to compute quadrature rules for significantly larger problem sizes. In 2014, Ignace Bogaert presented explicit asymptotic formulas for the Gauss–Legendre quadrature weights and nodes, which are accurate to within double-precision machine epsilon for any choice of n ≥ 21. [ 2 ]