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In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f is a number x such that f ( x ) = 0 . As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form , root-finding algorithms provide approximations to zeros.
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.
Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting). For finding one root, Newton's method and other general iterative methods work generally well.
The secant method is an iterative numerical method for finding a zero of a function f.Given two initial values x 0 and x 1, the method proceeds according to the recurrence relation
The input for the method is a continuous function f, an interval [a, b], and the function values f(a) and f(b). The function values are of opposite sign (there is at least one zero crossing within the interval). Each iteration performs these steps: Calculate c, the midpoint of the interval, c = a + b / 2 .
A linear fractional transformation of the variable makes it possible to use the rule of signs to count roots in any interval. This is the basic idea of Budan's theorem and the Budan–Fourier theorem. Repeated division of an interval in two results in a set of disjoint intervals, each containing one root, and together listing all the roots.
The bisection method consists roughly of starting from an interval containing all real roots of a polynomial, and divides it recursively into two parts until getting eventually intervals that contain either zero or one root. The starting interval may be of the form (-B, B), where B is an upper bound on the absolute values of the roots, such as ...
This yields 1.15448, which is not in the interval between (3a 3 + b 3) / 4 and b 3). Hence, it is replaced by the midpoint m = −2.71449. We have f(m) = 3.93934, so we set a 4 = a 3 and b 4 = −2.71449. In the fifth iteration, inverse quadratic interpolation yields −3.45500, which lies in the required interval.