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Halley's method. In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. Edmond Halley was an English mathematician and astronomer who introduced the method now called by his name. The algorithm is second in the class of Householder's methods, after Newton's ...
Cube root. In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 23 = 8, while the other ...
Fixed-point iteration. In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is. which gives rise to the sequence of iterated function applications which is ...
Use of Newton's method to compute square roots. Newton's method is one of many known methods of computing square roots. Given a positive number a, the problem of finding a number x such that x2 = a is equivalent to finding a root of the function f(x) = x2 − a. The Newton iteration defined by this function is given by.
The other roots of the equation are obtained either by changing of cube root or, equivalently, by multiplying the cube root by a primitive cube root of unity, that is . This formula for the roots is always correct except when p = q = 0 , with the proviso that if p = 0 , the square root is chosen so that C ≠ 0 .
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater ...
Tetration is iterated exponentiation (call this right-associative operation ^), starting from the top right side of the expression with an instance a^a (call this value c). Exponentiating the next leftward a (call this the 'next base' b), is to work leftward after obtaining the new value b^c. Working to the left, consume the next a to the left ...
For finding all the roots, arguably the most reliable method is the Francis QR algorithm computing the eigenvalues of the Companion matrix corresponding to the polynomial, implemented as the standard method [1] in MATLAB. The oldest method of finding all roots is to start by finding a single root. When a root r has been found, it can be removed ...