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Finding one root; Finding all roots; 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 ...
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.
In this case a and b are said to bracket a root since, by the intermediate value theorem, the continuous function f must have at least one root in the interval (a, b). At each step the method divides the interval in two parts/halves by computing the midpoint c = (a+b) / 2 of the interval and the value of the function f(c) at that point.
For finding real roots of a polynomial, the common strategy is to divide the real line (or an interval of it where root are searched) into disjoint intervals until having at most one root in each interval. Such a procedure is called root isolation, and a resulting interval that contains exactly one root is an isolating interval for this root.
For defining this starting interval, one may use bounds on the size of the roots (see Properties of polynomial roots § Bounds on (complex) polynomial roots). Then, one divides this interval in two, by choosing c in the middle of ( a , b ] . {\displaystyle (a,b].}
Laguerre's method may even converge to a complex root of the polynomial, because the radicand of the square root may be of a negative number, in the formula for the correction, , given above – manageable so long as complex numbers can be conveniently accommodated for the calculation. This may be considered an advantage or a liability ...
In this case cut it in half and consider separately the roots of p(x) inside the interval (0, 1 / 2 )—and that correspond to the roots of p orig (x) inside the interval (a, 1 / 2 (a + b)) from those inside the interval ( 1 / 2 , 1) and correspond to the roots of p orig (x) inside the interval ( 1 / 2 (a + b), b ...
Given a univariate polynomial p(x) with real coefficients, let us denote by # (ℓ,r] (p) the number of real roots, counted with their multiplicities, [1] of p in a half-open interval (ℓ, r] (with ℓ < r real numbers). Let us denote also by v h (p) the number of sign variations in the sequence of the coefficients of the polynomial p h (x ...