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The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. [3]
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 numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method , so it is considered a quasi-Newton method .
An important application is Newton–Raphson division, which can be used to quickly find the reciprocal of a number a, using only multiplication and subtraction, that is to say the number x such that 1 / x = a. We can rephrase that as finding the zero of f(x) = 1 / x − a. We have f ′ (x) = − 1 / x 2 . Newton's ...
One advantage of this proof over the others is that it shows not only that a polynomial must have a zero but the number of its zeros is equal to its degree (counting, as usual, multiplicity). Another use of Rouché's theorem is to prove the open mapping theorem for analytic functions. We refer to the article for the proof.
We can also define the multiplicity of the zeroes and poles of a meromorphic function. If we have a meromorphic function =, take the Taylor expansions of g and h about a point z 0, and find the first non-zero term in each (denote the order of the terms m and n respectively) then if m = n, then the point has non-zero value.
Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along Re(z) = 1/2. In a neighbourhood of a point z 0 , {\displaystyle z_{0},} a nonzero meromorphic function f is the sum of a Laurent series with at most finite principal part (the terms with negative index ...
Most root-finding algorithms can find some real roots, but cannot certify having found all the roots. Methods for finding all complex roots, such as Aberth method can provide the real roots. However, because of the numerical instability of polynomials (see Wilkinson's polynomial ), they may need arbitrary-precision arithmetic for deciding which ...