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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 ...
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.
The red curve shows the function f, and the blue lines are the secants. For this particular case, the secant method will not converge to the visible root. 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.
Root-finding of polynomials – Algorithms for finding zeros of polynomials; Square-free polynomial – Polynomial with no repeated root; Vieta's formulas – Relating coefficients and roots of a polynomial; Cohn's theorem relating the roots of a self-inversive polynomial with the roots of the reciprocal polynomial of its derivative.
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.
Let X be a Riemann surface.Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function :), we can associate a differential form of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X:
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.
Intuitively, the multiplicity of a common zero of several polynomials is the number of zeros into which the common zero can split when the coefficients are slightly changed. For example, a tangent to a curve is a line that cuts the curve at a point that splits in several points if the line is slightly moved.