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Finding the real roots of a polynomial with real coefficients is a problem that has received much attention since the beginning of 19th century, and is still an active domain of research. Most root-finding algorithms can find some real roots, but cannot certify having found all the roots.
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.
It follows that the roots of a polynomial with real coefficients are mirror-symmetric with respect to the real axis. This can be extended to algebraic conjugation: the roots of a polynomial with rational coefficients are conjugate (that is, invariant) under the action of the Galois group of the polynomial. However, this symmetry can rarely be ...
Finding roots of 3x 2 +5x−2. Lill's method can be used with Thales's theorem to find the real roots of a quadratic polynomial. In this example with 3x 2 +5x−2, the polynomial's line segments are first drawn in black, as above. A circle is drawn with the straight line segment joining the start and end points forming a diameter.
If x is a simple root of the polynomial , then Laguerre's method converges cubically whenever the initial guess, , is close enough to the root . On the other hand, when x 1 {\displaystyle \ x_{1}\ } is a multiple root convergence is merely linear, with the penalty of calculating values for the polynomial and its first and second derivatives at ...
Graeffe's method works best for polynomials with simple real roots, though it can be adapted for polynomials with complex roots and coefficients, and roots with higher multiplicity. For instance, it has been observed [ 2 ] that for a root x ℓ + 1 = x ℓ + 2 = ⋯ = x ℓ + d {\displaystyle x_{\ell +1}=x_{\ell +2}=\dots =x_{\ell +d}} with ...
For polynomials, more elaborate methods exist for testing the existence of a root in an interval (Descartes' rule of signs, Sturm's theorem, Budan's theorem). They allow extending the bisection method into efficient algorithms for finding all real roots of a polynomial; see Real-root isolation.
Bairstow's approach is to use Newton's method to adjust the coefficients u and v in the quadratic + + until its roots are also roots of the polynomial being solved. The roots of the quadratic may then be determined, and the polynomial may be divided by the quadratic to eliminate those roots.