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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 ...
Let k be an algebraic number field with ring of integers that contains a primitive n-th root of unity.. Let be a prime ideal and assume that n and are coprime (i.e. .). The norm of is defined as the cardinality of the residue class ring (note that since is prime the residue class ring is a finite field):
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.
Finding the root of a linear polynomial (degree one) is easy and needs only one division: the general equation + = has solution = /. For quadratic polynomials (degree two), the quadratic formula produces a solution, but its numerical evaluation may require some care for ensuring numerical stability .
Quadratic function#Upper bound on the magnitude of the roots; Real-root isolation – Methods for locating real roots of a polynomial; 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
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).
The original triple comprises the constant term in each of the respective quadratic equations. Below is a sample output from these equations. The effect of these equations is to cause the m -value in the Euclid equations to increment in steps of 4, while the n -value increments by 1.
The phenomenon is related to the Ulam spiral, which is also implicitly quadratic, and the class number; this polynomial is related to the Heegner number =. There are analogous polynomials for p = 2 , 3 , 5 , 11 and 17 {\displaystyle p=2,3,5,11{\text{ and }}17} (the lucky numbers of Euler ), corresponding to other Heegner numbers.