# Discriminant

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## discriminant

[di′skrim·ə·nənt]
(mathematics)
The quantity b 2- 4 ac, where a,b,c are coefficients of a given quadratic polynomial: ax 2+ bx + c.
More generally, for the polynomial equation a0 xn + a1 xn -1+···+ anx0= 0, a02 n-2times the product of the squares of all the differences of the roots of the equation, taken in pairs.

## Discriminant

The discriminant of a polynomial

P(x) = a0xn + a1xn−1 + … + an

is the expression

in which the product is distributed over all possible differences of the roots α1, β2, … , αn of the equation P (x) = 0. The discriminant vanishes if and only if there are equal roots among the roots of the polynomial. The discriminant can be expressed through the coefficients of the polynomial P(x) by representing it in the form of a determinant consisting of these coefficients. Thus, for the second-degree polynomial ax2 + bx + c, the discriminant is b2 − 4ac. For x3 + px + q, the discriminant is −4p3 −27q2. The discriminant differs only by a factor a0 from the resultant R(P, P′) of the polynomial P(x) and its derivative P′(x).

References in periodicals archive ?
with the module [DELTA] = [alpha][delta] - [beta][gamma], we get another quadratic form f', with its discriminant being D'.
When patterns are numeric, discriminants are based on one of the following concepts:
Solutions found by means of the use of neural networks allow to obtain discriminants of the first type.
The usual practice is to look for a pattern recognition system which is able to learn in an adaptive way from the experiences of several discriminants, each of them corresponding to a specific purpose.
Within the pattern recognition area, one of the most important concepts is that of discriminant.
In general, a discriminant is a function or operator that, when applied to a pattern, allows to obtain an output corresponding to an estimation of the class to which it belongs, or an estimation of one or more of the attributes of the pattern.
The discriminant is a surface dividing the space of input data from where patterns are classified according to the sector to which they belong.
The discriminant is a distance measurement, and patterns are classified according to the class to which their closest neighbor [Simp 92], [Simp 93], or the closest prototype [ Torb 98], or the closest class center [Mega 98], [Shie 95] belong.

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