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1: 1.11 Zeros of Polynomials
The discriminant of f ( z ) is defined by
1.11.9 D = a n 2 n - 2 j < k ( z j - z k ) 2 ,
The discriminant of g ( w ) is
1.11.12 D = - 4 p 3 - 27 q 2 .
The discriminant of g ( w ) is …
2: 27.14 Unrestricted Partitions
§27.14(vi) Ramanujan’s Tau Function
The discriminant function Δ ( τ ) is defined by
27.14.16 Δ ( τ ) = ( 2 π ) 12 ( η ( τ ) ) 24 , τ > 0 ,
27.14.17 Δ ( a τ + b c τ + d ) = ( c τ + d ) 12 Δ ( τ ) ,
3: 23.19 Interrelations
4: 23.3 Differential Equations
§23.3(i) Invariants, Roots, and Discriminant
The discriminant1.11(ii)) is given by
23.3.4 Δ = g 2 3 - 27 g 3 2 = 16 ( e 2 - e 3 ) 2 ( e 3 - e 1 ) 2 ( e 1 - e 2 ) 2 .
5: 28.29 Definitions and Basic Properties
§28.29(iii) Discriminant and Eigenvalues in the Real Case
is called the discriminant of (28.29.1). … For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of Q ( x ) ; see Magnus and Winkler (1966, §2.3, pp. 28–36). …
6: 23.1 Special Notation
𝕃

lattice in .

Δ

discriminant g 2 3 - 27 g 3 2 .

7: 23.22 Methods of Computation
  • (a)

    In the general case, given by c d 0 , we compute the roots α , β , γ , say, of the cubic equation 4 t 3 - c t - d = 0 ; see §1.11(iii). These roots are necessarily distinct and represent e 1 , e 2 , e 3 in some order.

    If c and d are real, and the discriminant is positive, that is c 3 - 27 d 2 > 0 , then e 1 , e 2 , e 3 can be identified via (23.5.1), and k 2 , k 2 obtained from (23.6.16).

    If c 3 - 27 d 2 < 0 , or c and d are not both real, then we label α , β , γ so that the triangle with vertices α , β , γ is positively oriented and [ α , γ ] is its longest side (chosen arbitrarily if there is more than one). In particular, if α , β , γ are collinear, then we label them so that β is on the line segment ( α , γ ) . In consequence, k 2 = ( β - γ ) / ( α - γ ) , k 2 = ( α - β ) / ( α - γ ) satisfy k 2 0 k 2 (with strict inequality unless α , β , γ are collinear); also | k 2 | , | k 2 | 1 .

    Finally, on taking the principal square roots of k 2 and k 2 we obtain values for k and k that lie in the 1st and 4th quadrants, respectively, and 2 ω 1 , 2 ω 3 are given by

    23.22.1 2 ω 1 M ( 1 , k ) = - 2 i ω 3 M ( 1 , k ) = π 3 c ( 2 + k 2 k 2 ) ( k 2 - k 2 ) d ( 1 - k 2 k 2 ) ,

    where M denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ( 2 ω 1 , 2 ω 3 ), corresponding to the 2 possible choices of the square root.

  • 8: Bibliography M
  • H. R. McFarland and D. St. P. Richards (2001) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.
  • H. R. McFarland and D. St. P. Richards (2002) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.