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Descartes’ rule of signs (for polynomials)

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1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
31.5.2 𝐻𝑝 n , m ( a , q n , m ; n , β , γ , δ ; z ) = H ( a , q n , m ; n , β , γ , δ ; z )
is a polynomial of degree n , and hence a solution of (31.2.1) that is analytic at all three finite singularities 0 , 1 , a . These solutions are the Heun polynomials. …
2: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
Normalization
Orthogonal Invariance
Summation
Mean-Value
3: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
Euler Numbers and Polynomials
The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
4: 18.3 Definitions
§18.3 Definitions
For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …For explicit power series coefficients up to n = 12 for these polynomials and for coefficients up to n = 6 for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …
Bessel polynomials
Bessel polynomials are often included among the classical OP’s. …
5: 1.11 Zeros of Polynomials
Every monic (coefficient of highest power is one) polynomial of odd degree with real coefficients has at least one real zero with sign opposite to that of the constant term. A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative.
DescartesRule of Signs
The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity. … Both polynomials have one change of sign; hence for each polynomial there is one positive zero, one negative zero, and six complex zeros. …
6: 3.8 Nonlinear Equations
§3.8(ii) Newton’s Rule
Newton’s rule is given by … Another iterative method is Halley’s rule: …The rule converges locally and is cubically convergent. …
7: 3.5 Quadrature
§3.5(i) Trapezoidal Rules
The composite trapezoidal rule is …
§3.5(ii) Simpson’s Rule
§3.5(iv) Interpolatory Quadrature Rules
For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. …
8: 29.20 Methods of Computation
A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. … A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. …
§29.20(ii) Lamé Polynomials
The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …
§29.20(iii) Zeros
9: 18.40 Methods of Computation
§18.40(i) Computation of Polynomials
Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). …
Derivative Rule Approach
An alternate, and highly efficient, approach follows from the derivative rule conjecture, see Yamani and Reinhardt (1975), and references therein, namely that … Further, exponential convergence in N , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate w ( x ) for these OP systems on x [ 1 , 1 ] and ( , ) respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …
10: 3.11 Approximation Techniques
§3.11(i) Minimax Polynomial Approximations
A sufficient condition for p n ( x ) to be the minimax polynomial is that | ϵ n ( x ) | attains its maximum at n + 2 distinct points in [ a , b ] and ϵ n ( x ) changes sign at these consecutive maxima. … The Padé approximants can be computed by Wynn’s cross rule. Any five approximants arranged in the Padé table as … If n = 2 m , then 𝛀 can be factored into m matrices, the rows of which contain only a few nonzero entries and the nonzero entries are equal apart from signs. …