Descartes’ rule of signs

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11: 7.22 Methods of Computation

… ►Additional references are Matta and Reichel (1971) for the application of the trapezoidal rule, for example, to the first of (7.7.2), and Gautschi (1970) and Cuyt et al. (2008) for continued fractions. …

12: 18.40 Methods of Computation

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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 … ►

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Figure 18.40.2: Derivative Rule inversions for $w^{\mathrm{RCP}}(x)$ carried out via Lagrange and PWCF interpolations. …For the derivative rule Lagrange interpolation (red points) gives $\sim 15$ digits in the central region, while PWCF interpolation (blue points) gives $\sim 25$. Magnify

►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 $x\in [-1,1]$ and $(-\mathrm{\infty },\mathrm{\infty })$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

13: David M. Bressoud

… ► 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. …

14: 1.2 Elementary Algebra

… ►This is the row times column rule. … ►

1.2.59 $$det(𝐀)=\sum _{\sigma \in {𝔖}_n}\mathrm{sign}\sigma \prod _{i=1}^n{a}_{i,\sigma (i)}.$$

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Symbols:

$det$: determinant, $\in$: element of, ${𝔖}_n$: set of permutations of $\{1,2,\mathrm{\dots },n\}$ , $\mathrm{sign}x$: sign of, $n$: nonnegative integer and $𝐀$: non-defective matrix

Permalink:

http://dlmf.nist.gov/1.2.E59

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

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15: 14.27 Zeros

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, $\mu \notin \mathbb{Z}$, $\nu \in \mathbb{Z}$, and $\sin \left((\mu -\nu )\pi \right)$ and $\sin \left(\mu \pi \right)$ have opposite signs.

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16: 10.74 Methods of Computation

… ►Newton’s rule (§3.8(i)) or Halley’s rule (§3.8(v)) can be used to compute to arbitrarily high accuracy the real or complex zeros of all the functions treated in this chapter. …Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …

17: 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. …

18: Bibliography R

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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

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Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.60(iii).

Encodings:

BibTeX

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19: 36.5 Stokes Sets

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36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.8 $$16u^5-\frac{Y^2}{10u}+4u^3\mathrm{sign}\left(z\right)-\frac{3}{10}|Y|\mathrm{sign}\left(z\right)+4t^5+2t^3\mathrm{sign}\left(z\right)+|Y|t^2=0,$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $t$: variable and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.9 $$t=-u+{\left(\frac{|Y|}{10u}-u^2-\frac{3}{10}\mathrm{sign}\left(z\right)\right)}^{1/2}.$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $t$: variable and $Y$: scaled coordinate

Referenced by:

§36.5(ii)

Permalink:

http://dlmf.nist.gov/36.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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20: 4.16 Elementary Properties

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Table 4.16.1: Signs of the trigonometric functions in the four quadrants.

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Quadrant	$\sin \theta ,\csc \theta$	$\cos \theta ,\sec \theta$	$\tan \theta ,\cot \theta$
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Table 4.16.2: Trigonometric functions: quarter periods and change of sign.

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$x$	$-\theta$	$\frac{1}{2}\pi \pm \theta$	$\pi \pm \theta$	$\frac{3}{2}\pi \pm \theta$	$2\pi \pm \theta$
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