Descartes’ rule of signs (for polynomials)

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21: Bibliography W

… ►

J. Waldvogel (2006) Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.

ⓘ

External Links:: ISSN 0006-3835, Document, MathReview Entry, ZentralBlatt
Referenced by:: §3.5(iv).
Encodings:: BibTeX

… ►

X.-S. Wang and R. Wong (2011) Global asymptotics of the Meixner polynomials. Asymptotic Analysis 75 (3-4), pp. 211–231.

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External Links:: ISSN 0921-7134, Document, MathReview (Mario Pérez Riera), ZentralBlatt
Referenced by:: §18.24, §18.24.
Encodings:: BibTeX

►

X.-S. Wang and R. Wong (2012) Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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External Links:: ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt
Referenced by:: §2.9(iii), §2.9(iii).
Encodings:: BibTeX

… ►

Z. Wang and R. Wong (2006) Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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External Links:: ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt
Referenced by:: §18.29.
Encodings:: BibTeX

… ►

J. A. Wilson (1978) Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:: §16.4(iii), §16.4(iii), §16.4(iii).
Encodings:: BibTeX

…

22: About MathML

… ►As a general rule, using the latest available version of your chosen browser, plugins and an updated operating system is helpful. …

23: 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. …

24: 36.5 Stokes Sets

… ►

36.5.7

X=\dfrac{9}{20}+20u^{4}-\frac{Y^{2}}{20u^{2}}+6u^{2}\operatorname{sign}\left(z% \right),

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Symbols:: $\operatorname{sign}\NVar{x}$ : 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

… ►

36.5.8

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

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Symbols:: $\operatorname{sign}\NVar{x}$ : 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

… ►

36.5.9

t=-u+\left(\dfrac{|Y|}{10u}-u^{2}-\dfrac{3}{10}\operatorname{sign}\left(z% \right)\right)^{1/2}.

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Symbols:: $\operatorname{sign}\NVar{x}$ : 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

… ►For

\left|Y\right|>Y_{1}

the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for

\left|Y\right|<Y_{1}

it is generated by the roots of the polynomial equation …

25: 15.13 Zeros

… ►where

S=\operatorname{sign}\left(\Gamma\left(a\right)\Gamma\left(b\right)\Gamma\left% (c-a\right)\Gamma\left(c-b\right)\right)

. … ►If

a

b

c

c-a

, or

c-b\in\{0,-1,-2,\dots\}

, then

F\left(a,b;c;z\right)

is not defined, or reduces to a polynomial, or reduces to

(1-z)^{c-a-b}

times a polynomial. …

26: 3.4 Differentiation

… ►The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2). … ►With the choice

r=k

(which is crucial when

k

is large because of numerical cancellation) the integrand equals

e^{k}

at the dominant points

\theta=0,2\pi

, and in combination with the factor

k^{-k}

in front of the integral sign this gives a rough approximation to

1/k!

. …As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands. …

27: 26.15 Permutations: Matrix Notation

… ►

26.15.1

\begin{bmatrix}0&0&1&0&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1\\ 1&0&0&0&0&0&0&0\\ 0&0&0&0&0&1&0&0\end{bmatrix}

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Permalink:: http://dlmf.nist.gov/26.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.15 and Ch.26

►The sign of the permutation

\sigma

is the sign of the determinant of its matrix representation. … ►The rook polynomial is the generating function for

r_{j}(B)

: ►

26.15.3

R(x,B)=\sum_{j=0}^{n}r_{j}(B)\,x^{j}.

ⓘ

Symbols:: $x$ : real variable, $j$ : nonnegative integer, $n$ : nonnegative integer, $B$ : subset, $r_{j}(B)$ : number of rook positions and $R(x,B)$ : rook polynomial
Permalink:: http://dlmf.nist.gov/26.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.15 and Ch.26

… ►

26.15.4

R(x,B)=R(x,B_{1})\,R(x,B_{2}).

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Symbols:: $x$ : real variable, $B$ : subset and $R(x,B)$ : rook polynomial
Permalink:: http://dlmf.nist.gov/26.15.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.15 and Ch.26

…

28: 3.3 Interpolation

… ►Here the prime signifies that the factor for

j=k

is to be omitted,

\delta_{k,j}

is the Kronecker symbol, and

\omega_{n+1}(z)

is the nodal polynomial … ►This represents the Lagrange interpolation polynomial in terms of divided differences: … ►The inverse interpolation polynomial is given by … ►and compute an approximation to

a_{1}

by using Newton’s rule (§3.8(ii)) with starting value

x=-2.5

. …Then by using

x_{3}

in Newton’s interpolation formula, evaluating

\left[x_{0},x_{1},x_{2},x_{3}\right]f=-0.26608\;28233

and recomputing

f^{\prime}(x)

, another application of Newton’s rule with starting value

x_{3}

gives the approximation

x=2.33810\;7373

, with 8 correct digits. …

29: 3.2 Linear Algebra

… ►Because of rounding errors, the residual vector

\mathbf{r}=\mathbf{b}-\mathbf{A}\mathbf{x}

is nonzero as a rule. … ►The polynomial …The multiplicity of an eigenvalue is its multiplicity as a zero of the characteristic polynomial (§3.8(i)). … … ►Its characteristic polynomial can be obtained from the recursion …

30: 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. …