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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: 10.74 Methods of Computation
Newton’s rule3.8(i)) or Halley’s rule3.8(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. …
13: 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. …
14: 34.7 Basic Properties: 9 j Symbol
This equation is the sum rule. It constitutes an addition theorem for the 9 j symbol. …
15: 1.11 Zeros of Polynomials
Descartes’ Rule of Signs
16: 3.7 Ordinary Differential Equations
The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation. … For w = f ( z , w ) the standard fourth-order rule reads … For w ′′ = f ( z , w , w ) the standard fourth-order rule reads …
17: 6.18 Methods of Computation
Zeros of Ci ( x ) and si ( x ) can be computed to high precision by Newton’s rule3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …
18: 3.10 Continued Fractions
We continue by means of the rhombus rule
19: 1.6 Vectors and Vector-Valued Functions
where 𝐧 is the unit vector normal to 𝐚 and 𝐛 whose direction is determined by the right-hand rule; see Figure 1.6.1.
See accompanying text
Figure 1.6.1: Vector notation. Right-hand rule for cross products. Magnify
20: Bibliography G
  • W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
  • A. Gil, J. Segura, and N. M. Temme (2003b) Computing special functions by using quadrature rules. Numer. Algorithms 33 (1-4), pp. 265–275.
  • G. H. Golub and J. H. Welsch (1969) Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.