chain rule
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11: 7.22 Methods of Computation
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►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.
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12: 10.74 Methods of Computation
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►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.
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13: 29.20 Methods of Computation
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►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.
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14: 34.7 Basic Properties: Symbol
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►This equation is the sum rule.
It constitutes an addition theorem for the symbol.
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15: 1.11 Zeros of Polynomials
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Descartes’ Rule of Signs
…16: 3.7 Ordinary Differential Equations
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►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.
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►For the standard fourth-order rule reads
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►For the standard fourth-order rule reads
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17: 6.18 Methods of Computation
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►Zeros of and can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations.
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18: 3.10 Continued Fractions
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►We continue by means of the rhombus rule
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19: 1.6 Vectors and Vector-Valued Functions
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►where is the unit vector normal to and whose direction is determined by the right-hand rule; see Figure 1.6.1.
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20: Bibliography G
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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.
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Computing special functions by using quadrature rules.
Numer. Algorithms 33 (1-4), pp. 265–275.
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Calculation of Gauss quadrature rules.
Math. Comp. 23 (106), pp. 221–230.
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