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11: 15.19 Methods of Computation

… ►The relations in §15.5(ii) can be used to compute $F(a,b;c;z)$, provided that care is taken to apply these relations in a stable manner; see §3.6(ii). …

12: 22.19 Physical Applications

… ► $\theta$ being the angular displacement from the point of stable equilibrium, $\theta =0$. … ►for the initial conditions $\theta (0)=0$, the point of stable equilibrium for $E=0$, and $d\theta (t)/dt=\sqrt{2E}$. … ►For an initial displacement with , bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\beta }$. …

13: 3.10 Continued Fractions

… ►A more stable version of the algorithm is discussed in Stokes (1980). … ►In general this algorithm is more stable than the forward algorithm; see Jones and Thron (1974). …

14: 18.40 Methods of Computation

… ►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let $N$ be a positive integer and define …See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby. …

15: 10.74 Methods of Computation

… ►For $z\in ℂ$ the function $H_{\nu }^{(1)}\left(z\right)$, for example, can always be computed in a stable manner in the sector $0\le \mathrm{ph}z\le \pi$ by integrating along rays towards the origin. …

16: 3.6 Linear Difference Equations

… ►This is of little consequence if the wanted solution is growing in magnitude at least as fast as any other solution of (3.6.3), and the recursion process is stable. … ►If, as $n\to \mathrm{\infty }$, the wanted solution $w_n$ grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable. …

17: 3.7 Ordinary Differential Equations

… ►If the solution $w(z)$ that we are seeking grows in magnitude at least as fast as all other solutions of (3.7.1) as we pass along $𝒫$ from $a$ to $b$, then $w(z)$ and $w^{\prime }(z)$ may be computed in a stable manner for $z=z_0,z_1,\mathrm{\dots },z_P$ by successive application of (3.7.5) for $j=0,1,\mathrm{\dots },P-1$, beginning with initial values $w(a)$ and $w^{\prime }(a)$. … ►Then to compute $w(z)$ in a stable manner we solve the set of equations (3.7.5) simultaneously for $j=0,1,\mathrm{\dots },P$, as follows. …

18: Bibliography G

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Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

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Referenced by:

§18.39(iii).

Encodings:

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19: Bibliography S

… ►

A stable quotient-difference algorithm. Math. Comp. 34 (150), pp. 515–519.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§3.10(ii).

Encodings:

BibTeX

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20: 18.39 Applications in the Physical Sciences

… ►The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the $x_i$ and $w_i$, from the J-matrix as in §3.5(vi), straightforward. …