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21: 3.2 Linear Algebra

… ►In practice, if any of the multipliers ${\mathrm{\ell }}_{jk}$ are unduly large in magnitude compared with unity, then Gaussian elimination is unstable. …

22: 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 $w^{\prime }(a)$. …

23: 3.8 Nonlinear Equations

… ►For moderate or large values of $n$ it is not uncommon for the magnitude of the right-hand side of (3.8.14) to be very large compared with unity, signifying that the computation of zeros of polynomials is often an ill-posed problem. …

24: 19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). …

25: 2.11 Remainder Terms; Stokes Phenomenon

… ►Hence from §7.12(i) $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ is of the same exponentially-small order of magnitude as the contribution from the other terms in (2.11.15) when $\rho$ is large. …

26: 1.9 Calculus of a Complex Variable

… ►If $f^{\prime }(z_0)\ne 0$, then the angle between $C_1$ and $C_2$ equals the angle between $C_1^{\prime }$ and $C_2^{\prime }$ both in magnitude and sense. …

27: Errata

… ►

Bounds have been sharpened. The second paragraph now reads, “The $n$th error term is bounded in magnitude by the first neglected term multiplied by $\chi (n+\sigma )+1$ where $\sigma =\frac{1}{6}$ for (9.7.7) and $\sigma =0$ for (9.7.8), provided that $n\ge 0$ in the first case and $n\ge 1$ in the second case.” Previously it read, “In (9.7.7) and (9.7.8) the $n$th error term is bounded in magnitude by the first neglected term multiplied by $2\chi (n)\exp \left(\sigma \pi /(72\zeta )\right)$ where $\sigma =5$ for (9.7.7) and $\sigma =7$ for (9.7.8), provided that $n\ge 1$ in both cases.” In Equation (9.7.16)

9.7.16

$$\mathrm{Bi}\left(x\right)\le \frac{{\mathrm{e}}^{\xi }}{\sqrt{\pi }{x}^{1/4}}\left(1+\left(\chi (\frac{7}{6})+1\right)\frac{5}{72\xi }\right),$$

$${\mathrm{Bi}}^{\prime }\left(x\right)\le \frac{x^{1/4}{\mathrm{e}}^{\xi }}{\sqrt{\pi }}\left(1+\left(\frac{\pi }{2}+1\right)\frac{7}{72\xi }\right),$$

the bounds on the right-hand sides have been sharpened. The factors $\left(\chi (\frac{7}{6})+1\right)\frac{5}{72\xi }$, $\left(\frac{\pi }{2}+1\right)\frac{7}{72\xi }$, were originally given by $\frac{5\pi }{72\xi }\exp \left(\frac{5\pi }{72\xi }\right)$, $\frac{7\pi }{72\xi }\exp \left(\frac{7\pi }{72\xi }\right)$, respectively.

►

Bounds have been sharpened. The first paragraph now reads, “The $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

9.7.17

provided that $n\ge 0$, $\sigma =\frac{1}{6}$ for (9.7.5) and $n\ge 1$, $\sigma =0$ for (9.7.6).” Previously it read, “When $n\ge 1$ the $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

9.7.17

Here $\sigma =5$ for (9.7.5) and $\sigma =7$ for (9.7.6).”

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