unstable pairs

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11: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. …

12: 27.19 Methods of Computation: Factorization

… ►Type II probabilistic algorithms for factoring $n$ rely on finding a pseudo-random pair of integers $(x,y)$ that satisfy $x^2\equiv y^2(modn)$. …

13: 9.2 Differential Equation

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§9.2(iii) Numerically Satisfactory Pairs of Solutions

►Table 9.2.1 lists numerically satisfactory pairs of solutions of (9.2.1) for the stated intervals or regions; compare §2.7(iv). ►

Table 9.2.1: Numerically satisfactory pairs of solutions of Airy’s equation.

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Pair	Interval or Region
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14: 3.6 Linear Difference Equations

… ►In this situation the unwanted multiples of $g_n$ grow more rapidly than the wanted solution, and the computations are unstable. … ►Then computation of $w_n$ by forward recursion is unstable. If it also happens that $f_n/w_n\to 0$ as $n\to \mathrm{\infty }$, then computation of $w_n$ by backward recursion is unstable as well. …

15: 22.19 Physical Applications

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Figure 22.19.1: Jacobi’s amplitude function $\mathrm{am}(x,k)$ for $0\le x\le 10\pi$ and $k=0.5,0.9999,1.0001,2$. …This corresponds to the pendulum being “upside down” at a point of unstable equilibrium. … Magnify

… ►As $a\to \sqrt{1/\beta }$ from below the period diverges since $a=\pm \sqrt{1/\beta }$ are points of unstable equilibrium. … ►As $a\to \sqrt{2/\beta }$ from below the period diverges since $x=0$ is a point of unstable equlilibrium. …

16: 13.28 Physical Applications

… ►and $V_{\kappa ,\mu }^{(j)}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). …

17: 15.17 Mathematical Applications

… ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL$(2,ℝ)$, and spherical functions on certain nonsymmetric Gelfand pairs. …

18: Bille C. Carlson

… ►In his paper Lauricella’s hypergeometric function $F_D$ (1963), he defined the $R$-function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. …

19: 10.2 Definitions

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§10.2(iii) Numerically Satisfactory Pairs of Solutions

►Table 10.2.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.2.1) for the stated intervals or regions in the case $\mathrm{\Re }\nu \ge 0$. … ►

Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.

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Pair	Interval or Region
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20: 3.10 Continued Fractions

… ►The quotient-difference algorithm is frequently unstable and may require high-precision arithmetic or exact arithmetic. … ►However, this may be unstable; also overflow and underflow may occur when evaluating $A_n$ and $B_n$ (making it necessary to re-scale from time to time). …