Romberg integration

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11: 9.17 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the defining differential equation (9.2.1) by direct numerical methods. As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation. …On the remaining rays, given by $\mathrm{ph}z=\pm \frac{1}{3}\pi$ and $\pi$, integration can proceed in either direction. … ►In the case of the Scorer functions, integration of the differential equation (9.12.1) is more difficult than (9.2.1), because in some regions stable directions of integration do not exist. … ►In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …

12: 31.18 Methods of Computation

… ►Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). …Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). …

13: 21.9 Integrable Equations

§21.9 Integrable Equations

►Riemann theta functions arise in the study of integrable differential equations that have applications in many areas, including fluid mechanics (Ablowitz and Segur (1981, Chapter 4)), magnetic monopoles (Ercolani and Sinha (1989)), and string theory (Deligne et al. (1999, Part 3)). … ►

14: 7.16 Generalized Error Functions

… ►These functions can be expressed in terms of the incomplete gamma function $\gamma (a,z)$ (§8.2(i)) by change of integration variable.

15: 16.25 Methods of Computation

… ►In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …

16: 32.17 Methods of Computation

… ►The Painlevé equations can be integrated by Runge–Kutta methods for ordinary differential equations; see §3.7(v), Hairer et al. (2000), and Butcher (2003). …

17: 33.23 Methods of Computation

… ►

§33.23(iii) Integration of Defining Differential Equations

►When numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. …

18: 11.13 Methods of Computation

… ►A comprehensive approach is to integrate the defining inhomogeneous differential equations (11.2.7) and (11.2.9) numerically, using methods described in §3.7. To insure stability the integration path must be chosen so that as we proceed along it the wanted solution grows in magnitude at least as rapidly as the complementary solutions. … ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. The solution ${𝐊}_{\nu }\left(x\right)$ needs to be integrated backwards for small $x$, and either forwards or backwards for large $x$ depending whether or not $\nu$ exceeds $\frac{1}{2}$. For ${𝐌}_{\nu }\left(x\right)$ both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). …

19: 19.15 Advantages of Symmetry

… ►Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration. …These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). …

20: 32.13 Reductions of Partial Differential Equations

… ►where $w(z)$ satisfies ${\text{P}}_{\text{II}}$ with $\alpha$ a constant of integration. … ►

32.13.10 $$v^{\prime \prime }=6v^2+(c^2-1)v+Az+B,$$

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Symbols:

$z$: real, $c$: constant, $v(z)$: solution, $A$: integration constant and $B$: integration constant

Permalink:

http://dlmf.nist.gov/32.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §32.13(iii), §32.13 and Ch.32

►with $A$ and $B$ constants of integration. Depending whether $A=0$ or $A\ne 0$, $v(z)$ is expressible in terms of the Weierstrass elliptic function (§23.2) or solutions of ${\text{P}}_{\text{I}}$, respectively. …