cubic equation

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1: 4.43 Cubic Equations

§4.43 Cubic Equations

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2: 1.11 Zeros of Polynomials

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Cubic Equations

… ►For the roots ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ of $g(w)=0$ and the roots ${\theta }_1,{\theta }_2,{\theta }_3$ of the resolvent cubic equation …

3: 23.22 Methods of Computation

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In the general case, given by $cd\ne 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^3-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_1$, $e_2$, $e_3$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^3-27d^2>0$, then $e_1$, $e_2$, $e_3$ can be identified via (23.5.1), and $k^2$, $k^{\prime }{}^2$ obtained from (23.6.16).

If , or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha ,\gamma ]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha ,\gamma )$. In consequence, $k^2=(\beta -\gamma )/(\alpha -\gamma )$, $k^{\prime }{}^2=(\alpha -\beta )/(\alpha -\gamma )$ satisfy $\mathrm{\Im }{k}^2\ge 0\ge \mathrm{\Im }k^{\prime }{}^2$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^2|$, $|k^{\prime }{}^2|\le 1$.

Finally, on taking the principal square roots of $k^2$ and $k^{\prime }{}^2$ we obtain values for $k$ and $k^{\prime }$ that lie in the 1st and 4th quadrants, respectively, and $2{\omega }_1$, $2{\omega }_3$ are given by

23.22.1 $$2{\omega }_{1}M(1,k^{\prime })=-2\mathrm{i}{\omega }_{3}M(1,k)=\frac{\pi }{3}\sqrt{\frac{c(2+k^{2}k^{\prime }{}^2)(k^{\prime }{}^2-k^2)}{d(1-k^{2}k^{\prime }{}^2)}},$$

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

Source:

Combine (23.6.16), (23.6.17), (22.20.6), (23.3.5)–(23.3.7), and use analytical continuation.

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 2 possible choices of the square root.

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4: 23.3 Differential Equations

… ►The lattice roots satisfy the cubic equation …

5: 3.8 Nonlinear Equations

… ►The equation to be solved is … ►Sometimes the equation takes the form … ►If $p=2$, then the convergence is quadratic; if $p=3$, then the convergence is cubic, and so on. … ►The rule converges locally and is cubically convergent. …

6: 23.21 Physical Applications

… ►The Weierstrass function $\mathrm{\wp }$ plays a similar role for cubic potentials in canonical form $g_3+g_{2}x-4x^3$. … ►

§23.21(ii) Nonlinear Evolution Equations

►Airault et al. (1977) applies the function $\mathrm{\wp }$ to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. For applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1). …

7: Bibliography R

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Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

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External Links:

MathReview (E. A. Boyd), ZentralBlatt

Referenced by:

(9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).

Encodings:

BibTeX

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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External Links:

MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

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External Links:

MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

BibTeX

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Asymptotics and Bounds of the Roots of Equations (Russian). Zinatne, Riga.

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External Links:

ISBN 5-7966-0570-4, MathReview (Guillermo López Lagomasino), ZentralBlatt

Referenced by:

§12.11(ii), §6.13.

Encodings:

BibTeX

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8: Bibliography J

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL.

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Notes:

Second edition of Jones and Sleeman (2003)

External Links:

ISBN 978-1-4200-8357-6, MathReview Entry, ZentralBlatt

Referenced by:

Profile Brian D. Sleeman.

Encodings:

BibTeX

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:

A second edition was published in 2010.

External Links:

ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt

Referenced by:

D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).

Encodings:

BibTeX

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On Boutroux’s tritronquée solutions of the first Painlevé equation. Stud. Appl. Math. 107 (3), pp. 253–291.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§32.12(i).

Encodings:

BibTeX

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On the simple cubic lattice Green function. Philos. Trans. Roy. Soc. London Ser. A 273, pp. 583–610.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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On the cubic lattice Green functions. Proc. Roy. Soc. London Ser. A 445, pp. 463–477.

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External Links:

ISSN 0962-8444, FullText, MathReview (Peter J. Forrester), ZentralBlatt

Referenced by:

§31.17(ii), §31.7(i).

Encodings:

BibTeX

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9: Bibliography W

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Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.

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External Links:

ISSN 0029-599X, Document, MathReview (Sergey M. Zagorodnyuk), ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

BibTeX

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Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

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External Links:

ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

BibTeX

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Solutions of the fifth Painlevé equation. I. Hokkaido Math. J. 24 (2), pp. 231–267.

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External Links:

ISSN 0385-4035, FullText, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.10(v).

Encodings:

BibTeX

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The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:

ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

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On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.

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External Links:

ISSN 0025-584X, MathReview, ZentralBlatt

Referenced by:

§31.12.

Encodings:

BibTeX

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10: 10.74 Methods of Computation

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§10.74(ii) Differential Equations

►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. … ►For further information, including parallel methods for solving the differential equations, see Lozier and Olver (1993). … ►Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …