quadratic equations

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1: 1.11 Zeros of Polynomials

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Quadratic Equations

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2: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. …

3: 19.36 Methods of Computation

… ►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. …

4: 15.8 Transformations of Variable

… ►A necessary and sufficient condition that there exists a quadratic transformation is that at least one of the equations shown in Table 15.8.1 is satisfied. …

5: 3.8 Nonlinear Equations

… ►If $p=2$, then the convergence is quadratic; if $p=3$, then the convergence is cubic, and so on. … ►If $\zeta$ is a simple zero, then the iteration converges locally and quadratically. … ►It converges locally and quadratically for both $ℝ$ and $ℂ$. … ►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of $q(z)$. … ►The quadratic nature of the convergence is evident. …

6: Bibliography M

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

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The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

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Quadratic relations for confluent hypergeometric functions. Tohoku Math. J. (2) 52 (4), pp. 489–513.

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ISSN 0040-8735, Document, MathReview (Takeshi Sasaki), ZentralBlatt

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§13.12.

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.

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ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.

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ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

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7: 3.11 Approximation Techniques

… ►The iterative process converges locally and quadratically (§3.8(i)). … ►Also, in cases where $f(x)$ satisfies a linear ordinary differential equation with polynomial coefficients, the expansion (3.11.11) can be substituted in the differential equation to yield a recurrence relation satisfied by the $c_n$. … ►With $b_0=1$, the last $q$ equations give $b_1,\mathrm{\dots },b_q$ as the solution of a system of linear equations. The first $p+1$ equations then yield $a_0,\mathrm{\dots },a_p$. … ►From the equations $\partial S/\partial a_k=0$, $k=0,1,\mathrm{\dots },n$, we derive the normal equations …

8: Bibliography W

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Extension of a quadratic transformation due to Whipple with an application. Adv. Difference Equ., pp. 2013:157, 8.

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ISSN 1687-1847, Document, FullText, MathReview (Daniele Ritelli)

Referenced by:

§16.6, §16.6.

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

Referenced by:

§2.9(iii).

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

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

Referenced by:

§2.9(iii).

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

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

Referenced by:

§32.10(v).

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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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ISSN 0025-584X, MathReview, ZentralBlatt

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§31.12.

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9: Bibliography C

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Théorie Générale de L’Équation de Mathieu et de quelques autres Équations différentielles de la mécanique. Masson et Cie, Paris (French).

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MathReview (A. Erdélyi), ZentralBlatt

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§28.1, Notations C, Notations I, Notations I, Notations J, Notations J, Notations S.

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Reduction theorems for elliptic integrands with the square root of two quadratic factors. J. Comput. Appl. Math. 118 (1-2), pp. 71–85.

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Notes:

Code written by the second author to compute symmetric elliptic integrals numerically in the Derive computer algebra system is available.

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ISSN 0377-0427, Document, Code, MathReview, ZentralBlatt

Referenced by:

§19.29(ii), §19.36(i), §19.36(i), §19.39(iv).

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Quadratic transformations of Appell functions. SIAM J. Math. Anal. 7 (2), pp. 291–304.

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Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.16(ii), §19.22(i).

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Elementary Differential Equations. Clarendon Press, Oxford.

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ISBN 0-19-853139-7, MathReview, ZentralBlatt

Referenced by:

§16.12.

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An extension of a Kummer’s quadratic transformation formula with an application. Proc. Jangjeon Math. Soc. 16 (2), pp. 229–235.

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ISSN 1598-7264, MathReview (Petr Blaschke), ZentralBlatt

Referenced by:

§16.6, §16.6.

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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. …