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: 18.7 Interrelations and Limit Relations

… ►Equations (18.7.13)–(18.7.20) are special cases of (18.2.22)–(18.2.23). …

6: 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. …

7: 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

Referenced by:

§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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External Links:

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

Referenced by:

§35.9.

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8: 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 …

9: 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), ZentralBlatt

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

Referenced by:

§31.12.

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10: Errata

… ►The specific updates to Chapter 18 include some results for general orthogonal polynomials including quadratic transformations, uniqueness of orthogonality measure and completeness, moments, continued fractions, and some special classes of orthogonal polynomials. … ►

Equation (16.16.5_5).

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These equations, originally added in Other Changes and Other Changes, respectively, have been assigned interpolated numbers.

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Four of the terms were rewritten for improved clarity.

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has been generalized to cover an additional case.

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