The Schwarzian-Newton method for solving nonlinear equations, with applications

The Schwarzian-Newton method can be defined as the minimal method for solving nonlinear equations f(x) = 0 which is exact for any function f with constant Schwarzian derivative; exactness means that the method gives the exact root in one iteration for any starting value in a neighborhood of the root. This is a fourth order method which has Halley?s method as limit when the Schwarzian derivative tends to zero. We obtain conditions for the convergence of the SNM in an interval and show how this methodcan be applied for a reliable and fast solution of some problems, like the inversion of cumulative distribution functions (gamma and beta distributions) and the inversion of elliptic integrals.The author acknowledges financial support from Ministerio de Economía y Competitividad (project MTM2012-34787

Simple bounds with best possible accuracy for ratios of modified Bessel functions

The best bounds of the form B(α, β, γ, x) = (α + ✓β2 + γ2 x2)/x for ratios of modified Bessel functions are characterized: if α, β and γ are chosen in such a way that B(α, β, γ, x) is a sharp approximation for Φν (x) = Iν−1(x)/Iν(x) as x → 0+ (respectively x → +∞) and the graphs of the functions B(α, β, γ, x) and Φν (x) are tangent at some x = x∗ > 0, then B(α, β, γ, x) is an upper (respectively lower) bound for Φν (x) for any positive x, and it is the best possible at x∗. The same is true for the ratio Φν (x) = Kν+1(x)/Kν (x) but interchanging lower and upper bounds (and with a slightly more restricted range for ν). Bounds with maximal accu- racy at 0+ and +∞ are recovered in the limits x∗ → 0+ and x∗ → +∞, and for these cases the coefficients have simple expressions. For the case of finite and positive x∗ we provide uniparametric families of bounds which are close to the optimal bounds and retain their confluence properties.The author acknowledges support from Ministerio de Ciencia e Innovación, projects PGC2018-098279-BI00 (MCIU/AEI/FEDER, UE) and PID2021-127252NB-I00 (MCIN/AEI/10.13039/501100011033/FEDER, UE

Uniform very sharp bounds for ratios of parabolic cylinder functions

Ministerio de Ciencia e Innovación, Grant/Award Number: PGC2018-098279-B-I00 (MCIU/AEI/FEDER UE

On bounds for solutions of monotonic first order difference-differential systems

Many special functions are solutions of first order linear systems y_ n(x) = an(x)yn(x) + dn(x)yn−1(x), y_n−1(x), = bn(x)yn−1(x) + en(x)yn(x) . We obtain bounds for the ratios yn(x)/yn-1(x) and the logarithmic derivatives of yn(x) for solutions of monotonic systems satisfying certain initial conditions. For the case dn(x)en(x) > 0, sequences of upper and lower bounds can be obtained by iterating the recurrence relation; for minimal solutions of the recurrence these are convergent sequences. The bounds are related to the Liouville-Green approximation for the associated second order ODEs as well as to the asymptotic behavior of the associated three-term recurrence relation as n ® +∞; the bounds are sharp both as a function of n and x. Many special functions are amenable to this analysis, and we give several examples of application: modified Bessel functions, parabolic cylinder functions, Legendre functions of imaginary variable and Laguerre functions. New Turán-type inequalities are established from the function ratio bounds. Bounds for monotonic systems with dn(x)en(x) < 0 are also given, in particular for Hermite and Laguerre polynomials of real positive variable; in that case the bounds can be used for bounding the monotonic region (and then the extreme zeros)

Monotonicity properties for ratios and products of modified bessel functions and sharp trigonometric bounds

Let Iv(x) and Kv(x) be the first and second kind modified Bessel functions. It is shown that the nullclines of the Riccati equation satisfied by x [phi] i,v(x) , i= 1 , 2 , with [PI]1,v = Iv-1(x) / Iv(x) and [PHI] 2,v(x) = - Kv-1(x) / Kv(x) , are bounds for x [PHI]+ i,v(x) , which are solutions with unique monotonicity properties; these bounds hold at least for ± [ épsilon] (0 , 1) and v -1 / 2. Properties for the product Pv(x) = Iv(x) Kv(x) can be obtained as a consequence; for instance, it is shown that Pv(x) is decreasing if v -1 (extending the known range of this result) and that xPv(x) is increasing for v 1 / 2. We also show that the double ratios Wi,v(x) = [PHI] i,v+1(x) / [PHI]+ i,v(x) are monotonic and that these monotonicity properties are exclusive of the first and second kind modified Bessel functions. Sharp trigonometric bounds can be extracted from the monotonicity of the double ratios. The trigonometric bounds for the ratios and the product are very accurate as x 0 +, x + and v + in the sense that the first two terms in the power series expansions in these limits are exact.The author acknowledges support from Ministerio de Ciencia e Innovación, Project PGC2018-098279-B-I00 (MCIU/AEI/FEDER, UE)

Sharp bounds for cumulative distribution functions

Ratios of integrals can be bounded in terms of ratios of integrands under certain mono- tonicity conditions. This result, related with L?H?opital?s monotone rule, can be used to obtain sharp bounds for cumulative distribution functions. We consider the case of non- central cumulative gamma and beta distributions. Three different types of sharp bounds for the noncentral gamma distributions (also called Marcum functions) are obtained in terms of modified Bessel functions and one additional type of function: a second modified Bessel function, two error functions or one incomplete gamma function. For the noncen- tral beta case the bounds are expressed in terms of Kummer functions and one additional Kummer function or an incomplete beta function. These bounds improve previous results with respect to their range of application and/or its sharpness.The author acknowledges financial support from Ministerio de Economía y Competitividad (project MTM2012-34787

Algorithm 939: Computation of the Marcum Q-function

Methods and an algorithm for computing the generalized Marcum Q.function (QƒÊ(x, y)) and the complementary function (PƒÊ(x, y)) are described. These functions appear in problems of different technical and scientific areas such as, for example, radar detection and communications, statistics and probability theory, where they are called the non-central chi-square or the non central gamma cumulative distribution functions. The algorithm for computing the Marcum functions combines different methods of evaluation in different regions: series expansions, integral representations, asymptotic expansions, and use of three-term homogeneous recurrence relations. A relative accuracy close to 10.12 can be obtained in the parameter region (x, y, ƒÊ) ¸ [0, A] ~ [0, A] ~ [1, A], A = 200, while for larger parameters the accuracy decreases (close to 10.11 for A = 1000 and close to 5 ~ 10.11 for A = 10000)

Asymptotic Approximations to the Nodes and Weights of Gauss-Hermite and Gauss-Laguerre Quadratures

Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a stand-alone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than 100, the asymptotic methods are enough for a double precision accuracy computation (15-16 digits) of the nodes and weights of the Gauss-Hermite and Gauss-Laguerre quadratures.The authors acknowledge financial support from Ministerio de Economía y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE)

Asymptotic Expansions of Jacobi Polynomials for Large Values of Beta

Asymptotic approximations of Jacobi polynomials are given for large values of the Beta-parameter and of their zeros. The expansions are given in terms of Laguerre polynomials and of their zeros. The levels of accuracy of the approximations are verified by numerical examples.This work was supported by Ministerio de Econom a y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE). NMT thanks CWI, Amsterdam, for scienti c support

On the computation and inversion of the cumulative noncentral beta distribution function

The computation and inversion of the noncentral beta distribution Bp,q(x, y) (or the noncentral F-distribution, a particular case of Bp,q(x, y)) play an important role in different applications. In this paper we study the stability of recursions satisfied by Bp,q(x, y) and its complementary function and describe asymptotic expansions useful for computing the function when the parameters are large. We also consider the inversion problem of finding x or y when a value of Bp,q(x, y) is given. We provide approximations to x and y which can be used as starting values of methods for solving nonlinear equations (such as Newton) if higher accuracy is needed