Minimax Rational Approximation Research Articles

Minimax rational approximation of the Fermi-Dirac distribution.

Accurate rational approximations of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known approximations use O(log(βΔ)log(ϵ-1)) poles to achieve an error tolerance ϵ at temperature β-1 over an energy interval Δ. We apply minimax approximation to reduce the number of poles by a factor of four and replace Δ with Δocc, the occupied energy interval. This is particularly beneficial when Δ ≫ Δocc, such as in electronic structure calculations that use a large basis set.

Padé Approximant and Minimax Rational Approximation in Standard Cosmology

The luminosity distance in the standard cosmology as given by ΛCDM and, consequently, the distance modulus for supernovae can be defined by the Padé approximant. A comparison with a known analytical solution shows that the Padé approximant for the luminosity distance has an error of 4 % at redshift = 10 . A similar procedure for the Taylor expansion of the luminosity distance gives an error of 4 % at redshift = 0 . 7 ; this means that for the luminosity distance, the Padé approximation is superior to the Taylor series. The availability of an analytical expression for the distance modulus allows applying the Levenberg–Marquardt method to derive the fundamental parameters from the available compilations for supernovae. A new luminosity function for galaxies derived from the truncated gamma probability density function models the observed luminosity function for galaxies when the observed range in absolute magnitude is modeled by the Padé approximant. A comparison of ΛCDM with other cosmologies is done adopting a statistical point of view.

Precise and fast computation of generalized Fermi–Dirac integral by parameter polynomial approximation

The generalized Fermi–Dirac integral, Fk(η, β), is approximated by a group of polynomials of β as Fk(η,β)≈∑j=0JgjβjFk+j(η) where J=1(1)10. Here Fk(η) is the Fermi-Dirac integral of order k while gj are the numerical coefficients of the single and double precision minimax polynomial approximations of the generalization factor as 1+x/2≈∑j=0Jgjxj. If β is not so large, an appropriate combination of these approximations computes Fk(η, β) precisely when η is too small to apply the optimally-truncated Sommerfeld expansion (Fukushima, 2014 [15]). For example, a degree 8 single precision polynomial approximation guarantees the 24 bit accuracy of Fk(η, β) of the orders, k=−1/2(1)5/2, when −∞<η≤8.92 and β ≤ 0.2113. Also, a degree 7 double precision polynomial approximation assures the 15 digit accuracy of Fk(η, β) of the same orders when −∞<η≤29.33 and 0≤β≤3.999×10−3. Thanks to the piecewise minimax rational approximations of Fk(η) (Fukushima, 2015 [18]), the averaged CPU time of the new method is roughly the same as that of single evaluation of the integrand of Fk(η, β). Since most of Fk(η) are commonly used in the approximation of Fk(η, β) of multiple contiguous orders, the simultaneous computation of Fk(η, β) of these orders is further accelerated by the factor 2–4. As a result, the new method runs 70–450 times faster than the direct numerical integration in practical applications requiring Fk(η, β).

Precise and fast computation of Fermi–Dirac integral of integer and half integer order by piecewise minimax rational approximation

Piecewise quadruple precision approximations of the Fermi–Dirac integral of integer order, 0(1)10, and half integer order, -9/2(1)21/2, are developed by combining (i) the optimally truncated Sommerfeld expansion, (ii) the piecewise truncated Chebyshev series expansion, and/or (iii) the reflection formula. They are used in constructing the double precision piecewise minimax rational approximations of the integral of the same orders. The relative errors of the new minimax approximations, which are all due to rounding off, are 3–13 machine epsilons at most and less than 4 machine epsilons typically while their CPU times are only 0.44–1.1 times that of the exponential function when -5⩽η⩽45. As a result, the new approximations run 16–31 times faster than the piecewise Chebyshev polynomial approximations (Macleod, 1998) for the physically important orders, -1/2(1)5/2, and 330–720 and 50–93 times faster than the combined series expansions (Goano, 1995) for the half integer orders, -1/2(1)21/2, and the integer orders, 1(1)9, respectively. A file of the Fortran codes of the obtained approximations and their test program and sample output is named xfdh.txt and located at: https://www.researchgate.net/profile/Toshio_Fukushima/.

One Method for Proving Inequalities by Computer

We consider a numerical method for proving a class of analytical inequalities via minimax rational approximations. All numerical calculations in this paper are given by Maple computer program.

Minimax rational approximations for the parabolic equation method

Higher‐order Padé approximations have been applied to reduce the limitations and to improve the efficiency of the parabolic equation (PE) method [M. D. Collins, J. Acoust. Soc. Am. 93, 1736–1742 (1993)]. Several of the derivatives are correct at the origin for Padé approximations. Improved efficiency may be achieved by using rational approximations designed using the minimax method, which involves minimizing the maximum error over a domain of interest. This approach has been applied for deriving a PE for fluid media that handles wide propagation angles [Vefring and Mjo/lsnes, J. Acoust. Soc. Am. 93, 1736–1742 (1993)]. Additional applications of the minimax approach include deriving improved approximations for the elastic PE (for which stability is an important issue), the two‐way PE (which may require a careful treatment of the evanescent spectrum), and the split‐step Padé solution (which involves that approximation of a relatively complicated function).

Best Rational Approximation and Strict Quasi-Convexity

If a continuous function is strictly quasi-convex on a convex set $\Gamma $, then every local minimum of the function must be a global minimum. Furthermore, every local maximum of the function on the interior of $\Gamma $ must also be a global minimum. Here, we prove that any minimax rational approximation problem defines a strictly quasi-convex function with the property that a best approximation (if one exists) is a minimum of that function. The same result is not true in general for best rational approximation in other norms.

A note on computing approximations to the exponential function

Two methods are discussed which result in near minimax rational approximations to the exponential function and at the same time retain the desirable property that the approximation for negative values of the argument is the reciprocal of the approximation for corresponding positive values. These methods lead to approximations which are much superior to the commonly used convergents of the Gaussian continued fraction for the exponential. Coefficients and errors are given for the intervals [-1/2 ln 2, 1/2 ln 2] and [-ln 2, ln 2]. Two methods are discussed which result in near minimax rational approximations to the exponential function and at the same time retain the desirable property that the approximation for negative values of the argument is the reciprocal of the approximation for corresponding positive values. These methods lead to approximations which are much superior to the commonly used convergents of the Gaussian continued fraction for the exponential. Coefficients and errors are given for the intervals [-1/2 ln 2, 1/2 ln 2] and [-ln 2, ln 2].

The Construction of Minimax Rational Approximations to Functions

The Construction of Minimax Rational Approximations to Functions