The Experts below are selected from a list of 3087 Experts worldwide ranked by ideXlab platform
Nico M. Temme - One of the best experts on this subject based on the ideXlab platform.
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fast and accurate computation of the weber Parabolic Cylinder function w a x
Ima Journal of Numerical Analysis, 2011Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:Methods for the numerical evaluation of the Weber Parabolic Cylinder functions $W(a,\pm x)$, which are independent solutions of the inverted harmonic oscillator $y''+(x^2/4-a)y=0$, are described. The functions appear in the solution of many physical problems, and notably in quantum mechanics. It is shown that the combined use of Maclaurin series, Chebyshev series, uniform asymptotic expansions for large $a$ and/or $x$ and the integration of the differential equation by local Taylor series are enough for computing accurately the functions in a wide rage of parameters. Differently from previous methods, the computational scheme is stable in the sense that high accuracy is retained: only 2 or 3 digits may be lost in double precision computations.
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Algorithm 914: Parabolic Cylinder function W ( a , x ) and its derivative
ACM Transactions on Mathematical Software, 2011Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:A Fortran 90 program for the computation of the real Parabolic Cylinder functions W(a, ± x), x ≥ 0 and their derivatives is presented. The code also computes scaled functions for a > 50. The functions W(a, p x) are a numerically satisfactory pair of solutions of the Parabolic Cylinder equation y′ p (x2/4 − a)y e 0, x ≥ 0. Using Wronskian tests, we claim a relative accuracy better than 5 10−13 in the computable range of unscaled functions, while for scaled functions the aimed relative accuracy is better than 5 10−14. This code, together with the algorithm and related software described in Gil et al. [2006a, 2006b], completes the set of software for Parabolic Cylinder Functions (PCFs) for real arguments.
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algorithm 850 real Parabolic Cylinder functions u a x v a x
ACM Transactions on Mathematical Software, 2006Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:Fortran 90 programs for the computation of real Parabolic Cylinder functions are presented. The code computes the functions U(a, x), V(a, x) and their derivatives for real a and x (x ≥ 0). The code also computes scaled functions. The range of computation for scaled PCFs is practically unrestricted. The aimed relative accuracy for scaled functions is better than 5 10−14. Exceptions to this accuracy are the evaluation of the functions near their zeros and the error caused by the evaluation of trigonometric functions of large arguments when vav G x. The routines always give values for which the Wronskian relation for scaled functions is verified with a relative accuracy better than 5 10−14. The accuracy of the unscaled functions is also better than 5 10−14 for moderate values of x and a (except close to the zeros), while for large x and a the error is dominated by exponential and trigonometric function evaluations. For IEEE standard double precision arithmetic, the accuracy is better than 5 10−13 in the computable range of unscaled PCFs (except close to the zeros).
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Computing the real Parabolic Cylinder functions U(a,x), V(a,x)
2006Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:htmlabstractMethods for the computation of real Parabolic Cylinder functions U(a,x) and V(a,x) and their derivatives are described. We give details on power series, asymptotic series, recursion and quadrature. A combination of these methods can be used for computing Parabolic Cylinder functions for unrestricted values of the order a and the variable x except for the overflow/underflow limitations. By factoring the dominant exponential factor, scaled functions can be computed without practical overflow/underflow limitations. In an accompanying paper we describe the precise domains for these methods and we present the Fortran 90 codes for the computation of these functions.
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algorithm 850 real Parabolic Cylinder functions u a x v a x
Modelling Analysis and Simulation [MAS], 2006Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:textabstractFortran 90 programs for the computation of real Parabolic Cylinder functions are presented. The code computes the functions U(a, x), V (a, x) and their derivatives for real a and $x(xgeq 0)$. The code also computes scaled functions. The range of computation for scaled PCFs is practically unrestricted. The aimed relative accuracy for scaled functions is better than 5 10^{-14}. Exceptions to this accuracy are the evaluation of the functions near their zeros and the error caused by the evaluation of trigonometric functions of large arguments when |a| >> x. The routines always give values for which the Wronskian relation for scaled functions is verified with a relative accuracy better than $5 10^{?14}$. The accuracy of the unscaled functions is also better than $5 10 {?14}$ for moderate values of x and a (except close to the zeros), while for large x and a the error is dominated by exponential and trigonometric function evaluations. For IEEE standard double precision arithmetic, the accuracy is better than $5 10^{?13}$ in the computable range of unscaled PCFs (except close to the zeros).
Javier Segura - One of the best experts on this subject based on the ideXlab platform.
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uniform very sharp bounds for ratios of Parabolic Cylinder functions
arXiv: Classical Analysis and ODEs, 2020Co-Authors: Javier SeguraAbstract:Parabolic Cylinder functions (PCFs) are classical special functions with applications in many different fields. However, there is little information available regarding simple uniform approximations and bounds for these functions. We obtain very sharp bounds for the ratio $\Phi_n(x)=U(n-1,x)/U(n,x)$ and the double ratio $\Phi_n(x)/\Phi_{n+1}(x)$ in terms of elementary functions (algebraic or trigonometric) and prove the monotonicity of these ratios; bounds for $U(n,z)/U(n,y)$ are also made available. The bounds are very sharp as $x\rightarrow \pm \infty$ and $n\rightarrow +\infty$, and this simultaneous sharpness in three different directions explains their remarkable global accuracy. Upper and lower elementary bounds are obtained which are able to produce several digits of accuracy for moderately large $|x|$ and/or $n$.
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Comments on the paper "Universal bounds and monotonicity properties of ratios of Hermite and Parabolic Cylinder functions".
arXiv: Classical Analysis and ODEs, 2020Co-Authors: Javier SeguraAbstract:In the abstract of [1] we read: "We obtain so far unproved properties of a ratio involving a classof Hermite and Parabolic Cylinder functions." However, we explain how some of the main results in that paper were already proved in [2], namely the `universal bounds'. An error in reference [2] was discussed in [1] which does not affect the proof given there for those `universal bounds'; we fix this erratum easily. We end this note proposing a conjecture regarding the best possible upper bound for a certain ratio of Parabolic Cylinder functions.
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fast and accurate computation of the weber Parabolic Cylinder function w a x
Ima Journal of Numerical Analysis, 2011Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:Methods for the numerical evaluation of the Weber Parabolic Cylinder functions $W(a,\pm x)$, which are independent solutions of the inverted harmonic oscillator $y''+(x^2/4-a)y=0$, are described. The functions appear in the solution of many physical problems, and notably in quantum mechanics. It is shown that the combined use of Maclaurin series, Chebyshev series, uniform asymptotic expansions for large $a$ and/or $x$ and the integration of the differential equation by local Taylor series are enough for computing accurately the functions in a wide rage of parameters. Differently from previous methods, the computational scheme is stable in the sense that high accuracy is retained: only 2 or 3 digits may be lost in double precision computations.
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Algorithm 914: Parabolic Cylinder function W ( a , x ) and its derivative
ACM Transactions on Mathematical Software, 2011Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:A Fortran 90 program for the computation of the real Parabolic Cylinder functions W(a, ± x), x ≥ 0 and their derivatives is presented. The code also computes scaled functions for a > 50. The functions W(a, p x) are a numerically satisfactory pair of solutions of the Parabolic Cylinder equation y′ p (x2/4 − a)y e 0, x ≥ 0. Using Wronskian tests, we claim a relative accuracy better than 5 10−13 in the computable range of unscaled functions, while for scaled functions the aimed relative accuracy is better than 5 10−14. This code, together with the algorithm and related software described in Gil et al. [2006a, 2006b], completes the set of software for Parabolic Cylinder Functions (PCFs) for real arguments.
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algorithm 850 real Parabolic Cylinder functions u a x v a x
ACM Transactions on Mathematical Software, 2006Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:Fortran 90 programs for the computation of real Parabolic Cylinder functions are presented. The code computes the functions U(a, x), V(a, x) and their derivatives for real a and x (x ≥ 0). The code also computes scaled functions. The range of computation for scaled PCFs is practically unrestricted. The aimed relative accuracy for scaled functions is better than 5 10−14. Exceptions to this accuracy are the evaluation of the functions near their zeros and the error caused by the evaluation of trigonometric functions of large arguments when vav G x. The routines always give values for which the Wronskian relation for scaled functions is verified with a relative accuracy better than 5 10−14. The accuracy of the unscaled functions is also better than 5 10−14 for moderate values of x and a (except close to the zeros), while for large x and a the error is dominated by exponential and trigonometric function evaluations. For IEEE standard double precision arithmetic, the accuracy is better than 5 10−13 in the computable range of unscaled PCFs (except close to the zeros).
Amparo Gil - One of the best experts on this subject based on the ideXlab platform.
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fast and accurate computation of the weber Parabolic Cylinder function w a x
Ima Journal of Numerical Analysis, 2011Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:Methods for the numerical evaluation of the Weber Parabolic Cylinder functions $W(a,\pm x)$, which are independent solutions of the inverted harmonic oscillator $y''+(x^2/4-a)y=0$, are described. The functions appear in the solution of many physical problems, and notably in quantum mechanics. It is shown that the combined use of Maclaurin series, Chebyshev series, uniform asymptotic expansions for large $a$ and/or $x$ and the integration of the differential equation by local Taylor series are enough for computing accurately the functions in a wide rage of parameters. Differently from previous methods, the computational scheme is stable in the sense that high accuracy is retained: only 2 or 3 digits may be lost in double precision computations.
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Algorithm 914: Parabolic Cylinder function W ( a , x ) and its derivative
ACM Transactions on Mathematical Software, 2011Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:A Fortran 90 program for the computation of the real Parabolic Cylinder functions W(a, ± x), x ≥ 0 and their derivatives is presented. The code also computes scaled functions for a > 50. The functions W(a, p x) are a numerically satisfactory pair of solutions of the Parabolic Cylinder equation y′ p (x2/4 − a)y e 0, x ≥ 0. Using Wronskian tests, we claim a relative accuracy better than 5 10−13 in the computable range of unscaled functions, while for scaled functions the aimed relative accuracy is better than 5 10−14. This code, together with the algorithm and related software described in Gil et al. [2006a, 2006b], completes the set of software for Parabolic Cylinder Functions (PCFs) for real arguments.
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algorithm 850 real Parabolic Cylinder functions u a x v a x
ACM Transactions on Mathematical Software, 2006Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:Fortran 90 programs for the computation of real Parabolic Cylinder functions are presented. The code computes the functions U(a, x), V(a, x) and their derivatives for real a and x (x ≥ 0). The code also computes scaled functions. The range of computation for scaled PCFs is practically unrestricted. The aimed relative accuracy for scaled functions is better than 5 10−14. Exceptions to this accuracy are the evaluation of the functions near their zeros and the error caused by the evaluation of trigonometric functions of large arguments when vav G x. The routines always give values for which the Wronskian relation for scaled functions is verified with a relative accuracy better than 5 10−14. The accuracy of the unscaled functions is also better than 5 10−14 for moderate values of x and a (except close to the zeros), while for large x and a the error is dominated by exponential and trigonometric function evaluations. For IEEE standard double precision arithmetic, the accuracy is better than 5 10−13 in the computable range of unscaled PCFs (except close to the zeros).
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Computing the real Parabolic Cylinder functions U(a,x), V(a,x)
2006Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:htmlabstractMethods for the computation of real Parabolic Cylinder functions U(a,x) and V(a,x) and their derivatives are described. We give details on power series, asymptotic series, recursion and quadrature. A combination of these methods can be used for computing Parabolic Cylinder functions for unrestricted values of the order a and the variable x except for the overflow/underflow limitations. By factoring the dominant exponential factor, scaled functions can be computed without practical overflow/underflow limitations. In an accompanying paper we describe the precise domains for these methods and we present the Fortran 90 codes for the computation of these functions.
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algorithm 850 real Parabolic Cylinder functions u a x v a x
Modelling Analysis and Simulation [MAS], 2006Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:textabstractFortran 90 programs for the computation of real Parabolic Cylinder functions are presented. The code computes the functions U(a, x), V (a, x) and their derivatives for real a and $x(xgeq 0)$. The code also computes scaled functions. The range of computation for scaled PCFs is practically unrestricted. The aimed relative accuracy for scaled functions is better than 5 10^{-14}. Exceptions to this accuracy are the evaluation of the functions near their zeros and the error caused by the evaluation of trigonometric functions of large arguments when |a| >> x. The routines always give values for which the Wronskian relation for scaled functions is verified with a relative accuracy better than $5 10^{?14}$. The accuracy of the unscaled functions is also better than $5 10 {?14}$ for moderate values of x and a (except close to the zeros), while for large x and a the error is dominated by exponential and trigonometric function evaluations. For IEEE standard double precision arithmetic, the accuracy is better than $5 10^{?13}$ in the computable range of unscaled PCFs (except close to the zeros).
L. C. Lew Yan Voon - One of the best experts on this subject based on the ideXlab platform.
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Confined states in Parabolic Cylinder quantum dots
Physica E: Low-dimensional Systems and Nanostructures, 2003Co-Authors: Morten Willatzen, L. C. Lew Yan VoonAbstract:Abstract An infinite-barrier Parabolic cylindrical quantum dot is solved using Parabolic Cylinder coordinates. An additional condition, beyond the conventional boundary condition, is required in order to solve the eigenvalue problem. The volume dependence of the energy spectrum is shown to be different from spheroidal and elliptical dots and to depend strongly on the shape.
Raimundas Vidunas - One of the best experts on this subject based on the ideXlab platform.
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Parabolic Cylinder functions examples of error bounds for asymptotic expansions
Analysis and Applications, 2003Co-Authors: Nico M. Temme, Raimundas VidunasAbstract:Several asymptotic expansions of Parabolic Cylinder functions are discussed and error bounds for remainders in the expansions are presented. In particular, Poincare-type expansions for large values of the argument z and uniform expansions for large values of the parameter are considered. It is shown how expansions can be derived by using the differential equation, and, for a special case, how an integral representation can be used. The expansions are based on those given in Olver (1959) and on modifications of these expansions given in Temme (2000). Computer algebra techniques are used for obtaining representations of the bounds and for numerical computations.
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Parabolic Cylinder functions examples of error bounds for asymptotic expansions
arXiv: Classical Analysis and ODEs, 2002Co-Authors: Raimundas Vidunas, Nico M. TemmeAbstract:Several asymptotic expansions of Parabolic Cylinder functions are discussed and error bounds for remainders in the expansions are presented. In particular Poincar{\'e}-type expansions for large values of the argument $z$ and uniform expansions for large values of the parameter are considered. The expansions are based on those given in Olver(1959) and on modifications of these expansions given in Temme(2000). Computer algebra techniques are used for obtaining representations of the bounds and for numerical computations.
