Numerical Calculation of Bessel Functions

Schwartz, Charles

doi:10.1142/s0129183112500842

Cited by 9 publications

(11 citation statements)

References 5 publications

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“…The best ECG-based wave function, namely, the 1200 term function is accurate to 4 Â 10 À12 hartree. This is still much less accurate than calculations based on Hylleraas-type wave functions with logarithmic factors which are converged to more than 40 significant digits (Schwartz, 2006a(Schwartz, , 2006b; Nakashima and Nakatsuji, 2007).…”

Section: The Hydrogen Atommentioning

confidence: 84%

“…The general properties of the CI expansion have been known since the work of Schwartz (1962), which provided the underlying foundation for later investigations (Carroll, Silverstone, and Metzger, 1979;Hill, 1985;Ottschofski and Kutzelnigg, 1997;Bromley and Mitroy, 2007a). The orbital CI expansion is slowly convergent with respect to L max , the maximum angular momentum of any orbital included.…”

Section: Expansions In Products Of Orbitalsmentioning

confidence: 98%

“…In particular, extremely high accuracy (more than 40 digits for the energy) has been achieved for helium (Schwartz, 2006a(Schwartz, , 2006b. Not all approaches to the determination of the helium ground state wave function are equally useful.…”

Section: Other Types Of Basis Functions 1 Hylleraas and Related Fmentioning

confidence: 99%

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Theory and application of explicitly correlated Gaussians

et al. 2013

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The variational method complemented with the use of explicitly correlated Gaussian basis functions is one of the most powerful approaches currently used for calculating the properties of few-body systems. Despite its conceptual simplicity, the method offers great flexibility, high accuracy, and can be used to study diverse quantum systems, ranging from small atoms and molecules to light nuclei, hadrons, quantum dots, and Efimov systems. The basic theoretical foundations are discussed, recent advances in the applications of explicitly correlated Gaussians in physics and chemistry are reviewed, and the strengths and weaknesses of the explicitly correlated Gaussians approach are compared with other few-body techniques.

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Section: The Hydrogen Atommentioning

confidence: 84%

Section: Expansions In Products Of Orbitalsmentioning

confidence: 98%

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Theory and application of explicitly correlated Gaussians

et al. 2013

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“…The error also depends on the accuracy of the algorithm used by the computing software to compute the Bessel functions in Eq. (4) [35,36]. Fortunately, as we will show shortly, a synthesized spectrum that accurately matches the ground truth can be achieved with very small ranges of the integer indices, i.e., with a very small number of spectral components or terms.…”

Section: Practicality Of Our Analytical Modelmentioning

confidence: 85%

Theoretical Spectral Analysis of the Ir-Uwb Radar Chest Reflection With Arbitrary Periodic Breathing- And Heart-Induced Displacements

Nguyễn¹,

Weitnauer²

2016

PIER B

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Abstract-The moving chest wall imparts a delay modulation onto the reflected IR-UWB radar signal, making the radar return a nonlinear function of chest wall displacement. Existing theoretical spectral models of the IR-UWB radar reflections from the human chest wall have restrictive assumptions that preclude realistic modeling of chest wall displacement and assume an aliased version of the radar signal. This paper presents novel theoretical analysis of the un-aliased spectrum, without the restrictive assumptions. Potential applications not specifically treated in this paper, but illustrative of the novelty and broader scope of the presented analysis, include heart-induced displacements that are realistically more bursty in nature and breathing-induced displacements consistent with, for example, non-unity inspiration-to-expiration ratios characteristic of asthma; neither of these could be analyzed using previous models. As well known, the un-aliased spectrum cannot generally be recovered from the aliased spectrum. In particular, the paper shows that the clusters of the non-aliased spectrum are not just scaled versions of each other; rather they have complex variations that if measurable, could enable estimation of parameters such as displacement amplitude; such variations are not apparent in the existing aliased spectrum model, which has just one cluster. This paper analyzes the degree to which the aliased model differs from the non-aliased model. The paper also addresses some practical aspects of the spectral model, such as the number of significant components in a spectral cluster and computational complexity of the theoretical model.

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“…applied to integral representations of the various Bessel functions. [2] Here, h is the mesh size, which will be systematically reduced to show the convergence of the method; and the sum over mesh points n, nominally from −∞ to +∞, will be truncated when the contributions to the sum are below the desired accuracy. Now that general technique is applied to other members of the venerable family called the Special Functions of Mathematical Physics.…”

Section: Introductionmentioning

confidence: 99%