Double precision approximations to the elementary functions using Jacobi-fractions

Using single precision floating point representation reduces the size of data and computation time by a factor of two relative to double precision conventionally used in electronic structure programs. For large-scale calculations, such as those encountered in many-body theories, reduced memory footprint alleviates memory and input/output bottlenecks. Reduced size of data can lead to additional gains due to improved parallel performance on CPUs and various accelerators. However, using single precision can potentially reduce the accuracy of computed observables. Here we report an implementation of coupled-cluster and equation-of-motion coupled-cluster methods with single and double excitations in single precision. We consider both standard implementation and one using Cholesky decomposition or resolution-of-the-identity of electron-repulsion integrals. Numerical tests illustrate that when single precision is used in correlated calculations, the loss of accuracy is insignificant and pure single-precision implementation can be used for computing energies, analytic gradients, excited states, and molecular properties. In addition to pure single-precision calculations, our implementation allows one to follow a single-precision calculation by clean-up iterations, fully recovering double-precision results while retaining significant savings.

Download Full-text

Second-order integrable Lagrangians and WDVV equations

Letters in Mathematical Physics ◽

10.1007/s11005-021-01403-3 ◽

2021 ◽

Vol 111 (2) ◽

Author(s):

E. V. Ferapontov ◽

M. V. Pavlov ◽

Lingling Xue

Keyword(s):

Lagrangian Density ◽

Theta Functions ◽

Second Order ◽

Elementary Functions ◽

Lagrange Equations ◽

Wdvv Equations ◽

Integrability Conditions ◽

Jacobi Theta Functions

AbstractWe investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

Download Full-text

Quantum periods and TBA-like equations for a class of Calabi-Yau geometries

Journal of High Energy Physics ◽

10.1007/jhep01(2021)002 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Bao-ning Du ◽

Min-xin Huang

Keyword(s):

Large Class ◽

Bethe Ansatz ◽

Difference Equations ◽

The Other ◽

Elementary Functions ◽

Quantum Dilogarithm ◽

Thermodynamic Bethe Ansatz ◽

Dilogarithm Function ◽

Quantum Operators

Abstract We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.

Download Full-text

Series and Products for Elementary Functions

Series and Products in the Development of Mathematics ◽

10.1017/9781108709453.016 ◽

2021 ◽

pp. 365-395

Keyword(s):

Elementary Functions

Download Full-text

Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Symmetry ◽

10.3390/sym13020354 ◽

2021 ◽

Vol 13 (2) ◽

pp. 354

Author(s):

Alexander Apelblat ◽

Francesco Mainardi

Keyword(s):

Laplace Transform ◽

Operational Calculus ◽

Inverse Laplace Transform ◽

Elementary Functions ◽

Hyperbolic Functions ◽

Error Functions ◽

Convolution Integrals ◽

Special Case ◽

Integral Identities ◽

Finite Integrals

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

Download Full-text