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With the IBM Blue Gene/P supercomputer JUGENE, inaugurated in May 2009 at the Julich Supercomputing Centre (JSC), the Forschungszentrum Julich has entered the era of petaflop supercomputer performance in Europe. The Julich Supercomputing Centre (JSC), as a part of the Institute for Advanced Simulation within the Forschungszentrum Julich, operates supercomputers of the highest performance class in Europe. When IBM Blue Gene technology became available in 2004/2005, Forschungszentrum Julich recognized the potential of this architecture as a leadership-class system for capability computing applications. The lifetime of the largest Blue Gene/P in the world ended in mid 2012 after having been in operation for over three years and being superseded by an IBM Blue Gene/Q system. Blue Gene/P is an MPP type system with the design goal of achieving an optimal Performance/Watt ratio and extreme scalability. The JUGENE system in Julich consists of 72 Blue Gene/P racks, configured as 9 rows with 8 racks each, which sums up to 73,728 nodes.

Herein, we present a sequel to earlier work on a generalization of the Lambert W function. In particular, we examine series expansions of the generalized version providing computational means for evaluating this function in various regimes and further confirming the notion that this generalization is a natural nextension of the standard Lambert W function.

The efficient and reliable evaluation of infinite series is a frequently occurring problem. In many cases one is confronted with series expansions which converge so slowly that an evaluation by a direct summation of their terms would not be feasible, or even with series expansions which diverge. Consequently, alternative methods for their evaluation have to be used. In this article we want to report our experiences with two nonlinear convergence accelerators, the Shanks transformation and the u transformation of Levin. The Shanks transformation was designed to accelerate linear convergence whereas the u transformation is especially powerful for slowly converging monotonic series and for some converging or diverging alternating series. The numerical examples presented are drawn from molecular multicenter integrals and from special function theory.

Many scientific applications require the computation of about 10–30 % of the eigenvalues and eigenvectors of large dense symmetric or complex hermitian matrices. In this paper we will present performance evaluation results of the eigensolvers of the three libraries Elemental, ELPA, and ScaLAPACK on the BlueGene/Q architecture. All libraries include solvers for the computation of only a part of the spectrum. The most time-consuming part of the eigensolver is the reduction of the full eigenproblem to a tridiagonal one. Whereas Elemental and ScaLAPACK only offer routines to directly reduce the full matrix to a tridiagonal one, which only allows the use of BLAS 2 matrix-vector operations and needs a lot of communication, ELPA also offers a two-step reduction routine, first transforming the full matrix to banded form and thereafter to tridiagonal form. This two-step reduction shortens the reduction time significantly but at the cost of a higher complexity of the back transformation step. We will show up to which part of the eigenspectrum the use of the two-step reduction pays off.

In this paper we calculate the rate of linear convergence of non-linear transformed sequences. Using symbolic computation new results are derived which quantify the convergence acceleration effects of the Aitken transformation S, the iterated Aitken transformation S(2) and the Shanks transformation S2 for subclasses of linearly convergent sequences. The results are applied to linearly convergent fixed-point iteration methods, i.e. sequences {xn}n≥0 generated by xn+1 = φ(xn), where the iteration function φ(x) possesses a certain number of continuous derivatives. We verify the findings by numerical examples.

In this paper, we study the classifications of unbalanced data sets of drugs. As an example we chose a data set of 2D6 inhibitors of cytochrome P450. The human cytochrome P450 2D6 isoform plays a key role in the metabolism of many drugs in the preclinical drug discovery process. We have collected a data set from annotated public data and calculated physicochemical properties with chemoinformatics methods. On top of this data, we have built classifiers based on machine learning methods. Data sets with different class distributions lead to the effect that conventional machine learning methods are biased toward the larger class. To overcome this problem and to obtain sensitive but also accurate classifiers we combine machine learning and feature selection methods with techniques addressing the problem of unbalanced classification, such as oversampling and threshold moving. We have used our own implementation of a support vector machine algorithm as well as the maximum entropy method. Our feature selection is based on the unsupervised McCabe method. The classification results from our test set are compared structurally with compounds from the training set. We show that the applied algorithms enable the effective high throughput in silico classification of potential drug candidates.

Herein, we present analytical solutions for the electronic energy eigenvalues of the hydrogen molecular ion H2+, namely the one-electron two-fixed-center problem. These are given for the homonuclear case for the countable infinity of discrete states when the magnetic quantum number m is zero. In this case, these solutions are the roots of a set of two coupled three-term recurrence relations. The eigensolutions are obtained from an application of EXPERIMENTAL MATHEMATICS using Computer Algebra as its principal tool and are vindicated by numerical and algebraic demonstrations. Finally, the mathematical nature of the eigenenergies is identified. The eigenenergies are related to a generalization of the Lambert W function., This is an analytical breakthrough for a special case of the quantum 3-body problem. The results have been published in Chem. Phys. and an internal report at the Forschungszentrum Juelich (Germany). (see references)

We have analyzed and reduced a general (quantum-mechanical) expression for the atom-atom exchange energy formulated as a five-dimensional surface integral, which arises in studying the charge exchange processes in diatomic molecules. It is shown that this five-dimensional surface integral can be decoupled into a three-dimensional integral and a two-dimensional angular integral which can be solved analytically using a special decomposition. Exact solutions of the two-dimensional angular integrals are presented and generalized. Algebraic aspects, invariance properties and exact solutions of integrals involving Legendre and Chebyshev polynomials are also discussed.

We present a general procedure. based on the Holstein-Herring method. for calculating exactly the leading term in the exponentially small exchange, energy splitting between two asymptotically degenerate states of a diatomic molecule or molecular ion. The general formulae we have derived are shown to reduce correctly to the previously known exact results for the specific cases of the lowest Sigma and Pi states of H-2(+). We then apply our general formulae 2 to calculate the exchange energy splittincys between the lowest states of the diatomic alkali cations K-2(+), Rb-2(+) and Cs-2, which are isovalent to H-2(+). Our results are found to be in very good agreement with the best available experimental data and ab initio calculations.

Different methods for accelerating convergence of known series expansions for the auxiliary functions Fm(t) encountered in molecular integrals over Gaussian‐type orbitals have been tested and compared. We have investigated the numerical properties of the Shanks transformation and the Levin transformations. It turns out that Levin’s t transformation is the best accelerator, whereas the Shanks transformation, which nearly yields the same results as the continued‐fraction method, is less effective.

We analyze the analytical and numerical properties of the hitherto known formulas of the Fourier transform of a two-center product of exponentially declining functions (exponential-type functions, ETFs) which are derived with the help of the Fourier convolution theorem and Feynman's identity. In detail, we consider B functions which are a special class of ETFs having advantageous properties under Fourier transformation. Other ETFs (orbitals) can be expressed in terms of B functions by linear combinations. In our derivations we use the properties of the differential operator Ylm(▽) specifying a solid harmonic whose argument is the nabla operator ∂/∂r instead of the vector r in order to generate multicenter integrals over nonscalar functions from integrals over scalar functions. Applying the generating differential operator Ylm(Λ) we obtain a recently derived new formula for the Fourier transform of a two-center product of B functions in a much more straightforward manner. Furthermore, we present an efficient procedure to compute this new formula which is valid for arbitrary quantum numbers and exponential parameters and report various numerical test values.

In this paper a unified analytical treatment of overlap, two-center nuclear attraction, and Coulomb integrals of B functions [E. Filter and E. O. Steinborn, Phys. Rev. A 18, 1 (1978)] via the Fourier-transform method is presented. B functions, which are a special class of exponentially decreasing functions (for large arguments), have a relatively complicated analytical structure. However, the Fourier transform of a B function is of exceptional simplicity. Consequently, it is relatively easy to express the Fourier integral representations of the two-center integrals mentioned above as finite sums or infinite series of Fourier integral representations for B functions and irregular solid harmonics which may be considered to be limiting cases of special B functions. The only advanced mathematical concepts which we need are the connection between B functions, classically divergent Fourier integrals, and derivatives of the three-dimensional \ensuremath{\delta} functions. The other mathematical tools---partial-fraction decompositions and Taylor expansions of rational functions---are fairly elementary. Our approach leads not only to a considerable simplification of the derivation of the previously known analytical representations for the two-center integrals but also to a large number of hitherto unknown representations.

The method of Bonham, Peacher, and Cox for computing molecular multicenter integrals for 1s Slater-type orbitals is generalized to include all states. This was possible by using B functions as basis functions which have the simplest structure under Fourier transformation, compared with other commonly used exponential-type orbitals (ETO's). Those ETO's which differ from B functions, like Slater-type orbitals (STO's), can be expressed by finite linear combinations of B functions. Therefore multicenter integrals occurring in molecular calculations with any of the commonly used ETO basis sets can be represented by integrals with B functions. In the present paper the three-center nuclear attraction integrals and the two-electron multicenter integrals with B functions are evaluated in a unified way via the Fourier-transform method and Feynman's identity. The resulting expressions require a two- or three-dimensional numerical integration, respectively. The numerical and computational properties of the resulting formulas are discussed and various test values are given. Comparison is made with some values of integrals with STO's which exist in the literature.