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This article considers the computation of the matrix exponential eA{{\rm{e}}}^{A} with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore, it is necessary to truncate the infinite interval. In this article, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates eA{{\rm{e}}}^{A} with the fixed mesh size, which is a parameter in the DE formula affecting the accuracy. The second one computes eA{{\rm{e}}}^{A} based on the first one with automatic selection of the mesh size depending on the given error tolerance.

In this paper the authors present highly accurate and remarkably efficient computational methods for fractional order derivatives and integrals applying Riemann-Liouville and Caputo formulae: the Gauss-Jacobi Quadrature with adopted weight function, the Double Exponential Formula, applying two arbitrary precision and exact rounding mathematical libraries (GNU GMP and GNU MPFR). Example fractional order derivatives and integrals of some elementary functions are calculated. Resulting accuracy is compared with accuracy achieved by applying widely known methods of numerical integration. Finally, presented methods are applied to solve Abel's Integral equation (in Appendix). [ABSTRACT FROM AUTHOR]

A financial institution that adopts an advanced measurement approach (AMA) as a method of computing operational risk capital has to measure 99.9 % value-at-risk (VaR) as the amount of an operational risk. The most popular method to satisfy the AMA standards requires the evaluation of aggregate (compound) loss distribution, which is called the loss distribution approach (LDA). The Monte Carlo (MC) method is a well known method for calculating VaR under the LDA. However, when using the MC method to calculate VaR, the statistical error of VaR for the fat-tailed distribution increases and the computation time increases in proportion to the expected value of frequency distribution. Since the MC method has these problems, this paper presents a new methodology to compute VaR under the LDA using fast wavelet expansion techniques. The key features of our algorithm are follows: (1) Scale transformation technique for loss distributions, (2) Double exponential transformation for oscillatory integrals, (3) Finite series expansion of the wavelet scaling coefficients, (4) Wynn's epsilon algorithm to accelerate the convergence of those series, (5) Efficient cubic spline interpolation method to calculate the moment generating function. We illustrate the effectiveness of our algorithms through numerical examples. [ABSTRACT FROM AUTHOR]

In the preceding study, we reported an application of the double exponential formula to the radial quadrature grid for numerical integration of the radial electron distribution function. Three-type new radial grids with the double exponential transformation were introduced. The performance of radial grids was compared between the double exponential grids and the grids proposed in earlier studies by applying to the electron-counting integrals of noble gas atoms and diatomic molecules including alkali metals, halogens, and transition metals. It was confirmed that the change in accuracy of the quadrature approximation depending on atomic or molecular species is not significant for the double exponential integration schemes rather than the other integration schemes. In the present study, we further investigate the accuracy of the double exponential formula for the electron-counting integrals of all the atoms from H to Kr in the periodic table to elucidate the stable performance of the double exponential radial grids. The electron densities of the atoms are calculated with the Gauss-type orbital basis functions at the B3LYP level. The quadrature accuracy and convergence behavior of numerical integration are compared among the double exponential formula and the formulas proposed by Treutler et al. and by Mura et al. The results reveal that the double exponential radial grids remarkably improve the convergence rate toward high accuracy compared with the previous radial grids, particularly for heavy elements in the 4th period, without fine tuning of the radial grids for each atom. [ABSTRACT FROM AUTHOR]

AC loses in high T~ superconductor (HIS) thin film are evaluated by element-free Galerkin method with double exponential formula. The shielding current density in an HTS is necessary information to calculate the AC loss. Formulation of the electromagnetic behavior of the shielding current density gives a system. However, improper integrals appear as coefficients of the system and the integrand have a stronger singularity with decreasing thickness of the HTS. The code for the shielding current analysis in HIS by element-free Galerkin method is developed. In order to evaluate the improper integrals accurately, the double exponential formula is adopted in the code. Results of the computation shows that the spatial distribution of the shielding current density is calculated with a high accuracy. [ABSTRACT FROM AUTHOR]

Abstract: We propose an IMT-type quadrature formula which achieves the same asymptotic error estimate as the DE formula. The point of the idea is to optimize the parameters of the IMT-type transformation depending on the number of sampling points. We also show the performance of our IMT-type quadrature formula by numerical examples. [Copyright &y& Elsevier]

A formula for numerical evaluation of two dimensional iterated integrals of the form I = ∫ dx ∫ f( x,y) dy where q( a) = a’, q( b) = b’ ( a < x < b) is derived by means of the sinc approximation based on the double exponential transformation. The integrand ƒ(x,y) is assumed to be analytic as a function of x on a < x < b and also of y on a’ < y < b’, and q(x) is assumed to be an analytic function of x on a < x < b. The order of the error of the formula derived is O (exp(−βN/ log(γN))) as a function of $$N = \sqrt {n_{total} } /2$$ where n total is the total number of function evaluations. Numerical examples also proves high efficiency of the formula. When the integrand is of a product type, i.e. ƒ(x,y) = X(x)Y(y), we can obtain an approximate value of I by evaluating only 2 × (2N + 1) values (X(xj),−N ≤ j ≤ N), (Y(yk), −N ≤ k ≤ N), and hence the number of function evaluations is reduced to O(4N). [ABSTRACT FROM AUTHOR]

We propose using the circle polynomials to describe a particle’s transmission function in a digital holography setup. This allows both opaque and phase particles to be determined. By means of this description, we demonstrate that it is possible to estimate the digital in-line hologram produced by a spherical particle. The experimental intensity distribution due to an opaque micro-inclusion is compared to the theoretical one obtained by our new model. Moreover, the simulated hologram and reconstructed image of the particle by an optimal fractional Fourier transformation under the opaque disk, quadratic phase, and quasi-spherical phase approximation are compared with the results obtained by simulating holograms by the Lorenz–Mie Theory (LMT). The Zernike coefficients corresponding to the considered particles are evaluated using the double exponential (DE) method which is optimal in various respects.

We consider the computation of the matrix logarithm by using numerical quadrature. The efficiency of numerical quadrature depends on the integrand and the choice of quadrature formula. The Gauss–Legendre quadrature has been conventionally employed; however, the convergence could be slow for ill-conditioned matrices. This effect may stem from the rapid change of the integrand values. To avoid such situations, we focus on the double exponential formula, which has been developed to address integrands with endpoint singularity. In order to utilize the double exponential formula, we must determine a suitable finite integration interval, which provides the required accuracy and efficiency. In this paper, we present a method for selecting a suitable finite interval based on an error analysis as well as two algorithms, and one of these algorithms is an adaptive quadrature algorithm. [ABSTRACT FROM AUTHOR]