Algorithms for geodesics

Algorithms for the computation of geodesics on an ellipsoid of revolution are given. These provide accurate, robust, and fast solutions to the direct and inverse geodesic problems and they allow differential and integral properties of geodesics to be computed.Comment: LaTex, 12 pages, 8 figures. Version 2 corrects some errors and adds numerical examples. Supplementary material is available at http://geographiclib.sourceforge.net/geod.htm

Generalized exponential and logarithmic functions

AbstractGeneralizations of the exponential and logarithmic functions are defined and an investigation is made of two possible versions of these functions. Some applications are described, including computer arithmetic, properties of very large and very small numbers, and the determination of functional roots

High order analysis of the limit cycle of the van der Pol oscillator

We have applied the Lindstedt-Poincaré method to study the limit cycle of the van der Pol oscillator, obtaining the numerical coefficients of the series for the period and for the amplitude to order 859. Hermite-Padé approximants have been used to extract the location of the branch cut of the series with unprecedented accuracy (100 digits). Both series have then been resummed using an approach based on Padé approximants, where the exact asymptotic behaviors of the period and the amplitude are taken into account. Our results improve drastically all previous results obtained on this subject.Fil: Amore, Paolo. Universidad de Colima; MéxicoFil: Boyd, John P.. University of Michigan; Estados UnidosFil: Fernández, Francisco Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas; Argentin

A Model of Temporal Intensity Modulation for Laser Generated Ultrasound

Q-switched lasers are often used as a non-contact ultrasound source in non-destructive evaluation (NDE) of materials [1]. Q-switched lasers typically have ns pulse durations and generate broadband ultrasound waves, though longer laser pulses, of 100 microseconds or greater, have also been used [2] for NDE. These longer pulses tend to produce somewhat lower center frequencies than do Q-switched pulses, though they are still a broadband source. But it would be desirable in some NDE applications to narrow the signal bandwidth to improve the signal to noise ration (SNR), and also to have direct control over the center frequency of the generated ultrasound. In principle, this may be achieved by temporal [3,4] or spatial modulation [5,6] of the laser pulse, or both [7]. The purpose of this work was to develop a numerical model of a single, temporally modulated laser source of ultrasound in the thermoelastic regime, for isotropic metals

Numerical Approximations Using Chebyshev Polynomial Expansions

We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the Chebyshev polynomial of order N (El-gendi's method). The solutions are exact at these points, apart from round-off computer errors and the convergence of other numerical methods used in connection to solving the linear system of equations. Applications to initial value problems in time-dependent quantum field theory, and second order boundary value problems in fluid dynamics are presented.Comment: minor wording changes, some typos have been eliminate

Semi-spectral Chebyshev method in Quantum Mechanics

Traditionally, finite differences and finite element methods have been by many regarded as the basic tools for obtaining numerical solutions in a variety of quantum mechanical problems emerging in atomic, nuclear and particle physics, astrophysics, quantum chemistry, etc. In recent years, however, an alternative technique based on the semi-spectral methods has focused considerable attention. The purpose of this work is first to provide the necessary tools and subsequently examine the efficiency of this method in quantum mechanical applications. Restricting our interest to time independent two-body problems, we obtained the continuous and discrete spectrum solutions of the underlying Schroedinger or Lippmann-Schwinger equations in both, the coordinate and momentum space. In all of the numerically studied examples we had no difficulty in achieving the machine accuracy and the semi-spectral method showed exponential convergence combined with excellent numerical stability.Comment: RevTeX, 12 EPS figure

The variable phase method used to calculate and correct scattering lengths

It is shown that the scattering length can be obtained by solving a Riccati equation derived from variable phase theory. Two methods of solving it are presented. The equation is used to predict how long-range interactions influence the scattering length, and upper and lower bounds on the scattering length are determined. The predictions are compared with others and it is shown how they may be obtained from secular perturbation theory.Comment: 7 pages including 3 figure

Ultrabithorax confers spatial identity in a context-specific manner in the Drosophila postembryonic ventral nervous system.

BACKGROUND: In holometabolous insects such as Drosophila melanogaster, neuroblasts produce an initial population of diverse neurons during embryogenesis and a much larger set of adult-specific neurons during larval life. In the ventral CNS, many of these secondary neuronal lineages differ significantly from one body segment to another, suggesting a role for anteroposterior patterning genes. RESULTS: Here we systematically characterize the expression pattern and function of the Hox gene Ultrabithorax (Ubx) in all 25 postembryonic lineages. We find that Ubx is expressed in a segment-, lineage-, and hemilineage-specific manner in the thoracic and anterior abdominal segments. When Ubx is removed from neuroblasts via mitotic recombination, neurons in these segments exhibit the morphologies and survival patterns of their anterior thoracic counterparts. Conversely, when Ubx is ectopically expressed in anterior thoracic segments, neurons exhibit complementary posterior transformation phenotypes. CONCLUSION: Our findings demonstrate that Ubx plays a critical role in conferring segment-appropriate morphology and survival on individual neurons in the adult-specific ventral CNS. Moreover, while always conferring spatial identity in some sense, Ubx has been co-opted during evolution for distinct and even opposite functions in different neuronal hemilineages

Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

Novel memory-efficient Arnoldi algorithms for solving matrix polynomial eigenvalue problems are presented. More specifically, we consider the case of matrix polynomials expressed in the Chebyshev basis, which is often numerically more appropriate than the standard monomial basis for a larger degree $d$. The standard way of solving polynomial eigenvalue problems proceeds by linearization, which increases the problem size by a factor $d$. Consequently, the memory requirements of Krylov subspace methods applied to the linearization grow by this factor. In this paper, we develop two variants of the Arnoldi method that build the Krylov subspace basis implicitly, in a way that only vectors of length equal to the size of the original problem need to be stored. The proposed variants are generalizations of the so called Q-Arnoldi and TOAR methods, which have been developed for the monomial case. We also show how the typical ingredients of a full implementation of the Arnoldi method, including shift-and-invert and restarting, can be incorporated. Numerical experiments are presented for matrix polynomials up to degree $30$ arising from the interpolation of nonlinear eigenvalue problems which stem from boundary element discretizations of PDE eigenvalue problems.Daniel Kressner; Román Moltó, JE. (2014). Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis. Numerical Linear Algebra with Applications. 21(4):569-588. doi:10.1002/nla.1913S569588214Mackey, D. S., Mackey, N., Mehl, C., & Mehrmann, V. (2006). Vector Spaces of Linearizations for Matrix Polynomials. SIAM Journal on Matrix Analysis and Applications, 28(4), 971-1004. doi:10.1137/050628350Mackey, D. S., Mackey, N., Mehl, C., & Mehrmann, V. (2006). Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations. SIAM Journal on Matrix Analysis and Applications, 28(4), 1029-1051. doi:10.1137/050628362Higham, N. J., Mackey, D. S., & Tisseur, F. (2006). The Conditioning of Linearizations of Matrix Polynomials. SIAM Journal on Matrix Analysis and Applications, 28(4), 1005-1028. doi:10.1137/050628283Adhikari, B., Alam, R., & Kressner, D. (2011). Structured eigenvalue condition numbers and linearizations for matrix polynomials. Linear Algebra and its Applications, 435(9), 2193-2221. doi:10.1016/j.laa.2011.04.020Bai, Z., & Su, Y. (2005). SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem. SIAM Journal on Matrix Analysis and Applications, 26(3), 640-659. doi:10.1137/s0895479803438523Meerbergen, K. (2009). The Quadratic Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem. SIAM Journal on Matrix Analysis and Applications, 30(4), 1463-1482. doi:10.1137/07069273xLin, Y., Bao, L., & Wei, Y. (2010). Model-order reduction of large-scalekth-order linear dynamical systems via akth-order Arnoldi method. International Journal of Computer Mathematics, 87(2), 435-453. doi:10.1080/00207160802130164Stewart, G. W. (2001). Matrix Algorithms. doi:10.1137/1.9780898718058Kamiya, N., Andoh, E., & Nogae, K. (1993). Eigenvalue analysis by the boundary element method: new developments. Engineering Analysis with Boundary Elements, 12(3), 151-162. doi:10.1016/0955-7997(93)90011-9Bindel D Hood A Localization theorems for nonlinear eigenvalues. arXiv: 1303.4668 2013Botchev, M. A., Sleijpen, G. L. G., & Sopaheluwakan, A. (2009). An SVD-approach to Jacobi–Davidson solution of nonlinear Helmholtz eigenvalue problems. Linear Algebra and its Applications, 431(3-4), 427-440. doi:10.1016/j.laa.2009.03.024Effenberger, C., & Kressner, D. (2012). Chebyshev interpolation for nonlinear eigenvalue problems. BIT Numerical Mathematics, 52(4), 933-951. doi:10.1007/s10543-012-0381-5Van Beeumen, R., Meerbergen, K., & Michiels, W. (2013). A Rational Krylov Method Based on Hermite Interpolation for Nonlinear Eigenvalue Problems. SIAM Journal on Scientific Computing, 35(1), A327-A350. doi:10.1137/120877556Sitton, G. A., Burrus, C. S., Fox, J. W., & Treitel, S. (2003). Factoring very-high-degree polynomials. IEEE Signal Processing Magazine, 20(6), 27-42. doi:10.1109/msp.2003.1253552Amiraslani, A., Corless, R. M., & Lancaster, P. (2008). Linearization of matrix polynomials expressed in polynomial bases. IMA Journal of Numerical Analysis, 29(1), 141-157. doi:10.1093/imanum/drm051Betcke, T., & Kressner, D. (2011). Perturbation, extraction and refinement of invariant pairs for matrix polynomials. Linear Algebra and its Applications, 435(3), 514-536. doi:10.1016/j.laa.2010.06.029Beyn, W. J., & Thümmler, V. (2010). Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications, 31(3), 1361-1381. doi:10.1137/080723107Kressner, D. (2009). A block Newton method for nonlinear eigenvalue problems. Numerische Mathematik, 114(2), 355-372. doi:10.1007/s00211-009-0259-xLehoucq, R. B., Sorensen, D. C., & Yang, C. (1998). ARPACK Users’ Guide. doi:10.1137/1.9780898719628Hernandez, V., Roman, J. E., & Vidal, V. (2005). SLEPc. ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019Clenshaw, C. W. (1955). A note on the summation of Chebyshev series. Mathematics of Computation, 9(51), 118-118. doi:10.1090/s0025-5718-1955-0071856-0Stewart, G. W. (2002). A Krylov--Schur Algorithm for Large Eigenproblems. SIAM Journal on Matrix Analysis and Applications, 23(3), 601-614. doi:10.1137/s0895479800371529Su Y A compact Arnoldi algorithm for polynomial eigenvalue problems 2008 http://math.cts.nthu.edu.tw/Mathematics/RANMEP%20Slides/Yangfeng%20Su.pdfSteinbach, O., & Unger, G. (2009). A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator. Numerische Mathematik, 113(2), 281-298. doi:10.1007/s00211-009-0239-1Effenberger, C., Kressner, D., Steinbach, O., & Unger, G. (2012). Interpolation-based solution of a nonlinear eigenvalue problem in fluid-structure interaction. PAMM, 12(1), 633-634. doi:10.1002/pamm.201210305Betcke, T., Higham, N. J., Mehrmann, V., Schröder, C., & Tisseur, F. (2013). NLEVP. ACM Transactions on Mathematical Software, 39(2), 1-28. doi:10.1145/2427023.2427024Grammont, L., Higham, N. J., & Tisseur, F. (2011). A framework for analyzing nonlinear eigenproblems and parametrized linear systems. Linear Algebra and its Applications, 435(3), 623-640. doi:10.1016/j.laa.2009.12.03

Polarizability of conducting sphere-doublets using series of images.

The classical electrostatic problem of two nonintersecting conducting spheres in a uniform incident electric field is considered. Starting from the basic Kelvin’s image principle, the two spheres are replaced with equivalent series of image sources, from which the polarizability is calculated. Explicit expressions for the axial and transversal components of the polarizability dyadic are found by solving the recurrence equations. Efficient numerical evaluation of the different series is also discussed.Peer reviewe