reconstruction in diffraction ultrasound tomography using nonuniform fft we show an iterative reconstruction framework for diffraction ultrasound tomography the use of broad band illumination allows significant reduction of the number of projections compared to straight ray tomography the proposed algorithm makes use of forward nonuniform fast fourier transform nufft for iterative fourier inversion incorporation of total variation regularization allows the reduction of noise and gibbs phenomena while preserving the edges the complexity of the nufft based reconstruction is comparable to the frequency domain interpolation gridding algorithm whereas the reconstruction accuracy in sense of the l sup 2 and the l sup spl infin norm is better

fast computation of the spectral differentiation by the fast multipole method abstract this paper is concerned with the application of the fast multipole method fmm to the pseudospectral methods the spectral differentiation in pseudospectral methods involves dense matrix vector products which if evaluated directly entail o n 2 operations the core of this work lies in the fact that the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products the fmm is exploited to estimate the barycentric weights and the pseudospectral differentiation which reduces the complexity to o n furthermore we show the convergence analysis and present the applications to the second order boundary value problems numerical examples confirm the advantages of the method

weighted inductive means abstract in this paper we present a unified framework for weighted inductive means on the cone p of positive definite hermitian matrices as natural multivariable extensions of two variable weighted means particularly of metric midpoint operations on p it includes some well known multivariable weighted matrix means the weighted arithmetic harmonic resolvent sturm s inductive geometric mean on the riemannian manifold p equipped with the trace metric log euclidean and spectral geometric means a recursion or weight additive formula is derived and applied to find a closed form and basic properties for a weighted inductive mean an upper bound on the sensitivity a metric characterization and min and max optimization problems over permutations for the inductive geometric mean are presented moreover we apply the obtained results to a class of midpoint operations of the non positively curved hadamard metrics on p parameterized over hermitian unitary matrices

some numerical considerations in h control abstract certain h control problems are considered and potential numerical difficulties are discussed an alternative test for checking that the solution to an algebraic riccati equation is positive semi definite which is robust to rounding error even when the matrix is singular is presented finally the h loop shaping method is summarised and its numerical properties are shown to be satisfactory

existence theorems for positive semidefinite and sign indefinite stabilizing solutions of h riccati equations the existence of both positive semidefinite and sign indefinite stabilizing solutions of h infty type riccati equations is determined via joint eigenvalue and hamiltonian tests these tests determine the inertia properties of the solutions they are dependent on a formula derived in this paper that shows how the solution of game riccati equations can be decomposed into a sum of two well known riccati equations namely the standard filter riccati equation and the bounded real riccati equation other properties of stabilizing solutions such as solution partial order spectral radius monotonicity and rank are also discussed

fast algorithms for discrete polynomial transforms on arbitrary grids abstract consider the vandermonde like matrix p p k x m l l k 0 m n where the polynomials p k satisfy a three term recurrence relation and x m l 1 1 are arbitrary nodes if p k are the chebyshev polynomials t k then p coincides with a t k x m l l 0 k 0 m n this paper presents a fast algorithm for the computation of the matrix vector product p a in o n log 2 n arithmetical operations the algorithm divides into a fast transform which replaces p a with a a and a fast cosine transform on arbitrary nodes ndct since the first part of the algorithm was considered in math comp 67 1998 1577 we focus on approximative algorithms for the ndct our considerations are completed by numerical tests

the inverse mean problem of geometric mean and contraharmonic means abstract in this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices proposed in w n anderson m e mays t d morley g e trapp the contraharmonic mean of hsd matrices siam j algebra disc meth 8 1987 674 682 a x y 2 x 1 y 1 1 b x y by proving its equivalence to the well known nonlinear matrix equation x t bx 1b where t 1 2 a a a 8 ba 1 b is the unique positive definite solution of x a 2bx 1b the inverse mean problem is solvable if and only if b a

rapid computation of the discrete fourier transform algorithms for the rapid computation of the forward and inverse discrete fourier transform for points which are nonequispaced or whose number is unrestricted are presented the computational procedure is based on approximation using a local taylor series expansion and the fast fourier transform fft the forward transform for nonequispaced points is computed as the solution of a linear system involving the inverse fourier transform this latter system is solved using the iterative method gmres with preconditioning numerical results are given to confirm the efficiency of the algorithms

an extended lie trotter formula and its applications abstract in this paper we present a class of lie trotter formulae for hermitian operators including the formulae derived by hiai petz and furuta a lie trotter formula for weighted log euclidean geometric means of several positive definite operators is given in terms of sagae tanabe geometric and spectral geometric means

random fibonacci sequences solutions to the random fibonacci recurrence xn 1 xn xn 1 decrease increase exponentially xn exp n for sufficiently small large in the limits 0 and we expand the lyapunov exponent in powers of and 1 respectively for the classical case of 1 we obtain exact non perturbative results in particular an invariant measure associated with ricatti variable rn xn 1 xn is shown to exhibit plateaux around all rational r

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

estimating generalized lyapunov exponents for products of random matrices we discuss several techniques for the evaluation of the generalized lyapunov exponents which characterize the growth of products of random matrices in the large deviation regime a monte carlo algorithm that performs importance sampling using a simple random resampling step is proposed as a general purpose numerical method which is both efficient and easy to implement alternative techniques complementing this method are presented these include the computation of the generalized lyapunov exponents by solving numerically an eigenvalue problem and some asymptotic results corresponding to high order moments of the matrix products taken together the techniques discussed in this paper provide a suite of methods which should prove useful for the evaluation of the generalized lyapunov exponents in a broad range of applications their usefulness is demonstrated on particular products of random matrices arising in the study of scalar mixing by complex fluid flows

a renormalization approach to the universality of scaling in phyllotaxis abstract phyllotaxis i e the arrangement of plant organs like leaves florets scales bracts etc around a shoot stem or cone is often highly regular across the plant kingdom phyllotaxis shows not only qualitatively but also quantitatively identical features like the occurrence of divergence angles close to noble irrationals in a previous study reick 2012 a mechanism has been identified that explains the selection of these particular divergence angles on the basis of self similarity and scaling numerically found in the bifurcation diagrams of simple dynamical models of phyllataxis in the present paper by constructing a renormalization theory the universality of this scaling is proved for a whole class of models prototypically represented by thornley s model of phyllotaxis thornley 1975 the renormalization is constructed from another self similarity found numerically for the fourier transform of the abstract potential governing the mutual inhibition of primordia surprisingly the resulting renormalization transformation is already known from the treatment of the quasiperiodic transition to chaos but operates here on a different function space it turns out that the fixed points of the renormalization transformation are characterized by divergences of the form 1 where written as continued fraction n to show the universality of the scaling it is demonstrated that the fixed points are unstable and that the associated scaling factors 2 and are exactly those that were numerically found in reick 2012 to rule the selfsimilarity of the bifurcation structure thereby the present paper puts forward an explanation for the universal appearance of certain phyllotactic patterns that is independent of physiological detail of plant growth

a fast algorithm for chebyshev fourier and sinc interpolation onto an irregular grid a chebyshev or fourier series may be evaluated on the standard collocation grid by the fast fourier transform fft unfortunately the fft does not apply when one needs to sum a spectral series at n points which are spaced irregularly the cost becomes o n2 operations instead of the ffts o n log n this sort of off grid interpolation is needed by codes which dynamically readjust the grid every few time steps to resolve a shock wave or other narrow features it is even more crucial to semi lagrangian spectral algorithms for solving convection diffusion and navier stokes problems because off grid interpolation must be performed several times per time step in this work we describe an alternative algorithm the first step is to pad the set of spectral coefficients an with zeros and then take an fft of length 3n to interpolate the chebyshev series to a very fine grid the second step is to apply either the mth order euler sum acceleration or 2m 1 point lagrangian interpolation to approximate the sum of the series on the irregular grid we show that both methods yield full precision with m n allowing an order of magnitude reduction in cost with no loss of accuracy

lie trotter means of positive definite operators abstract as extension of the lie trotter product formula we define the two variable and multivariate lie trotter means with several examples including the sagae tanabe and hansen inductive means we show that the weighted means satisfying the arithmetic geometric harmonic mean inequalities are the multivariate lie trotter means we finally give a few interesting versions of the extended lie trotter formula derived from multivariate lie trotter means

singular behaviour of certain infinite products of random 2 2 matrices the authors consider an infinite product of random matrices which appears in several physical problems in particular the ising chain in a random field the random matrices depend analytically on a parameter epsilon in such a way that for 0 they all commute under certain conditions they find that the lyapunov index of this product behaves approximately as c 2 for to 0 their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations they give an expression for the exponent alpha in terms of the probability distribution of the matrices and for the proportionality constant c in terms of the solutions of the reduced integral equations in cases where exact results are available agreement is obtained the physical consequences for disordered one dimensional systems are pointed out

computation of growth rates of random sequences with multi step memory we extend the generating function approach to the computation of growth rate of random fibonacci sequences with long memory functional iteration equations are obtained and its general form is conjectured and proved based on which an asymptotic representation of the growth rate is obtained the validity of both the derived and the conjectured formula are verified upon comparison with monte carlo simulation a numerical scheme of the functional iteration is designed and implemented successfully

computing yapunov constants for random recurrences with smooth coefficients in recent years there has been much interest in the growth and decay rates lyapunov constants of solutions to random recurrences such as the random fibonacci sequence x n 1 x n x n 1 many of these problems involve nonsmooth dynamics nondi1erentiable invariant measures making computations hard here however we consider recurrences with smooth random coe cients and smooth invariant measures by computing discretised invariant measures and applying richardson extrapolation we can compute lyapunov constants to 10 digits of accuracy in particular solutions to the recurrence x n 1 x n cn 1x n 1 where the cn are independent standard normal variables increase exponentially almost surely at the asymptotic rate 1 0574735537 n solutions to the related recurrences x n 1 cn 1x n x n 1 and x n 1 cn 1x n dn 1x n 1 where the dn are also independent standard normal variables increase decrease at the rates 1 1149200917 n and 0 9949018837 n respectively c 2001 elsevier science b v all rights reserved

strict contractions of symmetric cones let v be a euclidean jordan algebra with an associative inner product hji and let be the corresponding symmetric cone then the linear automorphism group h of the symmetric cone acts transitively on it let t v i be the symmetric tube domain associated to the symmetric cone and let g t be the biholomorphic automorphism group of t there exists a conformal compactification m g t p of v with respect to a parabolic subgroup p a lie semigroup that is naturally related to the symmetric cone occurs as the semigroup of compressionsof in the lie group g t fg 2 g t j g g it turns out that admits an ol shanskii decomposition h expc for an ad h invariant convex cone c in the lie algebra g t of g t a triple decomposition h 8 9 and it determines a globally hyperbolic causal order on the homogeneous space g t h defined by ah bh if and only if a 1 b 2 the family of bilinear forms x u v hp x 1 ujvi defines a hinvariant riemannian metric on where the map p is the quadratic representation of the jordan algebra v in 8 koufany showed that the elements of are contractions of for the riemannian metric the main result of this paper is that the elements of the interior of called strict compressions are strict contractions of for the riemannian distance induced by

a robust numerical method for the γ iteration in h control abstract we present a numerical method for the solution of the optimal h control problem based on the iteration and a novel extended matrix pencil formulation of the state space solution to the sub optimal h control problem in particular instead of algebraic riccati equations or unstructured matrix pencils our approach is solely based on solving even generalized eigenproblems the enhanced numerical robustness of the method is derived from the fact that using the structure of the problem spectral symmetries are preserved moreover these methods are also applicable even if the pencil has eigenvalues on the imaginary axis we compare the new method with conventional methods and present several examples

products of random matrices for disordered systems products of random transfer matrices are applied to low dimensional disordered systems to evaluate numerically extensive quantities such as entropy and overlap probability distribution the main advantage is the possiblity to avoid numerical differentiation the method works for arbitrary disorder distributions at any temperature

nonuniform fast fourier transforms using min max interpolation the fast fourier transform fft is used widely in signal processing for efficient computation of the ft of finite length signals over a set of uniformly spaced frequency locations however in many applications one requires nonuniform sampling in the frequency domain i e a nonuniform ft several papers have described fast approximations for the nonuniform ft based on interpolating an oversampled fft this paper presents an interpolation method for the nonuniform ft that is optimal in the min max sense of minimizing the worst case approximation error over all signals of unit norm the proposed method easily generalizes to multidimensional signals numerical results show that the min max approach provides substantially lower approximation errors than conventional interpolation methods the min max criterion is also useful for optimizing the parameters of interpolation kernels such as the kaiser bessel function

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

sparse fourier transform via butterfly algorithm this paper introduces a fast algorithm for computing sparse fourier transforms with spatial and fourier data supported on curves or surfaces this problem appears naturally in several important applications of wave scattering digital signal processing and reflection seismology the main idea of the algorithm is that the interaction between a frequency region and a spatial region is approximately low rank if the product of their widths are bounded by the maximum frequency based on this property we can approximate the interaction between these two boxes accurately and compactly using a small number of equivalent sources the overall structure of the algorithm follows the butterfly algorithm the computation is further accelerated by exploiting the tensor product property of the fourier kernel in two and three dimensions the proposed algorithm is accurate and has the optimal complexity we present numerical results in both two and three dimensions

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

novel computation of the growth rate of generalized random fibonacci sequences a new formulation is proposed for the computation of average growth rates of generalized random fibonacci sequences based on the new formula a novel numerical scheme is designed and successfully implemented and interesting analytic asymptotic expansions are obtained for several examples

growth and structure of stochastic sequences we introduce a class of stochastic integer sequences in these sequences every element is a sum of two previous elements at least one of which is chosen randomly the interplay between randomness and memory underlying these sequences leads to a wide variety of behaviours ranging from stretched exponential to log normal to algebraic growth interestingly the set of all possible sequence values has an intricate structure

fast ewald summation based on nfft with mixed periodicity in this paper we develop new fast fourier based methods for the coulomb problem we combine the ewald summation formulas and the fast summation approach based on the nonequispaced fast fourier transform nfft in order to develop efficient methods for calculating the coulomb energies as well as the acting forces in charged particle systems subject to mixed periodic boundary conditions therewith we extend the applicability of nfft based methods which already exist for open as well as for 3d periodic boundary conditions to arbitrary combinations of periodic and open boundary conditions we reconsider the derivation of the ewald formulas for 2d and 1d periodic systems introduce the new algorithms and present high precision numerical results

high accuracy approximation techniques for analytic functions a generalization of the familiar mesh point technique for numerical approximation of functions is presented high accuracy and very rapid convergence may be obtained by thoughtful choice of the reference function chosen for interpolation between the mesh points in particular derivative operators are represented by highly nonlocal matrices but this is no drawback when one has computing machines to perform the algebraic manipulations some examples are given from familiar quantum mechanical problems

generalized discrete spherical harmonic transforms two generalizations of the spherical harmonic transforms are provided first they are generalized to an arbitrary distribution of latitudinal points i this unifies transforms for gaussian and equally spaced distributions and provides transforms for other distributions commonly used to model geophysical phenomena the discrete associated legendre functions pnm i are shown to be orthogonal to within roundoff error with respect to a weighted inner product thus providing the forward transform to spectral space second the representation of the transforms is also generalized to rotations of the discrete basis set pnm i a discrete function basis is defined that provides an alternative to pnm i on a grid with n latitudes the new basis requires o n2 memory compared to the usual o n3 the resulting transforms differ in spectral space but provide identical results for certain applications for example a forward transform followed immediately by a backward transform projects the original discrete function in a manner identical to the existing transforms namely they both project the original function onto the same smooth least squares approximation without the high frequencies induced by the closeness of the points in the neighborhood of the poles finally a faster projection is developed based on the new transforms

multipole expansions and pseudospectral cardinal functions a new generalization of the fast fourier transform the polynomial or trigonometric interpolant of an arbitrary function f x may be presented as a critical function series whose coefficients are the values of f x at the interpolation points we show that the cardinal series is identical to the sum of the forces due to a set of n point charges with appropriate force laws it follows that the cardinal series can be summed via the fast multipole method fmm in o nlog sub 2 n operations which is much cheaper than the o n sup 2 cost of direct summation the fmm is slower than the fast fourier transform fft so the latter should always be used where applicable however the multipole expansion succeeds where the fft fails in particular the fmm can be used to evaluate fourier and chebyshev series on an irregular grid as is needed when adaptively regridding in a time integration also the multipole expansion can be applied to basis sets for which the fft is inapplicable even on the canonical grid including legendre polynomials hermite and laguerre functions spherical harmonics and zinc functions 12 refs

a spectral method of characteristics for hyperbolic problems a new numerical technique is introduced for first order hyperbolic equations based on combining the spectral method with explicit timestepping along the characteristics which permits the use of large timesteps at no cost in accuracy the method is unconditionally stable and spectrally accurate the theoretical results are illustrated by numerical experiments

a nonuniform fast fourier transform based on low rank approximation by viewing the nonuniform discrete fourier transform nudft as a perturbed version of a uniform discrete fourier transform we propose a fast and quasi optimal algorithm for computing the nudft based on the fast fourier transform fft our key observation is that an nudft and dft matrix divided entry by entry is often well approximated by a low rank matrix allowing us to express a nudft matrix as a sum of diagonally scaled dft matrices our algorithm is simple to implement automatically adapts to any working precision and is competitive with state of the art algorithms in the fully uniform case our algorithm is essentially the fft we also describe quasi optimal algorithms for the inverse nudft and two dimensional nudfts

direct operatorial tau method for pantograph type equations abstract the lanczos tau method is applied to find chebyshev polynomial approximations for the solutions of pantograph differential equations the results are accompanied by an error analysis numerical examples calculated by our matlab package chebpack confirm the theory and prove the importance for practice of this approach

weak disorder in fibonacci sequences we study how weak disorder affects the growth of the fibonacci series we introduce a family of stochastic sequences that grow by the normal fibonacci recursion with probability 1 but follow a different recursion rule with a small probability we focus on the weak disorder limit and obtain the lyapunov exponent that characterizes the typical growth of the sequence elements using perturbation theory the limiting distribution for the ratio of consecutive sequence elements is obtained as well a number of variations to the basic fibonacci recursion including shift doubling and copying are considered

approximate factorizations of fourier matrices with nonequispaced knots abstract we propose a new algorithm for the multiplication of vectors with fourier matrices of the form a tf e 2 i x k v j n n 2 1 j k n 2 where both x k and v j are arbitrary knots in n 2 n 2 the algorithm is based on an approximate factorization of the transform matrix a tf into sparse matrices which entries are chosen to minimize the frobenius norm of certain error matrices numerical experiments demonstrate that the approximation error introduced by our new algorithm is about 10 2 times smaller than the approximation error of previously reported algorithms with the gaussian or b splines as window functions and as good as a previous algorithm with the kaiser bessel function as window function

fast approximate fourier transforms for irregularly spaced data several algorithms for efficiently evaluating trigonometric polynomials at irregularly spaced points are presented and analyzed the algorithms can be viewed as approximate generalizations of the fast fourier transform fft and they are compared with regard to their accuracy and their computational efficiency

reaction diffusion model for phyllotaxis abstract this paper presents a numerical study of a reaction diffusion model with three chemical substances the model makes possible the breaking of the bilateral symmetry and presents a new kind of solution with a two time periodic asymptotic behavior these solutions are of particular interest for the phyllotaxis because they correspond to propagating structures peaks which locally appear and disappear on a ring

random fibonacci sequences and the number 1 13198824 begin abstract for the familiar fibonacci sequence defined by f 1 f 2 1 and f n f n 1 f n 2 for n 2 f n increases exponentially with n at a rate given by the golden ratio 1 sqrt 5 2 1 61803398 ldots but for a simple modification with both additions and subtractions the it random fibonacci sequences defined by t 1 t 2 1 and for n 2 t n pm t n 1 pm t n 2 where each pm sign is independent and either or with probability 1 2 it is not even obvious if abs t n should increase with n our main result is that begin equation sqrt n abs t n rightarrow 1 13198824 ldots text as n rightarrow infty end equation with probability 1 finding the number 1 13198824 ldots involves the theory of random matrix products stern brocot division of the real line a fractal like measure a computer calculation and a rounding error analysis to validate the computer calculation end abstract

some geometric interpretations of quantum fidelity abstract we consider quantum fidelity between two states and where we fix and allow to be sent through a quantum channel we determine the minimal fidelity where one minimizes over a all unital channels b all mixed unitary channels and c arbitrary channels we derive results involving the minimal eigenvalue of which we can interpret as a convex combination coefficient as a consequence we give a new geometric interpretation of the minimal fidelity with respect to the closed convex set of density matrices and with respect to the closed convex set of quantum channels we further investigate the geometric nature of fidelity by considering density matrices arising as normalized projections onto subspaces in this way fidelity can be viewed as a geometric measure of distance between two spaces we give a connection between fidelity and the canonical principal angles between the subspaces

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

non hermitian localization and delocalization we study localization and delocalization in a class of non hermitian hamiltonians inspired by the problem of vortex pinning in superconductors in various simplified models we are able to obtain analytic descriptions in particular of the nonperturbative emergence of a forked structure the appearance of wings in the density of states we calculate how the localization length diverges at the localization delocalization transition we map some versions of this problem onto a random walker problem in two dimensions for a certain model we find an intricate structure in its density of states

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

fast spherical fourier algorithms spherical fourier series play an important role in many applications a numerically stable fast transform analogous to the fast fourier transform is of great interest for a standard grid of o n2 points on the sphere a direct calculation has computational complexity of o n4 but a simple separation of variables reduces the complexity to o n3 here we improve well known fast algorithms for the discrete spherical fourier transform with a computational complexity of o n2 log2 n furthermore we present for the first time a fast algorithm for scattered data on the sphere for arbitrary o n2 points on the sphere a direct calculation has a computational complexity of o n4 but we present an approximate algorithm with a computational complexity of o n2 log2 n

dynamical models of phyllotaxis leave patterns on plants phyllotaxis present remarkable universal features the underlying morphogenesis phenomenon can be modelled by a class of dynamical systems referred to as inhibitory field models numerical studies have shown that surprisingly often the fixed points of such systems closely correspond to noble numbers this is exactly what is observed in nature in the present paper a proof of this fact is given that applies in particular to the model of thornley but the analysis is general enough to embrace all models usually considered an explicit criterion is given to explain under which circumstances noble numbers are actually selected a stability analysis of the fixed points is also performed explaining observed bifurcations

rapid prolate pseudospectral differentiation and interpolation with the fast multipole method pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials however the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix vector products which if evaluated directly entail o n2 memory requirement and computational complexity where n is the number of unknowns utilized for discretization and interpolation in this work we show that the fast multipole method fmm can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to o n example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver

on the cycle expansion for the lyapunov exponent of a product of random matrices the cycle expansion of the thermodynamical zeta function for the lyapunov exponent of a product of random matrices typically converges exponentially with the maximal cycle length mainieri 1992 phys rev lett 68 1965 in this paper we show that the convergent exponents are given by the spectrum of a properly defined evolution operator which describes how a steady distribution of vector direction is established under the action of random matrices the exponential decay terms are automatically eliminated in the cycle expansion of the spectral determinant which greatly accelerates the convergence provided all matrix elements are positive numbers as a marginal case the random fibonacci series is studied in detail and it is shown that this method is helpful

a new positive definite geometric mean of two positive definite matrices abstract we introduce and study a new positive definite in certain singular cases positive semidefinite geometric mean of two positive definite under certain conditions positive semidefinite matrices

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

growth and decay of random fibonacci sequences for 0 they grow exponentially by formulating the problem as a markov chain involving random matrix products and computing its invariant measure a fractal the lyapunov constant limn x n 1 n is determined numerically for a wide range of values and its dependence on is observed to be non smooth the limit is defined in the almost sure sense this generalizes recent work of viswanath who proved 1 1 131 988 24 by a simple rescaling these results also apply to the more general random recurrence x n 1 x n x n 1 for fixed and these random recurrence relations have links with many fields including ergodic theory dynamical systems heavy tailed statistics spectral theory continued fractions and condensed matter physics

distribution of eigenvalues in non hermitian anderson models we develop a theory which describes the behavior of eigenvalues of a class of one dimensional random non hermitian operators introduced recently by hatano and nelson we prove that the eigenvalues are distributed along a curve in the complex plane an equation for the curve is derived and the density of complex eigenvalues is found in terms of spectral characteristics of a reference hermitian disordered system the generic properties of the eigenvalue distribution are discussed

fast computation of fourier integral operators we introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation general hyperbolic equations and curvilinear tomography the problem is to numerically evaluate a so called fourier integral operator fio of the form int e 2 pi i phi x xi a x xi hat f xi d xi at points given on a cartesian grid here xi is a frequency variable hat f xi is the fourier transform of the input f a x xi is an amplitude and phi x xi is a phase function which is typically as large as xi hence the integral is highly oscillatory because a fio is a dense matrix a naive matrix vector product with an input given on a cartesian grid of size n by n would require o n 4 operations this paper develops a new numerical algorithm which requires o n 2 5 log n operations and as low as o sqrt n in storage space the constants in front of these estimates are small it operates by localizing the integral over polar wedges with small angular aperture in the frequency plane on each wedge the algorithm factorizes the kernel e 2 pi i phi x xi a x xi into two components 1 a diffeomorphism which is handled by means of a nonuniform fft and 2 a residual factor which is handled by numerical separation of the spatial and frequency variables the key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology we also discuss the potential of our ideas for various applications such as reflection seismology

fast evaluation of quadrature formulae on the sphere recently a fast approximate algorithm for the evaluation of expansions in terms of standard l 2 s 2 orthonormal spherical harmonics at arbitrary nodes on the sphere s 2 has been proposed in s kunis and d potts fast spherical fourier algorithms j comput appl math 161 75 98 2003 the aim of this paper is to develop a new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules we give a formulation in matrix vector notation and an explicit factorisation of the spherical fourier matrix based on the former algorithm starting from this we obtain the corresponding factorisation of the adjoint spherical fourier matrix and are able to describe the associated algorithm for the adjoint transformation which can be employed to evaluate quadrature rules for arbitrary weights and nodes on the sphere we provide results of numerical tests showing the stability of the obtained algorithm using as examples classical gaus legendre and clenshaw curtis quadrature rules as well as the healpix pixelation scheme and an equidistribution

timestepping lagrangian particles in two dimensional eulerian flow fields abstract the speed and accuracy of different methods for the interpolation and timestepping of lagrangian particles in eulerian velocity fields are examined two circumstances are considered 1 a steady flow field in which particles trace closed orbits along constant stream function values and 2 decaying two dimensional turbulence in which vorticity is conserved within a correction for dissipation a fourth order runge kutta timestepping is shown to give best results for the steady flow field tracking an invariant along particle paths in a fully eddy active case is shown to be relatively insensitive with regard to choice of interpolation or timestepping method the cause of this insensitivity is shown to be a resolution problem of the eulerian field dynamics and methods are outlined to ameliorate the problem

a preconditioned implementation of pseudospectral methods on arbitrary grids in this article a preconditioner for pseudospectral methods in terms of expansion coefficients on arbitrary grids is presented the preconditioner is first developed for application in ordinary differential equations and then is generalized for partial differential equations a fully discrete pseudospectral approximation is then used to solve bivariate problems this preconditioner in effect makes the pseudospectral methods insensitive to specific grid points the numerical results show the efficiency of the method

a framework for discrete integral transformations i the pseudopolar fourier transform the fourier transform of a continuous function evaluated at frequencies expressed in polar coordinates is an important conceptual tool for understanding physical continuum phenomena an analogous tool suitable for computations on discrete grids could be very useful however no exact analogue exists in the discrete case in this paper we present the notion of pseudopolar grid pp grid and the pseudopolar fourier transform ppft which evaluates the discrete fourier transform at points of the pp grid the pp grid is a type of concentric squares grid in which the radial density of squares is twice as high as usual the pp grid consists of equally spaced samples along rays where different rays are equally spaced in slope rather than angle we develop a fast algorithm for the ppft with the same complexity order as the cartesian fast fourier transform the algorithm is stable invertible requires only one dimensional operations and uses no approximate interpolations we prove that the ppft is invertible and develop two algorithms for its inversion iterative and direct both with complexity o n 2 log n where n times n is the size of the reconstructed image the iterative algorithm applies conjugate gradients to the gram operator of the ppft since the transform is ill conditioned we introduce a preconditioner which significantly accelerates the convergence the direct inversion algorithm utilizes the special frequency domain structure of the transform in two steps first it resamples the pp grid to a cartesian frequency grid and then recovers the image from the cartesian frequency grid

factorizations and geometric means of positive definite matrices abstract in this paper we provide a new class of metric geometric means of positive definite matrices varying over hermitian unitary matrices we show that each hermitian unitary matrix induces a factorization of the cone p m of m m positive definite hermitian matrices into geodesically convex subsets and a hadamard metric structure on p m an explicit formula for the corresponding metric midpoint operation is presented in terms of the geometric and spectral geometric means and show that the resulting two variable mean is different to the standard geometric mean some basic properties comparable to those of the geometric mean and its extensions to finite number of positive definite matrices are studied

strong majorization for hermitian matrices we introduce and study an additive relation and a multiplicative relation for hermitian matrices the relation g a f means that f is the real part of some matrix similar to g the relation g m f expresses the property that the positive definite matrix f is the modulus of some matrix similar to the positive definite matrix g there is a close connection with the additive and multiplicative majorization relations for the corresponding spectra

γ commuting triples of positive definite matrices and midpoint equations on unitary matrices triples of positive definite matrices that can be transformed to a commuting triple by a congruence transformation are called commuting in this paper we give various descriptions and characterizations of commuting triples and particularly focus on those involving the geometric mean and midpoint equations on a generating set of the special unitary group

on the computation of the optimal h norms for two feedback control problems abstract numerical computation of the optimal h norms for the state feedback and the output feedback control problems is considered in 1989 doyle et al derived for a given parameter r 0 necessary and sufficient conditions under which an admissible output feedback controller exists as well as conditions for the existence of a state feedback controller we explore numerical methods to find the infimum of parameter r that satisfies those conditions

plants and fibonacci the universality of many features of plant patterns and phyllotaxis has mystified and intrigued natural scientists for at least four hundred years it is remarkable that to date there is no widely accepted theory to explain the observations we hope that the ideas explained below lead towards increased understanding

phyllotaxis an interdisciplinary phenomenon abstract regular phyllotaxis is largely non uniform in the sense that 1 2 and the golden mean are the most frequently observed values of the characteristic parameter divergence the different approaches to this enigmatic phenomenon form a fascinating chapter in the history of science more recently different models have been independently suggested in an attempt to recreate the phenomenon these are dynamic systems in discrete time founded on a simple inhibiting hypothesis ih we analyse the divergences selected by these models and consequently we investigate the ih capability to account for regular phyllotaxis this capability is conditional it depends on the convex concave shape of the elementary inhibitions and involves considering the ih from a statistical standpoint our conclusions are essentially new and they partly conflict with previous predictions because of its generality our analysis applies to botanical as well as to physical systems such as vortex networks

linearly parameterized bandits we consider bandit problems involving a large possibly infinite collection of arms in which the expected reward of each arm is a linear function of an r dimensional random vector z rr where r 2 the objective is to minimize the cumulative regret and bayes risk when the set of arms corresponds to the unit sphere we prove that the regret and bayes risk is of order r t by establishing a lower bound for an arbitrary policy and showing that a matching upper bound is obtained through a policy that alternates between exploration and exploitation phases the phase based policy is also shown to be effective if the set of arms satisfies a strong convexity condition for the case of a general set of arms we describe a near optimal policy whose regret and bayes risk admit upper bounds of the form o r t log3 2t

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

kaiser bessel basis for particle mesh interpolation in this work we introduce the kaiser bessel interpolation basis for the particle mesh interpolation in the fast ewald method a reliable a priori error estimate is developed to measure the accuracy of the force computation in correlated charge systems and is shown to be effective in optimizing the shape parameter of the kaiser bessel basis in terms of accuracy by comparing the optimized kaiser bessel basis with the traditional b spline basis we demonstrate that the former is more accurate than the latter in part of the working parameter space say a relatively small real space cutoff a relatively small reciprocal space mesh and a relatively large truncation of basis in some cases the kaiser bessel basis is found to be more than one order of magnitude more accurate

a fast spherical harmonics transform algorithm the spectral method with discrete spherical harmonics transform plays an important role in many applications in spite of its advantages the spherical harmonics transform has a drawback of high computational complexity which is determined by that of the associated legendre transform and the direct computation requires time of o n3 for cut off frequency n in this paper we propose a fast approximate algorithm for the associated legendre transform our algorithm evaluates the transform by means of polynomial interpolation accelerated by the fast multipole method fmm the divide and conquer approach with split legendre functions gives computational complexity o n2 log n experimental results show that our algorithm is stable and is faster than the direct computation for n 511

a double exponential fast gauss transform algorithm for pricing discrete path dependent options this paper develops algorithms for the pricing of discretely sampled barrier lookback and hindsight options and discretely exercisable american options under the black scholes framework the pricing of these options can be reduced to evaluation of a series of convolutions of the gaussian distribution and a known function we compute these convolutions efficiently using the double exponential integration formula and the fast gauss transform the resulting algorithms have computational complexity of o nn where the number of monitoring exercise dates is n and the number of sample points at each date is n and our results show the error decreases exponentially with n we also extend the approach and provide results for merton s lognormal jump diffusion model

zeta function for the lyapunov exponent of a product of random matrices a cycle expansion for the lyapunov exponent of a product of random matrices is derived the formula is nonperturbative and numerically effective which allows the lyapunov exponent to be computed to high accuracy in particular the free energy and heat capacity are computed for the one dimensional ising model with quenched disorder the formula is derived by using a bernoulli dynamical system to mimic the randomness

hybrid chebyshev function bases for sparse spectral methods in parity mixed pdes on an infinite domain abstract we present a numerical spectral method to solve systems of differential equations on an infinite interval y in presence of linear differential operators of the form q y y b where q y is a rational fraction and b a positive integer even when these operators are not parity preserving we demonstrate how a mixed expansion in interleaved chebyshev rational functions t b n y and s b n y preserves the sparsity of their discretization this paves the way for fast o n ln n and spectrally accurate mixed implicit explicit time marching of sets of linear and nonlinear equations in unbounded geometries

time varying discrete riccati equation some monotonicity results abstract using a fre chet derivative based approach some monotonicity and comparison results concerning the solutions of the time varying discrete time riccati equation are obtained connections with the existence of the semistabilizing solution are made explicit as well

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

a fast spherical filter with uniform resolution this paper introduces a fast algorithm for obtaining a uniform resolution representation of a function known at a latitude longitude grid on the surface of a sphere equivalent to a triangular isotropic truncation of the spherical harmonic coefficients for the function the proposedspectral truncation method which is based on the fast multipole method and the fast fourier transform projects the function to a space with uniform resolution while avoiding surface harmonic transformations the method requireso n2logn operations foro n2 grid points as opposed too n3 operations for the standard spectral transform method providing a reduced complexity spectral method obviating the pole problem in the integration of time dependent partial differential equations on the sphere the filter s performance is demonstrated with numerical examples

some analytical results about two physical models of phyllotaxis we present analytical results about a generalization of the variational problem of levitov and the experiment of douady and couder we show that the hierarchical selection of noble numbers in these systems occurs with a precision depending on the range of the interaction this precision can be infinite if the interaction is scale invariant

congruence of hermitian matrices by hermitian matrices two hermitian matrices a b mn c are said to be hermitian congruent if there exists a nonsingular hermitian matrix c mn c such that b cac in this paper we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable hermitian matrices a and b to be hermitian congruent moreover when a and b are hermitian congruent we describe the possible inertias of the hermitian matrices c that carry the congruence we also give necessary and sufficient conditions for any 2 by 2 nonsingular hermitian matrices to be hermitian congruent in both of the studied cases we show that if a and b are real and hermitian congruent then they are congruent by a real symmetric matrix finally we note that if a and b are 2 by 2 nonsingular real symmetric matrices having the same sign pattern then there is always a real symmetric matrix c satisfying b cac moreover if both matrices are positive then c can be picked with arbitrary inertia

a penalty method for american options with jump diffusion processes the fair price for an american option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem we develop an implicit discretization method for pricing such american options the jump diffusion correlation integral term is computed using an iterative method coupled with an fft while the american constraint is imposed by using a penalty method we derive sufficient conditions for global convergence of the discrete penalized equations at each timestep finally we present numerical tests which illustrate such convergence

discrete time h algebraic riccati equation and parametrization of all h filters this paper is concerned with the algebraic riccati equations ares related to the h filtering problem a necessary and sufficient condition for the h problem to be solvable is that the h are has a positive semidefinite stabilizing solution with an additional condition that a certain matrix is positive definite it is shown that such a stabilizing solution is a monotonically non increasing convex function of the prescribed h norm bound this property of the h are is very important for the analysis of the performance of the h filter in this paper the size of the set of all h filters is considered on the basis of the monotonicity of the above riccati solution it turns out that under a certain condition the degree of freedom of the h filter reduces a1 the optimal h norm bound these results provide a guideline for selecting the value of some numerical examples are included