halley s method for the matrix sector function the matrix n sector function is a generalization of the matrix sign function it can be used to determine the number of eigenvalues of a matrix in a specific sector of the complex plane and to extract the eigenpairs belonging to this sector without explicitly computing the eigenvalues it is known that newton s method which can be used for computing the matrix sign function is not globally convergent for the matrix sector function the only existing algorithm for computing the matrix sector function is based on the continued fraction expansion approximation to the principal nth root of an arbitrary complex matrix in this paper we introduce a new algorithm based on halley s generalized iteration formula for solving nonlinear equations it is shown that the iteration has good error propagation properties and high accuracy finally we give two application examples and summarize the results of our numerical experiments comparing newton s the continued fraction and halley s method

the matrix sign function a survey of the matrix sign function is presented including some historical background definitions and properties approximation theory and computational methods and condition theory and estimation procedures applications to areas such as control theory eigendecompositions and roots of matrices are outlined and some new theoretical results are also given

families of explicit two step methods for integration of problems with oscillating solutions we study a new sixth algebraic order explicit numerov type family of methods using every free parameter of the family even its nodes we manage to derive two methods the first with phase lag of order 12 while the other method has one stage less this is a considerable improvement over the 10th order phase lag order methods found in the literature until now numerical experiments confirm the superiority of our new methods over older methods

a generator of hybrid symmetric four step methods for the numerical solution of the schrodinger equation in this paper a generator of hybrid explicit four step methods with minimal phase lag is developed the methods are of sixth algebraic order and have large intervals of periodicity the coefficients of the methods are determined in order to have minimal phase lag the efficiency of the new methods is showed by their application to the schrodinger equation and by their comparison with other well known methods

high orderp stable formulae for the numerical integration of periodic initial value problems recently there has been considerable interest in the approximate numerical integration of the special initial value problemy f x y for cases where it is known in advance that the required solution is periodic the well known class of stormer cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems an appropriate stability requirement for the numerical integration of periodic problems is that ofp stability however lambert and watson have shown that ap stable linear multistep method cannot have an order of accuracy greater than 2 in the present paper a class of 2 step methods of runge kutta type is discussed for the numerical solution of periodic initial value problems p stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods

an arithmetic for matrix pencils theory and new algorithms this paper introduces arithmetic like operations on matrix pencils the pencil arithmetic operations extend elementary formulas for sums and products of rational numbers and include the algebra of linear transformations as a special case these operations give an unusual perspective on a variety of pencil related computations we derive generalizations of monodromy matrices and the matrix exponential a new algorithm for computing a pencil arithmetic generalization of the matrix sign function does not use matrix inverses and gives an empirically forward numerically stable algorithm for extracting deflating subspaces

parallel algorithms for algebraic riccati equations the matrix sign function is the basis of a parallel algorithm for solving the generalized algebraic riccati equation three forms of the algorithm were implemented and tested on a distributed memory hypercube multiprocessor performance results indicate that the method is an excellent means of solving large scale problems on a parallel computer

algorithms for model reduction of large dynamical systems three algorithms for the model reduction of large scale continuous time time invariant linear dynamical systems with a sparse or structured transition matrix and a small number of inputs and outputs are described they rely on low rank approximations to the controllability and observability gramians which can efficiently be computed by adi based iterative low rank methods the first two model reduction methods are closely related to the well known square root method and schur method which are balanced truncation techniques the third method is a heuristic balancing free technique the performance of the model reduction algorithms is studied in numerical experiments

traveling instability waves in a mach 8 flow over an elliptic cone simultaneous measurements were carried out with three hot film probes in the mach 8 flow over an elliptic cone of 2 1 aspect ratio and the data obtained were compared to the results of computations using the parabolized navier stokes equations and linear stability theory the elliptic cone flow was found to be significantly different from the flows studied in previous hypersonic flow stability experiments which have focused exclusively on wind tunnel models with two dimensional planar or axial symmetry at least two instability mechanisms appear to be active in the present flow one associated with the region of maximum crossflow in the vicinity of the shoulder of the cone and the other associated with the inflectional velocity profiles on the top centerline between the shoulder and leading edge of the cone the dominant flow instability occurred at relatively low frequency and the direction of the phase velocity was significantly skewed from that of the boundary layer edge streamlines the results were found to be in rough agreement with linear stability calculations and are suggestive of a traveling crossflow instability mode which apparently has not heen observed before in hypersonic flow

space time correlation measurements in a hypersonic transitional boundary layer an experimental investigation of the spatial structure of second mode instability waves was carried out in the boundary layer of a 7 deg half angle sharp nosed cone at an edge mach number of 6 8 measurements were made at reynolds numbers of 2 3 x 10 6 to 9 1 x 10 6 based on boundary edge conditions spanning the range from unstable laminar flow to nearly turbulent flow simultaneous measurements with two hot film probes in the boundary layer comprise the primary data set the mean boundary layer state was measured with pitot and total temperature surveys correlations were taken with circumferential streamwise and vertical probe separations the second mode waves preceding transition are relatively limited in their circumferential dimension typically less than four boundary layer thicknesses based on a coherence level of 20

a two step method for the numerical solution of the radial schrodinger equation abstract a two step method is developed for the numerical solution of the radial schrodinger equation numerical results obtained for the integration of resonance problem and eigenvalue problem show that this new method is better than other similar methods

an explicit four step phase fitted method for the numerical integration of second order initial value problems abstract an explicit four step method with phase lag of infinite order is developed for the numerical integration of second order initial value problems extensive numerical testing indicates that this new method can be generally more efficient than other four step methods

damping and phase analysis for some methods for solving second order ordinary differential equations we consider numerical methods for initial value problems for second order systems of ordinary differential equations analysing them by applying them to the test equation we discuss conditions which ensure an oscillatory numerical solution and the desirability of such a property we also use a slightly more general test equation to derive conditions which ensure that the numerical forced oscillation is in phase with the true forced oscillation to illustrate the theory we consider the damping and phase properties of some frequently used methods

block runge kutta methods for periodic initial value problems abstract block runge kutta methods with minimal phase lag for first order periodic initialvalue problems are developed it should be noted that the new methods are based on the runge kutta methods of algebraic order three and on a new error estimate introduced in this paper the numerical results indicate that these new methods are efficient for the numerical solution of differential equations with periodic solutions using variable step size

an extended numerov type method for the numerical solution of the schrodinger equation abstract a new method which is based on the new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases 1 i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x 2 ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new method is more efficient compared with previously derived methods

scheifele two step methods for perturbed oscillators two step methods specially adapted to the numerical integration of perturbed oscillators are obtained the formulation of the methods is based on a refinement of classical taylor expansions due to scheifele g scheifele on the numerical integration of perturbed linear oscillating systems z angew math phys 22 1971 186 210 the key property is that those algorithms are able to integrate exactly harmonic oscillators with frequency w the methods depend on a parameter email protected where h is the stepsize based on the b2 series theory of coleman j p coleman order conditions for a class of two step methods for y f x y ima j numer anal 23 2003 197 220 we derive the order conditions of this new type of method the linear stability and phase properties are examined the theory is illustrated with some fourth and fifth order explicit schemes numerical results carried out on an assortment of test problems such as the integration of the orbital motion of earth satellites show the relevance of the theory

a p stable complete in phase obrechkoff trigonometric fitted method for periodic initial value problems in this paper we derive a p stable trigonometric fitted obrechkoff method with phase lag frequency distortion infinity it is easy to see from numerical results presented that the new method is much more accurate than previous methods

the matrix sign decomposition and its relation to the polar decomposition the sign function of a square matrix was introduced by roberts in 1971 we show that it is useful to regard s sign a as being part of a matrix sign decomposition a sn where n a2 1 2 this decomposition leads to the new representation sign a a a2 1 2 most results for the matrix sign decomposition have a counterpart for the polar decomposition a uh and vice versa to illustrate this we derive best approximation properties of the factors u h and s determine bounds for a s and a u and describe integral formulas for s and u we also derive explicit expressions for the condition numbers of the factors s and n an important equation expresses the sign of a block 2 2 matrix involving a in terms of the polar factor u of a we apply this equation to a family of iterations for computing s by pandey kenney and laub to obtain a new family of iterations for computing u the iterations have some attractive properties including suitability for parallel computation

a runge kutta fehlberg method with phase lag of order infinity for initial value problems with oscillating solution abstract a runge kutta method is developed for the numerical solution of initial value problems with oscillating solution based on the runge kutta fehlberg 2 3 method a runge kutta method with phase lag of order infinity is developed based on these methods we produce a new embedded runge kutta fehlberg 2 3 method with phase lag of order infinity this method is called as runge kutta fehlberg phase fitted method rkfpf the numerical results indicate that this new method is much more efficient compared with other well known runge kutta methods for the numerical solution of differential equations with oscillating solution using variable step size

laminar turbulent transition in a mach 8 elliptic cone flow to examine the influence of crossflow on transition in a hypersonic boundary layer an experimental and computational study was made of a mach 8 flow over a sharp nosed elliptic cone of 2 1 cross section extensive flow visualization was carried out experimentally including schlieren photography and surface oil flow visualization mean pressure and heat flux were measured at the wall of the wind tunnel model for comparison with the experiments a computation of the basic state flow was carried out using the parabolized navier stokes equations the basic state laminar flow was found to be relatively complex with highly skewed velocity profiles and a pair of symmetric vortices on the top centerline of the cone transition in the elliptic cone flow was found to be significantly different than transition in hypersonic flow over planar and axisymmetric configurations which is driven by the second mode instability in the elliptic cone flow the transition front was nonuniform with early transition near the top centerline and delayed transition near the leading edge rope waves were observed on both the leading edge and top centerline unusual elongated streaks were observed on the top centerline

a jacobi like method for solving algebraic riccati equations on parallel computers an algorithm to solve continuous time algebraic riccati equations through the hamiltonian schur form is developed it is an adaption for hamiltonian matrices of an asymmetric jacobi method of eberlein 1987 it uses unitary symplectic similarity transformations and preserves the hamiltonian structure of the matrix each iteration step needs only local information about the current matrix thus admitting efficient parallel implementations on certain parallel architectures convergence performance of the algorithm is compared with the hamiltonian jacobi algorithm of byers 1990 the numerical experiments suggest that the method presented here converges considerably faster for non hermitian hamiltonian matrices than byers hamiltonian jacobi algorithm besides that numerical experiments suggest that for the method presented here the number of iterations needed for convergence can be predicted by a simple function of the matrix size

a finite difference method for the numerical solution of the schro dinger equation a new approach which is based on a new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x ii the general case of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient compared to previously derived methods

explicit numerov type methods with reduced number of stages we present in this paper a new approach for the derivation of hybrid explicit numerov type methods the new methodology does not require the intermediate use of high accuracy inter polatory nodes since we only need the taylor expansion of the internal points as a consequence a sixth order method is produced at a cost of only four stages per step instead of six stages needed for the methods which have appeared in the literature until now numerical results over some well known problems in physics and mechanics indicate the superiority of the new method 2003 elsevier science ltd all rights reserved

an explicit almost p stable two step method with phase lag of order infinity for the numerical integration of second order pacific initial value problems an explicit almost p stable two step method with phase lag of order infinity is developed for the numerical integration of second order periodic initial value problems the method has algebraic order four extensive numerical testing indicates that the new method is generally more accurate than other two step methods

a new hybrid imbedded variable step procedure for the numerical integration of the schrodinger equation a new imbedded variable step procedure is developed for the numerical integration of the radial schrodinger equation the new imbedded method is based on p stable methods of exponential order eight ten 12 and 14 numerical results indicate that the new procedure is more efficient than similar variable step procedures

phase fitted numerov type methods we present a hybrid two step method for the solution of second order initial value problem it costs only five function evaluations per step and attains sixth algebraic order the coefficients depend on the frequency of the problem and satisfy zero dissipation and phase fitted properties so the method is well suited for facing problems with oscillatory solutions numerical tests justify our effort

an improved trigonometrically fitted p stable obrechkoff method for periodic initial value problems in this paper we present an improved p stable trigonometrically fitted obrechkoff method with phase lag frequency distortion infinity compared with the previous p stable trigonometrically fitted obrechkoff method developed by simos our new method is simpler in structure and more stable in computation we have also improved the accuracy of the first order derivative formula from the numerical illustration presented we can show that the new method is much more accurate than the previous methods

positive and negative solution of dual riccati equations by matrix sign function iteration abstract for a matrix algebraic riccati equation it is shown that the positive definite stabilizing solution the negative definite anti stabilizing solution and the positive and negative solutions of the corresponding dual equation may all be computed from a single matrix sign function iteration this is potentially far more efficient than four separate schur vector solutions moreover necessary and sufficient conditions are given for the existence of unique stabilizing or antistabilizing solutions of an algebraic riccati equation in terms of the matrix sign of an associated hamiltonian matrix

a runge kutta nystro m method for the numerical integration of special second order periodic initial value problems abstract a new runge kutta nystrom method is developed to integrate second order differential equations of the form u t f t u when they posses an oscillatory solution through an appropriate definition of the parameters of the method a fourth algebraic order is obtained and the phase lag is reduced significantly

embedded methods for the numerical solution of the schrodinger equation abstract new methods for the approximate numerical integration of the one dimensional schrodinger equation are developed in this paper complete phase lag analysis of the new methods is included these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for a one dimensional schrodinger equation show the validity of the developed theory

explicit two step high accuracy hybrid methods with minimal phase lag for y f x y 0 and their application to the one dimensional schrodinger equation abstract in this paper two families of explicit two step sixth and eighth algebraic order hybrid methods with minimal phaselag are developed for the numerical integration of special second order periodic initial value problems these methods have the advantage of higher algebraic accuracy and minimal phase lag compared with some methods in 1 2 4 8 11 14 the methods proposed in this paper may be considered as a generalization of some methods in 1 5 7 8 12 an application to the one dimensional schrodinger equation on the resonance problem indicates that these new methods are generally more accurate than some methods in 5 8 2 11 13 14

phase fitted and amplification fitted two step hybrid methods for y f x y this paper provides a theoretical framework for a new type of phase fitted and amplification fitted two step hybrid ftsh methods which is introduced by the author in h van de vyver a phase fitted and amplification fitted explicit two step hybrid method for second order periodic initial value problems internat j modern phys c 17 2006 663 675 the methods constitute a modification of dissipative two step hybrid methods in the sense that two free parameters are added to eliminate the phase lag and the amplification error the methods are useful only when a good estimate of the frequency of the problem is known in advance the parameters depend on the product of the estimated frequency and the stepsize the algebraic order zero stability stability and phase properties are examined the theory is illustrated with sixth order explicit ftsh methods numerical results carried out on an assortment of test problems show the relevance of the theory

a low order embedded runge kutta method for periodic initial value problems sideridis a b and t e simos a low order embedded runge kutta method for periodic initial value problems journal of computational and applied mathematics 44 1992 23 5 244 an embedded runge kutta fehlberg method is developed it should be noted that this embedded method is procuced using the runge kutta fehlberg method with algebraic order four to estimate a truncation phase lag error of algebraic order three the numerical results indicate that this new method is efficient for the numerical solution of differential equations with periodic solution using variable stepsize

a noumerov type method with minimal phase lag for the integration of second order periodic initial value abstract in a recent paper 2 we gave a noumerov type method with minimal phase lag for the integration of second order initial value problems y f t y y t0 y0 y t0 y 0 however the method given there is implicit we show here the interesting result that if the noumerov type methods of 2 are made explicit with the help of the classical second order method then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size 1 40320 h 6 and also a slightly larger interval of periodicity of size 2 75 than the phase lag of size 1 12096 h 6 and interval of periodicity of size 2 71 for the implicit method 2 more interestingly it turns out that noumerov made explicit of chawla 3 also has less frequency distortion than the implicit noumerov method we shall assume a familiarity with the notation and discussion given in 2

on the matrix sign function method for solving algebraic riccati equations this paper gives an improvement to the method of solving algebraic riccati equations via use of the matrix sign function we show that a 2n x n or n x 2n system can be truncated to an n x n system and the solution of the truncated system can be guaranteed to be hermitian we show also with standard numerical examples that although the truncated equation has a larger condition number its solution may actually make the algebraic riccati equation have smaller residual in our numerical experiment using matlab the solution of the truncated system needed only about 26 of the number of floating point operations that the original system did

an explicit sixth order method with phase lag of order eight for y f t y abstract recently thomas 6 gave a two step sixth order method with phase lag 1 101478 67 h8 for the numerical integration of periodic initial value problems y t y y t 0 y 0 y t 0 y 0 however thomas method is implicit it possesses an interval of periodicity of size 2 77 and for nonlinear problems her method requires 6i 1 function evaluations for i modified newton iterations for the solution of the resulting implicit equations at each step in the present paper we present a new two step sixth order method which also has phase lag of order eight but with a smaller constant given by 1 3 628 800 h8 in contrast with thomas method our method is explicit possesses a larger interval of periodicity of size 4 63 and it is decidedly more economical since it involves only six function evaluations per step numerical experiments confirm the superiority of our present method over thomas method