on continuous approximate solutions of nonlinear differential equations abstract two recently developed methods for continuous approximate solution of equations involving differential operators the adomian decomposition method and the sarafyan method are compared

high order methods for state dependent delay differential equations with nonsmooth solutions this work presents a theoretical basis for high order numerical methods to solve state dependent delay differential equations of the form begin gathered dot x t f t x t x alpha t x t quad text for t in a b hfill alpha t x t leqq t hfill x t phi t quad text for t in bar a a hfill end gathered where bar a min alpha t x t for t in a b the solutions to such equations typically have derivative jump discontinuities jump points which propagate from the initial jump point t a thus high order methods require an accurate determination of the location of jump discontinuities in lower order derivatives of the solution x t the locations of these jump points are characterized as the zeros of certain nonlinear equations which themselves depend on x t because of this interdependence the following nontrivial question arises can these unknown jump points be determined accurately enough to develop high order methods this question

interpolation for runge kutta methods runge kutta methods provide a popular way to solve the initial value problem for a system of ordinary differential equations in contrast to the adams methods there is no natural way to approximate the solution between mesh points a way to accomplish this is proposed which is applicable to some important formulas its theoretical support is much better than that of interpolation in the popular variable order variable step adams codes

approximate solution of ordinary differential equations and their systems through discrete and continuous embedded runge kutta formulae and upgrading of their order abstract the eight main contributions of the author to the field of approximate solutions of ordinary differential equations described herein are all application oriented with the purposes of simplification and the increase in efficiency and effectiveness of the runge kutta processes generated they range from the determination of an initial trial step size to be adopted to expedite the approximation process through embedded runge kutta algorithms to a more recent procedure for upgrading the order of runge kutta processes these contributions encompass all classes of differential equations of all orders such as explicit implicit single or systems and their treatment by runge kutta processes of scalar or vector type with the related equivalence conditions of discrete or continuous kind including the computer derivations of nonlinear algebraic equations associated with the runge kutta processes specifically the author developed the first fifth order runge kutta formulae with fourth order embedded and the first c 1 approximate solution through interpolation and runge kutta formulae which he improved by developing c 1 embeddings with runge kutta formulae without the use of interpolative techniques

flexible codebook design for limited feedback systems via sequential smooth optimization on the grassmannian manifold grassmannian quantization codebooks play a central role in a number of limited feedback schemes for single and multi user multiple input multiple output mimo communication systems in practice it is often desirable that these codebooks possess additional properties that facilitate their implementation beyond the provision of good quantization performance although some good codebooks exist their design tends to be a rather intricate task the goal of this paper is to suggest a flexible approach to the design of grassmannian codebooks based on sequential smooth optimization on the grassmannian manifold and the use of smooth penalty functions to obtain additional desirable properties as one example the proposed approach is used to design rank 2 codebooks that have a nested structure and elements from a phase shift keying psk alphabet in some numerical comparisons codebooks designed using the proposed methods have larger minimum distances than some existing codebooks and provide tangible performance gains when applied to a simple mimo downlink scenario with zero forcing beamforming per user unitary beamforming and rate control pu 2rc and block diagonalization signalling furthermore the proposed approach yields codebooks that attain desirable additional properties without incurring a substantial degradation in performance

dirty paper coding versus tdma for mimo broadcast channels we compare the capacity of dirty paper coding dpc to that of time division multiple access tdma for a multiple antenna multiple input multiple output mimo gaussian broadcast channel bc we find that the sum rate capacity achievable using dpc of the multiple antenna bc is at most min m k times the largest single user capacity i e the tdma sum rate in the system where m is the number of transmit antennas and k is the number of receivers this result is independent of the number of receive antennas and the channel gain matrix and is valid at all signal to noise ratios snrs we investigate the tightness of this bound in a time varying channel assuming perfect channel knowledge at receivers and transmitters where the channel experiences uncorrelated rayleigh fading and in some situations we find that the dirty paper gain is upper bounded by the ratio of transmit to receive antennas we also show that min m k upper bounds the sum rate gain of successive decoding over tdma for the uplink channel where m is the number of receive antennas at the base station and k is the number of transmitters

computer solutions of state dependent delay differential equations abstract an adaptation of the runge kutta verner 5 6 formula pair is used to construct a numerical method for the solution of state dependent delay differential equations with nonvanishing lag a fifth degree divided difference newton backward interpolation polynomial is used to find the location of the derivative jump discontinuities of the solution in order to maintain the sixth order accuracy of the rkv pair the value of the solution at the delay is approximated by a three point hermite polynomial this new method is tested on some real life problems a fortran program called sysdel is available from the authors

the capacity region of the gaussian multiple input multiple output broadcast channel the gaussian multiple input multiple output mimo broadcast channel bc is considered the dirty paper coding dpc rate region is shown to coincide with the capacity region to that end a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality to show that a superposition of gaussian codes is optimal for the degraded vector broadcast channel and that dpc is optimal for the nondegraded case furthermore the capacity region is characterized under a wide range of input constraints accounting as special cases for the total power and the per antenna power constraints

design modeling and performance analysis of multi antenna heterogeneous cellular networks this paper presents a stochastic geometry based framework for the design and analysis of downlink multi user multiple input multiple output mimo heterogeneous cellular networks with linear zero forcing transmit precoding and receive combining assuming rayleigh fading channels and perfect channel state information the generalized tiers of base stations may differ in terms of their poisson point process spatial density number of transmit antennas transmit power artificial biasing weight and number of user equipments served per resource block the spectral efficiency of a typical user equipped with multiple receive antennas is characterized using a non direct moment generating function based methodology with closed form expressions of the useful received signal and aggregate network interference statistics systematically derived in addition the area spectral efficiency is formulated under different space division multiple access and single user beamforming transmission schemes we examine the impact of different cellular network deployments propagation conditions antenna configurations and mimo setups on the achievable performance through theoretical and simulation studies based on the state of the art system parameters the results highlight the inherent limitations of baseline single input single output transmission and conventional sparse macro cell deployment as well as the promising potential of multi antenna communications and small cell solution in interference limited cellular environments

duality achievable rates and sum rate capacity of gaussian mimo broadcast channels we consider a multiuser multiple input multiple output mimo gaussian broadcast channel bc where the transmitter and receivers have multiple antennas since the mimo bc is in general a nondegraded bc its capacity region remains an unsolved problem we establish a duality between what is termed the dirty paper achievable region the caire shamai see proc ieee int symp information theory washington dc june 2001 p 322 achievable region for the mimo bc and the capacity region of the mimo multiple access channel mac which is easy to compute using this duality we greatly reduce the computational complexity required for obtaining the dirty paper achievable region for the mimo bc we also show that the dirty paper achievable region achieves the sum rate capacity of the mimo bc by establishing that the maximum sum rate of this region equals an upper bound on the sum rate of the mimo bc

high snr analysis for mimo broadcast channels dirty paper coding versus linear precoding in this correspondence we compare the achievable throughput for the optimal strategy of dirty paper coding dpc to that achieved with suboptimal and lower complexity linear precoding techniques zero forcing and block diagonalization both strategies utilize all available spatial dimensions and therefore have the same multiplexing gain but an absolute difference in terms of throughput does exist the sum rate difference between the two strategies is analytically computed at asymptotically high snr furthermore the difference is not affected by asymmetric channel behavior when each user has a different average snr weighted sum rate maximization is also considered in the process it is shown that allocating user powers in direct proportion to user weights asymptotically maximizes weighted sum rate

on the geometry of classifying spaces and horizontal slices we study the local properties of the moduli space of a polarized calabi yau manifold let u be a neighborhood of the moduli space then we know the universal covering space v of u is a smooth manifold suppose d is the classifying space of a polarized calabi yau manifold with the automorphism group g then we prove that the map from v to d induced by the period map is a pluriharmonic map we also give a kahler metric on v which is called the hodge metric we prove that the ricci curvature of the hodge metric is negative away from zero we also proved the nonexistence of the kahler metric on the classifying space of a calabi yau threefold which is invariant under a cocompact lattice of g 1 introduction let x be a polarized simply connected calabi yau manifold that is x is a simply connected compact kahler manifold of dimension n with zero first chern class and is akform of x such that h 2 x z in this paper we study the local properties of the moduli space of the polarized calabi yau manifold x by definition is the parameter space of the complex structures over x for the fixed polarization is a quasi projective variety by a theorem of viehweg 17 suppose x is a calabi yau manifold let n d im h 1 x tx 0

on the weil petersson volume and the first chern class of the moduli space of calabi yau manifolds in this paper we proved that the weil petersson volume of calabi yau moduli is a rational number we also proved that the integrations of the invariants of the ricci curvature of the weli petersson metric with respect to the weil petersson volume form are all rational numbers

effect of transmit correlation on the sum rate capacity of two user broadcast channels this letter investigates the effect of the transmit correlation on the sum rate capacity of a two user broadcast channel bc with the use of equal power allocation in high signal to noise ratio snr environments it is shown using an upper bound analysis that the sum rate capacity of a correlated two user bc at high snr highly depends on the phase difference between the transmit correlation coefficients of two users as well as the magnitude of the transmit correlation coefficients and it is maximized or minimized when the principal eigenvectors of the transmit correlation matrix of two users are orthogonal or parallel to each other the analytic results are verified by computer simulation

dpc rates and multiplexing gains for mimo broadcast systems with multi dimensional space time modulation this paper considers multiple antenna broadcast channels bcs with multi dimensional space time modulation schemes created by various allocations of signal dimensions to transmit antennas the signal dimensions are divided into disjoint subsets where one is shared between all users and each of the others are assigned to different users this model encompasses several practical single point to multi point wireless communication systems employing multiple antennas assuming space and time dispersive channels we consider the transmission rates that are achieved by applying dirty paper coding dpc at the transmitter present inner and outer bounds for the dpc rate region and study the effects of space time modulation formats at high snr we consider the limit form of these bounds derive an achievable multiplexing gain region and study the effects of sharing signal dimensions between users

implicit single step methods by spline like functions for solution of ordinary differential equations abstract the main aim of our work is to create an approximate solution to the ordinary differential equations ode s which is continuously differentiable for this purpose we add an extra term to any one step explicit method for solution of ode s and this new method gives a continuously differentiable solution and by stealth the order of the solution s accuracy increases by one

families of continuous approximate processes for the solution of ordinary differential equations as is well known the solution of the initial value problem y f x y y x0 y0 is a contiuously differentiable function having this property in mind the author in 1984 developed a polynomial function of fourth degree p4 x x x0 ch 0 c 1 where h is the chosen step size such that for any c 1 p4 x0 ch would constitute a fourth order runge kutta approximation to y x0 ch and for c 1 the obtained approximation is upgraded to fifth order the author will show that there is a family of such polynomial approximants and that for any chosen fixed value of c 0 1 not just for c 1 p4 x0 ch constitutes a fifth order runge kutta approximation all these p4s necessitate six evaluations of f i e they constitute six stage processes however the first four stages are common to all members of the family the latter property may be used advantageously for step size adjustment and for error estimation the entire process is valid for scalar as well as vector initial value problems

outage constrained robust transmit optimization for multiuser miso downlinks tractable approximations by conic optimization in this paper we study a probabilistically robust transmit optimization problem under imperfect channel state information csi at the transmitter and under the multiuser multiple input single output miso downlink scenario the main issue is to keep the probability of each user s achievable rate outage as caused by csi uncertainties below a given threshold as is well known such rate outage constraints present a significant analytical and computational challenge indeed they do not admit simple closed form expressions and are unlikely to be efficiently computable in general assuming gaussian csi uncertainties we first review a traditional robust optimization based method for approximating the rate outage constraints and then develop two novel approximation methods using probabilistic techniques interestingly these three methods can be viewed as implementing different tractable analytic upper bounds on the tail probability of a complex gaussian quadratic form and they provide convex restrictions or safe tractable approximations of the original rate outage constraints in particular a feasible solution from any one of these methods will automatically satisfy the rate outage constraints and all three methods involve convex conic programs that can be solved efficiently using off the shelf solvers we then proceed to study the performance complexity tradeoffs of these methods through computational complexity and comparative approximation performance analyses finally simulation results are provided to benchmark the three convex restriction methods against the state of the art in the literature the results show that all three methods offer significantly improved solution quality and much lower complexity

geodesics distance and the cat 0 property for the manifold of riemannian metrics given a fixed closed manifold m we exhibit an explicit formula for the distance function of the canonical l 2 riemannian metric on the manifold of all smooth riemannian metrics on m additionally we examine the metric completion of the manifold of metrics with respect to the l 2 metric and show that there exists a unique minimal path between any two points this path is also given explicitly as an application of these formulas we show that the metric completion of the manifold of metrics is a cat 0 space

higher order continuous solution of y f x y y and y f x y initial value problems abstract sarafyan s continuous method for approximate solution of initial value problem is applied to the case of systems of the form y f x y y and y f x y three formulas are presented here containing polynomials in c 0 1 such that y and y are continuously imbedded with x x0 ch as valid approximations of runge kutta type the first is fifth order four stage with continuous fourth order y and third order y the second is fifth order six stage with continuous fifth order y and fourth order y the third is sixth order seven stage with conntinuous fifth order y and fourth oder y at c 1 the order for the continuous y improves by 1 the formulas may even be used advantageously in the extended interval c 0 5 1 5 values of c greater than 1 can be used to extend the approximation for the current step and to see whether h can be increased for the next step

zero forcing methods for downlink spatial multiplexing in multiuser mimo channels the use of space division multiple access sdma in the downlink of a multiuser multiple input multiple output mimo wireless communications network can provide a substantial gain in system throughput the challenge in such multiuser systems is designing transmit vectors while considering the co channel interference of other users typical optimization problems of interest include the capacity problem maximizing the sum information rate subject to a power constraint or the power control problem minimizing transmitted power such that a certain quality of service metric for each user is met neither of these problems possess closed form solutions for the general multiuser mimo channel but the imposition of certain constraints can lead to closed form solutions this paper presents two such constrained solutions the first referred to as block diagonalization is a generalization of channel inversion when there are multiple antennas at each receiver it is easily adapted to optimize for either maximum transmission rate or minimum power and approaches the optimal solution at high snr the second known as successive optimization is an alternative method for solving the power minimization problem one user at a time and it yields superior results in some e g low snr situations both of these algorithms are limited to cases where the transmitter has more antennas than all receive antennas combined in order to accommodate more general scenarios we also propose a framework for coordinated transmitter receiver processing that generalizes the two algorithms to cases involving more receive than transmit antennas while the proposed algorithms are suboptimal they lead to simpler transmitter and receiver structures and allow for a reasonable tradeoff between performance and complexity

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

an investigation of runge kutta processes and equivalence of scalar and vector cases the number p 1 of ki s or stages must be chosen large enough and the parameters must be determined so that the taylor series of y z h matches that of y i h through the terms involving h these requirements generate a system of nonlinear algebraic equations in the parameters as indicated in the following table the number of equations grows rapidly with m and much more rapidly for the vector case k 1 the formula is to be used for systems of differential equations than for the scalar case k 1 for a single differential equation

on vector perturbation precoding for the mimo gaussian broadcast channel precoding schemes in the framework of vector perturbation vp for the multiple input multiple output mimo gaussian broadcast channel gbc are investigated the vp scheme originally a one shot technique is generalized to encompass processing over multiple time instances using lattice based extended alphabets perturbations and considering the infinite time span extension limit a lower bound on the achievable sum rate using the generalized vp scheme is analytically obtained the lower bound is shown to asymptotically achieve the optimum sum rate in the high signal to noise ratio snr regime both in terms of degrees of freedom and power offset for any number of users and transmit antennas for the two user cases it is shown that the lower bound coincides with the sum capacity for low snr the above lower bound is constructively obtained by means of an efficient practically oriented suboptimal transmit energy minimization algorithm which exhibits a polynomial complexity in the number of users the proposed precoding scheme demonstrates that the shaping gain is achievable for vp schemes when employing good multidimensional lattices it is also shown that the suboptimum algorithm has its merits even when processing over multiple time instances is not employed for the 2 times 2 mimo gbc the vp scheme is generalized further and an inner bound for the entire achievable rate region is obtained by which an interesting correspondence is identified with the ultimate capacity region as obtained by dirty paper coding

stability of some continuously imbedded runge kutta methods of sarafyan abstract sarafyan and others have recently developed novel explicit runge kutta methods associated with each method is an imbedded polynomial which interpolates the runge kutta method and which is itself a runge kutta approximation at non meshpoints in this paper we show that the interpolation polynomials possess the desirable stability properties of the runge kutta methods

a user selection algorithm using angle between subspaces for downlink mu mimo systems one of major issues in the efficient use of radio resource for multiuser multiple input multiple output mu mimo systems is the selection of users to achieve the maximum system throughput the optimal user selection algorithm which requires exhaustive search is prohibitive due to its high computational complexity where block diagonalization bd method is applied and known as a suboptimal precoding technique for downlink mu mimo systems which intents to perfectly eliminate inter user interference in this paper we propose efficient iterative user selection algorithms with low complexity where the product of eigenvalues of effective channels is utilized as a selection metric by applying the concept of principal angles between subspaces and we further examined the applicability of the proposed algorithms to limited feedback systems and proportional fair pf scheduling through computational complexity analysis we show that the proposed algorithm has low complexity with a little loss in throughput simulation results validate that the proposed algorithm achieves almost the same system throughput by a capacity based algorithm under high snr regime with considerable reduction in complexity

a sixth order imbedded runge kutta algorithm with continuously variable weights abstract sarafyan s continuous method for approximate solution of initial value problems is extended to the sixth order case the approximations obtained are continuous throughout an interval the main formula is a polynomial of fifth degree in c 0 c c a valid fifth order approximation of runge kutta type is obtained for the solution at x x 0 ch at c 1 the approximation improves to sixth order lower order continuous formulas are imbedded using polynomials of lesser degree which may be used for error estimation or step size selection these formulas are differentiated to give continuous approximations to the first and second derivatives of the solution the formulas are valid for systems as well as for a single differential equation

a new method for derivation of continuous runge kutta formulas abstract a technique is introduced for the derivation of continuously embedded runge kutta formulas the method is illustrated in detail for the fourth order case a sixth order formula is presented which was derived by the technique it has two fifth order formulas continuously embedded one of which is c 1 the other is for error estimation an additional fifth order approximation is provided which usually gives error bounds for sufficiently small step sizes

sum capacity of gaussian vector broadcast channels this paper characterizes the sum capacity of a class of potentially nondegraded gaussian vector broadcast channels where a single transmitter with multiple transmit terminals sends independent information to multiple receivers coordination is allowed among the transmit terminals but not among the receive terminals the sum capacity is shown to be a saddle point of a gaussian mutual information game where a signal player chooses a transmit covariance matrix to maximize the mutual information and a fictitious noise player chooses a noise correlation to minimize the mutual information the sum capacity is achieved using a precoding strategy for gaussian channels with additive side information noncausally known at the transmitter the optimal precoding structure is shown to correspond to a decision feedback equalizer that decomposes the broadcast channel into a series of single user channels with interference pre subtracted at the transmitter

a close to capacity dirty paper coding scheme the writing on dirty paper channel model offers an information theoretic framework for precoding techniques for canceling arbitrary interference known at the transmitter it indicates that lossless precoding is theoretically possible at any signal to noise ratio snr and thus dirty paper coding may serve as a basic building block in both single user and multiuser communication systems we design an end to end coding realization of a system materializing a significant portion of the promised gains we employ multidimensional quantization based on trellis shaping at the transmitter coset decoding is implemented at the receiver using virtual bits combined with iterative decoding of capacity approaching codes we achieve an improvement of 2db over the best scalar quantization scheme code design is done using the exit chart technique

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

stepsize control for delay differential equations using continuously imbedded runge kutta methods of sarafyan the use of continuously imbedded runge kutta sarafyan methods for the solution of ordinary differential equations with either time dependent or state dependent delays is discussed it is shown how to get reliable solutions for such problems in a manner that does not require that the effect of the local approximation error be considered separately from the local integration error it is also shown how to reliably handle derivative discontinuities that arise in the solution of differential equations with delays

capacity scaling in mimo systems with general unitarily invariant random matrices we investigate the capacity scaling of multiple input multiple output systems with the system dimensions to that end we quantify how the mutual information varies when the number of antennas at either the receiver or transmitter side is altered for a system comprising r receive and t transmit antennas with r t we find the following by removing as many receive antennas as needed to obtain a square system provided the channel matrices before and after the removal have full rank the maximum resulting loss of mutual information over all signal to noise ratios snrs depends only on r t and the matrix of left singular vectors of the initial channel matrix but not on its singular values in particular if the latter matrix is haar distributed the ergodic rate loss is given by sum t 1 t sum r t 1 r frac 1 r t nats under the same assumption if t r to infty with the ratio phi triangleq t r fixed the rate loss normalized by r converges almost surely to h phi bits with h cdot denoting the binary entropy function we also quantify and study how the mutual information as a function of the system dimensions deviates from the traditionally assumed linear growth in the minimum of the system dimensions at high snr

generation of log aesthetic curves using adaptive runge kutta methods log aesthetic curve lac has been explored extensively by many researchers since 2005 at first gaussian kronrod has been proposed to evaluate lac as the formulation of lac involves double integration recently incomplete gamma function igf has been proposed to represent lac analytically which decreases the computation time up to 13 times this paper embarks on the representation of lac using adaptive runge kutta methods to decrease the lac computation time the famous adaptive methods such as runge kutta fehlberg dormand prince sarafyan and kutta merson are employed to evaluate lac so that a desired accuracy can be achieved this paper ends with a detailed investigation on performance metric of igf and adaptive rk methods to compute lac these methods will be compared in terms of computation time and truncation error numerical results indicate that the computation time of lac can be greatly improved and at the same time preserving the lac s family

new algorithms for the continuous approximate solutions of ordinary differential equations and the upgrading of the order of the processes three different continuous processes are described the main algorithms in all the three methods are the embedded type six stage fifth order runge kutta formulas the principal objective of this work is the continuous approximate solution of second and higher orderdifferential equations and implicit differential equations of any orders a method for upgrading the orders of the approximate solutions will be established thus in the case of the explicit differential equations of order n and the implicit equations of order n 1 it is possible to generate with the upgraded processes based upon a main fifth order six stage algorithm continuous approximations of order n 3 over the entire interval of application including the endpoints additionally they have continuous derivatives through order n 1 the upgraded formulas provide excellent error approximations

joint precoding optimization for multiuser multi antenna relaying downlinks using quadratic programming this paper studies the optimization problem for joint precoding design in a multi antenna downlink channel using relaying we formulate the joint source and relay precoding design by aiming at sum capacity maximization since this problem is in general nonconvex we first convert this problem into standard convex quadratic programs and then propose an iterative joint precoding optimization algorithm by utilizing efficient quadratic programming approaches we observe that the iterative method always yields optimal precoding matrices which diagonalize the compound channel of the backward source to relay and the forward relay to destination links at high snr regimes motivated by this observation we further develop an efficient structured precoding design simulation results are presented to verify the effectiveness of our proposed precoding schemes

stepsize control for delay differential equations using a pair of formulae abstract the effect of the local approximation error on the stepsize control at one step methods which are used for the numerical solution of delay differential equations are considered it is shown how to get a reliable estimate for the local error and so a working stepsize control by using a pair of formulae

continuous approximate solution of ordinary differential equations and their systems abstract the solution of initial value problems in ordinary differential equations are continuously differentiable functions yet all effective methods presently used for the numerical solution of these problems either in scalar or vector form are discrete processes they yield a finite set of approximate values of the solution the process derived in this paper yields a continuously differentiable function a polynomial which gives fourth order runge kutta approximations at all interior points of each interval of application and a fifth order approximation at each end point like conventional fifth order runge kutta algorithms the process requires only six evaluations of the given directional function per application or step it is endowed with an imbedding property for step size control and provides valuable information about committed local truncation errors it may be called a continuous imbedded runge kutta process of maximum order five it is valid for both scalar and vector cases

runge kutta triples abstract the addition of a third runge kutta rk formula to an embedded pair forming an rk triple can yield an algorithm which gives solutions of appropriate asymptotic accuracy at points within the normal step intervals in the numerical solution of the initial value problem two modes of implementation are possible and it is shown that these can be regarded as equivalent when certain conditions are satisfied an optimized rk5 4 triple based on a particularly efficient embedded pair is presented and tested the results indicate the efficiency of the dense output technique