Abstract.A two-parameter family of approximations to the exponential function is considered. Constraints on the parameters are determined which guarantee the approximations are ^-acceptable. The suitability of these approximations for 2-point vl-stable exponential fitting is established. Several numerical methods, which produce these approximations when solving y = \v, are presented.
Abstract.Let y =f (y,t) with y(to) =Y0 possess a solution y(t) for t~to. Set t~ =to +nh, n = 1, 2 ...... Let Yo denote the approximate solution of y(tn) defined by the composite multistep methodwith .//n--f(yn,tn) and _N = 1,2 ...... It is conjectured that the method is stiffly stable with order p=l for all l:>l and shown to be so for l= 1 ..... 25. The method is intrinsically efficient in that l future approximate solution values are established simultaneously in an iterative solution process with only one function evaluation per iteration for each of the I future time points.Step and order control are easily implemented, in that the approximate solution at only one past point appears in each component multistep formula of the method and in that the local truncation error for the first component multistep formula of the method is easily evaluated as h kU~l --YlVlf , PREn denotes the value at tiv~ of tile Lagrange interpolating polynomial where u~v~ passing through the points Y(N-1)z+j at t(~-l)~+j with j = -1, 0,..., l-1. Introduction.Many dynamical systems are characterized by stiff differential equations. For example, a linear electrical network with a large ratio of maximum to minimum moduli of natural frequencies is stiff. To achieve efficient numerical integration of stiff differential equations, it is necessary to choose a numerical integration method with care. One property sought in selecting a method is stiff-stability.Gear [1] having originated the definition of stiff stability went on to
Abstract.A two-parameter family of approximations to the exponential function is considered. Constraints on the parameters are determined which guarantee the approximations are ^-acceptable. The suitability of these approximations for 2-point vl-stable exponential fitting is established. Several numerical methods, which produce these approximations when solving y = \v, are presented.
This paper introduces a flexible design method of computationally efficient uniform filter bank where each filter channel is obtained by frequency translation of a prototype base-band filter.It is shown that the original signal decomposed by the analysis filter bank into N adjacent uniform subbands subsampled by N, can be reconstructed with negligible distortion. We givea practical implementation which allows an important reduction in computational complexity.It can be therefore applied to such signal processing tasks as speech coding or spectral parametrization of signals.
A set of stiffly stable, cychc composite multlstep methods, expressed as l t a,~ymt+j-h ~ fl,jYmt+j=O with ~= 1,. ,l, j=--k+~ ]--I of orders 3 through 7 is established Each method exhibits better stability properties than those of the backward dffferentmtlon formula of the same order. A new varmble-step-slze, variable-order integration algorithm incorporating these new cyclic methods is described This algorithm, realized as a Fortran program, is ideally suited for the numemcal mtegratmn of stiff systems of first order ordinary differential equations y = f(y, t) wherein the Jacobian matrix fy(y, t) has complex eigenvalues near the Imaginary axis Numermal results to support th~s clmm are presented, mcludmg a comparison of the new program with GEAR, Hmdmarsh's tuned versmn of Gear's widely accepted computer program DIFSUB, and with EPISODE, Byrne and Hmdmarsh's counterpart of DIFSUB and GEAR using a varmble-step form of the backward dffferentmuon formulas.Key Words and Phrases stiff dffferentml equatmns, stiffly stable methods, composite multlstep methods, cyclic methods, numerical mtegratlon, ordinary differential equatmns, initial value problems, multlstep formulas, numermal mtegratmn program, Fortran code STINT CR Categorms 5 16, 5 17 The Algorithm. STINT. STiff {differential equatmns) INTegrator. ACM Trans Math. Software 4, 4 {Dec. 1978), 399-403.
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