(r∞, n12, s12) measurements on the 2-CPU CRAY X-MP

We report performance measurements made on the 2-CPU CRAY X-MP at ECMWF, Reading. Vector (SIMD) performance on one CPU is interpreted by the two parameters ($\text{r}{}_{\mathrm{\infty }}\text{, n}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$), and we find for dyadic operations using FORTRAN $\text{r}{}_{\mathrm{\infty }}\text{= 70}\text{Mflop/s}\text{, n}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 53}\text{flop}$. All vector triadic operations produce $\text{r}{}_{\mathrm{\infty }}\text{= 107}\text{Mflop/s}\text{, n}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 45}\text{flop}$; and a triadic operation with two vectors and one scalar gives r_∞ = 148 Mflop/s and $\text{n}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 60}\text{flop}$. MIMD performance using both CPUs on one job is interpreted with the two parameters ($\text{r}{}_{\mathrm{\infty }}\text{, s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$), where $\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is the amount of arithmetic that could have been done during the time taken to synchronize the two CPUs. We find, for dyadic operations using the TSKSTART and TSKWAIT synchronization primitives, that r_∞ = 130 Mflop/s and $\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 5700}\text{flop}$. This means that a job must contain more than ～ 6000 floating-point operations if it is to run at more than 50% of the maximum performance when split between both CPUs by this method. Less expensive synchronization methods using LOCKS and EVENTS reduces $\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ to 4000 flop and 2000 flop respectively. A simplified form of LOCK synchronization written in CAL code further reduces $\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ to 220 flop. This is probably the minimum possible value for synchronization overhead on the CRAY X-MP.     

Off-line feed-forward and H∞ feedback control on a vibration rig

This paper discusses the improvement of the tracking accuracy on an automotive test rig by extending the industrial available off-line controller with an $\text{H}{}_{\mathrm{\infty }}$ feedback controller. The improvement is shown by comparing the tracking error for both control schemes: the industrial controller on the one hand and the industrial controller with $\text{H}{}_{\mathrm{\infty }}$ feedback controller on the other hand. The $\text{H}{}_{\mathrm{\infty }}$ controller is designed based on a mixed sensitivity approach. After a theoretical analysis, the results are illustrated by simulations and experimental results on one axis of a hydraulic test rig which is used in the automotive industry for vibration comfort evaluations of new vehicle prototypes.     

Averaging shifted histograms

A method which consists in shifting different histograms of the same spectrum and then taking their average is presented in order to smooth the data and to increase the localization accuracy and separation of the peaks. The statistical properties of this method are investigated. The average of two histograms with shifted bin limits is studied. It is shown that for histograms with random bin limits, distributed according to

$\text{F}{}_{\mathrm{i}}\text{(x)=}\text{∫}\text{?∞}\text{x}\text{?}{}_{\mathrm{i}}\text{(ξ, μ}{}_{\mathrm{i}}\text{, σ)}\text{d}\text{ξ}$

; where the standard deviation σ is very small compared to the difference of the means (μ_i+1 ? μ_i) for ll i the zero order approximation to the variance of this histogram is given by:

$\text{var}\text{∽}\text{(H)=}\text{∑}\text{i=0}\text{m}\text{(A}{}_{\mathrm{i+1}}\text{?a}{}_{\mathrm{i}}\text{)}{}^2\text{F}{}_{\mathrm{i+1}}\text{(x)(1?F}{}_{\mathrm{i+1}}\text{(x))}$

, where

$\text{a}{}_{\mathrm{i}}\text{=}\text{1}\text{x}{}_{\mathrm{i=1}}\text{?x}{}_{\mathrm{i}}\text{∫}\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i+1}}\text{g(ξ)}\text{d}\text{ξ}$

and g is an unknown function fitted by the histogram. Formula (1) gives also the relation:

$\text{v}\text{a}\text{?}\text{r}\text{(}\text{(H}{}_1\text{+ H}{}_2\text{)}\text{2}\text{) =}\text{1}\text{4}\text{(}\text{v}\text{a}\text{?}\text{r}\text{(H}{}_1\text{(x)) +}\text{v}\text{a}\text{?}\text{r}\text{(H}{}_2\text{(x))}$

, when H₁ and H₂ have stochastically independent bin limits.When the histogram H is considered as a spline function S of order one it is shown that for the minimization criterion with respect to the coefficient of the spline:

$\text{M}{}_1\text{=}\text{min}\text{∫}\text{x}{}_1\text{x}{}_{\mathrm{m+1}}\text{(g(x) ? S}{}_1\text{(x))}{}^2\text{d}\text{x}$

, the following result holds: $\text{M}{}_{\mathrm{a}}\text{?}{}^1{}_2\text{(M}{}_1\text{+ M}{}_2\text{)}$, where $\text{S}{}_{\mathrm{a}}\text{(x) =}{}^1{}_2\text{(S}{}_1\text{(x) + S}{}_2\text{(x))}$. If the number of shifted histograms tends to infinity, then

$\text{S}{}_{\mathrm{\infty }}\text{(x) = [Γ(x + h) + Γ(x ? h) ? 2Γ(x)]/h}{}^2$

, where $\text{Γ(x) =}\text{∫}\text{?∞}\text{x}\text{∫}\text{?∞}\text{η}\text{g(ξ)}\text{d}\text{ξ}\text{d}\text{η}$, and h is a constant bin size. Then

$\text{M}{}_{\mathrm{\infty }}\text{≈}\text{h}{}^4\text{144}\text{∫}\text{x}{}_1\text{x}{}_{\mathrm{m+1}}\text{g″}{}^2\text{(x)}\text{D}\text{x}$

. Extensions to two-dimensional histograms and to higher order (empirical distributions) are presented.     

Computation of subcritical compressible flows

A finite element analysis and iterative solution of steady plane inviscid compressible flows is developed. Specific attention is directed to subcritical flows in which the nonlinear governing equation is elliptic, and to slightly supercritical mixed flows. The underlying variational theory for this nonlinear flow problem and a corresponding finite element formulation are presented. A Newton-Raphson iteration is introduced within this approximate analysis to provide efficient solution of subcritical flows and also slightly supercritical flows in which the nonlinear flow equation is of mixed type. The performance of the algorithm is studied with particular reference to the effect of a two-parameter scheme in which incident Mach number $\text{M}{}_{\mathrm{x}}$ and specific heat ratio γ may be varied. By thus regularizing the operator during the early iteration history, more efficient computations can be realized in the mildly-transonic flow regime of higher incident Mach numbers.     

Determination of a general economic location constant by monte carlo simulation

The problem of estimating the number of markets (or plants) to serve a set of sources in a given geographical area was considered. Markets were located so as to minimize total assembly cost which was considered a linear function of the weighted Euclidean distances between sources and markets. The following predictive function C_m was proposed for estimating the minimum total assembly cost for a given number of markets: $\text{C}{}_{\mathrm{m}}\text{= C}{}_1\text{?(}\text{m?1}\text{m}\text{)}{}^{\mathrm{k}}\text{(}\text{M}\text{M ? 1}\text{)}{}^{\mathrm{k}}\text{(C}{}_1\text{? C}{}_{\mathrm{M}}\text{),m = 1, 2, 3, …, M}$ where m = number of markets being located. M = maximum number of potential market sites. C₁ = minimum assembly cost when only one market is located. C_M = minimum assembly cost when all M markets are located. k = an undetermined constant which specifies the shape of the function.The validity of the C_m function and the range of the k constant were determined by computer Monte Carlo experimentation. The constant k, to a sufficient degree of approximation and ordinary use, was found independent of the number of sources and their distribution. A general economic location co     

Nonsimilar boundary layers on the leeside of cones at incidence

It is known that the similarity equations for the leeward line of symmetry of a cone have no solutions for $\text{?1 < k}{}_2{}^1\text{< k < k}{}_1\text{1 < 0}$, where k is the incidence parameter of Moore, and $\text{k}{}_1{}^1\text{, k}{}_2{}^1$ depend on the external Mach number and enthalpy ratio. In this paper we present evidence that a leeside integration for such k terminates at a finite distance from the vertex in a singularity of the type analyzed by Stewartson and Simpson for entry flow in a curved pipe, and compare the theory with a representative numerical solution at $\text{k = ?}\text{1}\text{2}$. A possible interpretation of this singularity is that the boundary layers growing from the windward line of symmetry have collided, and support for this view is given by windward to leeward integrations for a range of values of k. For k such that the similarity equations have solutions, these are interpreted as limits of more general solutions.     

Crosscorrelation between linearly and nonlinearly distorted versions of a given signal

A given deterministic signal x(.) is distorted by passing it through a linear time-invariant filter and also by subjecting it to the action of an instantaneous nonlinearity. The resulting time crosscorrelation of the two distorted versions of the original signal is expressed by the function

$\text{R}{}_2\text{(s)?∫}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{?∫}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{g[x(t)]k(t?t′)x(t?s)dt dt′}$

, where x(.) is the given signal, k(.) is the nonnegative definite impulse response of the linear filter, and g(.) is the output-input characteristic of the zero-memory nonlinear device. The problem considered is that of determining conditions on the pair (x,g) such that R₂(s) ? R₂(0) for all s and any choice of nonnegative definite filter function k; the principal result is the formulation of a necessary and sufficient condition for R₂ to have a global maximum at the origin. In particular, the peak value occurs at the origin if and only if $\text{G}{}_{\mathrm{x}}{}^1\text{(ω)X(ω)}$ is real and nonnegative for all ω ? 0, where G_x(.) and X(.) are the Fourier transforms of g[x(.)] and x(.), respectively. An equivalent condition is that the correlation function

$\text{R}{}_2\text{(s)?∫}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{g[x(t)]x(t?s)dt}$

, previously studied by Richardson, be nonnegative definite.Several examples are given, and it is shown that, unlike the case for R₁(.), monotonicity of g(.) is not a sufficient condition for R₂(.) to have a global maximum at s = 0 independently of the choice of filter characteristic k. Certain extensions of these results are given for the case when x(.) is a Gaussian random input.     

An optimal triangulation for second-order elliptic problems

Let Ω be a polygonal domain in $\text{R}{}^{\mathrm{n}}$, τ_h an associated triangulation and u_h the finite element solution of a well-posed second-order elliptic problem on (Ω, τ_h). Let M = {M_i}^{p + q}_{i = 1} be the set of nodes which defines the vertices of the triangulation τ_h: for each i,$\text{M}{}_{\mathrm{i}}\text{= {x}{}_{\mathrm{i}}{}^{\mathrm{l}}\text{¦1 ? l ?n}}$ in Rⁿ. The object of this paper is to provide a computational tool to approximate the best set of positions M? of the nodes and hence the best triangulation $\text{}\text{?}\text{gt}{}_{\mathrm{h}}$ which minimizes the solution error in the natural norm associated with the problem.The main result of this paper are theorems which provide explicit expressions for the partial derivatives of the associated energy functional with respect to the coordinates x_i^l, 1 ? l ? n, of each of the variable nodes M_i, i = 1,…, p.     

Uniform computational complexity of the derivatives of C∞-functions

We discuss the uniform computational complexity of the derivatives of C^∞-functions in the model of Ko and Friedman (Ko, Complexity Theory of Real Functions, Birkhäuser, Basel, 1991; Ko, Friedman, Theor. Comput. Sci. 20 (1982) 323–352). We construct a polynomial time computable real function g∈C^∞[−1,1] such that the sequence $\text{{|g}{}^{\mathrm{(n)}}\text{(0)|}}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is not bounded by any recursive function. On the other hand, we show that if f∈C^∞[−1,1] is polynomial time computable and the sequence of the derivatives of f is uniformly polynomially bounded, i.e., |f⁽ⁿ⁾(x)| is bounded by 2^p(n) for all x∈[−1,1] for some polynomial p, then the sequence $\text{{f}{}^{\mathrm{(n)}}\text{}}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is uniformly polynomial time computable.     

Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on ƒ involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem $\text{y}{}_{\mathrm{<mtext>k</mtext><mtext>+1</mtext>}}\text{− 2y}{}_{\mathrm{k}}\text{+ y}{}_{\mathrm{<mtext>k</mtext><mtext>-1</mtext>}}\text{+ ƒ(k,y}{}_{\mathrm{k}}\text{,v}{}_{\mathrm{k}}\text{) = 0, for k = 1,…, n − 1, y}{}_0\text{= 0 = y}{}_{\mathrm{n}}$, where ƒ is continuous and v_k = y_k − y_k−1, for k = 1,…,n. In the special case $\text{ƒ(k,t,p) = ƒ(t) ≥ 0}$, we give growth conditions on ƒ and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and Peterson and are in the spirit of our results for the continuous analogue.     

On the stability of convolution feedback systems with dynamical feedback

This paper considers distributed n-inputn-output convolution feedback systems characterized by $\text{y = G}{}_1\text{1e}$, $\text{z = G}{}_2\text{1y}$ and e = u ? z, where the forward path transfer function $\text{G}\text{?}{}_1$ and the feedback path transfer function $\text{G}\text{?}{}_2$ both contain a real single unstable pole at different locations. Theorem 1 gives necessary and sufficient conditions for both input-error and input-output stability. In addition to usual conditions that guarantee input-error stability a new condition is found which results in the fact that input-error stability will guarantee input-output stability. These conditions require to investigate only the open-loop characteristics. A basic device is the consideration of the residues of different transfer functions at the open-loop unstable poles. An example is given.     

An iterative method for determining the current-potential curves of a spherical probe in a slightly ionized continuum plasma with 5 ⩽ ρp ⩽ 50, yp < 7

A desk analogue computer has been used to draw normalized curves for positive and negative particle concentrations $\text{(n}{}_{\mathrm{+}}\text{,n}{}_{\mathrm{?}}\text{)}$ and potential $\text{(y)}$ vs the distance from a spherical probe surface for a slightly ionized continuum plasma. The governing equations are three simultaneous differential equations leading to a two-point boundary condition problem. An iterative analogue method has been developed which enables the positive and negative particle currents $\text{(J}{}_{\mathrm{+}}\text{,J}{}_{\mathrm{?}}\text{)}$ to the probe, for particular $\text{y}{}_{\mathrm{p}}\text{< 7}$ and $\text{ρ}{}_{\mathrm{p}}\text{< 50}$, to be found to a numerical accuracy of about 1 digit in the second decimal place.     

The queueing system EkMa,b1 and its numerical analysis

In this paper steady-state random-point results for single-server bulk-service system ${}_{\mathrm{k}}\text{M}{}^{\mathrm{a,b}}\text{1}$ are related to imbedded Markov Chain results using the Erlangian technique. Steady-state results, including the expected number in queue at random and arrival epochs, and waiting time distribution are found in terms of the unique root of the characteristic equation. Examples of extensive numerical work are presented in tabular forms.     

Gibbs energies of formation of Mn3C,M(Fe,Mn)3C and Mn23C6 from the ternary phase equilibria in the FeMnC system

The high temperature phase relations in the FeMnC system have been analyzed in light of the recently developed thermodynamic method by the authors to obtain the Gibbs energies of formation of Mn₂₃C₆ and Mn,C. A new thermodyn/amic treatment is outlined and applied to obtain the stability of the ternary carbide M(Fe,Mn)₃C without any a priori assumption of a solution model for the M₃C phase. The recommended Gibbs energies of formation for the Mn carbides, Mn₃C and Mn₂₃C₆ With γ-Mn (graphite) as the Standard states are:

The present method can be extended to obtain a consistent set of thermodynamic data for binary and ternary carbides from various ternary metal-metal-carbon phase relations.     

Normal forms for nonautonomous difference equations

We extend Henry Poincaré's normal form theory for autonomous difference equations χ_{k + 1} = f(χ_k) to nonautonomous difference equations χ_{k + 1} = f_k(χ_k). Poincaré's nonresonance condition $\text{α}{}_{\mathrm{j}}\text{-П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{=1α}{}^{\mathrm{qi}}{}_{\mathrm{i}}\text{≠0}$ for eigenvalues is generalized to the new nonresonance condition λ_j∪$\text{α}{}_{\mathrm{j}}\text{⊔П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{α}{}^{\mathrm{qi}}{}_{\mathrm{i}}\text{≠0}$ for spectral intervals.