Showing papers on "Bilinear interpolation published in 1977"
TL;DR: A system of bilinear equations is derived which under appropriate conditions model the microbial cellgrowth and product formation of various waste treatment and fermentation systems and are significant for the on-line control of these biological processes.
200 citations
TL;DR: The bilinear model explains the particular effect that in homologous series the relationship between biological activity and hydrophobic character is strictly linear for the lower members, while for higher members this relationship is nonlinear.
Abstract: The bilinear model, log 1/C =a log P-b log (betaP+1) +C, a new model for nonlinear dependence of biological activity on hydrophobic character, is applied to 57 data sets of biological activity values in homologous series. From a comparison of the statistical parameters and the residuals obtained with the bilinear model and the parabolic model, the superiority of the bilinear model for a precise quantitative description of both linear and nonlinear parts of sturcture-activity relationships can be derived; the bilinear model explains the particular effect that in homologous series the relationship between biological activity and hydrophobic character is strictly linear for the lower members, while for higher members this relationship is nonlinear.
190 citations
15 Aug 1977
TL;DR: The problem of mathematically defining a smooth surface, passing through a finite set of given points, is studied andoretical results are presented that provide valuable guidance in the development of algorithms for constructing triangular grids.
Abstract: The problem of mathematically defining a smooth surface, passing through a finite set of given points is studied. Literature relating to the problem is briefly reviewed. An algorithm is described that first constructs a triangular grid in the (x,y) domain, and first partial derivatives at the modal points are estimated. Interpolation in the triangular cells using a method that gives C sup.1 continuity overall is examined. Performance of software implementing the algorithm is discussed. Theoretical results are presented that provide valuable guidance in the development of algorithms for constructing triangular grids.
89 citations
TL;DR: In this article, it is shown that broken line interpolation as a scheme for piecewise monotone interpolation is hard to improve upon, and that a family of smooth piecewise polynomial interpolants, introduced by Swartz and Varga and noted by Passow to be piecewise non-linear, converges to a piecewise constant interpolant as the degree goes to infinity.
74 citations
TL;DR: The stabilization algorithm is applied to the design of a control system for the Large-Space Telescope using a linear feedback scheme to establish a finite region of stability for the overall system.
Abstract: Two classes of large-scale systems are identified, which can always be stabilized by decentralized feedback control. For the class of systems composed of interconnected linear subsystems, we can choose local controllers for the subsystems to achieve stability of the overall system. The same linear feedback scheme can be used to stabilize a class of linear systems with bilinear interconnections. In this case, however, the scheme is used to establish a finite region of stability for the overall system. The stabilization algorithm is applied to the design of a control system for the Large-Space Telescope.
58 citations
TL;DR: In this paper, the authors discuss the properties of a Bilinear Programming problem and develop a convergent cutting plane algorithm, which involves only a subset of the variables and preserves the special structure of the constraints involving the remaining variables.
Abstract: In this paper we discuss the properties of a Bilinear Programming problem, and develop a convergent cutting plane algorithm. The cuts involve only a subset of the variables and preserve the special structure of the constraints involving the remaining variables. The cuts are deeper than other similar cuts.
51 citations
TL;DR: A quadratic application of binary search yields a variant of conventional interpolation search that is easily shown to use about 2.4 lg lg n probes on the average, andine analysis of this variant requires only a simple argument, and the most
49 citations
TL;DR: In this article, a class of methods for local interpolation at irregularly spaced points for functions of two or more variables is developed, based on a weighted average of the values of local interpolating functions.
Abstract: : A class of methods for local interpolation at irregularly spaced points for functions of two or more variables is developed. The methods are based on a weighted average of the values of local interpolating functions, with the local interpolating functions and the weighting functions chosen so as to incorporate the desired smoothness. Numerical results for several interpolation functions from this class are compared with global approximations, some of which are local when implemented on a computer.
45 citations
TL;DR: A method based on 2-dimensional, bivariate interpolation is described for use in evaluating the Sommerfeld integrals which arise in an integral-equation solution of antennas which interact with a halfspace.
Abstract: A method based on 2-dimensional, bivariate interpolation is described for use in evaluating the Sommerfeld integrals which arise in an integral-equation solution of antennas which interact with a halfspace. This approach provides the rigour of the Sommerfeld formulation with a computer time comparable to the perfect-ground problem.
25 citations
TL;DR: In this article, the set of values taken on by the conjugate bilinear functional (A x, y) asx andy range over all unit vectors with prescribed inner product is analyzed.
Abstract: LetA be an operator on a finite dimensional unitary space. This paper contains results on the set of values taken on by the conjugate bilinear functional (A x, y) asx andy range over all unit vectors with prescribed inner product. By analyzing the same problem for the induced functional on the Grassmannian, results on non-principal subdeterminants are also obtained.
19 citations
TL;DR: In this article, the Lagrange interpolating projection on the nodes t 0,. •., tn is the operator which takes any function ƒ continuous on [a, b] to the polynomial S?=oA )̂V/«.
Abstract: On an interval [a, b], we may place points f0,. * . , tn such that a = 10 < t! < * • * < tn = b. Using these points, called nodes, we may construct unique polynomials y0, . . . ,yn of degree n, such that, for 1 < i, ƒ < n>y£tj) = 1 and yt(tj) = 0 for ƒ =£ i. The Lagrange interpolating projection on the nodes t0,. • . , tn is the operator which takes any function ƒ continuous on [a, b] to the polynomial S?=oA )̂V/« It is easily seen that this projection is bounded for any degree n, for any interval [a, b], and for any set of nodes in [a, b]. The norm is easily shown to be the sup norm of A = SJLQI^I, called the Lebesgue function of the projection, and thus the norm depends exclusively on the placement of tv . . . , tn_x. It is irrelevant, in attempting to minimize the norm, to move tQ or tn. Of the function A, it is true that A(^) = 1 for 0 < i < n, while if n > 2 and if t is not a node, then A(f) > 1. Let \\x, . • . , \
be the values given by
01 Jan 1977
TL;DR: By consideration of special ideals a n-dimensional generalization of Max Noether's theorem is obtained, which enables us to answer questions arising in the constructive theory of functions as it is shown by three examples.
Abstract: The ideal-theoretic concept of the Hermite interpolation was presented in [9]. Some of its results are summarized in this paper. By consideration of special ideals a n-dimensional generalization of Max Noether's theorem is obtained. This generalization enables us to answer questions arising in the constructive theory of functions as it is shown by three examples.
TL;DR: In this paper, a series of numerical tests have been made to assess the size of errors (relative to a fine-group standard) in eigenvalue, reaction rate ratios, isotopic worth components, and spectral shapes resulting from the use of the two energy collapse procedures.
Abstract: Flux weighting for the generation of broad-group cross sections is designed to preserve eigenvalue, flux spectrum, and reaction rates; however, it will not preserve adjoint spectrum and reactivity worths. Bilinear (flux-adjoint) weighting preserves all of the above quantities except reaction rates. Bilinear weighting also makes the eigenvalue of the broad-group calculation less sensitive to distortions of the spectrum away from the fundamental mode over which the cross sections were collapsed than is the case when flux weighting is used. A series of 29- and 11-group numerical tests has been made to assess the size of errors (relative to a fine-group standard) in eigenvalue, reaction rate ratios, isotopic worth components, and spectral shapes resulting from the use of the two energy collapse procedures. The errors in adjoint spectrum and calculated worth of scattering materials are found to be large when flux weighting is used.
TL;DR: In this paper, a formula for interpolation between output samples of a fast Fourier transform (FFT) is derived for obtaining greater frequency resolution when two coarse FFT outputs are available.
Abstract: A formula is derived for interpolation between output samples of a fast Fourier transform (FFT), i.e., in the frequency domain. Such a formula is useful for obtaining greater frequency resolution when two coarse FFT outputs are available. Consideration is also given to the effect of such interpolation on a weighted FFT.
TL;DR: The experimental results show that the method proposed can give an accurate estimation to the bilinear distributed parameter orocess by a simple device.
TL;DR: In this article, a complete derivation of Romberg integration for an arbitrary sequence of integration steplenghts, using classical interpolation theory only, is given, and an explicit expression for the error is derived using Lagrange interpolation.
Abstract: A complete derivation of Romberg integration for an arbitrary sequence of integration steplenghts, using classical interpolation theory only, is given. An explicit expression for the error is derived using Lagrange interpolation. From the general theory developed, several previous known results may be derived as special cases.
TL;DR: In this article, the formal equivalence of Rauhala's method for dimensionwise factorization to conventional least squares using rectangular calibration grids is proven, and numerical stability is analyzed in terms of the condition numbers of normal datamatrices.
TL;DR: In this article, a necessary and sufficient condition for a general class of neutral differential-difference equations to be exponentially stable is given in terms of certain bilinear functionals which are the equivalent of quadratic Liapunov functions for finite-dimensional systems.
01 Dec 1977
TL;DR: In this article, the augmented error signal method developed for adaptive control of continuous and discrete, linear, single input, single output plants is extended to a class of dynamical processes described by a bilinear equations.
Abstract: The augmented error signal method developed for adaptive control of continuous and discrete, linear, single input, single output plants is extended to a class of dynamical processes described by a bilinear equations. The approach combines the adaptation algorithms given in [1] and respectively [2] with the stability conditions for certain classes of bilinear systems presented in [8].
TL;DR: In this paper, it was shown that Cauchy interpolation may be used together with Pade approximation to construct reduced-order models from large-order linear time-invariant systems.
Abstract: It is shown in this correspondence that Cauchy interpolation may be used together with Pade approximation to construct reduced-order models from large-order linear time-invariant systems.