Abstract: A method of blvariate interpolation and smooth surface fitting is developed for z values given at points irregularly distributed in the x-y plane. The interpolating function is a fifth-degree polynomial in x and y defined in each triangular cell whmh has projections of three data points in the x-y plane as its vertexes. Each polynomial is determined by the given values of z and estimated values of partial derivatives at the vertexes of the triangle. Procedures for dividing the x-y plane into a number of triangles, for estimating partial derivatives at each data point, and for determining the polynomial in each triangle are described A simple example of the application of the proposed method is shown. Key W6rds and Phrases bivariate interpolation, interpolation, partial derivative, polynomial, smooth surface fitting CR Categories: 5.13 The Algorithm Bivariate Interpolation and Smooth Surface Fitting for Irregularly Distributed Data Points ACM Trans. Math. Software 2, 1(June 1978), 160-164.

Abstract: Shepard developed a scheme for interpolation to arbitrarily spaced discrete bivariate data. This scheme provides an explicit global representation for an interpolant which satisfies a maximum principle and which reproduces constant functions. The interpolation method is basically an inverse distance formula which is generalized to any Euclidean metric. These techniques extend to include interpolation to partial derivative data at the interpolation points.

Abstract: Data from a boundary-layer experiment conducted over a flat, uniform site in Minnesota provide a clue to the behavior of the low-frequency peak in surface-layer horizontal velocity spectra This portion of the spectra shows systematic behavior only when plotted in dimensionless coordinates appropriate to the mixed layer, whereas the inertial subrange frequencies follow Monin-Obukhov similarity Based on this observation, interpolation formulas for both the longitudinal and the lateral velocity components, are derived The expressions involve the boundary-layer depth in addition to the usual surface-layer parameters

Abstract: The collection contains 12 articles (with Russian-language abstracts) concerned with the theory and application of spline functions as used for solving problems of interpolation, smoothing, function approximations, and solving differential and integral equations. Also presented are analysis and correction of boundary problems.

Abstract: For many sonar applications, the sensor outputs of a hydrophone array are sampled at a rate significantly higher than that required for waveform reconstruction when digital beamforming is used. The reason for this is that the number of synchronous, or ’’natural,’’ beampointing directions is proportional to the beamformer input rate. This paper presents an implementation of a digital beamformer that achieves the desired synchronous beams while minimizing the sensor channel sampling rate requirement. The technique employs zero padding of sensor data followed by digital interpolation filters to achieve vernier beamformer delays. Interpolation filtering can be done either at the beamformer input or output to minimize processing requirements. The resulting structure realizes a hardware savings since both A/D converter and cable bandwith requirements can be traded off against digital processing complexity to achieve an optimal partitioning.

Abstract: Algorithms are presented for computing a quadratic spline interpolant with variable knots which preserves the monotonicity and convexity of the data. It is also shown that such a spline may not exist for fixed knots.

Abstract: This paper presents the use of B-splines as a tool in various digital signal processing applications. The theory of B-splines is briefly reviewed, followed by discussions on B-spline interpolation and B-spline filtering. Computer implementation using both an efficient software viewpoint and a hardware method are discussed. Finally, experimental results are presented for illustrative purposes in two-dimensional image format. Applications to image and signal processing include interpola- tion, smoothing, filtering, enlargement, and reduction.

Abstract: In this note we will present the most general linear form of a Neville-Aitken-algorithm for interpolation of functions by linear combinations of functions forming a Cebysev-system. Some applications are given. Expecially we will give simple new proofs of the recurrence formula for generalized divided differences [5] and of the author's generalization of the classical Neville-Aitkena-algorithm[8]applying to complete Cebysev-systems. Another application of the general Neville-Aitken-algorithm deals with systems of linear equations. Also a numerical example is given.

Abstract: The twin-cone beam geometry has been used to reconstruct a three-dimensional object from projections. The reconstruction has been performed by means of the additive algebraic reconstruction technique (ART) extended to three dimensions. Several simulation experiments have been undertaken to explore the effect of pseudo-projections, analytic projections, iteration-dependent damping and interpolation schemes on the performance of the reconstruction. The results indicate that the ART-reconstruction from twin-cone beam projections can be achieved without blurring artefacts. The measured reconstruction error is of the same order of magnitude as the error of a ART-multislice reconstruction from coaxial projections.

Abstract: : A new hydrodynamical interpolation technique has been developed and tested to construct a model of global ocean tides with the support of empirical tidal constants collected around the world. The discrete tide model features a 1 deg by 1 deg graded grid system in connection with a hydrodynamically defined bathymetry. The Laplace tidal equations are augmented by turbulent friction terms with novel mesh-area (latitude and depth) dependent eddy-viscosity and bottom-friction coefficients. The well-known astronomical tide-generating forces are modified by effects due to solid earth tides and ocean-tidal loading. New averaged finite differences in time are used to enhance stability characteritics and to facilitate the hydrodynamical interpolation of empirical data. This unique interpolation is accomplished by a controlled adjustment of the bottom-friction coefficient and by redefining a more physical shoreline. Extensive computer experiments were conducted to study the characteristics of the novel friction laws and hydrodynamical interpolation methods. The computed M2 tide data along with all (specially labeled) empirical constants are tabulated in map form for four typical 30 by 50 deg ocean areas. A complete tabulation and discussion of the computed M2 tide will be published in Part II of this report. It is estimated that the tabulated tidal charts permit a prediction of the M2-tide elevation of the ocean surface over the geoidal level with an accuracy of better than 5 cm anywhere in the open ocean and with somewhat less accuracy near rough shorelines.

Abstract: The various ‘universal’ matrices from which finite element matrices for triangular elements are assembled in many electromagnetics and acoustics problems, can all be derived from a basic set of three fundamental matrices. These represent, respectively, the metric of the linear manifold spanned by the triangle interpolation polynominals, the finite differentiation operator on that same manifold, and a product-embedding operator for the corresponding manifold for interpolation polynomials one order higher. Two of these have already been tabulated and published; the required method for computing the third is given in this paper, along with tables of low-order matrices.

Abstract: An interpolation process is described for use with an incremental position measuring device which produces a periodic analog output signal The output signal is digitized and then applied as an input to a digital computer programmed to calculate the interpolation value of the signal In the preferred embodiment the computer is also programmed to apply a number of corrections to the digitized signal prior to interpolation

Abstract: Disclosed is a parameter interpolator for a speech synthesis circuit. Using a parameter interpolator permits the data rate to the speech synthesis circuit to be lowered inasmuch as the incoming speech data is used to slowly charge the data previously inputted to the values of the incoming data. The speech synthesis circuit includes an input circuit for receiving the target values of the speech data and a memory for stored interpolated values of the speech data. The interpolator includes a circuit coupled to the input circuit and the memory which calculates the difference between the target values and the stored values. Another circuit is used to add a portion of the difference to the values stored in the memory; the particular portion of the difference is equal to 1/2N where N=0, 1, 2 . . . Further, the interpolator is arranged to inhibit the normal interpolation upon certain conditions, such as changes from voiced speech to unvoiced speech, and visa versa.

Abstract: It has been known for some time that certain least-squares problems are “ill-conditioned”, and that it is therefore difficult to compute an accurate solution. The degree of ill-conditioning depends on the basis chosen for the subspace in which it is desired to find an approximation. This paper characterizes the degree of ill-conditioning, for a general inner-product space, in terms of the basis.The results are applied to least-squares polynomial approximation. It is shown, for example, that the powers {1, z, z2,…} are a universally bad choice of basis. In this case, the condition numbers of the associated matrices of the normal equations grow at least as fast as 4n, where n is the degree of the approximating polynomial.Analogous results are given for the problem of finite interpolation, which is closely related to the least-squares problem.Applications of the results are given to two algorithms—the Method of Moments for solving linear equations and Krylov's Method for computing the characteristic polynomial of a matrix.

Abstract: A method for the numerical solution of initial value problems for systems of retarded delay-differential equations is described. It uses a variable step Runge-Kutta Fehlberg 4/5 method combined with fourth degree Hermite-Birkhoff interpolation, which makes the usual local error estimator asymptotically correct. To obtain good numerical stability, the basic interpolation is modified for small delays. The possible initial discontinuities are algorithmically treated by using a very short stepsize in the critical step, and to this end the stepsize control has been somewhat modified. Numerical examples are included.

Abstract: Apparatus for calculating a plurality of interpolation values is adapted to calculate linear interpolation values, consisting of a second data train, from a first data train and includes a memory for storing the first data train and a calculator for calculating the interpolation value from the corresponding two data in the first data train read out of the memory. The calculator comprises an n-bit register for designating those addresses of the memory where data to be read out of the upper m-bit section of the n-bit register is stored and for determining weighted factor data for calculating the interpolation value at the lower (n-m) bit section of the register, a calculating unit for calculating the interpolation value from the data read out of the memory and the weighting coefficient data, an adder for adding a position increment value for designating the adjacent interpolation value to the register each time each interpolation value is calculated at the calculating unit, and a counter stepped one count for each calculation of each interpolation value and adapted to send an end signal to a central processing unit when a predetermined number of counts are completed. The memory, register, adder and counter are controlled by the central processing unit.

Abstract: In [3] Korevaar and Dixon have considered an interpolation problem for entire functions (stemming from work by Pavlov [4]), which is connected with Muntz — approximation on arcs and Macintyre's conjecture. In this note we give a characterization of interpolation sequences in terms of a “uniform separation condition”. We also give an explicit example which shows that a condition for interpolation sequences in [3] is essentially best possible of its kind.