Abstract: We consider a class of iterative algorithms for solving systems of linear equations where the coefficient matrix is nonsymmetric with positive-definite symmetric part. The algorithms are modelled after the conjugate gradient method, and are well suited for large sparse systems. They do not make use of any associated symmetric problems. Convergence results and error bounds are presented.

Abstract: In this paper, we present an algorithm to estimate a signal from its modified short-time Fourier transform (STFT). This algorithm is computationally simple and is obtained by minimizing the mean squared error between the STFT of the estimated signal and the modified STFT. Using this algorithm, we also develop an iterative algorithm to estimate a signal from its modified STFT magnitude. The iterative algorithm is shown to decrease, in each iteration, the mean squared error between the STFT magnitude of the estimated signal and the modified STFT magnitude. The major computation involved in the iterative algorithm is the discrete Fourier transform (DFT) computation, and the algorithm appears to be real-time implementable with current hardware technology. The algorithm developed in this paper has been applied to the time-scale modification of speech. The resulting system generates very high-quality speech, and appears to be better in performance than any existing method.

Abstract: In this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities. This iterative scheme not only contains, as special cases the projection, linear approximation and relaxation methods but also induces new algorithms. Then, we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in En induced by symmetric positive definite matrices Gm. Thus, in contrast to the above mentioned methods, this technique allows the possibility of adjusting the norm at each step of the algorithm. This flexibility will generally yield convergence under weaker assumptions.

Abstract: A method for establishing connections by automatically routing a plurality of paths between individual components using initially simple connection path shapes. The method is used to create an interconnection package with better use of wiring space. Each connection, in turn, is removed if previously routed, rerouted and evaluated according to specified penalty costs to minimize undesirable routing characteristics. This method is particularly advantageous in providing automatic path routing in directionally uncommitted planes for wiring highly integrated electric circuits, or the like.

Abstract: An iterative algorithm for the inversion of a Toeplitz-block Toeplitz matrix consisting of m × m blocks of size p × p is described. The algorithm presented exploits the structure of the Toeplitz-block Toeplitz matrix and outperforms Akaike's algorithm by a factor of \max {2(p/m), 2} . The use of this algorithm for an iterative solution of a Toeplitz-block Toeplitz set of linear equations is also presented.

Abstract: Using the theory of Euler methods from summability theory, we investigate general iterative methods for solving linear systems of equations. In particular, for a given Euler method, a regionS of the complex plane is determined such that ak-step iterative method converges if the eigenvalues of an iteration operatorT are contained inS. For a givenS, optimal methods are described, and upper and lower bounds are derived for the associated asymptotic rate of convergence. Special attention is given to two-step methods with complex parameters.

Abstract: While many iterative signal restoration methods have been shown to converge in the mathematical sense, a practical criterion is needed to indicate when the numerical algorithm has converged. The use of the residual signal in defining such a criterion is investigated. The advantage of this criterion over the criterion of examining successive iterations is demonstrated.

Abstract: This paper is concerned with iterative methods for the linear complementarity problem (LCP) of findingx and y in Rn such thatc+Dx+yź0, b-xź0, andxT(c+Dx+y)=yT(b-x)=0 whenb>0. This is the LCP (M, q) withM=(l 0D t), which is in turn equivalent to a linear variational inequality over a rectangle. This type of problem arises, for example, from quadratic programming (QP) problems with upper and lower bounds ifD is symmetric, and from multicommodity market equilibrium problems with institutional price controls imposed ifD is not necessarily symmetric. Iterative methods for the LCP (M, q) with symmetricM are well developed and studied using QP applications. For nonsymmetric cases withM being either an H-matrix with positive diagonals, or a Z-matrix, there exists a robust iterative method with guaranteed convergence. This paper extends this algorithm so that the LCP (M, q) withM=(l 0D t) which is neither symmetric, nor an H-matrix with positive diagonals, nor a Z-matrix, can be processed when onlyD notM satisfies such properties. The case whereD is nonsymmetric is explicitly discussed.

Abstract: A robust algorithm for flash calculations that uses an equation of state (EOS) is presented. It first uses a special version of the successive substitution (SS) method and switches to Powell's method if poor convergence is observed. Criteria are established for an efficient switch from one method to the other. Experience shows that this method converges near the critical point and also detects the single-phase region without computing the saturation pressure. The Soave-Redlich-Kwong (SRK) and the Peng-Robinson (PR) EOS's are used in this work, but the method is general and applies to any EOS.

Abstract: Two techniques are described which correct scattering parameter data taken on an N-port device measured with N-2 imperfect terrniuations and a two-port network analyzer. The first technique rises a simple iterative algorithm and may be easily implemented in software. Each iteration reduces the error due to imperfect terminations typically by one decade. The second, more complicated, technique uses a general closed-form solution which requires specially developed Gamma-R parameters of which S-, Y-, and Z-parameters are particular cases. The closed-form solution is completely valid for any termination. The closed-form solution is the limit to which the iterative solution converges. The iterative technique has been implemented in software controlling an HP 8409 automated microwave network analyzer.

Abstract: We deal with iterative least-squares solutions of the linear signal-restoration problem g = Af. First, several existing techniques for solving this problem with different underlying models are unified. Specifically, the following are shown to be special cases of a general iterative procedure [ BialyH., Arch. Ration. Mech. Anal.4, 166 ( 1959)] for solving linear operator equations in Hilbert spaces: (1) a Van Cittert-type algorithm for deconvolution of discrete and continuous signals; (2) an iterative procedure for regularization when g is contaminated with noise; (3) a Papoulis–Gerchberg algorithm for extrapolation of continuous signals [ PapoulisA., IEEE Trans. Circuits Syst.CAS-22, 735 ( 1975); GerchbergR. W., Opt. Acta21, 709 ( 1974)]; (4) an iterative algorithm for discrete extrapolation of band-limited infinite-extent discrete signals {and the minimum-norm property of the extrapolation obtained by the iteration [ JainA.RanganathS., IEEE Trans. Acoust. Speech Signal Process. ASSP-29, ( 1981)]}; and (5) a certain iterative procedure for extrapolation of band-limited periodic discrete signals [ TomV., IEEE Trans. Acoust. Speech Signal Process.ASSP-29, 1052 ( 1981)]. The Bialy algorithm also generalizes the Papoulis–Gerchberg iteration to cases in which the ideal low-pass operator is replaced by some other operators. In addition a suitable modification of this general iteration is shown. This technique leads us to new iterative algorithms for band-limited signal extrapolation. In numerical simulations some of these algorithms provide a fast reconstruction of the sought signal.

Abstract: We present some theoretical results on the band-limited signal extrapolation problem. In Section I we describe four basic models for the extrapolation problem. These models are useful in understanding the relationship between the continuous extrapolation problem and some discrete algorithms given in [1] and [2]. One of these models was shown to approximate the continuous band-limited extrapolation problem [3]. Another model is obtained when the discrete Fourier transform (DFT) is used to implement the well-known iterative algorithm given in [4] and [5] which was designed for solving the continuous extrapolation problem; in Section II this model is related to the continuous model by means of an interesting approximation theorem. Also, an important conjecture is presented. Section III shows some approximation results. Specifically, we prove that some discrete-discrete and discrete-continuous extrapolations of noisy signals converge to solutions of a certain continuous-continuous noisy extrapolation problem when the noise η is bounded by a known number, max \eta(x)| leq \epsilon . This convergence is obtained by using normal families of entire functions in ¢nand some other complex analysis tools. We also show that the extrapolation problem is very sensitive to noise even in cases where only small amounts of extrapolation are desired. This result indicates that in the presence of noise, extrapolation techniques should be used judiciously in order to obtain reasonable results.

Abstract: This paper describes the procedure for analyzing nonlinear transient electromagnetic phenomena in electrical machines and devices with the finite element method. Two time integration methods are used which are based on (1) an implicit forward difference scheme and (2) the Crank-Nicholson method. The quasilinearization of the nonlinear matrix equation is handled by a simple chord iteration method in the former and the Newton-Raphson scheme in the latter. The associated variational expressions for the time-dependent diffusion equation are obtained in terms of energy-related functionals. The paper includes illustrative examples of application of the methods to one-and two-dimensional time-dependent eddy current problems in a conducting slab, a rotating machine under asynchronous operation, and a three- phase bus-bar enclosure.

Abstract: In computing a scene description from an image, a useful intermediate representation of a scene object is given by the orientation and area of the constituent surface facets, termed the Extended Gaussian Image (EGI) of the object The EGI of a convex object uniquely represents that object We are concerned with the computational task of reconstructing the shape of scene objects from their Extended Gaussian Images, where the objects are restricted to convex polyhedra We present an iterative method for reconstructing convex polyhedra from their Extended Gaussian Images

Abstract: Nonlinear stationary fixed point iterations inR n are considered. The Perron-Ostrowski theorem [23] guarantees convergence if the iteration functionG possesses an isolated fixed pointu. In this paper a sufficient condition for convergence is given ifG possesses a manifold of fixed points. As an application, convergence of a nonlinear extension of the method of Kaczmarz is proved. This method is applicable to underdetermined equations; it is appropriate for the numerical treatment of large and possibly ill-conditioned problems with a sparse, nonsquare Jacobian matrix. A practical example of this type (nonlinear image reconstruction in ultrasound tomography) is included.

Abstract: The application of discretizatio n techniques frequently leads to a system of algebraic equations having a welldefined coefficient structure. A modified strongly implicit procedure for solving the system resulting from the modeling of heat conduction in three dimensions is presented in this work. The method is derived for a 19 point scheme with the more common 7 point scheme emerging as a special case of the procedure. In this way, the asymmetric influence of the additional terms in the LU matrix product is weakened. As a consequence, the method is less sensitive to the iteration parameter and mesh aspect ratio and, in addition, provides considerably more rapid convergence than does the strongly implicit procedure. The increased convergence is exhibited by a significant reduction in the computational cost. The characteristics of the method are examined through application to several model problems and application is made to a more complex three-dimensional problem. Comparisons with the SIP (strongly implicit) and ADI (alternating direction implicit) methods are provided.

Abstract: This paper presents a novel seven-point finite difference approximation for simulations of adverse mobility ratio displacements. The method is based on partitioning of a two-dimensional flow domain into regular or nearly regular hexagons. The accuracy of the method for pattern steam floods of heavy oil reservoirs is compared to five- and nine-point approximations. For five-spot floods, the accuracy of the seven-point method is good and comparable to that of the nine-point scheme. For seven-spot floods, the seven-point method provides good numerical accuracy at substantially less computational work than five- or nine-point methods. For nine-spot floods, only the nine-point method is found to give accurate results. 16 references, 10 figures, 3 tables.

Abstract: A mathematical model is presented which describes the salt water–fresh water motion with a sharp interface, assuming the validity of the Dupuit approximation. This model is used as a base to derive a numerical model (finite difference method) which is unconditionally convergent and stable. A method for solving the equations is selected together with a convergence accelerating procedure. The treatment of the boundary conditions in the interface is discussed, and a general and automatic solution for that problem is presented. Several tests with analytical solutions have been performed with good results.

Abstract: In this paper we analyze the numerical aspects of the various methods that have been utilized to analyze thin wire antennas. First, we derive the properties of the operators for Pocklington's and Hallen's integral equation. On the basis of these properties, we discuss the various iterative methods used to find current distribution on thin wire structures. An attempt has been made to resolve the question of numerical stability associated with various entire domain and subdomain expansion functions in Galerkin's method. It has been shown that the sequence of solutions generated by the iterative methods monotonically approaches the exact solution provided the excitations chosen for these problems are in the range of the operator. Such a statement may not hold for Galerkin's methods if the inverse operator is unbounded. Moreover, if the excitation function is not in the range of the operator, then the sequence of solutions forms an asymptotic series. Examples have been presented to illustrate this point.

Abstract: The tomography problem is investigated when the available projections are restricted to a limited angular domain. It is shown that a previous algorithm proposed for extrapolating the data to the missing cone in Fourier space is unstable in the presence of noise because of the ill-posedness of the problem. A regularized algorithm is proposed, which converges to stable solutions. The efficiency of both algorithms is tested by means of numerical simulations.

Abstract: We present a new approach to the problem of estimating multiple signal and parameter unknowns given noisy and incomplete data. Using cross-entropy, we fit a separable density to the given model density, then use this separable density to estimate each unknown independently. Not only does this method include all the various MAP methods as degenerate cases, but it also directly leads to a simple iterative algorithm which can solve either the cross-entropy method or any of the MAP methods. This algorithm is particularly effective for exponential families of densities. Applications include estimation using grouped or quantized data, and a wide variety of reconstruction, smoothing, interpolation, extrapolation and modeling problems involving linear Gaussian systems.