Mean CPU Time Research Articles

An iterative method for solving singular linear systems with index one

An iterative method along with its convergence analysis is developed for solving singular linear systems with index one. Necessary and sufficient conditions along with the estimation of error bounds for the unique solution are derived. Four numerical examples including singular square M-matrix, randomly generated singular square matrices, sparse symmetric and nonsymmetric singular matrices obtained from discretization of the special partial differential equations are worked out. A comparison between proposed method and method from Chen (Appl Math Comput 86:171–184, 1997) is given in terms of number of iterations, mean CPU time and error bounds. It is observed that proposed iterative method is superior and gives improved performance when compared with the method from Chen (Appl Math Comput 86:171–184, 1997).

An iterative method for solving general restricted linear equations

The Newton iterative method for computing outer inverses with prescribed range and null space is used in the non-stationary Richardson iterative method to develop an iterative method for solving general restricted linear equations. Starting with any suitably chosen initial iterate, our method generates a sequence of iterates converging to the solution. The necessary and sufficient conditions for the convergence along with the error bounds are established. The applications of the iterative method for solving some special linear equations are also discussed. A number of numerical examples are worked out. They include singular square, rectangular, randomly generated rank deficient matrices, full rank matrices and a set of singular matrices given in Matrix Computation Toolbox (mctoolbox) with the condition numbers ranging from order 1016 to 1050. The mean CPU time (MCT) and the error bounds are the performance measures used. Our results when compared with the results obtained by Chen (1997)leads to substantial improvement in terms of both computational speed and accuracy.

AN ITERATIVE METHOD FOR ORTHOGONAL PROJECTIONS OF GENERALIZED INVERSES

This paper describes an iterative method for orthogonal projections <TEX>$AA^+$</TEX> and <TEX>$A^+A$</TEX> of an arbitrary matrix A, where <TEX>$A^+$</TEX> represents the Moore-Penrose inverse. Convergence analysis along with the first and second order error estimates of the method are investigated. Three numerical examples are worked out to show the efficacy of our work. The first example is on a full rank matrix, whereas the other two are on full rank and rank deficient randomly generated matrices. The results obtained by the method are compared with those obtained by another iterative method. The performance measures in terms of mean CPU time (MCT) and the error bounds for computing orthogonal projections are listed in tables. If <TEX>$Z_k$</TEX>, k = 0,1,2,... represents the k-th iterate obtained by our method then the sequence of the traces {trace(<TEX>$Z_k$</TEX>)} is a monotonically increasing sequence converging to the rank of (A). Also, the sequence of traces {trace(<TEX>$I-Z_k$</TEX>)} is a monotonically decreasing sequence converging to the nullity of <TEX>$A^*$</TEX>.

A higher order iterative method for $A^{(2)}_{T,S}$

The aim of this paper is to propose a higher order iterative method for computing the outer inverse \(A^{(2)}_{T,S}\) for a given matrix A. Convergence analysis along with the error bounds of the proposed method is established. A number of numerical examples including singular square, rectangular, randomly generated rank deficient matrices and a set of singular matrices obtained from the Matrix Computation Toolbox (mctoolbox) are worked out. The performance measures used are the number of iterations, the mean CPU time (MCT) and the error bounds. The results obtained are compared with some of the existing methods. It is observed that our method gives improved results in terms of both computational speed and accuracy.

Fast GPU-based CT reconstruction applied in ablation treatment for hepatocellular carcinoma

Objective: To develop an image visualization system based on graphic processing unit (GPU) hardware acceleration for clinical use in hepatocellular carcinoma (HCC) interventional planning.Methods: We developed a liver tumor planning tool to assist the physician in providing patient-specific analysis and visualization. We employed a spatial distance computation algorithm to determine the spatial location of tumors and their relation to the main hepatic vessels. GPU hardware acceleration was implemented for rapid calculation of the spatial distance from the tumor surface to the surrounding vascular territories.Results: The algorithm for spatial distance provided an accurate minimum value for the distance from the tumor surface to the surrounding duct system as well as the region of interest (ROI). Analyzing the data (mean CPU time = 43.14 ± 29.34; mean GPU time = 0.41 ± 0.38) using an independent samples t-test, the result showed a remarkable difference (p < 0.001). Thus, GPU hardware acceleration performed the distance arithmetic at higher rates than conventional CPUs.Conclusions: The visual assistance tool performs as an intuitive and objective module in clinical cases, and is expected to help physicians achieve a more reliable treatment in liver tumor patients. As such, we believe it represents an improvement in image guided preoperative planning.

An operational policy for an integrated inventory system under consignment stock policy with controllable lead time and buyers’ space limitation

This paper aims to enable the decision maker of an integrated vendor–buyer system, under Consignment Stock (CS) policy, to make the optimal/sub-optimal production/replenishment decisions where the buyer places a space limitation to the vendor and the lead-time is controllable with an extra investment. Within any production cycle, the vendor produces at a finite rate and ships the outputs to the buyer with a number of equal-sized lots. With a long-term consignment stock agreement, the vendor takes responsibility to maintain a certain inventory level in the buyer's warehouse. Some of the shipments are delayed so that the buyer's inventory does not go beyond the capacity limitation. The buyer compensates the vendor after the complete consumption of the products. The holding cost consists of a storage component and a financial component. Two constraint four-variable non-linear integer optimization models are established wherein the buyer space limitation is considered. Because the developed models are mathematically very difficult to solve, three doubly hybrid meta-heuristic algorithms are employed to solve the models. The computational results show that one of these three algorithms works very well both in the sense of the success rate and the mean CPU time. The analysis of the computational example also reveals the quantitative effects the buyer space limitation may have to the annual joint total expected cost (JTEC) of the integrated system.

High throughput docking for library design and library prioritization.

The prioritization of the screening of combinatorial libraries is an extremely important task for the rapid identification of tight binding ligands and ultimately pharmaceutical compounds. When structural information for the target is available, molecular docking is an approach that can be used for prioritization. Here, we present the initial validation of a new rapid approach to molecular docking developed for prioritizing combinatorial libraries. The algorithm is tested on 103 individual cases from the protein data bank and in nearly 90% of these cases docks the ligand to within 2.0 A of the observed binding mode. Because the mean CPU time is <5 s/mol, this approach can process hundreds of thousands of compounds per week. Furthermore, if a somewhat less thorough search is performed, the search time drops to 1 s/mol, thus allowing millions of compounds to be docked per week and tested for potential activity. Proteins 2001;43:113-124.

Fiducial intervals for the waiting time in batch and time-sharing systems

In this paper error-estimates for fiducial intervals for the waiting time are analyzed, the intensity and the mean CPU-time being submitted to variations; this is done for a batch-system withn CPU's. Moreover we consider fiducial intervals for the conditional exspectationW(x)=E(waiting time‖CPU-time=x) in the case of the Round-Robin algorithm. Especially we focus on the dependence of the fiducial intervals forW(x) upon the distribution of the CPU-time (Erlang- versus exponential distribution). The results are derived as equations or inequalities including the possibility of numerical computation.