Search

A new one-point k-order iterative method for finding zeros of nonlinear equations having unknown multiplicity is introduced. In terms of computational cost the new iterative method requires k+1 evaluations of functions per iteration. It is shown that the new iterative method has a convergence of order k.

In this paper we define two new fourth -order Schroder -type methods for finding zeros of nonlinear equations having unknown multiplicity. In terms of computational cost the new iterative methods requires six evaluations of functions per iteration. It is proved tha t the new methods have a convergence of order four. Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed methods.

In th is paper, we present a fifth-order method for find ing mult iple zeros of nonlinear equations. Per iteration, the new method requires two evaluations of functions and two of its first derivative. It is proved that the method has a convergence of order five. Finally, som e nu merical examp les are g iven to show the performance of the presented method, and compared with some known methods.

New four-point derivative-free sixteenth-order iterative methods for solving nonlinear equations are con- structed. It is proved that these methods have the convergence order of sixteen requiring only five function evaluations per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve optimal convergence order 1 2. n− Thus, we present new derivative-free methods which agree with the Kung and Traub conjecture for 5. n = Numerical comparisons are made with other existing methods to show the performance of the presented methods.

Background: Hip replacement following failed internal fixation (dynamic hip screw for intertrochanteric fractures) or previous hip arthroplasty presents a major surgical challenge. Proximal fitting revision stems do not achieve adequate fixation. Distal fixation with long-stemmed extensively coated cementless implants (like the Solution™ system) affords a suitable solution. We present our early results of 15 patients treated with extensively coated cementless revision stems. Materials and Methods: Fifteen patients with severely compromised proximal femora following either failed hip arthroplasty or failed internal fixation (dynamic hip screw fixation for intertrochanteric fractures) were operated by the senior author over a two-year period. Eight patients had aseptic loosening of their femoral stems following cemented hip replacements, with severe thinning of their proximal cortices and impending stress fractures. Seven had secondary hip arthritis following failure of long implants for comminuted intertrochanteric or subtrochanteric femoral fractures. All patients were treated by removal of implant (cemented stems/DHS implants) and insertion of long-stemmed extensively coated cementless revision (' Solution™; DePuy, Warsaw (IN), US' ) stems along with press-fit acetabular component (Duraloc Cup, DePuy, Warsaw (IN), US). All eight hip revisions needed extended trochanteric osteotomies. Results: All patients were primarily kept in bed on physiotherapy for six weeks and then gradually progressed to weight-bearing walking over the next six to eight weeks. The Harris Hip Scores and patient satisfaction were used for final evaluation. We achieved good results in the short term studied. In our first three patients (all following failed cemented total hip replacements), we resorted to cerclage wiring to hold osteotomised segments (done to facilitate stem removal). The subsequent 12 proceeded without the need for cerclage wiring. One patient had a intraoperative severe comminuted fracture extending into the supracondylar region while hammering in the stem. Post cerclage wiring, she was put on a long knee brace and her mobilization was delayed to 12 weeks. Conclusions: The extensively coated cementless ( 'Solution™' ) femoral stem provides a reasonable 'solution' to the deficient femur in hip revision. The proximal femoral deficiences can be relatively easily bypassed and distal fixation can be achieved with this stem. Extreme care needs to be taken to avoid fractures and penetration of the femoral shaft, which can, however, be managed by cerclage wiring. Principles of a successful outcome include preservation of the functional continuity of the abduction apparatus, care to recognize and prevent distal extension of fracture while inserting the stem (preemptive cerclage wiring) and supervised rehabilitation.

There are two aims of this paper. Firstly we shall introduce the determinantal representations of the new Levin-type algorithms and secondly we shall demonstrate further development of the Levin-type algorithms. We consider the use of the Levin-type algorithms to accelerate the convergence of scalar sequence and their effectiveness for approximating the solution of a given power series is illustrated. In process we shall demonstrate the convergence of each of the methods considered. The approximate solution of the super enhanced Levin algorithm and the efficient Levin algorithm are found to be substantially more accurate than the Cizek, Zamastil and Skala transformation and the iterated Aitken Δ 2 algorithm.

A collection of new algorithms for accelerating the convergence of sequence of functions was described. The definitions and connections of these new algorithms with the improved functional epsilon algorithms are given. The effectiveness of these new algorithms was examined, namely the α-algorithms, the β-algorithms, the γ-algorithms, the δ-algorithms and the improved functional epsilon algorithms, for approximating solutions of a given power series. The estimates derived using the δ-algorithms are found to be substantially more accurate than other similar algorithms.

We describe a collection of Pade-type methods for accelerating the convergence of sequence of functions. The construction and connections of Pade's methods with other similar methods are given. We examine the effectiveness of these new methods, namely integral Pade approximant, modified Pade approximant and squared Pade approximant together with the well-established methods, namely functional Pade approximant and classical Pade approximant, for approximating the characteristic value and corresponding characteristic function. Estimates of characteristic value and characteristic function derived using integral Pade approximants are found to be substantially more accurate than other similar methods.

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on 𝑛 evaluations, could achieve optimal convergence order 2 𝑛 − 1 . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for 𝑛 = 4 . Numerical comparisons are made to demonstrate the performance of the methods presented.

Embedded memories are indispensable components in today's system-on-chips. High-speed memories are highly valued by the ASIC designers. Designing a flexible, yet high speed (780 MHz), SRAM compiler faces many challenges. We have managed to develop such a memory compiler without penalizing area and power. Moreover, we extended our compiler to support the common design views needed in typical ASIC design flow. In this paper, we describe some of the critical issues and techniques in the development of a high-speed memory compiler.

In this paper, new three-point eighth-order iterative methods for solving nonlinear equations are constructed. It is proved that these methods have the convergence order of eight requiring only four function evaluations per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on \(n\) evaluations, could achieve optimal convergence order \(2^{n-1} .\) Thus, we present new iterative methods which agree with the Kung and Traub conjecture for \(n=4\). Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed methods using only a few function evaluations.