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Multiplication is considered one of the most time-consuming and a key operation in wide variety of embedded applications. Speeding up this operation has a significant impact on the overall performance of these applications. A vast number of multiplication approaches are found in the literature where the goal is always to achieve a higher performance. One of these approaches relies on using smaller multiplier blocks which are built based on direct Boolean algebra equations to build large multipliers. In this work, we present a methodology for designing binary multipliers where different sizes customized partial products generation (CPPG) cells are designed and used as smaller building blocks. The sizes of the designed CPPG cells are 2×2, 3×3, 4×4, 5×5, and 6×6. We use these cells to build 8×8, 16×16, 32×32, 64×64, and 128×128 binary multipliers. All of the CPPG cells and the binary multipliers are described using the VHDL language, tested, and implemented using XILINX ISE 14.6 tools targeting different FPGA families. The implementation results show that the best performance is achieved when cell 3×3 is used and Virtex-7 FPGA is targeted. The binary multipliers that are designed using the proposed CPPG cells achieve better performance when compared with the binary multipliers presented in the literature. As an application that utilizes the proposed multiplier, a Multiply-Accumulate (MAC) unit is designed and implemented in Spartan-3E. The implementation results of the MAC unit demonstrate the effectiveness of the proposed multiplier. [ABSTRACT FROM AUTHOR]

The Schwarz waveform relaxation (SWR) algorithms have many favorable properties and are extensively studied and investigated for solving time dependent problems mainly at a continuous level. In this paper, we consider a semi-discrete level analysis and we investigate the convergence behavior of what so-called semi-discrete SWR algorithms combined with discrete transmission conditions instead of the continuous ones. We shall target here the hyperbolic problems but not the parabolic problems that are usually considered by most of the researchers in general when investigating the properties of the SWR methods. We first present the classical overlapping semi-discrete SWR algorithms with different partitioning choices and show that they converge very slow. We then introduce optimal, optimized, and quasi optimized overlapping semi-discrete SWR algorithms using new transmission conditions also with different partitioning choices. We show that the new algorithms lead to a much better convergence through using discrete transmission conditions associated with the optimized SWR algorithms at the semi-discrete level. In the performed semi-discrete level analysis, we also demonstrate the fact that as the ratio between the overlap size and the spatial discretization size gets bigger, the convergence factor gets smaller which results in a better convergence. Numerical results and experiments are presented in order to confirm the theoretical aspects of the proposed algorithms and providing an evidence of their usefulness and their accuracy. [ABSTRACT FROM AUTHOR]

We use in this paper the concept of cyclic (φψ )-Kannan and (φψ )-Chatterjea contractions to study new extensions of the Kannan and Chatterjea fixed point theorems. We give some generalized versions of the fixed point results proved in the literature. The analysis and theory are illustrated by some examples. [ABSTRACT FROM AUTHOR]

The optimized waveform relaxation (OWR) methods, benefiting from intelligent information exchange between subsystems – the so-called transmission conditions (TCs), are recognized as efficient solvers for large scale circuits and get a lot of attention in recent years. The TCs contain a free parameter, namely α, which has a significant influence on the convergence rates. So far, the analysis of finding the best parameter is merely performed at the continuous level and such an analysis does not take into account the influence of temporal discretizations. In this paper, we show that the temporal discretizations do have an important effect on the OWR methods. Precisely, for the Backward–Euler method, compared to the parameter αcopt from the continuous analysis, we show that the convergence rates can be further improved by using the one αdopt analyzed at the discrete level, while for the Trapezoidal rule, it is better to use αcopt. This conclusion is confirmed by numerical results. [ABSTRACT FROM AUTHOR]