The problem of the evaluation in floating-point arithmetic of a polynomial with floating-point coefficients at a point which is a finite sum of floating-point numbers is studied. The solution is obtained as an infinite convergent series of floating-point numbers. The algorithm requires a precise scalar product, but this can always be implemented by software in a high-level language without assembly language routines as we indicate. A convergence result is proved under a very weak restriction on the size of the degree of the polynomial in terms of the unit roundoff u; roughly speaking, the degree should not be larger than the square root of (1 + u)(2u). Even in the particular case when the point at which to evaluate the polynomial reduces to one floating-point number, we find a new simplified algorithm among the whole family that the preceding convergence result allows.
This problem occurs, among others, in the convergence of the Newton method to some real root of the given polynomial p. If we simply use the Horner scheme to evaluate the polynomial p in a neighbourhood of the root, in some cases the evaluation will contain no correct digits and will prevent us from getting convergence even to machine accuracy. The convergence of iterative methods, among which the Newton method, with added perturbations was the central theme of my talk given at the ICCAM'92. The second part will appear in a forthcoming paper. These added perturbations can represent for example forward or backward errors occurring in finite-precision computations.
The problem discussed here appears in validating some hypotheses of these general convergence results (see the forthcoming paper). 相似文献
Even though NMR has found countless applications in the field of small molecule characterization, there is no standard file format available for the NMR data relevant to structure characterization of small molecules. A new format is therefore introduced to associate the NMR parameters extracted from 1D and 2D spectra of organic compounds to the proposed chemical structure. These NMR parameters, which we shall call NMReDATA (for nuclear magnetic resonance extracted data), include chemical shift values, signal integrals, intensities, multiplicities, scalar coupling constants, lists of 2D correlations, relaxation times, and diffusion rates. The file format is an extension of the existing Structure Data Format, which is compatible with the commonly used MOL format. The association of an NMReDATA file with the raw and spectral data from which it originates constitutes an NMR record. This format is easily readable by humans and computers and provides a simple and efficient way for disseminating results of structural chemistry investigations, allowing automatic verification of published results, and for assisting the constitution of highly needed open‐source structural databases. 相似文献
We perform MD simulations of the nanoindentation on (001) and (111) surfaces of Ag–Ni multilayers with different modulation periods, and find that both the hardness and maximum force increase with the increase of modulation period, in agreement with the inverse Hall–Petch relation. A prismatic partial dislocation loop is observed in the Ni(111)/Ag(111) sample when the modulation period is relatively large. We also find that misfit dislocation network shows a square shape for the Ni(111)/Ag(111) interface, while a triangle shape for the Ni(001)/Ag(001) interface. The pyramidal defect zones are also observed in Ni(001)/Ag(001) sample, while the intersecting stacking faults are observed in Ni(111)/Ag(111) sample after dislocation traversing interface. The results offer insights into the nanoindentation behaviors in metallic multilayers, which should be important for clarifying strengthening mechanism in many other multilayers. 相似文献
This article investigates the performance of various sophisticated channel coding and transmission schemes for achieving reliable transmission of a highly compressed video stream. Novel error protection schemes including Non-Convergent Coding (NCC) scheme, Non-Convergent Coding assisted with Differential Space Time Spreading (DSTS) and Sphere Packing (SP) modulation (NCDSTS-SP) scheme and Convergent Coding assisted with DSTS and SP modulation (CDSTS-SP) are analyzed using Bit Error Ratio (BER) and Peak Signal to Noise Ratio (PSNR) performance metrics. Furthermore, error reduction is achieved using sophisticated transceiver comprising SP modulation technique assisted by Differential Space Time Spreading. The performance of the iterative Soft Bit Source Decoding (SBSD) in combination with channel codes is analyzed using various error protection setups by allocating consistent overall bit-rate budget. Additionally, the iterative behavior of SBSD assisted RSC decoder is analyzed with the aid of Extrinsic Information Transfer (EXIT) Chart in order to analyze the achievable turbo cliff of the iterative decoding process. The subjective and objective video quality performance of the proposed error protection schemes is analyzed while employing H.264 advanced video coding and H.265 high efficient video coding standards, while utilizing diverse video sequences having different resolution, motion and dynamism. It was observed that in the presence of noisy channel the low resolution videos outperforms its high resolution counterparts. Furthermore, it was observed that the performance of video sequence with low motion contents and dynamism outperforms relative to video sequence with high motion contents and dynamism. More specifically, it is observed that while utilizing H.265 video coding standard, the Non-Convergent Coding assisted with DSTS and SP modulation scheme with enhanced transmission mechanism results in gain of 20 dB with reference to the Non-Convergent Coding and transmission mechanism at the objective PSNR value of 42 dB. It is important to mention that both the schemes have employed identical code rate. Furthermore, the Convergent Coding assisted with DSTS and SP modulation mechanism achieved superior performance with reference to the equivalent rate Non-Convergent Coding assisted with DSTS and SP modulation counterpart mechanism, with a performance gain of 16 dB at the objective PSNR grade of 42 dB. Moreover, it is observed that the maximum achievable PSNR gain through H.265 video coding standard is 45 dB, with a PSNR gain of 3 dB with reference to the identical code rate H.264 coding scheme. 相似文献