The solution of transcendental equations

Some of the existing methods to globally approximate the roots of transcendental equations namely, Graeffe's method, are studied. Summation of the reciprocated roots, Whittaker-Bernoulli method, and the extension of Bernoulli's method via Koenig's theorem are presented. The Aitken's delta squared process is used to accelerate the convergence. Finally, the suitability of these methods is discussed in various cases

PSO based Neural Networks vs. Traditional Statistical Models for Seasonal Time Series Forecasting

Seasonality is a distinctive characteristic which is often observed in many practical time series. Artificial Neural Networks (ANNs) are a class of promising models for efficiently recognizing and forecasting seasonal patterns. In this paper, the Particle Swarm Optimization (PSO) approach is used to enhance the forecasting strengths of feedforward ANN (FANN) as well as Elman ANN (EANN) models for seasonal data. Three widely popular versions of the basic PSO algorithm, viz. Trelea-I, Trelea-II and Clerc-Type1 are considered here. The empirical analysis is conducted on three real-world seasonal time series. Results clearly show that each version of the PSO algorithm achieves notably better forecasting accuracies than the standard Backpropagation (BP) training method for both FANN and EANN models. The neural network forecasting results are also compared with those from the three traditional statistical models, viz. Seasonal Autoregressive Integrated Moving Average (SARIMA), Holt-Winters (HW) and Support Vector Machine (SVM). The comparison demonstrates that both PSO and BP based neural networks outperform SARIMA, HW and SVM models for all three time series datasets. The forecasting performances of ANNs are further improved through combining the outputs from the three PSO based models.Comment: 4 figures, 4 tables, 31 references, conference proceeding

A double peaked pulse profile observed in GX 1+4

The hard X-ray pulsar GX 1+4 was observed several times in the last few years with a pair of balloon-borne Xenon filled Multi-cell Proportional Counters (XMPC). In a balloon flight made on 22 March 1995, the source was detected in a bright state, the average observed source count rate being $8.0\pm0.2/s$ per detector. X-ray pulsations with a period of $121.9\pm0.1$ s were detected in the source with a broad double peak pulse feature. When observed in December 1993 with the same instrument, the pulse profile of GX 1+4 showed a single peak. This change in the pulse profile to a double pulse structure in about 15 months indicates either activation of the opposite pole of the neutron star if the magnetic field is asymmetric or possibly a change in the beam pattern, from a pencil beam to a fan beam. Assuming a fan beam configuration, the pulse profile is used to find the inclinations of the magnetic axis and the viewing axis with the spin axis. The derived angles support the GINGA observations of a dip in the pulse profile which was resolved to have a local maximum in one of the observations and was explained with resonance scattering of cyclotron line energy photons by the accretion column (Makishima et al., \markcite{maki1988}, Dotani et al., \markcite{dotani1989}.). Compared to our previous observation of the same source with the same telescope (Rao et al., \markcite{rao1994}) a period change rate of $0.72 \pm 0.40 s/yr$ is obtained which is the lowest rate of change of period for this source since its discovery. Average pulse fraction in the hard X-ray range is low (30%), consistent with its anti correlation with luminosity as reported by us earlier (Rao et al., \markcite{rao1994}) and the observed spectrum is very hard (power law photon index $1.67\pm0.12$).Comment: 10 pages, to appear in A&

Anti-correlated hard X-ray time lags in Galactic black hole sources

We investigate the accretion disk geometry in Galactic black hole sources by measuring the time delay between soft and hard X-ray emissions. Similar to the recent discoveries of anti-correlated hard X-ray time lags in Cyg X-3 and GRS 1915+105, we find that the hard X-rays are anti-correlated with soft X-rays with a significant lag in another source: XTE J1550-564. We also find the existence of pivoting in the model independent X-ray spectrum during these observations. We investigate time-resolved X-ray spectral parameters and find that the variation in these parameters is consistent with the idea of a truncated accretion disk. The QPO frequency, which is a measure of the size of truncated accretion disk, too changes indicating that the geometric size of the hard X-ray emitting region changes along with the spectral pivoting and soft X-ray flux. Similar kind of delay is also noticed in 4U 1630-47.Comment: 14 pages, 7 figures, accepted for publication in Ap