The shifted classical circulant and skew circulant splitting iterative methods for Toeplitz matrices

It is known that every Toeplitz matrix T enjoys a circulant and skew circulant splitting (denoted by CSCS), i.e., T=C-S with C a circulant matrix and S a skew circulant matrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss-Seidel (GS) iterative methods if the CSCS is convergent and that there is always a constant $\alpha$ such that the shifted CSCS iteration converges much faster than the Gauss-Seidel iteration, no matter whether the CSCS itself is convergent or not.National Natural Science Foundation of China No. 11371075The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the research innovation program of Hunan province for postgraduate students under Grant No. CX2015B374, the Portuguese Funds through FCT–Fundac˜ao para a Ciˆencia, within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Trigonometric transform splitting methods for real symmetric Toeplitz systems

In this paper, we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix A. Theoretical analyses show that if the generating function f of the n × n Toeplitz matrix A is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large n. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved. Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix A is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method.National Natural Science Foundation of China under Grant No. 11371075info:eu-repo/semantics/publishedVersio

Organosulphide profile and hydrogen sulphide-releasing activity of garlic fermented by Lactobacillus plantarum

Blanched and unblanched garlic were fermented using L. plantarum for investigation of organosulphide profiles, hydrogen sulphide-releasing activity, pH, titratable activity and microbial growth. Both raw and blanched garlic preparations allowed growth of L. plantarum with corresponding lowering of pH below 4.0 and an increase in titratable acidity from an initial value of less 0.05% to 0.3 and 0.5%, respectively. Fermentation, alone, decreased allicin and vinyl dithiins, but increased the concentration of DATS. The H2S-releasing activity, expressed as DATS-E (mmol DATS/g oil), of raw-fermented (2.91) garlic was not significantly different to that of raw (4.74) garlic, but values in blanched (0.41) and blanched-fermented (0.71) samples significantly decreased. Reductions in DATS-E values in blanched and blanched-fermented garlic corresponded well with the negative effect of blanching on the organosulphide concentrations of the products. Fermentation with L. plantarum retains H2S-releasing activity by increasing DATS, despite notable losses in allicin and allicin transformation products