• Journal of applied mathematics & informatics
• /
• v.11 no.1_2
• /
• pp.305-316
• /
• 2003

Recently, an iterative algorithm for finding the interior eigenvalues of a definite matrix by CG-type method has been proposed. This method compares to the inverse power method. The given matrices A, and B are assumed to be large and sparse, and SPD( Symmetric Positive Definite) The CG scheme for the optimization of the Rayleigh quotient has been proven a very attractive and promising technique for large sparse eigenproblems for smallest eigenvalue. Also, it is very amenable to parallel computations, like the CG method for the linear systems. A proper choice of the preconditioner significantly improves the convergence of the CG scheme. But for parallel computations we need to find an efficient parallel preconditioner. Our candidates we ILU(0) in the wave-front order, ILU(0) in the multi-coloring order, Point-SSOR(Symmetric Successive Overrelaxation), and Multi-Color Block SSOR preconditioner. Wavefront order is a simple way to increase parallelism in the natural order, and Multi-coloring realizes a parallelism of order(N), where N is the order of the matrix. Another choice is the Multi-Color Block SSOR(Symmetric Successive OverRelaxation) preconditioning. Block SSOR is a symmetric preconditioner which is expected to minimize the interprocessor communication due to the blocking. We implemented the results on the CRAY-T3E with 128 nodes. The MPI (Message Passing Interface) library was adopted for the interprocessor communications. The test problem was drawn from the discretizations of partial differential equations by finite difference methods. The results show that for small number of processors Multi-Color ILU(0) has the best performance, while for large number of processors Multi-Color Block SSOR performs the best.

• Journal of applied mathematics & informatics
• /
• v.25 no.1_2
• /
• pp.435-443
• /
• 2007

Computing the interior spectrum of large sparse generalized eigenvalue problems $Ax\;=\;{\lambda}Bx$, where A and b are large sparse and SPD(Symmetric Positive Definite), is often required in areas such as structural mechanics and quantum chemistry, to name a few. Recently, CG-type methods have been found useful and hence, very amenable to parallel computation for very large problems. Also, as in the case of linear systems proper choice of preconditioning is known to accelerate the rate of convergence. After the smallest eigenpair is found we use the orthogonal deflation technique to find the next m-1 eigenvalues, which is also suitable for parallelization. This offers advantages over Jacobi-Davidson methods with partial shifts, which requires re-computation of preconditioner matrx with new shifts. We consider as preconditioners Incomplete LU(ILU)(0) in two variants, ever-relaxation(SOR), and Point-symmetric SOR(SSOR). We set m to be 5. We conducted our experiments on matrices from discretizations of partial differential equations by finite difference method. The generated matrices has dimensions up to 4 million and total number of processors are 32. MPI(Message Passing Interface) library was used for interprocessor communications. Our results show that in general the Multi-Color ILU(0) gives the best performance.

• KOREAN JOURNAL OF CROP SCIENCE
• /
• v.63 no.3
• /
• pp.175-185
• /
• 2018

The aim of this study was to evaluate the variation of growth and quality characteristics in colored rice from 178 accessions and to develop useful, basic rice breeding data by classifying these germplasm characteristics via principal component (PC) analysis. The coefficient of variation of the 178colored rice accessions were the highest for panicle length (PL) and protein contents, followed by length-width ratio (LWR), 1000-grain weight (TGW), culm length (CL), and amylose contents, whereas the lowest was for the number of panicles per hill (NP), which is a yield component. The results from the PC analysis exhibited eigenvalues and contributions respective to each PC as follows: PC1, 2.06 and 29.49%; PC2, 1.31 and 18.75%; PC3, 1.21 and 17.36%; PC4, 1.01 and 14.38%. The eigenvalues of four PCs were over 1.0, and their cumulative contributions were 79.98%, which completes the necessary condition for evaluation of the 178 colored rice accessions. Cluster analysis showed cluster I as the largest, which included 79 accessions, while clusters II, III, IV, V, VI, and VII comprised 46, 19, 13, 4, 8, and 9 accessions, respectively. Moreover, dark brown accessions were dispersed in clusters I and II, and many resources of purple seed coat color were found in clusters V, VI, and VII. Particularly, cluster V had resources of only black and purple seed coat colors. Resources of cluster VII were found to have a relatively small average CL, PL, and LWR; notably, cluster V had the smallest average TGW, and cluster IV the lowest NP but the highest TGW. Finally, considering the yield potential, growth characteristics, heading stage, and color during breeding of colored rice, we obtained the following conclusions: cluster VII is suitable for breeding of colored rice; cross breeding among clusters I, II, and VII has a high yield potential; and it is possible to produce a superior color by cross breeding plants from cluster V and VI.