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This paper discusses the eigenvalue problem for the real symmetric matrix, especially the determination of the largest eigenvalue. The largest eigenvalue is the maximum extremum of the objective function called the Rayleigh quotient and can be determined by the steepest descent method. It is known, however, that the steepest descent method suffers from slow convergence because it converges linearly. Especially, when the largest and the nest largest eigenvalues have very close values, the convergence is particularly slow. This paper analyzes this situation and shows that the convergence can be accelerated by combining the steepest descent method with a technique called shaking. Finally, it is demonstrated by a numerical example that the convergence is accelerated drastically by the proposed technique. [ABSTRACT FROM AUTHOR]

Abstract: Protein phosphorylation plays a key regulatory role in a variety of cellular processes. To better understand the function of protein phosphorylation in seed maturation, a PCR-based cloning method was employed and five cDNA clones (pvcipk1–5) for protein kinases were isolated from a cDNA library prepared from immature seeds of kidney bean (Phaseolus vulgaris L.). The deduced amino acid sequences showed that the five protein kinases (PvCIPK1–5) are members of the sucrose non-fermenting 1-related protein kinase type 3 (SnRK3) family, which interacts with calcineurin B-like proteins (CBLs). Two cDNA clones (pvcbl1 and 2) for CBLs were further isolated from the cDNA library. The predicted primary sequences of the proteins (PvCBL1 and 2) displayed significant identity (more than 90%) with those of other plant CBLs. Semi-quantitative RT-PCR analysis showed that the isolated genes, except pvcbl1, are expressed in leaves and early maturing seeds, whereas pvcbl1 is constitutively expressed during seed development. Yeast two-hybrid assay indicated that among the five PvCIPKs, only PvCIPK1 interacts with both PvCBL1 and PvCBL2. These results suggest that calcium-dependent protein phosphorylation-signaling via CBL–CIPK complexes occurs during seed development. [Copyright &y& Elsevier]