Basins of attraction for several methods to find simple roots of nonlinear equations

The article of record as published may be located at http://dx.doi.org/10.1016/j.amc.2010.04.017There are many methods for solving a nonlinear algebraic equation. The methods are clas- sified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss several third and fourth order methods to find simple zeros. The relationship between the basins of attraction and the corresponding conjugacy maps will be discussed in numerical experiments. The effect of the extraneous roots on the basins is also discussed

An initial-value method for Fredholm integral equations with generalized degenerate kernels

AbstractAn initial-value method is derived for integral equations with generalized degenerate kernels. Both the necessity and sufficiency of the Cauchy system are demonstrated

Complex Transitions Between C3 and C4 Photosynthesis During the Evolution of Paniceae: A Phylogenetic Case Study Emphasizing the Position of Steinchisma Hians (Poaceae), A C3-C4 Intermediate

A two-tiered, nested molecular phylogenetic study of panicoid grasses to explore character state transitions between the C3 and C4 adaptive syndromes is presented. A broad survey of 92 panicoid species was sampled for the grass-specific insert sequence in the chloroplast RNA polymerase locus (rpoC2), combining published and unpublished sequences. This portion of the study also included an intensive phylogenetic investigation of one clade of seven species that included Steinchisma hians, which is notable for exhibiting intermediacy between the C3 and C4 photosynthetic types. Both rpoC2 data and previously published sequences of the F subunit of an NADH-dependent dehydrogenase were analyzed together for this small group. A rigorous phylogenetic investigation of S. hians and 13 other species of Panicoideae included in the broad survey was then performed with sequences of both rpoC2 and the externally transcribed spacer region of the nuclear ribosomal repeat. These 14 species were selected to maximize representation among photosynthetic subtypes. Combined analysis resolved single origins of two photosynthetic subtypes. A reversion of C4 to C3 photosynthesis during the evolution of the lineage that includes S. hians is identified. These and other recent results indicate that repeated reversions from C4 to C3 have occurred. The C3 species Panicum laxum has a strongly supported sister group relationship to S. hians (C3-C4). The most parsimonious interpretation is that S. hians represents an incipient reversal from C3 to C4 photosynthesis, beginning with the capacity to compartmentalize photorespiratory metabolism in the bundle sheath tissue