We consider a rate of convergence of the Levenberg-Marquardt method (LMM) for solving a system of nonlinear equations F(x) = 0, where F is a mapping from R" into Rm. It is well-known that LMM has a quadratic rate of convergence when m = n, the Jacobian matrix of F is nonsingular at a solution x and an initial point is chosen sufficiently close to x. In this paper, we show that if IIF(x) II provides a local error bound for the system of nonlinear equations, then a sequence generated by LMM converges to the solution set quadratically.AMS Subject Classifications: 65HIO, 49M15, 90C33.

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Exogenous thiamine regulates Aspergillus oryzae thiA, which is involved in thiamine synthesis. One of the two introns in its 5P P-untranslated region (5P P-UTR) contains motifs (regions A and B) highly conserved among fungal thiamine biosynthesis genes. Deletion of either region relieved the repression by thiamine and thiamine inhibited intron splicing, suggesting that regions A and B are required for e⁄cient splicing. Furthermore, transcript splicing was essential for thiA gene expression. These observations suggest a novel gene expression regulatory mechanism in ¢lamentous fungi, in which exogenous thiamine controls intron splicing to regulate gene expression. Interestingly, regions A and B constitute a part of a thiamine pyrophosphate-binding riboswitch-like domain that has been quite recently found in the 5P P-UTR of thiA.

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