منابع مشابه
Nilpotent Symmetric Jacobian Matrices and the Jacobian Conjecture Ii
It is shown that the Jacobian Conjecture holds for all polynomial maps F : k → k of the form F = x + H , such that JH is nilpotent and symmetric, when n ≤ 4. If H is also homogeneous a similar result is proved for all n ≤ 5. Introduction Let F := (F1, . . . , Fn) : C → C be a polynomial map i.e. each Fi is a polynomial in n variables over C. Denote by JF := (i ∂xj )1≤i,j≤n, the Jacobian matrix ...
متن کاملOptimal direct determination of sparse Jacobian matrices
It is well known that a sparse Jacobian matrix can be determined with fewer function evaluations or automatic differentiation passes than the number of independent variables of the underlying function. In this paper we show that by grouping together rows into blocks one can reduce this number further. We propose a graph coloring technique for row partitioned Jacobian matrices to efficiently det...
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The Jacobian Conjecture would follow if it were known that real polynomial maps with a unipotent Jacobian matrix are injective. The conjecture that this is true even for C maps is explored here. Some results known in the polynomial case are extended to the C context, and some special cases are resolved.
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This paper is concerned with the efficient computation of sparse Jacobian matrices of nonlinear vector maps using automatic differentiation (AD). Specifically, we propose the use of a graph coloring technique, bicoloring, to exploit the sparsity of the Jacobian matrix J and thereby allow for the efficient determination of J using AD software. We analyze both a direct scheme and a substitution p...
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The computation of large sparse Jacobian matrices is required in many important large-scale scienti c problems. We consider three approaches to computing such matrices: hand-coding, di erence approximations, and automatic di erentiation using the ADIFOR (Automatic Di erentiation in Fortran) tool. We compare the numerical reliability and computational e ciency of these approaches on applications...
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ژورنال
عنوان ژورنال: Advances in Applied Mathematics
سال: 2020
ISSN: 0196-8858
DOI: 10.1016/j.aam.2019.101987