We describe a code transformation technique that, given code for a vector function F , produces code suitable for computing collections of Jacobian-vector products F (x)ẋ or Jacobiantranspose-vector products F (x) ȳ. Exploitation of scarcity a measure of the degrees of freedom in the Jacobian matrix means solving a combinatorial optimization problem that is believed to be hard. Our heuristics t...

TM2 is a global three-dimensional model of the atmospheric transport of passive tracers. The adjoint of TM2 is a model that allows the efficient evaluation of derivatives of the simulated tracer concentration at observational locations with respect to the tracer’s sources and sinks. We describe the generation of the adjoint model by applying the Tangent linear and Adjoint Model Compiler in the ...

We review the extended Jacobian approach to automatic differentiation of a user-supplied function and highlight the Schur complement form’s forward and reverse variants. We detail a Matlab operator overloaded approach to construct the extended Jacobian that enables the function Jacobian to be computed using Matlab’s sparse matrix operations. Memory and runtime costs are reduced using a variant ...

Any endomorphism of Cn defined by n polynomials with everywhere non-vanishing Jacobian is an automorphism. The Jacobian conjecture originated fromKeller ([5]). Let F1, . . . , Fn ∈ C[x1, . . . , xn] be a set of n polynomials in n variables with n ≥ 1 such that the Jacobian of these polynomials is a nonzero constant. The Jacobian conjecture says that the subalgebra C[F1, . . . , Fn] of C[x1, . ....

The problems of radiative transfer give rise to interesting integral equations that must be faced with efficient numerical solver. Very often the integral equations are discretized to large-scale nonlinear equations and solved by Newton’s-like methods. Generally, these kind of methods require the computation and storage of the Jacobian matrix or its approximation. In this paper, we present a ne...

In this paper we present the results obtained in solving consistent sparse systems of n nonlinear equations F(x) = 0; by a Quasi-Newton method combined with a p block iterative row-projection linear solver of Cimmino-type, 1 p n: Under weak regularity conditions for F; it is proved that this Inexact Quasi-Newton method has a local, linear convergence in the energy norm induced by the preconditi...

ADIFOR is a source translator that, given a collection of Fortran subroutines for the computation of a \function," produces Fortran code for the computation of the derivatives of this function. More speciically, ADIFOR produces code to compute the matrix-matrix product JS, where J is the Jacobian of the \function" with respect to the user-deened independent variables, and S is the composition o...

Suppose f = p/q is a quotient of two polynomials and that p has degree rp and q has degree rq . Assume that f(A) and f(A+ uv ) are defined where A ∈ R, u ∈ R, and v ∈ R are given and set r = max{rp, rq}. We show how to compute f(A+uv ) in O(rn2) flops assuming that f(A) is available together with an appropriate factorization of the “denominator matrix” q(A). The central result can be interprete...