By H. Martin Bücker, George Corliss, Paul Hovland, Uwe Naumann, Boyana Norris

ISBN-10: 3540284036

ISBN-13: 9783540284031

ISBN-10: 3540284389

ISBN-13: 9783540284383

This assortment covers the state-of-the-art in computerized differentiation conception and perform. Practitioners and scholars will know about advances in automated differentiation concepts and techniques for the implementation of strong and strong instruments. Computational scientists and engineers will enjoy the dialogue of purposes, which offer perception into powerful thoughts for utilizing automated differentiation for layout optimization, sensitivity research, and uncertainty quantification.

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**Extra info for Automatic Differentiation: Applications, Theory, and Implementations **

**Sample text**

Solutions of ODEs with Removable Singularities 39 Theorem 3. There does not exist a rational function R(T, X) such that the ODE x˙ = R(t, x) is satisﬁed by x(t) = sin t or by x(t) = cos t. Proof. We use the notation s = sin t, c = cos t, and as above, y = x. ˙ If x = sin t, then c = R(t, s) , which is impossible by Lemma 3. A similar argument works for x(t) = cos t. Theorem 4. The only irreducible polynomial P (T, X, Y ) such that P (t, x(t), x(t)) ˙ =0 is for x(t) = sin t t (x(0) = 1) P (T, X, Y ) = (X + T Y )2 + T 2 X 2 − 1 .

There may be more than one output, but still it is an ordered system. com). Each circle represents the calculation of an intermediate variable, and the arrows ﬂowing into any circle show us which earlier results are directly used in that calculation. ) The idea of recurrent or recursive neural networks was known back in Minsky’s time [370]. The commonest form of recurrence is a loop from neuron number k back to itself. The literature on recurrent networks has become very confused and often inaccurate, in part because there are diﬀerent interpretations of what it means when people insert a backwards arrow into the computational graph.

Each circle represents the calculation of an intermediate variable, and the arrows ﬂowing into any circle show us which earlier results are directly used in that calculation. ) The idea of recurrent or recursive neural networks was known back in Minsky’s time [370]. The commonest form of recurrence is a loop from neuron number k back to itself. The literature on recurrent networks has become very confused and often inaccurate, in part because there are diﬀerent interpretations of what it means when people insert a backwards arrow into the computational graph.

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