By Christian H. Bischof, H. Martin Bücker, Paul Hovland, Uwe Naumann, Jean Utke

ISBN-10: 3540689354

ISBN-13: 9783540689355

ISBN-10: 3540689427

ISBN-13: 9783540689423

This assortment covers advances in automated differentiation idea and perform. desktop scientists and mathematicians will find out about contemporary advancements in automated differentiation concept in addition to mechanisms for the development of sturdy and robust computerized differentiation instruments. Computational scientists and engineers will enjoy the dialogue of assorted functions, which supply perception into potent options for utilizing automated differentiation for inverse difficulties and layout optimization.

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**Extra info for Advances in Automatic Differentiation (Lecture Notes in Computational Science and Engineering)**

**Example text**

SIAM (2000) 9. : The scaling and squaring method for the matrix exponential revisited. SIAM Journal on Matrix Analysis and Applications 26(4), 1179–1193 (2005) 10. : Matrix inversion algorithms by means of automatic differentiation. Applied Mathematics Letters 7(4), 19–22 (1994) 11. : Matrix differential calculus with applications in statistics and econometrics. John Wiley & Sons (1988) 12. : Jacobians of matrix transformations and functions of matrix argument. World Scientific, New York (1997) 13.

A classical example is the identity function y = f (x) = x coded as if x == 0 then y = 0 else y = x endif. Applying AD to this code will give f (0) = 0 in lieu of f (0) = 1. This unfortunate scenario can happen whenever a control variable in a guard (logical expression) of an IF construct or a loop is active. These scenarios can be tracked by computing the intersection between the set V (e) of variables in each guard e and the set A of active variables in the program. If V (e) ∩ A = 0/ for each guard e in the program, then the PD hypothesis holds, otherwise the PD hypothesis may be violated, in which case an AD tool should, at least, issue a warning to the user that an identified construct in the program may cause non-differentiability of the input program.

One must ensure that C1 ∼ C2 meaning C1 and C2 are semantically equivalent. To this end, we use a variant of Hoare logic called relational Hoare logic [3]. The inference rules are given in Fig. 3 and are similar to Benton’s [3]. The judgement C1 ∼ C2 : φ ⇒ ψ means simply {φ }C1 {ψ } ⇒ {φ }C2 {ψ }. In the assignment rule (asgn), the lhs variable may be different. Also, notice that the same conditional branches must be taken (see the if rule) and that loops be executed the same number of times (see the while rule) on the source and target to guarantee their semantics equivalence.

### Advances in Automatic Differentiation (Lecture Notes in Computational Science and Engineering) by Christian H. Bischof, H. Martin Bücker, Paul Hovland, Uwe Naumann, Jean Utke

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