# Doubly Recursive Multivariate Automatic Differentiation

```@article{Kalman2002DoublyRM,
title={Doubly Recursive Multivariate Automatic Differentiation},
author={Dan Kalman},
journal={Mathematics Magazine},
year={2002},
volume={75},
pages={187 - 202}
}```
• D. Kalman
• Published 1 June 2002
• Computer Science
• Mathematics Magazine
system. That's right. The automatic differentiation system never formulates a symbolic expression for the derivative. Automatically calling on something like Mathematica to produce a symbolic derivative, and then plugging in a value for x is the wrong image entirely. Automatic differentiation is something completely different. Well OK, but so what? Symbolic algebra systems are so prevalent and powerful today, why should we be concerned with avoiding symbolic methods? There are two answers. The…

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## References

SHOWING 1-10 OF 18 REFERENCES

### The Arithmetic of Differentiation

Abstract : This report describes automatic differentiation, which is neither symbolic nor approximate, for single functions of one real variable. The rules of evaluation and differentiation are

### Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition

• Biology
Frontiers in applied mathematics
• 2000
This second edition has been updated and expanded to cover recent developments in applications and theory, including an elegant NP completeness argument by Uwe Naumann and a brief introduction to scarcity, a generalization of sparsity.

### An efficient method for the numerical evaluation of partial derivatives of arbitrary order

The key ideas are a hyperpyramid data structure and a generalized Leibniz's rule which produces any partial derivative by forming the minimum number of products (between two lower partials) together with a product of binomial coefficients.

### A recursive approach to multivariate automatic differentiation

• Mathematics
• 1995
In one approach to automatic differentiation, the range of a function is generalized from a single real value to an aggregate representing the values of the function and one or more derivatives. The

### and A

• Griewank, Structured second- and higher-order derivatives through univariate Taylor series, preprint MCS-P296-0392, Argonne National Laboratory, Argonne, Illinois, May
• 1992

### Recursive multivariate automatic differentiation, Optimization

• Methods and Software
• 1995

### Automatic differentiation of composite functions, in Automatic Differentiation of Algorithms: Theory, Implementation, and Application, A

• Griewank and G. F. Corliss, eds., SIAM, Philadelphia,
• 1991

### Structured second-and higher-order derivatives through univariate Taylor series

• Computer Science
• 1993
This work computes derivatives in a fashion that parallelizes well, exploits sparsity or other structure frequently found in Hessian matrices, can compute only selected elements of a Hessian matrix, and computes Hessian × vector products.

• 1992