Abstract
In mathematics and computer algebra, automatic differentiation (AD) is a set of techniques to evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.), elementary functions (exp, log, sin, cos, etc.) and control flow statements. AD takes source code of a function as input and produces source code of the derived function. By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program. This paper presents AD techniques available in ROOT, supported by Cling, to produce derivatives of arbitrary C/C++ functions through implementing source code transformation and employing the chain rule of differential calculus in both forward mode and reverse mode. We explain its current integration for gradient computation in TFormula. We demonstrate the correctness and performance improvements in ROOT’s fitting algorithms.
Highlights
Accurate and efficient computation of derivatives is vital for a wide variety of computing applications, including numerical optimization, solution of nonlinear equations, sensitivity analysis, and nonlinear inverse problems
We empirically compare automatic differentiation (AD, our implementation based on Clad) and numerical differentiation (ND, based on finite difference method) in ROOT
We show that AD can drastically improve accuracy and performance of derivative evaluation, compared to ND
Summary
Accurate and efficient computation of derivatives is vital for a wide variety of computing applications, including numerical optimization, solution of nonlinear equations, sensitivity analysis, and nonlinear inverse problems. Numerical and symbolic differentiation methods are slow at computing gradients of functions with many input variables, as is often needed for gradient-based optimization algorithms. Both methods have problems calculating higher-order derivatives, where the complexity and errors due to numerical precision increase. We describe the implementation of automatic differentiation techniques in ROOT, which is the data analysis framework broadly used High-Energy Physics [4] This implementation is based on Clad [5, 6], which is an automatic differentiation plugin for computation expressed in C/C++
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