propagate (version 1.0-4)

makeDerivs: Utility functions for creating Gradient- and Hessian-like matrices with symbolic derivatives and evaluating them in an environment

Description

These are three different utility functions that create matrices containing the symbolic partial derivatives of first (makeGrad) and second (makeHess) order and a function for evaluating these matrices in an environment. The evaluations of the symbolic derivatives are used within the propagate function to calculate the propagated uncertainty, but these functions may also be useful for other applications.

Usage

makeGrad(expr, order = NULL)
makeHess(expr, order = NULL)
evalDerivs(deriv, envir)

Arguments

expr

an expression, such as expression(x/y).

order

order of creating partial derivatives, i.e. c(2, 1). See 'Examples'.

deriv

a matrix returned from makeGrad or makeHess.

envir

an environment to evaluate in. By default the workspace.

Value

The symbolic or evaluated Gradient/Hessian matrices.

Details

Given a function \(f(x_1, x_2, \ldots, x_n)\), the following matrices containing symbolic derivatives of \(f\) are returned:

makeGrad: $$\nabla(f) = \left[\frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n}\right]$$ makeHess:

$$H(f) = \left[ \begin{array}{cccc} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1\,\partial x_n} \\ \frac{\partial^2 f}{\partial x_2\,\partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2\,\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_n\,\partial x_1} & \frac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{array} \right] $$

References

The Matrix Cookbook (Version November 2012). Petersen KB & Pedersen MS. http://ais.informatik.uni-freiburg.de/teaching/ws12/mapping/pdf/imm3274.pdf

Examples

# NOT RUN {
EXPR <- expression(a^b + sin(c))
ENVIR <- list(a = 2, b = 3, c = 4)

## First-order partial derivatives: Gradient.
GRAD <- makeGrad(EXPR) 

## This will evaluate the Gradient.
evalDerivs(GRAD, ENVIR)

## Second-order partial derivatives: Hessian.
HESS <- makeHess(EXPR) 

## This will evaluate the Hessian.
evalDerivs(HESS, ENVIR)

## Change derivatives order.
GRAD <- makeGrad(EXPR, order = c(2,1,3)) 
evalDerivs(GRAD, ENVIR)
# }