propagate (version 1.3-0)

numDerivs: Functions for creating Gradient and Hessian matrices by numerical differentiation (Richardson's method) of the partial derivatives

Description

These two functions create Gradient and Hessian matrices by Richardson's central finite difference method of the partial derivatives for any expression.

Usage

numGrad(expr, envir = parent.frame())
numHess(expr, envir = parent.frame())

Value

The numeric Gradient/Hessian matrices.

Arguments

expr

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

envir

the environment in which the variables of expr are found (default: the calling environment, so that it also works inside functions).

Author

Andrej-Nikolai Spiess

Details

Calculates first- and second-order numerical approximations of the partial derivatives at the current values \(x = (x_1, \ldots, x_N)\) of the variables, using the central difference formulas $$f'_i(x) \approx \frac{f(x_1, \ldots, x_i + d_i, \ldots, x_N) - f(x_1, \ldots, x_i - d_i, \ldots, x_N)}{2d_i}$$ $$f''_{ii}(x) \approx \frac{f(x_1, \ldots, x_i + d_i, \ldots, x_N) - 2f(x_1, \ldots, x_N) + f(x_1, \ldots, x_i - d_i, \ldots, x_N)}{d_i^2}$$ and, for the mixed partial derivatives (\(i \neq j\)), with \(x^{\pm}\) = \(x\) with \(x_i \pm d_i\) and \(x_j \pm d_j\), $$f''_{ij}(x) \approx \frac{f(x^{+}) - 2f(x) + f(x^{-}) - f''_{ii}(x)\,d_i^2 - f''_{jj}(x)\,d_j^2}{2 d_i d_j}.$$ These approximations are refined by Richardson extrapolation, as in genD of the 'numDeriv' package: each is computed for \(r = 4\) step sizes, starting with \(d_i = 0.1|x_i|\) (plus \(10^{-4}\) if \(x_i\) is close to 0) and halving the step each time, and the \(r\) values \(D_1, \ldots, D_r\) are combined successively for \(m = 1, \ldots, r - 1\) by \(D_k \leftarrow (4^m D_{k+1} - D_k)/(4^m - 1)\).

Examples

## Check for equality of symbolic  
## and numerical derivatives.
EXPR <- expression(2^x + sin(2 * y) - cos(z))
x <- 5
y <- 10
z <- 20

symGRAD <- evalDerivs(makeGrad(EXPR))
numGRAD <- numGrad(EXPR)
all.equal(symGRAD, numGRAD)

symHESS <- evalDerivs(makeHess(EXPR))
numHESS <- numHess(EXPR)
all.equal(symHESS, numHESS)