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##### Numerical Gradient of a Function

Calculate the gradient of a function by numerical approximation.

Keywords
multivariate
##### Usage
grad(func, x, method="Richardson", side=NULL, method.args=list(), ...)     # S3 method for default
method.args=list(), ...)
##### Arguments
func

a function with a scalar real result (see details).

x

a real scalar or vector argument to func, indicating the point(s) at which the gradient is to be calculated.

method

one of "Richardson", "simple", or "complex" indicating the method to use for the approximation.

method.args

arguments passed to method. Arguments not specified remain with their default values as specified in details

side

an indication of whether one-sided derivatives should be attempted (see details).

...

an additional arguments passed to func. WARNING: None of these should have names matching other arguments of this function.

##### Details

The function grad calculates a numerical approximation of the first derivative of func at the point x. Any additional arguments in … are also passed to func, but the gradient is not calculated with respect to these additional arguments. It is assumed func is a scalar value function. If a vector x produces a scalar result then grad returns the numerical approximation of the gradient at the point x (which has the same length as x). If a vector x produces a vector result then the result must have the same length as x, and it is assumed that this corresponds to applying the function to each of its arguments (for example, sin(x)). In this case grad returns the gradient at each of the points in x (which also has the same length as x -- so be careful). An alternative for vector valued functions is provided by jacobian.

If method is "simple", the calculation is done using a simple epsilon difference. For method "simple" method.args=list(eps=1e-4) is the default. Only eps is used by this method.

If method is "complex", the calculation is done using the complex step derivative approach of Lyness and Moler, described in Squire and Trapp. This method requires that the function be able to handle complex valued arguments and return the appropriate complex valued result, even though the user may only be interested in the real-valued derivatives. It also requires that the complex function be analytic. (This might be thought of as the complex equivalent of the requirement for continuity and smoothness of a real valued function.) So, while this method is extremely powerful it is applicable to a very restricted class of functions. Avoid this method if you do not know that your function is suitable. Your mistake may not be caught and the results will be spurious. For cases where it can be used, it is faster than Richardson's extrapolation, and it also provides gradients that are correct to machine precision (16 digits). For method "complex", method.args is ignored. The algorithm uses an eps of .Machine$double.eps which cannot (and should not) be modified. If method is "Richardson", the calculation is done by Richardson's extrapolation (see e.g. Linfield and Penny, 1989, or Fornberg and Sloan, 1994.) This method should be used if accuracy, as opposed to speed, is important (but see method "complex" above). For this method method.args=list(eps=1e-4, d=0.0001, zero.tol=sqrt(.Machine$double.eps/7e-7), r=4, v=2, show.details=FALSE) is set as the default. d gives the fraction of x to use for the initial numerical approximation. The default means the initial approximation uses 0.0001 * x. eps is used instead of d for elements of x which are zero (absolute value less than zero.tol). zero.tol tolerance used for deciding which elements of x are zero. r gives the number of Richardson improvement iterations (repetitions with successly smaller d. The default 4 general provides good results, but this can be increased to 6 for improved accuracy at the cost of more evaluations. v gives the reduction factor. show.details is a logical indicating if detailed calculations should be shown.

The general approach in the Richardson method is to iterate for r iterations from initial values for interval value d, using reduced factor v. The the first order approximation to the derivative with respect to $x_{i}$ is

$$f'_{i}(x) = <f(x_{1},\dots,x_{i}+d,\dots,x_{n}) - f(x_{1},\dots,x_{i}-d,\dots,x_{n})>/(2*d)$$

This is repeated r times with successively smaller d and then Richardson extraplolation is applied.

If elements of x are near zero the multiplicative interval calculation using d does not work, and for these elements an additive calculation using eps is done instead. The argument zero.tol is used determine if an element should be considered too close to zero. In the iterations, interval is successively reduced to eventual be d/v^r and the square of this value is used in second derivative calculations (see genD) so the default zero.tol=sqrt(.Machine$double.eps/7e-7) is set to ensure the interval is bigger than .Machine$double.eps with the default d, r, and v.

If side is NULL then it is assumed that the point at which the calculation is being done is interior to the domain of the function. If the point is on the boundary of the domain then side can be used to indicate which side of the point x should be used for the calculation. If not NULL then it should be a vector of the same length as x and have values NA, +1, or -1. NA indicates that the usual calculation will be done, while +1, or -1 indicate adding or subtracting from the parameter point x. The argument side is not supported for all methods.

Since usual calculation with method "simple" uses only a small eps step to one side, the only effect of argument side is to determine the direction of the step. The usual calculation with method "Richardson" is symmetric, using steps to both sides. The effect of argument side is to take a double sized step to one side, and no step to the other side. This means that the center of the Richardson extrapolation steps is moving slightly in the reduction, and is not exactly on the boundary. (Warning: I am not aware of theory or published experimental evidence to support this, but the results in my limited testing seem good.)

##### Value

A real scalar or vector of the approximated gradient(s).

##### References

Linfield, G. R. and Penny, J. E. T. (1989) Microcomputers in Numerical Analysis. New York: Halsted Press.

Fornberg, B. and Sloan, D, M. (1994) A review of pseudospectral methods for solving partial differential equations.'' Acta Numerica, 3, 203-267.

Lyness, J. N. and Moler, C. B. (1967) Numerical Differentiation of Analytic Functions.'' SIAM Journal for Numerical Analysis, 4(2), 202-210.

Squire, William and Trapp, George (1998) Using Complex Variables to Estimate Derivatives of Real Functions.'' SIAM Rev, 40(1), 110-112.

jacobian, hessian, genD, numericDeriv

##### Examples
# NOT RUN {
func0 <- function(x){ sum(sin(x))  }

func1 <- function(x){ sin(10*x) - exp(-x) }

curve(func1,from=0,to=5)

x <- 2.04
exact <- 10*cos(10*x) + exp(-x)
c(numd1, exact, (numd1 - exact)/exact)

x <- c(1:10)
exact <- 10*cos(10*x) + exp(-x)
cbind(numd1, numd2, exact, (numd1 - exact)/exact, (numd2 - exact)/exact)

sc2.f <- function(x){
n <- length(x)
sum((1:n) * (exp(x) - x)) / n
}

sc2.g <- function(x){
n <- length(x)
(1:n) * (exp(x) - 1) / n
}

x0 <- rnorm(100)
exact <- sc2.g(x0)

max(abs(exact - g)/(1 + abs(exact)))