Generate Bates and Watts D Matrix

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Description

Generate a matrix of function derivative information.

Usage

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    genD(func, x, method="Richardson",
                   method.args=list(), ...)
    ## Default S3 method:
genD(func, x, method="Richardson",
      method.args=list(), ...)

Arguments

func

a function for which the first (vector) argument is used as a parameter vector.

x

The parameter vector first argument to func.

method

one of "Richardson" or "simple" indicating the method to use for the aproximation.

method.args

arguments passed to method. See grad. (Arguments not specified remain with their default values.)

...

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

Details

The derivatives are calculated numerically using Richardson improvement. Methods "simple" and "complex" are not supported in this function. The "Richardson" method calculates a numerical approximation of the first and second derivatives of func at the point x. For a scalar valued function these are the gradient vector and Hessian matrix. (See grad and hessian.) For a vector valued function the first derivative is the Jacobian matrix (see jacobian). For the Richardson method method.args=list(eps=1e-4, d=0.0001, zero.tol=sqrt(.Machine$double.eps/7e-7), r=4, v=2) is set as the default. See grad for more details on the Richardson's extrapolation parameters.

A simple approximation to the first order derivative with respect to x_i is

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

A simple approximation to the second order derivative with respect to x_i is

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

The second order derivative with respect to x_i, x_j is

f''_{i,j}(x) = <f(x_{1},…,x_{i}+d,…,x_{j}+d,…,x_{n}) - 2 *f(x_{1},…,x_{n}) +

f(x_{1},…,x_{i}-d,…,x_{j}-d,…,x_{n})>/(2*d^2) - (f''_{i}(x) + f''_{j}(x))/2

Richardson's extrapolation is based on these formula with the d being reduced in the extrapolation iterations. In the code, d is scaled to accommodate parameters of different magnitudes.

genD does 1 + r (N^2 + N) evaluations of the function f, where N is the length of x.

Value

A list with elements as follows: D is a matrix of first and second order partial derivatives organized in the same manner as Bates and Watts, the number of rows is equal to the length of the result of func, the first p columns are the Jacobian, and the next p(p+1)/2 columns are the lower triangle of the second derivative (which is the Hessian for a scalar valued func). p is the length of x (dimension of the parameter space). f0 is the function value at the point where the matrix D was calculated. The genD arguments func, x, d, method, and method.args also are returned in the list.

References

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

Bates, D.M. & Watts, D. (1980), "Relative Curvature Measures of Nonlinearity." J. Royal Statistics Soc. series B, 42:1-25

Bates, D.M. and Watts, D. (1988) "Non-linear Regression Analysis and Its Applications." Wiley.

See Also

hessian, grad

Examples

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    func <- function(x){c(x[1], x[1], x[2]^2)}
    z <- genD(func, c(2,2,5))