Calculate Hessian Matrix
Calculate a numerical approximation to the Hessian matrix of a function at a parameter value.
hessian(func, x, method="Richardson", method.args=list(), ...)
## Default S3 method:
hessian(func, x, method="Richardson",
method.args=list(), ...)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 |
method.args |
arguments passed to method. See |
... |
an additional arguments passed to |
The function hessian calculates an numerical approximation to
the n x n second derivative of a scalar real valued function with n-vector
argument.
The argument method can be "Richardson" or "complex".
Method "simple" is not supported.
For method "complex" the Hessian matrix is calculated as the Jacobian
of the gradient. The function grad with method "complex" is used,
and method.args is ignored for this (an eps of
.Machine$double.eps is used).
However, jacobian is used in the second step, with method
"Richardson" and argument method.args is used for this.
The default is
method.args=list(eps=1e-4, d=0.1, zero.tol=sqrt(.Machine$double.eps/7e-7),
r=4, v=2, show.details=FALSE). (These are the defaults for hessian
with method "Richardson", which are slightly different from the defaults
for jacobian with method "Richardson".)
See addition comments in grad before choosing
method "complex".
Methods "Richardson" uses genD and extracts the
second derivative. For this method
method.args=list(eps=1e-4, d=0.1, zero.tol=sqrt(.Machine$double.eps/7e-7),
r=4, v=2, show.details=FALSE) is set as the default. hessian does
one evaluation of func in order to do some error checking before
calling genD, so the number of function evaluations will be one more
than indicated for genD.
The argument side is not supported for second derivatives and since
... are passed to func there may be no error message if it is
specified.
An n by n matrix of the Hessian of the function calculated at the
point x.
sc2.f <- function(x){
n <- length(x)
sum((1:n) * (exp(x) - x)) / n
}
sc2.g <- function(x){
n <- length(x)
(1:n) * (exp(x) - 1) / n
}
x0 <- rnorm(5)
hess <- hessian(func=sc2.f, x=x0)
hessc <- hessian(func=sc2.f, x=x0, "complex")
all.equal(hess, hessc, tolerance = .Machine$double.eps)
# Hessian = Jacobian of the gradient
jac <- jacobian(func=sc2.g, x=x0)
jacc <- jacobian(func=sc2.g, x=x0, "complex")
all.equal(hess, jac, tolerance = .Machine$double.eps)
all.equal(hessc, jacc, tolerance = .Machine$double.eps)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.