A formula. The right side of a formula specifies
the variable(s) with which to
carry out the integration or differentiation. On the left side should be
an expression or a function that returns a numerical vector
of the same length as its argument.
The expression can contain unbound variables. Functions
will be differentiated as if the formula f(x) ~ x were specified
but with x replaced by the first argument of f.
.hstep
horizontal distance between points used for secant slope
calculation in numerical derivatives.
add.h.control
logical indicating whether the returned derivative function
should have an additional parameter for setting .hstep. Meaningful only for numerical
derivatives.
lower.bound
for numerical integration only, the lower bound used
force.numeric
If TRUE, a numerical integral is performed even when a
symbolic integral is available.
.function
function to be integrated
.wrt
character string naming the variable of integration
from
default value for the lower bound of the integral region
.tol
Numerical tolerance. See stats::integrate
f
a function
wrt
character string naming a variable: the var. of integration
av
a list of the arguments passed to the function calling this
args
default values (if any) for parameterss
vi.from
the the lower bound of the interval of integration
ciName
character string giving the name of the symbol for the constant of integration
...
Default values to be given to unbound variables in the expression expr.
See examples.#' Note that in creating anti-derivative functions,
default values of "from" and "to" can be assigned. They are to be written with
the name of the variable as a prefix, e.g. y.from.
Details
D attempts to find a symbolic derivative for simple expressions, but
will provide a function that is a numerical derivative if the attempt at
symbolic differentiation is unsuccessful. The symbolic derivative can be of
any order (although the expression may become unmanageably complex). The
numerical derivative is limited to first or second-order partial derivatives
(including mixed partials).
antiD will attempt simple symbolic integration but if it fails
it will return a numerically-based anti-derivative.
antiD returns a function with the same arguments as the
expression passed to it. The returned function is the anti-derivative
of the expression, e.g., antiD(f(x)~x) -> F(x).
To calculate the integral of f(x), use F(to) - F(from).
Value
For derivatives, the return value is a function of the variable(s)
of differentiation, as well as any other symbols used in the expression. Thus,
D(A*x^2 + B*y ~ x + y) will compute the mixed partial with respect to x
then y (that is, d2f/dydx). The returned value will be a function of x and y,
as well as A and B. In evaluating the returned function, it's best to use the
named form of arguments, to ensure the order is correct.
a function of the same arguments as the original expression with a
constant of integration set to zero by default, named "C", "D", ... depending on the first
such letter not otherwise in the argument list.
Note
numerical_integration is not intended for direct use. It packages
up the numerical anti-differentiation process so that the contents
of functions produced by antiD look nicer to human readers.
Examples
D(sin(t) ~ t)
D(A*sin(t) ~ t )
D(A*sin(2*pi*t/P) ~ t, A=2, P=10) # default values for parameters.
f <- D(A*x^3 ~ x + x, A=1) # 2nd order partial -- note, it's a function of x
f(x=2)
f(x=2,A=10) # override default value of parameter A
g <- D(f(x=t, A=1)^2 ~ t) # note: it's a function of t
g(t=1)
gg <- D(f(x=t, A=B)^2 ~ t, B=10) # note: it's a function of t and B
gg(t=1)
gg(t=1, B=100)
f <- makeFun(x^2~x)
D(f(cos(z))~z) #will look in user functions also
antiD( a*x^2 ~ x, a = 3)
antiD( A/x~x ) # This gives a warning about no default value for A
F <- antiD( A*exp(-k*t^2 ) ~ t, A=1, k=0.1)
F(t=Inf)
one = makeFun(1 ~ x + y)
by.x = antiD(one(x=x, y=y) ~ x, y=1)
by.xy = antiD(by.x(x = sqrt(1-y^2), y = y) ~ y)
4 * by.xy(y = 1) # area of quarter circle