mosaic (version 0.5-1)

D: Derivative and Anti-derivative operators

Description

Operators for computing derivatives and anti-derivatives as functions.

Usage

D(formula, ..., .hstep = NULL, add.h.control = FALSE)
  antiD(formula, ..., Const = 0)
  makeAntiDfun(.function, .wrt, from, to, .tol, Const)
  numerical.integration(f, wrt, av, args, vi.from)

Arguments

formula

A formula. The right side 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 expres

...

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 v

Const

Numerical value for the constant of integration.

.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.

.function

function to be integrated

.wrt

character string naming the variable of integration

from

default value for the lower bound of the integral region

to

default value for the upper bound of the integral region

.tol

tolerance of the numerical integrator (not yet implemented)

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

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, $\frac{d^2 f}{dy\;dx}$). 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.

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 always does numerical integration.

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). This feature may be deprecated in an upcoming release: There is also an argument, initVal, that plays the role of the constant of integration.

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
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)
by.xy = antiD(by.x(x=sqrt(1-y^2), y=y)~y)
4*by.xy(y=1) #area of quarter circle
vel <- antiD( -9.8 ~ t  )
pos <- antiD( vel( t=t, initVal=v0)~t, Const=50)
pos(0:5, v0=10)
pos(0:5, v0=10, initVal=100)