[R] Numerical Derivative / Numerical Differentiation of unknown funct ion
Berton Gunter
gunter.berton at gene.com
Fri May 6 00:34:06 CEST 2005
But...
See ?numericDeriv which already does it via a C call and hence is much
faster (and probably more accurate,too).
-- Bert Gunter
Genentech Non-Clinical Statistics
South San Francisco, CA
"The business of the statistician is to catalyze the scientific learning
process." - George E. P. Box
> -----Original Message-----
> From: r-help-bounces at stat.math.ethz.ch
> [mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Uzuner, Tolga
> Sent: Thursday, May 05, 2005 3:21 PM
> To: 'r-help at stat.math.ethz.ch'
> Subject: [R] Numerical Derivative / Numerical Differentiation
> of unknown funct ion
>
> Hi,
>
> I have been trying to do numerical differentiation using R.
>
> I found some old S code using Richardson Extrapolation which
> I managed to get
> to work.
>
> I am posting it here in case anyone needs it.
>
>
> ##############################################################
> ##########
> richardson.grad <- function(func, x, d=0.01, eps=1e-4, r=6, show=F){
> # This function calculates a numerical approximation of the first
> # derivative of func at the point x. The calculation
> # is done by Richardson's extrapolation (see eg. G.R.Linfield and
> J.E.T.Penny
> # "Microcomputers in Numerical Analysis"). The method
> should be used if
> # accuracy, as opposed to speed, is important.
> #
> # * modified by Paul Gilbert from orginal code by XINGQIAO LIU.
> # CALCULATES THE FIRST ORDER DERIVATIVE
> # VECTOR using a Richardson extrapolation.
> #
> # GENERAL APPROACH
> # -- GIVEN THE FOLLOWING INITIAL VALUES:
> # INTERVAL VALUE D, NUMBER OF ITERATIONS R, AND
> # REDUCED FACTOR V.
> # - THE FIRST ORDER aproximation to the DERIVATIVE WITH
> RESPECT TO Xi IS
> #
> # F'(Xi)={F(X1,...,Xi+D,...,Xn) -
> F(X1,...,Xi-D,...,Xn)}/(2*D)
> #
> # -- REPEAT r TIMES with successively smaller D and
> # then apply Richardson extraplolation
> #
> # INPUT
> # func Name of the function.
> # x The parameters of func.
> # d Initial interval value (real) by default set
> to 0.01*x or
> # eps if x is 0.0.
> # r The number of Richardson improvement iterations.
> # show If T show intermediate results.
> # OUTPUT
> #
> # The gradient vector.
>
> v <- 2 # reduction factor.
> n <- length(x) # Integer, number of variables.
> a.mtr <- matrix(1, r, n)
> b.mtr <- matrix(1, (r - 1), n)
> #-------------------------------------------------------------
> -----------
> # 1 Calculate the derivative formula given in 'GENERAL
> APPROACH' section.
> # -- The first order derivatives are stored in the matrix
> a.mtr[k,i],
> # where the indexing variables k for rows(1 to r), i
> for columns
> # (1 to n), r is the number of iterations, and n is
> the number of
> # variables.
> #-------------------------------------------------------------
> ------------
>
> h <- abs(d*x)+eps*(x==0.0)
> for(k in 1:r) { # successively reduce h
> for(i in 1:n) {
> x1.vct <- x2.vct <- x
> x1.vct[i] <- x[i] + h[i]
> x2.vct[i] <- x[i] - h[i]
> if(k == 1) a.mtr[k,i] <- (func(x1.vct) -
> func(x2.vct))/(2*h[i])
> else{
> if(abs(a.mtr[(k-1),i])>1e-20)
> # some functions are unstable near 0.0
> a.mtr[k,i] <- (func(x1.vct)-func(x2.vct))/(2*h[i])
> else a.mtr[k, i] <- 0
> }
> }
> h <- h/v # Reduced h by 1/v.
> }
> if(show) {
> cat("\n","first order approximations", "\n")
> print(a.mtr, 12)
> }
>
> #-------------------------------------------------------------
> -----------
> # 1 Applying Richardson Extrapolation to improve the accuracy of
> # the first and second order derivatives. The algorithm as follows:
> #
> # -- For each column of the 1st and 2nd order derivatives
> matrix a.mtr,
> # say, A1, A2, ..., Ar, by Richardson Extrapolation, to
> calculate a
> # new sequence of approximations B1, B2, ..., Br used
> the formula
> #
> # B(i) =( A(i+1)*4^m - A(i) ) / (4^m - 1) , i=1,2,...,r-m
> #
> # N.B. This formula assumes v=2.
> #
> # -- Initially m is taken as 1 and then the process is repeated
> # restarting with the latest improved values and increasing the
> # value of m by one each until m equals r-1
> #
> # 2 Display the improved derivatives for each
> # m from 1 to r-1 if the argument show=T.
> #
> # 3 Return the final improved derivative vector.
> #-------------------------------------------------------------
> ------------
>
> for(m in 1:(r - 1)) {
> for(i in 1:(r - m)) b.mtr[i,]<-
> (a.mtr[(i+1),]*(4^m)-a.mtr[i,])/(4^m-1)
> # a.mtr<- b.mtr
> # a.mtr<- (a.mtr[2:(r+1-m),]*(4^m)-a.mtr[1:(r-m),])/(4^m-1)
> if(show & m!=(r-1) ) {
> cat("\n","Richarson improvement group No. ", m, "\n")
> print(a.mtr[1:(r-m),], 12)
> }
> }
> a.mtr[length(a.mtr)]
> }
>
> ## try it out
> richardson.grad(function(x){x^3},2)
> ##############################################################
> ##########################
>
>
> Regards,
> Tolga Uzuner
>
>
> ==============================================================
> ================
> This message is for the sole use of the intended recipient...{{dropped}}
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