[R] Numerical Derivative / Numerical Differentiation of unknown funct ion

From: Uzuner, Tolga <tolga.uzuner_at_csfb.com>
Date: Fri 06 May 2005 - 08:20:57 EST


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



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