Numerical Derivative / Numerical Differentiation of unknown funct ion

-----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


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