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

Uzuner, Tolga tolga.uzuner at csfb.com
Fri May 6 00:20:57 CEST 2005

```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
########################################################################################

Regards,
Tolga Uzuner

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