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Approximation of a function

Description
Approximating an exponential function by a quadratic polynomial.


Source Files

Data Files





expfita.mos

(!*********************************************************************
   Mosel NL examples
   =================
   file expfita.mos
   ````````````````
   Approximating an exponential function by a quadratic polynomial.

   Convex NLP problem
 
   Based on AMPL model expfita.mod by Hande Y. Benson
   Source: http://www.orfe.princeton.edu/~rvdb/ampl/nlmodels/cute/
   Reference:
     M.J.D. Powell, "A tolerant algorithm for linearly constrained optimization 
     calculations", Mathematical Programming 45(3), pp.561--562, 1989.

   (c) 2008 Fair Issac Corporation
       author: S. Heipcke, Sep. 2008, rev. Mar. 2013
*********************************************************************!)

model "expfita"
 uses "mmxnlp"

 parameters
  DATAFILE = "expfita.dat"
  R = 15                             ! Number of points
 end-parameters

 declarations
  RR = 1..R                          ! Set of points defining the function to approximate
  T: array(RR) of real               ! x-coordinates of given points
  ET: array(RR) of real              ! y-coordinates of given points
  RP: range
  PInit: array(RP) of integer        ! Start values for coefficients of the polynom
 end-declarations

 forall(i in RR) T(i):= 5*(i-1)/(R-1)
 forall(i in RR) ET(i):= exp(T(i))   ! Function to approximate

 initialisations from DATAFILE
  PInit
 end-initialisations

 declarations
  pcoeff: array(RP) of mpvar         ! Coefficients of the polynom sought
  RQ = 1..2
  qcoeff: array(RQ) of mpvar         ! Coefficients
 end-declarations

 forall(i in RP) do
  pcoeff(i) is_free
  setinitval(pcoeff(i), PInit(i))
 end-do

 forall(i in RQ) do
  qcoeff(i) is_free
  setinitval(qcoeff(i),0)
 end-do
 
! Objective function
 ErrF:= sum(i in RR) ((pcoeff(0)+pcoeff(1)*T(i)+pcoeff(2)*T(i)^2) / 
                   (ET(i)*(1+qcoeff(1)*(T(i)-5)+qcoeff(2)*(T(i)-5)^2)) - 1)^2

 forall(i in RR) 
  Cons1(i):= pcoeff(0) + pcoeff(1)*T(i) + pcoeff(2)*T(i)^2 -   
             (T(i)-5)*ET(i)*qcoeff(1) - (T(i)-5)^2*ET(i)*qcoeff(2)-ET(i) >= 0
 forall(i in RR) 
  Cons2(i):= (T(i)-5)*qcoeff(1) + (T(i)-5)^2*qcoeff(2) >= -1

! Solve the problem
 setparam("XNLP_verbose", true)
 
 minimise(ErrF)

! Solution printing
 declarations
   SolP: array(RR) of real	
 end-declarations
 
 forall(i in RR) SolP(i):= pcoeff(0).sol+pcoeff(1).sol*T(i)+pcoeff(2).sol*T(i)^2
 writeln("Solution: ", ErrF.sol)
 write("Polynomials: \n P: ")
 forall(i in RP) write(if(pcoeff(i).sol>0, " +"," "), strfmt(pcoeff(i).sol,5,3), "*x^", i)
 write("\n Q: ")
 forall(i in RQ) write(if(qcoeff(i).sol>0, " +"," "), strfmt(qcoeff(i).sol,5,3), "*x^", i)
 writeln("\nEvaluation of P at data points:")
 forall(i in RR) writeln(" (",strfmt(T(i),6,3), ",", strfmt(ET(i),6,3) ,") ", SolP(i)) 

end-model

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