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### 1.15 A conjugate gradient method

Function File: [x0,v,nev] cg_min ( f,df,args,ctl )

  NonLinear Conjugate Gradient method to minimize function f.

  #### Arguments

  - f : string : Name of function. Return a real value
  - df : string : Name of f’s derivative. Returns a (R*C) x 1 vector
  - args: cell : Arguments passed to f.
  - ctl : 5-vec : (Optional) Control variables, described below

  #### Returned values

  - x0 : matrix : Local minimum of f
  - v : real : Value of f in x0
  - nev : 1 x 2 : Number of evaluations of f and of df

  #### Control Variables

  - ctl(1) : 1 or 2 : Select stopping criterion amongst :
  - ctl(1)==0 : Default value
  - ctl(1)==1 : Stopping criterion : Stop search when value doesn’t improve, as tested by ctl(2) > Deltaf/max(|f(x)|,1) where Deltaf is the decrease in f observed in the last iteration (each iteration consists R*C line searches).
  - ctl(1)==2 : Stopping criterion : Stop search when updates are small, as tested by ctl(2) > max { dx(i)/max(|x(i)|,1) | i in 1..N } where dx is the change in the x that occured in the last iteration.
  - ctl(2) : Threshold used in stopping tests. Default=10*eps
  - ctl(2)==0 : Default value
  - ctl(3) : Position of the minimized argument in args Default=1
  - ctl(3)==0 : Default value
  - ctl(4) : Maximum number of function evaluations Default=inf
  - ctl(4)==0 : Default value
  - ctl(5) : Type of optimization:
  - ctl(5)==1 : "Fletcher-Reves" method
  - ctl(5)==2 : "Polak-Ribiere" (Default)
  - ctl(5)==3 : "Hestenes-Stiefel" method

  ctl may have length smaller than 4. Default values will be used if ctl is not passed or if nan values are given.

  #### Example:

  function r=df( l ) b=[1;0;-1]; r = -( 2*l{1} - 2*b + rand(size(l{1}))); endfunction
  function r=ff( l ) b=[1;0;-1]; r = (l{1}-b)’ * (l{1}-b); endfunction
  ll = { [10; 2; 3] };
  ctl(5) = 3;
  [x0,v,nev]=cg_min( "ff", "df", ll, ctl )

  Comment: In general, BFGS method seems to be better performin in many cases but requires more computation per iteration See also http://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient.

  See also: bfgsmin.

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