Homework "Minimization"

1. (6 points) Quasi-Newton method with numerical gradient, back-tracking linesearch, and rank-1 update

   Implement the quasi-Newton minimization method with numerical gradient, back-tracking linesearch, and a rank-1 update.

   The interface to the function could be like this,

   public static vector qnewton(
   	Func<vector,double> f, /* objective function */
   	vector start, /* starting point */
   	double acc /* accuracy goal, on exit |gradient| should be < acc */
   )
   

   Start with the unity inverse Hessian matrix and then apply the updates. If the update diverges or the linesearch fails reset the inverse Hessian matrix to unity and continue.

   Find a minimum of the Rosenbrock's valley function, f(x,y)=(1-x)2+100(y-x2)2.

   Find a minimum of the Himmelblau's function, f(x,y)=(x2+y-11)2+(x+y2-7)2.

   Record the number of steps it takes for the algorithm to reach the minimum.

2. (3 points) Higgs discovery

   In 2012 CERN announced the discovery of a previously unknown boson with mass 125.3±0.6 GeV/c² [Wikipedia].

   In one of the experiments a certain cross-section involving Higgs was measured and the data (with background subtracted – that's why some data are negative) are given as [arxiv:1207.7235]

   # energy E[GeV], cross-section σ(E)[certain units], experimental uncertainty Δσ [same units]
   101 -0.25 2.0
   103 -0.30 2.0
   105 -0.15 1.9
   107 -1.71 1.9
   109  0.81 1.9
   111  0.65 1.9
   113 -0.91 1.9
   115  0.91 1.9
   117  0.96 1.6
   119 -2.52 1.6
   121 -1.01 1.6
   123  2.01 1.6
   125  4.83 1.6
   127  4.58 1.6
   129  1.26 1.3
   131  1.01 1.3
   133 -1.26 1.3
   135  0.45 1.3
   137  0.15 1.3
   139 -0.91 1.3
   141 -0.81 1.1
   143 -1.41 1.1
   145  1.36 1.1
   147  0.50 1.1
   149 -0.45 1.1
   151  1.61 1.1
   153 -2.21 1.1
   155 -1.86 0.9
   157  1.76 0.9
   159 -0.50 0.9
   

   Fit the Breit-Wigner function

   F(E|m,Γ,A)=A/((E-m)²+Γ²/4)
   

   (where A is a scale-factor, m is the mass and Γ is the widths of the sought resonance) to the data and determine the mass and the experimetnal width of the Higgs boson (the experimental width is only the upper limit on the physical width).

   One fits the data by minimizing the deviation function

   D(m,Γ,A)=Σi[(F(Ei|m,Γ,A)-σi)/Δσi]2
   

   You can send the data into the standard input stream of your program like this,

   mono main.exe < higgs.data.txt 1> out.txt 2> log
   

   and you can read the data table into generic lists (either our genlist or the System.Collections.Generic.List) from the standard input like this,

   var energy = new genlist<double>();
   var signal = new genlist<double>();
   var error  = new genlist<double>();
   var separators = new char[] {' ','\t'};
   var options = StringSplitOptions.RemoveEmptyEntries;
   do{
           string line=Console.In.ReadLine();
           if(line==null)break;
           string[] words=line.Split(separators,options);
           energy.add(double.Parse(words[0]));
           signal.add(double.Parse(words[1]));
           error .add(double.Parse(words[2]));
   }while(true);
   

3. (1 point) Implement the downhill simplex method