c++ - Souriau method for Characteristic Polynomial -

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does know souriau method finding characteristic polynomial of n × n matrix? found out first coefficient, obvious, how can find out other coefficients? after need inverse matrix know how.

#include <iostream>  #include <fstream> using namespace std;  double trace(double a[5][5],int n){          int i;         double trace=0;         for(i=0;i<n;i++)              trace+=a[i][i];          return trace; } double prod(double a[5][5],double b[5][5],int n) {     double c[5][5];     int i,j,k;     cout << "\nprod:\n";      for(i=0;i<n;++i){          for(j=0;j<n;++j){              c[i][j]=0;              for(k=0;k<n;++k)                  c[i][j]=c[i][j]+(a[i][k]*b[k][j]);                 cout << c[i][j] << " ";          }          cout << "\n";      }      return c[i][j];  } double theta(double a[5][5], int n){     int i;     double theta[5];     theta[1]=-trace(a,n);     for(i=0;i<n;i++)         cout << "theta[" << i+1 << "]=" << theta[i+1] << "\n";     return theta[i+1]; }  int main(){      ifstream f("a.txt");      ifstream g("b.txt");     double a[5][5],b[5][5];     int i,j,n;      f >> n;      g >> n;      for(i=0;i<n;++i)          for(j=0;j<n;++j)              f >> a[i][j];      cout << "matrix a:"<<endl;     for(i=0;i<n;++i){         for(j=0;j<n;++j)              cout << a[i][j] << " ";             cout << endl;     }     cout << endl;     for(i=0;i<n;++i)          for(j=0;j<n;++j)              g >> b[i][j];      cout << "matrix b:" << endl;     for(i=0;i<n;++i){          for(j=0;j<n;++j)              cout << b[i][j] << " ";             cout << endl;     }          cout << endl;         cout << "trace = ";         cout << trace(a,n);         cout << endl;         prod(a,b,n);         cout << endl;         theta(a,n);     }

taken https://math.stackexchange.com/a/405975/115115 Ｊ. Ｍ.

c=a; k=1,…,n      if k>1         c=a*(c+c[n−k+1]*i);      c[n−k]=−tr(c)/k;  end

if can read german, there wiki page extended pseudo-code algortihm @ https://de.wikipedia.org/wiki/algorithmus_von_faddejew-leverrier (add 2017: or equally english version https://en.wikipedia.org/wiki/faddeev%e2%80%93leverrier_algorithm)

if want directly compute inverse matrix, have, in wiki page, use matrix b related matrix c above via c=ab. gives, can seen in wiki page, more complicated algorithm. however, last matrix b satisfies ab=-c[0]*i, inverse matrix, if there one, can directly computed.


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