/*--------------------------------------------------------------------------*/
/* Linear least squares fit                                                 */
/* In Linux compile with                                                    */
/*    gcc -o lsm lsm.c -lm                                                  */
/* Usage:                                                                   */
/*    lsm < datafile                                                        */
/* where datafile is a file containing two columns                          */
/* e.g. time and altitude  as real numbers                                  */
/* The program will fit a parabola y=a+bx+cx^2 to the data                  */
/*--------------------------------------------------------------------------*/
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <ctype.h>
#include <math.h>
#include <float.h>

#define N 100

main()
{
  double  a[N][3], b[N], s[3][3], c[3], x0, x1, x2,
          d, d0, d1, d2, tim, alt, tmp, x, y ;
  int n, i, j, k ;

  n = -1 ;
  while ((scanf(" %lf %lf", &tim, &alt)) != EOF)
  {   n++ ;
      a[n][0] = 1.0 ;
      a[n][1] = tim ;
      a[n][2] = tim * tim ;
      b[n] = alt ;
  }

  /* a transpose a */
  for (i=0; i<3; i++)
  for (j=0; j<3; j++)
  { tmp = 0.0;
  for (k=0; k<=n; k++) tmp += a[k][i] * a[k][j] ;
    s[i][j] = tmp ;
  }

  /* a transpose b */
  for (i=0; i<3; i++)
  { tmp = 0;
    for (k=0; k<=n; k++) tmp += a[k][i] * b[k] ;
    c[i] = tmp ;
  }

  /* determinant */
  d =  s[0][0] * (s[1][1] * s[2][2] - s[1][2] * s[2][1]) 
     - s[1][0] * (s[0][1] * s[2][2] - s[0][2] * s[2][1]) 
     + s[2][0] * (s[0][1] * s[1][2] - s[0][2] * s[1][1]) ;

  /* cofactors */
  d0 =  c[0] * (s[1][1] * s[2][2] - s[1][2] * s[2][1]) 
      - c[1] * (s[0][1] * s[2][2] - s[0][2] * s[2][1]) 
      + c[2] * (s[0][1] * s[1][2] - s[0][2] * s[1][1]) ;
  d1 =  s[0][0] * (c[1] * s[2][2] - s[1][2] * c[2]) 
      - s[1][0] * (c[0] * s[2][2] - s[0][2] * c[2]) 
      + s[2][0] * (c[0] * s[1][2] - s[0][2] * c[1]) ;
  d2 =  s[0][0] * (s[1][1] * c[2] - c[1] * s[2][1]) 
      - s[1][0] * (s[0][1] * c[2] - c[0] * s[2][1]) 
      + s[2][0] * (s[0][1] * c[1] - c[0] * s[1][1]) ;
  
  x0 = d0 / d ;  x1 = d1 / d ; x2 = d2 / d ;

  printf("y = %f + %f * x + %f * x**2\n", x0, x1, x2) ;

  x = -0.5 * x1 / x2 ;
  y = x0 + x1 * x + x2 * x * x ;

  printf("coordinates of the apex: x=%f y=%f\n", x, y) ;
  
}