Linear Algebra and the C Language/a040

/* ------------------------------------ */
/*  Save as:   c00f.c                  */
/* ------------------------------------ */
#include "v_a.h"
/* ------------------------------------ */
/* ------------------------------------ */
#define   RA R5
#define   CA C5
/* ------------------------------------ */
#define FACTOR_E        +1.E-2         
/* ------------------------------------ */
int main(void)
{
double tA[RA*CA]={
/*  x**4   x**3   x**2    x**1    x**0     */
  +1,     +1,     +1,     +1,     +1,       
 +16,     +8,     +4,     +2,     +1,        
 +81,    +27,     +9,     +3,     +1,        
+256,    +64,    +16,     +4,     +1,         
+625,   +125,    +25,     +5,     +1,       
};

double tb[RA*C1]={
/*    y  */
   -2,
   -2,
    3,
   -9,
    4,
};

double **A       = ca_A_mR(tA, i_mR(RA,CA));
double **b       = ca_A_mR(tb, i_mR(RA,C1));
double **Pinv    =             i_mR(CA,RA);          
double **Pinvb   =             i_mR(CA,C1);          

  clrscrn();
  printf(" Fitting a linear Curve to Data:\n\n");
  printf(" A:");
  p_mR(A, S10,P2,C7);
  printf(" b:");
  p_mR(b, S10,P2,C7);
  stop();
  
  clrscrn();   
  printf(" Pinv = V invS_T U_T ");
  Pinv_Rn_mR(A,Pinv,FACTOR_E); 
  pE_mR(Pinv, S12,P4,C10); 
  
  printf(" Pinv b ");   
  mul_mR(Pinv,b,Pinvb); 
  p_mR(Pinvb, S10,P4,C10);
  printf(" The Quartic equation Curve to Data: \n\n"
         "  y = %+.9f*x^4 %+.9f*x^3 %+.9f*x^2 %+.9f*x %+.9f\n\n"
            ,Pinvb[R1][C1],Pinvb[R2][C1],Pinvb[R3][C1],
             Pinvb[R4][C1],Pinvb[R5][C1]);        
  stop();  

  f_mR(b);  
  f_mR(A); 
  f_mR(Pinv);
  f_mR(Pinvb); 

  return 0;
}
/* ------------------------------------ */
/* ------------------------------------ */

  Let's calculate the coefficients of a polynomial.
 
              y =  ax**4 + bx**3 + cx**2 + dx + e        
  
         Which passes through these five points.    
          
       x[1],  y[1] 
       x[2],  y[2] 
       x[3],  y[3] 
       x[4],  y[4] 
       x[5],  y[5] 
       
  Using the points we obtain the matrix:

      x**4         x**3        x**2      x**1      x**0     y

     x[1]**4     x[1]**3     x[1]**2   x[1]**1   x[1]**0   y[1]
     x[2]**4     x[2]**3     x[2]**2   x[2]**1   x[2]**0   y[2]
     x[3]**4     x[3]**3     x[3]**2   x[3]**1   x[3]**0   y[3]
     x[4]**4     x[4]**3     x[4]**2   x[4]**1   x[4]**0   y[4]
     x[5]**4     x[5]**3     x[5]**2   x[5]**1   x[5]**0   y[5]

  That we can write:

       x**4          x**3        x**2     x     1    y

      x[1]**4       x[1]**3    x[1]**2   x[1]   1   y[1]
      x[2]**4       x[2]**3    x[2]**2   x[2]   1   y[2]
      x[3]**4       x[3]**3    x[3]**2   x[3]   1   y[3]
      x[4]**4       x[4]**3    x[4]**2   x[4]   1   y[4]
      x[5]**4       x[5]**3    x[5]**2   x[5]   1   y[5]   

     Let's use the Pinv_Rn_mR() function to solve
     the system that will give us the coefficients a, b, c, d, e

 Fitting a linear Curve to Data :

 A :
     +1.00      +1.00      +1.00      +1.00      +1.00 
    +16.00      +8.00      +4.00      +2.00      +1.00 
    +81.00     +27.00      +9.00      +3.00      +1.00 
   +256.00     +64.00     +16.00      +4.00      +1.00 
   +625.00    +125.00     +25.00      +5.00      +1.00 

 b :
     -2.00 
     -2.00 
     +3.00 
     -9.00 
     +4.00 

 Press return to continue. 


 Pinv = V * invS_T * U_T 
 +4.1667e-02  -1.6667e-01  +2.5000e-01  -1.6667e-01  +4.1667e-02 
 -5.8333e-01  +2.1667e+00  -3.0000e+00  +1.8333e+00  -4.1667e-01 
 +2.9583e+00  -9.8333e+00  +1.2250e+01  -6.8333e+00  +1.4583e+00 
 -6.4167e+00  +1.7833e+01  -1.9500e+01  +1.0167e+01  -2.0833e+00 
 +5.0000e+00  -1.0000e+01  +1.0000e+01  -5.0000e+00  +1.0000e+00 

 Pinv * b 
   +2.6667 
  -30.3333 
 +117.8333 
 -181.1666 
  +89.0000 

 The Quartic equation Curve to Data : 

  y = +2.666665810*x^4 -30.333323719*x^3 +117.833298608*x^2 -181.166625268*x +88.999995106

 Press return to continue.