Mathc complexes/a246

Application

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	c03a.c

/* ------------------------------------ */
/*  Save as :   c03a.c                  */
/* ------------------------------------ */
#include "w_a.h"
/* ------------------------------------ */
/* ------------------------------------ */
#define   RA R3
#define   CA C2
/* ------------------------------------ */
/* ------------------------------------ */
int main(void)
{
double a[RA*(CA*C2)]={
 -2,-3, -4,-2,
  1, 0, -3,-5,
  0, 1, -6,-4,
};

double x[RA*(C1*C2)]={
 -1,-3, 
  2,-4, 
 -3,-5,
};

double **A           = ca_A_mZ(a, i_mZ(RA,CA));
double **AT          =            i_mZ(CA,RA);
double **ATA         =            i_mZ(CA,CA); //         AT*A
double **invATA      =            i_mZ(CA,CA); //     inv(AT*A)
double **invATA_AT   =            i_mZ(CA,RA); //     inv(AT*A)*AT
double **V           =            i_mZ(RA,RA); //     inv(AT*A)*AT

double **X           = ca_A_mZ(x, i_mZ(RA,C1));
double **VX          =            i_mZ(RA,C1);

  clrscrn();
  printf(" A is subspace of R%d              \n\n"
         " Find a transformation matrix for    \n"
         " a projection onto R%d  :          \n\n"
         " Proj(x) =  A * inv(AT*A) * AT * x \n\n",RA,RA);
  printf(" A :");
  p_mZ(A,S5,P1,S5,P1,C7);
  stop();
  
  clrscrn();
  printf(" AT :");
  p_mZ(ctranspose_mZ(A,AT),S5,P1,S5,P1,C7);
  printf(" ATA :");
  p_mZ(mul_mZ(AT,A,ATA),S5,P1,S5,P1,C7);
  printf(" inv(AT*A) :");
  p_mZ(invgj_mZ(ATA,invATA),S5,P4,S5,P4,C7);  
  printf(" inv(AT*A)*AT :");
  p_mZ(mul_mZ(invATA,AT,invATA_AT),S5,P4,S5,P4,C7); 
  printf(" V = A*inv(AT*A)*AT :");
  p_mZ(mul_mZ(A,invATA_AT,V),S5,P4,S5,P4,C7);    
  stop();  
  
  clrscrn();
  printf(" V is transformation matrix for      \n"
         " a projection onto a subspace R%d :\n\n",RA);
  p_mZ(V,S5,P4,S5,P4,C7); 
  
  printf(" X :");
  p_mZ(X,S5,P1,S5,P1,C7);
  
  printf(" Proj(x) =  A * inv(AT*A) * AT * x \n\n"); 
  printf(" Proj(x) =  V                  * x :");  
  p_mZ(mul_mZ(V,X,VX),S5,P4,S5,P4,C7); 
  stop();    
  
  
  f_mZ(A);
  f_mZ(AT);
  f_mZ(ATA);       //         AT*A
  f_mZ(invATA);    //     inv(AT*A)
  f_mZ(invATA_AT); //     inv(AT*A)*AT
  f_mZ(V);         //   A*inv(AT*A)*AT  
  
  f_mZ(X); 
  f_mZ(VX);         

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

Trouver une projection sur un sous-espace vectoriel par une application linéaire :

• A est un sous espace de R3. Trouver une matrice V qui projette un vecteur x sur R3.

          Proj(x) =  V * x
                V =  A * inv(AT*A) * AT 
       

Exemple de sortie écran :

 ------------------------------------ 
 A is subspace of R3              

 Find a transformation matrix for    
 a projection onto R3  :          

 Proj(x) =  A * inv(AT*A) * AT * x 

 A :
 -2.0 -3.0i  -4.0 -2.0i 
 +1.0 +0.0i  -3.0 -5.0i 
 +0.0 +1.0i  -6.0 -4.0i 

 Press return to continue. 


 ------------------------------------ 
 AT :
 -2.0 +3.0i  +1.0 -0.0i  +0.0 -1.0i 
 -4.0 +2.0i  -3.0 +5.0i  -6.0 +4.0i 

 ATA :
+15.0 +0.0i  +7.0 -7.0i 
 +7.0 +7.0i +106.0 +0.0i 

 inv(AT*A) :
+0.0710+0.0000i -0.0047+0.0047i 
-0.0047-0.0047i +0.0101+0.0000i 

 inv(AT*A)*AT :
-0.1327+0.1850i +0.0617-0.0375i +0.0094-0.1180i 
-0.0168+0.0154i -0.0349+0.0456i -0.0650+0.0449i 

 V = A*inv(AT*A)*AT :
+0.9182+0.0000i -0.0054-0.2225i -0.0228+0.1582i 
-0.0054+0.2225i +0.3941+0.0000i +0.4290+0.0724i 
-0.0228-0.1582i +0.4290-0.0724i +0.6877-0.0000i 

 Press return to continue. 


 ------------------------------------ 
 V is transformation matrix for      
 a projection onto a subspace R3 :


+0.9182+0.0000i -0.0054-0.2225i -0.0228+0.1582i 
-0.0054+0.2225i +0.3941+0.0000i +0.4290+0.0724i 
-0.0228-0.1582i +0.4290-0.0724i +0.6877-0.0000i 

 X :
 -1.0 -3.0i 
 +2.0 -4.0i 
 -3.0 -5.0i 

 Proj(x) =  A * inv(AT*A) * AT * x 

 Proj(x) =  V                  * x :
-0.9598-3.5389i 
+0.5362-4.1448i 
-1.9464-5.0724i 

 Press return to continue.