Mathc matrices/c21u
Application
Installer et compiler ces fichiers dans votre répertoire de travail.
/* ------------------------------------ */
/* Save as : c00b.c */
int main(void)
double ab[R4*C7]={
 +1, 3, -2, 0, +2, +0, 0,
 0, 0, -5, -3, +4, -3, 0,
 0, 0, +10, 6, -8, +6, 0, 
 1, 0, 0, 0, 4, 8, 0, 
double **Ab = ca_A_mR(ab,i_Abr_Ac_bc_mR(R4,C6,C1));
double **A = c_Ab_A_mR(Ab,i_mR(R4,C6));
double **b = c_Ab_b_mR(Ab,i_mR(R4,C1));
double **Ab_free = i_Abr_Ac_bc_mR(csize_A_R(Ab),csize_A_R(Ab),C1+C3);
double **b_free = i_mR(rsize_R(Ab_free),csize_b_R(Ab_free)); 
double **A_bfree = i_mR(rsize_R(A),csize_R(b_free)) ;
int r;
 clrscrn();
 printf("Find a basis for the orthogonal complement of A :\n\n");
 printf(" A :");
 p_mR(A,S6,P1,C10);
 printf(" b :");
 p_mR(b,S6,P1,C10);
 printf(" Ab :");
 p_mR(Ab,S6,P1,C10);
 stop();
 clrscrn();
 printf(" Ab : gj_PP_mR(Ab,NO) :");
 gj_PP_mR(Ab,NO);
 p_mR(Ab,S7,P3,C10);
 put_var_mR(Ab,Ab_free); 
// printf(" Ab_free : put_var_mR(Ab,Ab_free);"); 
// p_mR(Ab_free,S7,P3,C10); 
 put_freeV_mR(Ab_free);
// printf(" Ab_free : put_freeV_mR(Ab_free);"); 
// p_mR(Ab_free,S7,P3,C10); 
 stop();
 clrscrn(); 
 r = rsize_R(Ab_free);
 while(r>R1) 
 zero_below_pivot_gj1Ab_mR(Ab_free,r--);
 printf(" Ab_free : zero_below_pivot_gj1Ab_mR(Ab_free,r--);"); 
 p_mR(Ab_free,S7,P3,C10); 
 c_Ab_b_mR(Ab_free,b_free);
// printf(" b_free :"); 
// p_mR(b_free,S10,P3,C7);
 printf(" b_free : free variables");
 p_freeV(b_free,S6,P3);
 stop(); 
 clrscrn();
 printf(" A :");
 p_mR(A,S10,P3,C10);
 printf(" b_free :"); 
 p_mR(b_free,S10,P3,C7);
 printf(" A * bfree :"); 
 p_mR(mul_mR(A,b_free,A_bfree),S10,P3,C7);
 stop();
 f_mR(Ab);
 f_mR(Ab_free);
 f_mR(b_free);
 f_mR(b);
 f_mR(A);
 f_mR(A_bfree);
 return 0;
/* ------------------------------------ */
On commence par calculer les variables libres. 
Les colonnes de b_free sont une base pour le complément orthogonal de A. 
 A * b_free = 0
Cela prouve que les vecteurs lignes de A sont orthogonaux aux vecteurs colonnes de b_free.
 Exemple de sortie écran :
Find a basis for the orthogonal complement of A :
 A :
 +1.0 +3.0 -2.0 +0.0 +2.0 +0.0 
 +0.0 +0.0 -5.0 -3.0 +4.0 -3.0 
 +0.0 +0.0 +10.0 +6.0 -8.0 +6.0 
 +1.0 +0.0 +0.0 +0.0 +4.0 +8.0 
 b :
 +0.0 
 +0.0 
 +0.0 
 +0.0 
 Ab :
 +1.0 +3.0 -2.0 +0.0 +2.0 +0.0 +0.0 
 +0.0 +0.0 -5.0 -3.0 +4.0 -3.0 +0.0 
 +0.0 +0.0 +10.0 +6.0 -8.0 +6.0 +0.0 
 +1.0 +0.0 +0.0 +0.0 +4.0 +8.0 +0.0 
 Press return to continue. 
 Ab : gj_PP_mR(Ab,NO) :
 +1.000 +3.000 -2.000 +0.000 +2.000 +0.000 +0.000 
 -0.000 +1.000 -0.667 -0.000 -0.667 -2.667 -0.000 
 +0.000 +0.000 +1.000 +0.600 -0.800 +0.600 +0.000 
 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 
 Press return to continue. 
 ------------------------------------ 
 Ab_free : zero_below_pivot_gj1Ab_mR(Ab_free,r--);
 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 -4.000 -8.000 
 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 -0.400 +1.200 +2.267 
 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 -0.600 +0.800 -0.600 
 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 
 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +1.000 +0.000 
 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +1.000 
 b_free : free variables
 x1 = +0.000 +0.000*s -4.000*t -8.000*u
 x2 = +0.000 -0.400*s +1.200*t +2.267*u
 x3 = +0.000 -0.600*s +0.800*t -0.600*u
 x4 = +0.000 +1.000*s +0.000*t +0.000*u
 x5 = +0.000 +0.000*s +1.000*t +0.000*u
 x6 = +0.000 +0.000*s +0.000*t +1.000*u
 Press return to continue. 
 ------------------------------------ 
 A :
 +1.000 +3.000 -2.000 +0.000 +2.000 +0.000 
 +0.000 +0.000 -5.000 -3.000 +4.000 -3.000 
 +0.000 +0.000 +10.000 +6.000 -8.000 +6.000 
 +1.000 +0.000 +0.000 +0.000 +4.000 +8.000 
 b_free :
 +0.000 +0.000 -4.000 -8.000 
 +0.000 -0.400 +1.200 +2.267 
 +0.000 -0.600 +0.800 -0.600 
 +0.000 +1.000 +0.000 +0.000 
 +0.000 +0.000 +1.000 +0.000 
 +0.000 +0.000 +0.000 +1.000 
 A * bfree :
 +0.000 +0.000 +0.000 +0.000 
 +0.000 +0.000 +0.000 +0.000 
 +0.000 -0.000 +0.000 -0.000 
 +0.000 +0.000 +0.000 +0.000 
 Press return to continue.

Mathc matrices/c21v
Application 
Installer et compiler ces fichiers dans votre répertoire de travail.
/* ------------------------------------ */
/* Save as : c00c.c */
int main(void)
double ab[R4*C6]={
 +1, 4, 5 , 6, 9, 0,
 +3, -2, 1, 4, -1, 0,
 -1, 0, -1, -2, -1, 0, 
 2, 3, 5, 7, 8, 0, 
double **Ab = ca_A_mR(ab,i_Abr_Ac_bc_mR(R4,C5,C1));
double **A = c_Ab_A_mR(Ab,i_mR(R4,C5));
double **b = c_Ab_b_mR(Ab,i_mR(R4,C1));
double **Ab_free = i_Abr_Ac_bc_mR(csize_A_R(Ab),csize_A_R(Ab),C1+C3);
double **b_free = i_mR(rsize_R(Ab_free),csize_b_R(Ab_free)); 
double **A_bfree = i_mR(rsize_R(A),csize_R(b_free)) ;
int r;
 clrscrn();
 printf("Find a basis for the orthogonal complement of A :\n\n");
 printf(" A :");
 p_mR(A,S6,P1,C10);
 printf(" b :");
 p_mR(b,S6,P1,C10);
 printf(" Ab :");
 p_mR(Ab,S6,P1,C10);
 stop();
 clrscrn();
 printf(" Ab : gj_PP_mR(Ab,NO) :");
 gj_PP_mR(Ab,NO);
 p_mR(Ab,S7,P3,C10);
 put_var_mR(Ab,Ab_free); 
// printf(" Ab_free : put_var_mR(Ab,Ab_free);"); 
// p_mR(Ab_free,S7,P3,C10); 
 put_freeV_mR(Ab_free);
// printf(" Ab_free : put_freeV_mR(Ab_free);"); 
// p_mR(Ab_free,S7,P3,C10); 
 stop();
 clrscrn(); 
 r = rsize_R(Ab_free);
 while(r>R1) 
 zero_below_pivot_gj1Ab_mR(Ab_free,r--);
 printf(" Ab_free : zero_below_pivot_gj1Ab_mR(Ab_free,r--);"); 
 p_mR(Ab_free,S7,P3,C10); 
 c_Ab_b_mR(Ab_free,b_free);
// printf(" b_free :"); 
// p_mR(b_free,S10,P3,C7);
 printf(" b_free : free variables");
 p_freeV(b_free,S6,P3);
 stop(); 
 clrscrn();
 printf(" A :");
 p_mR(A,S10,P3,C10);
 printf(" b_free :"); 
 p_mR(b_free,S10,P3,C7);
 printf(" A * bfree :"); 
 p_mR(mul_mR(A,b_free,A_bfree),S10,P3,C7);
 stop();
 f_mR(Ab);
 f_mR(Ab_free);
 f_mR(b_free);
 f_mR(b);
 f_mR(A);
 f_mR(A_bfree);
 return 0;
/* ------------------------------------ */
On commence par calculer les variables libres. 
Les colonnes de b_free sont une base pour le complément orthogonal de A. 
 A * b_free = 0
Cela prouve que les vecteurs lignes de A sont orthogonaux aux vecteurs colonnes de b_free.
 Exemple de sortie écran :
Find a basis for the orthogonal complement of A :
 A :
 +1.0 +4.0 +5.0 +6.0 +9.0 
 +3.0 -2.0 +1.0 +4.0 -1.0 
 -1.0 +0.0 -1.0 -2.0 -1.0 
 +2.0 +3.0 +5.0 +7.0 +8.0 
 b :
 +0.0 
 +0.0 
 +0.0 
 +0.0 
 Ab :
 +1.0 +4.0 +5.0 +6.0 +9.0 +0.0 
 +3.0 -2.0 +1.0 +4.0 -1.0 +0.0 
 -1.0 +0.0 -1.0 -2.0 -1.0 +0.0 
 +2.0 +3.0 +5.0 +7.0 +8.0 +0.0 
 Press return to continue. 
 ------------------------------------ 
 Ab : gj_PP_mR(Ab,NO) :
 +1.000 -0.667 +0.333 +1.333 -0.333 +0.000 
 +0.000 +1.000 +1.000 +1.000 +2.000 +0.000 
 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 
 +0.000 +0.000 -0.000 -0.000 +0.000 +0.000 
 Press return to continue. 
 ------------------------------------ 
 Ab_free : zero_below_pivot_gj1Ab_mR(Ab_free,r--);
 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 -1.000 -2.000 -1.000 
 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 -1.000 -1.000 -2.000 
 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 
 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +1.000 +0.000 
 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +1.000 
 b_free : free variables
 x1 = +0.000 -1.000*t -2.000*u -1.000*v
 x2 = +0.000 -1.000*t -1.000*u -2.000*v
 x3 = +0.000 +1.000*t +0.000*u +0.000*v
 x4 = +0.000 +0.000*t +1.000*u +0.000*v
 x5 = +0.000 +0.000*t +0.000*u +1.000*v
 Press return to continue. 
 ------------------------------------ 
 A :
 +1.000 +4.000 +5.000 +6.000 +9.000 
 +3.000 -2.000 +1.000 +4.000 -1.000 
 -1.000 +0.000 -1.000 -2.000 -1.000 
 +2.000 +3.000 +5.000 +7.000 +8.000 
 b_free :
 +0.000 -1.000 -2.000 -1.000 
 +0.000 -1.000 -1.000 -2.000 
 +0.000 +1.000 +0.000 +0.000 
 +0.000 +0.000 +1.000 +0.000 
 +0.000 +0.000 +0.000 +1.000 
 A * bfree :
 +0.000 +0.000 +0.000 +0.000 
 +0.000 +0.000 +0.000 +0.000 
 +0.000 +0.000 +0.000 +0.000 
 +0.000 +0.000 +0.000 +0.000 
 Press return to continue.