Exemple de sortie écran :
 Cosine of the Angle Between Vectors U and V :
 u :
 -7 
 -6 
 +8 
 v :
 -5 
 -4 
 +7 
 <u,v> 
 cos(alpha) = ----------- = +0.9931 
 ||u|| ||v|| 
 Press return to continue. 
 --------------------------------
 Q :
+0.4819 +0.4615 +0.7448 
+0.6196 -0.7806 +0.0828 
+0.6196 +0.4216 -0.6621 
 Press return to continue. 
 --------------------------------
 Cosine of the Angle Between Vectors U and V :
 Qu :
-0.1839 
+1.0088 
-12.1636 
 Qv :
+0.9581 
+0.6040 
-9.4191 
 <Qu,Qv> 
 cos(alpha) = ----------- = +0.9931 
 ||u|| ||v|| 
 Press return to continue. 
 --------------------------------
 Cosine of the Angle Between Vectors U and V :
 <u,v> 
 cos(alpha) = ----------- = +0.9931 
 ||u|| ||v|| 
 <Qu,Qv> 
 cos(alpha) = ----------- = +0.9931 
 ||u|| ||v||

Mathc matrices/c25l
Application
Installer et compiler ces fichiers dans votre répertoire de travail.
/* Save as : c06a.c */
/* ------------------------------------ */
/* ------------------------------------ */
void fun()
double **Q = r_Q_mR(i_mR(RCA,RCA), 9);
double **Qu = i_mR(RCA,C1);
double **Qv = i_mR(RCA,C1);
double **u = r_mR(i_mR(RCA,C1),9.);
double **v = r_mR(i_mR(RCA,C1),9.);
 clrscrn(); 
 printf(" Q : an orthogonal matrix ");
 p_mR(Q,S3,P4,C6);
 printf(" u :");
 p_mR(u,S3,P1,C6);
 printf(" v :");
 p_mR(v,S3,P1,C6);
 printf(" Q u . Q v\t = u . v \n\n ");
 printf(" %.4f\t = %.4f \n",
 dot_uv_R(mul_mR(Q,u,Qu), mul_mR(Q,v,Qv)), 
 dot_uv_R(u,v));
 f_mR(u);
 f_mR(v); 
 f_mR(Qu);
 f_mR(Qv);
 f_mR(Q);
/* ------------------------------------ */
int main(void)
time_t t;
 srand(time(&t));
do
 fun();
} while(stop_w());
 return 0;
/* ------------------------------------ */
 Si Q est une matrice orthonormale alors : Qu.Qv = u.v :
 Pour changer la taille des vecteurs il suffit de modifier la valeur de :

Exemple de sortie écran :
 Q : an orthogonal matrix 
-0.0937 +0.4562 +0.8849 
-0.7493 +0.5530 -0.3644 
+0.6556 +0.6972 -0.2900 
 u :
-7.0 
-8.0 
+6.0 
 v :
+6.0 
-1.0 
-1.0 
 Q u . Q v = u . v 
 -40.0000 = -40.0000 
 Press return to continue
 Press X to stop

Mathc matrices/c25m
Application
Installer et compiler ces fichiers dans votre répertoire de travail.
/* Save as : c06b.c */
/* ------------------------------------ */
/* ------------------------------------ */
void fun()

double tu[RCA*C1]={
 -7, 
 -6, 
 +8 
double tv[RCA*C1]={
 -5, 
 -4, 
 +7 
double tQ[RCA*RCA]={
+0.301511344578, +0.275240941282, -0.912870929175, 
+0.301511344578, -0.935819200357, -0.182574185835, 
-0.904534033733, -0.220192753025, -0.365148371670,
double **u = ca_A_mR(tu,i_mR(RCA,C1));
double **v = ca_A_mR(tv,i_mR(RCA,C1));
double **Q = ca_A_mR(tQ,i_mR(RCA,RCA));
double **Qu = i_mR(RCA,C1);
double **Qv = i_mR(RCA,C1);
 clrscrn();
 printf(" Q : an orthogonal matrix ");
 p_mR(Q,S3,P5,C6);
 printf(" u :");
 p_mR(u,S3,P1,C6);
 printf(" v :");
 p_mR(v,S3,P1,C6);
 printf(" Q u . Q v\t = u . v \n\n");
 printf(" %.4f\t = %.4f \n\n",
 dot_uv_R(mul_mR(Q,u,Qu), mul_mR(Q,v,Qv)), 
 dot_uv_R(u,v));
 stop();
 f_mR(u);
 f_mR(v); 
 f_mR(Qu);
 f_mR(Qv);
 f_mR(Q);
/* ------------------------------------ */
int main(void)
 fun();
 return 0;
/* ------------------------------------ */
 Si Q est une matrice orthonormale alors : Qu.Qv = u.v :
 Pour changer la taille des vecteurs il suffit de modifier la valeur de :
 et de rajouter éventuellement de nouvelles lignes et colonnes.

Exemple de sortie écran :
 Q : an orthogonal matrix 
+0.30151 +0.27524 -0.91287 
+0.30151 -0.93582 -0.18257 
-0.90453 -0.22019 -0.36515 
 u :
-7.0 
-6.0 
+8.0 
 v :
-5.0 
-4.0 
+7.0 
 Q u . Q v = u . v 
 115.0000 = 115.0000 
 Press return to continue.

Mathc matrices/c25n
Application ou Pseudo Inverse 
Installer et compiler ces fichiers dans votre répertoire de travail.
/* Save as : c01h.c */
/* ------------------------------------ */
void fun(void)
 double xy[6] ={
 1, -2,
 2, -3,
 3, 6 };
double tA[RA*CA]={
/* x**2 y**2 x y e */
 +1.00, +0.00, +0.00, +0.00, +0.00, 
 +0.00, +1.00, +0.00, +0.00, +0.00, 
 +1.00, +4.00, +1.00, -2.00, +1.00, 
 +4.00, +9.00, +2.00, -3.00, +1.00, 
 +9.00, +36.00, +3.00, +6.00, +1.00 
double tb[RA*C1]={
/* = 0 */
 +1.00, 
 +1.00, 
 +0.00, 
 +0.00, 
 +0.00
double **XY = ca_A_mR(xy,i_mR(R3,C2));
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("\n");
 printf(" Find the coefficients a, b, c, d, of a circle \n\n");
 printf(" ax**2 + ay**2 + bx + cy + d = 0 \n\n");
 printf(" that passes through these three XY. \n\n");
 printf(" x y");
 p_mR(XY,S5,P0,C6);
 stop();
 clrscrn(); 
 printf(" Using the given XY, we obtain this matrix.\n");
 printf(" (a = 1. This is my choice)\n\n");
 printf(" A :");
 p_mR(A,S10,P2,C7);
 printf(" b :");
 p_mR(b,S10,P2,C7);
 printf(" Pinv = V * invS_T * U_T ");
 pseudo_Rn_uv_mR(A,Pinv,FACTOR_E); 
 pE_mR(Pinv,S12,P4,C10); 
 stop();
 clrscrn(); 
 printf(" Pinv = V * invS_T * U_T "); 
 p_mR(Pinv,S10,P4,C10); 
 printf(" x = Pinv * b "); 
 mul_mR(Pinv,b,Pinvb); 
 p_mR(Pinvb,S10,P4,C10);
 printf(" The coefficients a, b, c, d, e, of the curve are : \n\n"
 " %+.2fx**2 %+.2fy**2 %+.2fx %+.2fy %+.2f = 0\n\n"
 ,Pinvb[R1][C1],Pinvb[R2][C1],Pinvb[R3][C1],
 Pinvb[R4][C1],Pinvb[R5][C1]); 
 stop(); 
 f_mR(XY); 
 f_mR(A);
 f_mR(b); 
 f_mR(Pinv);
 f_mR(Pinvb); 
/* ------------------------------------ */
int main(void)

Exemple de sortie écran :
 Find the coefficients a, b, c, d, of a circle 
 ax**2 + ay**2 + bx + cy + d = 0 
 that passes through these three XY. 
 x y
 +1 -2 
 +2 -3 
 +3 +6 
 Press return to continue. 
 -----------------------------------
 Using the given XY, we obtain this matrix.
 (a = 1. This is my choice)
 A :
 +1.00 +0.00 +0.00 +0.00 +0.00 
 +0.00 +1.00 +0.00 +0.00 +0.00 
 +1.00 +4.00 +1.00 -2.00 +1.00 
 +4.00 +9.00 +2.00 -3.00 +1.00 
 +9.00 +36.00 +3.00 +6.00 +1.00 
 b :
 +1.00 
 +1.00 
 +0.00 
 +0.00 
 +0.00 
 Press return to continue. 
 -----------------------------------
 Pinv = V * invS_T * U_T 
 +1.0000 +0.0000 +0.0000 -0.0000 -0.0000 
 -0.0000 +1.0000 -0.0000 +0.0000 +0.0000 
 -3.2000 -7.2000 -0.9000 +0.8000 +0.1000 
 -0.2000 -2.2000 +0.1000 -0.2000 +0.1000 
 +1.8000 -1.2000 +2.1000 -1.2000 +0.1000 
 x = Pinv * b 
 +1.0000 
 +1.0000 
 -10.4000 
 -2.4000 
 +0.6000 
 The coefficients a, b, c, d, e, of the curve are : 
 +1.00x**2 +1.00y**2 -10.40x -2.40y +0.60 = 0
 Press return to continue.