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 
 Using the given XY, we obtain this matrix.
 (a = 1. This is my choice)
 x**2 y**2 x y 
 +1.00 +0.00 +0.00 +0.00 +0.00 +1.00 
 +0.00 +1.00 +0.00 +0.00 +0.00 +1.00 
 +1.00 +4.00 +1.00 -2.00 +1.00 +0.00 
 +4.00 +9.00 +2.00 -3.00 +1.00 +0.00 
 +9.00 +36.00 +3.00 +6.00 +1.00 +0.00 
 Press return to continue. 
 -----------------------------------
 Q :
 +0.1005 -0.4928 -0.5780 -0.3248 +0.5544 
 +0.0000 +0.1340 +0.0472 -0.9182 -0.3696 
 +0.1005 +0.0433 +0.7229 -0.2169 +0.6468 
 +0.4020 -0.7650 +0.3370 +0.0544 -0.3696 
 +0.9045 +0.3899 -0.1659 +0.0360 +0.0308 
 R :
 +9.9499 +36.5834 +3.6181 +4.0202 +1.4071 
 -0.0000 +7.4603 -0.3168 +4.5480 -0.3317 
 -0.0000 -0.0000 +0.8993 -3.4522 +0.8940 
 +0.0000 +0.0000 -0.0000 +0.4863 -0.1264 
 +0.0000 +0.0000 +0.0000 +0.0000 +0.3080 
 Press return to continue. 
 -----------------------------------
 Q_T :
 +1.0050e-01 +0.0000e+00 +1.0050e-01 +4.0202e-01 +9.0453e-01 
 -4.9284e-01 +1.3404e-01 +4.3327e-02 -7.6499e-01 +3.8994e-01 
 -5.7800e-01 +4.7225e-02 +7.2290e-01 +3.3703e-01 -1.6589e-01 
 -3.2485e-01 -9.1825e-01 -2.1687e-01 +5.4449e-02 +3.5992e-02 
 +5.5444e-01 -3.6962e-01 +6.4684e-01 -3.6962e-01 +3.0802e-02 
 invR :
 +1.0050e-01 -4.9284e-01 -5.7800e-01 -3.2485e-01 +5.5444e-01 
 +0.0000e+00 +1.3404e-01 +4.7225e-02 -9.1825e-01 -3.6962e-01 
 -0.0000e+00 +0.0000e+00 +1.1120e+00 +7.8932e+00 +1.2321e-02 
 -0.0000e+00 +0.0000e+00 -0.0000e+00 +2.0561e+00 +8.4398e-01 
 -0.0000e+00 +0.0000e+00 -0.0000e+00 -0.0000e+00 +3.2465e+00 
 Press return to continue. 
 -----------------------------------
 Solving this system yields a unique
 least squares solution, namely 
 x = invR * Q_T * b :
 +1.00 
 +1.00 
 -10.40 
 -2.40 
 +0.60 
 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.

Mathc matrices/c23q
Application ou Résoudre avec ...
Installer et compiler ces fichiers dans votre répertoire de travail.
/* Save as : c00s.c */
/* ------------------------------------ */
/* ------------------------------------ */
void fun(void)
double xy[6] ={
 1, -2,
 2, -3,
 3, 6 };
double ab[RA*(CA+Cb)]={
/* x**2 y**2 x y e = 0 */
 +1.00, +0.00, +0.00, +0.00, +0.00, +1.00, 
 +0.00, +1.00, +0.00, +0.00, +0.00, +1.00, 
 +1.00, +4.00, +1.00, -2.00, +1.00, +0.00, 
 +4.00, +9.00, +2.00, -3.00, +1.00, +0.00, 
 +9.00, +36.00, +3.00, +6.00, +1.00, +0.00, 
double **XY = ca_A_mR(xy,i_mR(R3,C2));
double **Ab = ca_A_mR(ab,i_Abr_Ac_bc_mR(RA,CA,Cb));
double **A = c_Ab_A_mR(Ab,i_mR(RA,CA));
double **b = c_Ab_b_mR(Ab,i_mR(RA,Cb));
double **A_T = i_mR(CA,RA);
double **A_TA = i_mR(CA,CA); // A_T*A
double **invA_TA = i_mR(CA,CA); // inv(A_T*A)
double **invA_TAA_T = i_mR(CA,RA); // inv(A_T*A)*A_T
double **x = i_mR(CA,Cb); // x = inv(A_T*A)*A_T*b
 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);
 printf("\n");
 printf(" Using the given XY, we obtain this matrix.\n");
 printf(" (a = 1. This is my choice)\n\n");
 printf(" x**2 y**2 x y ");
 p_mR(Ab,S7,P2,C6);
 stop();
 clrscrn();
 printf(" A_T :");
 p_mR(transpose_mR(A,A_T),S10,P2,C7);
 printf(" A_TA :");
 p_mR(mul_mR(A_T,A,A_TA),S10,P2,C7);
 stop();
 clrscrn();
 printf(" inv(A_TA) :");
 p_mR(inv_mR(A_TA,invA_TA),S10,P4,C7); 
 printf(" inv(A_TA)*A_T :");
 p_mR(mul_mR(invA_TA,A_T,invA_TAA_T),S10,P4,C7);
 printf("\n x = inv(A_TA)*A_T*b :");
 p_mR(mul_mR(invA_TAA_T,b,x),S10,P4,C7);
 stop();
 clrscrn();
 printf("\n x = inv(A_TA)*A_T*b :");
 p_mR(x,S10,P2,C7); 
 printf(" The coefficients a, b, c, d, e, of the curve are : \n\n"
 " %+.2fx**2 %+.2fy**2 %+.2fx %+.2fy %+.2f = 0\n\n"
 ,x[R1][C1],x[R2][C1],x[R3][C1],x[R4][C1],x[R5][C1]); 
 stop(); 
 f_mR(XY); 
 f_mR(A);
 f_mR(b);
 f_mR(Ab);
 f_mR(A_T);
 f_mR(A_TA); // A_T*A
 f_mR(invA_TA); // inv(A_T*A)
 f_mR(invA_TAA_T); // inv(A_T*A)*A_T
 f_mR(x); 
/* ------------------------------------ */
int main(void)

fun();
 return 0;
/* ------------------------------------ */
 Calculons les coefficients a, b, c, d d'un cercle,
 ax**2 + ay**2 + bx + cy + d = 0 
 Qui passe par ces trois points. 
 (x[1],y[1]) (x[2],y[2]) (x[3],y[3]) 
 En utilisant ces trois points nous avons cette matrice.
 (a)x**2 (a)y**2 (b)x (c)y (d) = 0 
 x[1]**2 y[1]**2 x[1] y[1] 1 0
 x[2]**2 y[2]**2 x[2] y[2] 1 0
 x[3]**2 y[3]**2 x[3] y[3] 1 0
 Ce système a trois lignes et quatre inconnues.
 Il est homogène, donc il a une infinité de solution.
 Pour trouver une solution j'ai choisi que a = 1.
 Nous obtenons cette matrice.
 (a)x**2 (a)y**2
 1 0 0 0 0 1 
 0 1 0 0 0 1 
 x[1]**2 y[1]**2 x[1] y[1] 1 0 
 x[2]**2 y[2]**2 x[2] y[2] 1 0 
 x[3]**2 y[3]**2 x[3] y[3] 1 0 
 Il suffit de resoudre le système.

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 
 Using the given XY, we obtain this matrix.
 (a = 1. This is my choice)
 x**2 y**2 x y 
 +1.00 +0.00 +0.00 +0.00 +0.00 +1.00 
 +0.00 +1.00 +0.00 +0.00 +0.00 +1.00 
 +1.00 +4.00 +1.00 -2.00 +1.00 +0.00 
 +4.00 +9.00 +2.00 -3.00 +1.00 +0.00 
 +9.00 +36.00 +3.00 +6.00 +1.00 +0.00 
 Press return to continue. 
 -----------------------------------
 A_T :
 +1.00 +0.00 +1.00 +4.00 +9.00 
 +0.00 +1.00 +4.00 +9.00 +36.00 
 +0.00 +0.00 +1.00 +2.00 +3.00 
 +0.00 +0.00 -2.00 -3.00 +6.00 
 +0.00 +0.00 +1.00 +1.00 +1.00 
 A_TA :
 +99.00 +364.00 +36.00 +40.00 +14.00 
 +364.00 +1394.00 +130.00 +181.00 +49.00 
 +36.00 +130.00 +14.00 +10.00 +6.00 
 +40.00 +181.00 +10.00 +49.00 +1.00 
 +14.00 +49.00 +6.00 +1.00 +3.00 
 Press return to continue. 
 -----------------------------------
 inv(A_TA) :
 +1.0000 -0.0000 -3.2000 -0.2000 +1.8000 
 -0.0000 +1.0000 -7.2000 -2.2000 -1.2000 
 -3.2000 -7.2000 +63.5400 +16.2400 +0.0400 
 -0.2000 -2.2000 +16.2400 +4.9400 +2.7400 
 +1.8000 -1.2000 +0.0400 +2.7400 +10.5400 
 inv(A_TA)*A_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 = inv(A_TA)*A_T*b :
 +1.0000 
 +1.0000 
 -10.4000 
 -2.4000 
 +0.6000 
 Press return to continue. 
 -----------------------------------
 x = inv(A_TA)*A_T*b :
 +1.00 
 +1.00 
 -10.40 
 -2.40 
 +0.60 
 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.