Mathc matrices/c23o
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c00p.c |
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/* ------------------------------------ */
/* Save as : c00p.c */
/* ------------------------------------ */
#include "v_a.h"
/* ------------------------------------ */
/* ------------------------------------ */
#define RA R5
#define CA C5
#define Cb C1
/* ------------------------------------ */
/* ------------------------------------ */
void fun(void)
{
double xy[8] ={
1, 1,
2, 4,
3, 9,
4, 16 };
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,
+1.00, +1.00, +1.00, +1.00, +1.00, +0.00,
+4.00, +16.00, +2.00, +4.00, +1.00, +0.00,
+9.00, +81.00, +3.00, +9.00, +1.00, +0.00,
+16.00, +256.00, +4.00, +16.00, +1.00, +0.00
};
double **XY = ca_A_mR(xy,i_mR(R4,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, e, of the curve \n\n");
printf(" ax**2 + by**2 + cx + dy + e = 0 \n\n");
printf(" that passes through these four points.\n\n");
printf(" x y");
p_mR(XY,S5,P0,C6);
printf(" Using the given points, we obtain this matrix.\n");
printf(" (a = 1. This is my choice)\n\n");
printf(" Ab :\n");
printf(" x**2 y**2 x y e = 0 ");
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 = 0\n\n"
,x[R1][C1],x[R4][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;
}
/* ------------------------------------ */
/* ------------------------------------ */
Trouver les coefficients a, b, c, d, e du conique,
ax**2 + by**2 + cx + dy + e = 0
qui passe par ces quatre points.
(x[1],y[1]) (x[2],y[2]) (x[3],y[3]) (x[4],y[4])
En utilisant les quatre points nous obtenons la matrice.
(a)x**2 (b)y**2 (c)x (d)y (e) = 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
x[4]**2 y[4]**2 x[4] y[4] 1 0
Ce système à quatre lignes et cinq inconnus (a, b, c, d, e).
C'est un système homogène, il a donc une infinité de solution.
Pour trouver une solution j'ai choisi de poser que a = 1.
Nous avons donc cinq lignes et cinq inconnus.
1 0 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
x[4]**2 y[4]**2 x[4] y[4] 1 0
Il suffit maintenant de résoudre le système. Exemple de sortie écran :
-----------------------------------
Find the coefficients a, b, c, d, e, of the curve
ax**2 + by**2 + cx + dy + e = 0
that passes through these four points.
x y
+1 +1
+2 +4
+3 +9
+4 +16
Using the given points, we obtain this matrix.
(a = 1. This is my choice)
Ab :
x**2 y**2 x y e = 0
+1.00 +0.00 +0.00 +0.00 +0.00 +1.00
+1.00 +1.00 +1.00 +1.00 +1.00 +0.00
+4.00 +16.00 +2.00 +4.00 +1.00 +0.00
+9.00 +81.00 +3.00 +9.00 +1.00 +0.00
+16.00 +256.00 +4.00 +16.00 +1.00 +0.00
Press return to continue.
-----------------------------------
A_T :
+1.00 +1.00 +4.00 +9.00 +16.00
+0.00 +1.00 +16.00 +81.00 +256.00
+0.00 +1.00 +2.00 +3.00 +4.00
+0.00 +1.00 +4.00 +9.00 +16.00
+0.00 +1.00 +1.00 +1.00 +1.00
A_TA :
+355.00 +4890.00 +100.00 +354.00 +30.00
+4890.00 +72354.00 +1300.00 +4890.00 +354.00
+100.00 +1300.00 +30.00 +100.00 +10.00
+354.00 +4890.00 +100.00 +354.00 +30.00
+30.00 +354.00 +10.00 +30.00 +4.00
Press return to continue.
-----------------------------------
inv(A_TA) :
+1.0000 +0.0000 +0.0000 -1.0000 -0.0000
+0.0000 +0.0056 +0.6500 -0.2222 -0.4500
+0.0000 +0.6500 +82.5000 -27.2500 -59.4000
-1.0000 -0.2222 -27.2500 +10.1389 +19.2500
-0.0000 -0.4500 -59.4000 +19.2500 +44.2000
inv(A_TA)*A_T :
+1.0000 +0.0000 +0.0000 +0.0000 +0.0000
+0.0000 -0.0167 +0.0500 -0.0500 +0.0167
+0.0000 -3.5000 +7.0000 -4.5000 +1.0000
-1.0000 +0.9167 -2.2500 +1.7500 -0.4167
+0.0000 +3.6000 -4.8000 +2.8000 -0.6000
x = inv(A_TA)*A_T*b :
+1.0000
+0.0000
+0.0000
-1.0000
+0.0000
Press return to continue.
-----------------------------------
x = inv(A_TA)*A_T*b :
+1.00
+0.00
+0.00
-1.00
+0.00
The coefficients a, b, c, d, e, of the curve are :
+1.00x**2 -1.00y = 0
Press return to continue.