Mathc matrices/c22v
Application ou QR décomposition
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c00e.c |
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/* ------------------------------------ */
/* Save as : c00e.c */
/* ------------------------------------ */
#include "v_a.h"
/* ------------------------------------ */
/* ------------------------------------ */
#define RA R5
#define CA C3
#define Cb C1
/* ------------------------------------ */
/* ------------------------------------ */
void fun(void)
{
double ab[RA*(CA+Cb)]={
/* x**0 x**1 x**2 y */
1, .1, .01, -0.19,
1, .2, .04, 0.32,
1, .3, .09, 1.04,
1, .4, .16, 2.47,
1, .5, .25, 3.74,
};
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 **Q = i_mR(RA,CA);
double **R = i_mR(CA,CA);
double **invR = i_mR(CA,CA);
double **Q_T = i_mR(CA,RA);
double **invR_Q_T = i_mR(CA,RA);
double **x = i_mR(CA,Cb); // x invR * Q_T * b
clrscrn();
printf(" Fitting a Quadratic Curve to Data :\n\n");
printf(" A :");
p_mR(A,S5,P2,C7);
printf(" b :");
p_mR(b,S5,P2,C7);
printf(" Ab :");
p_mR(Ab,S5,P2,C7);
stop();
clrscrn();
QR_mR(A,Q,R);
printf(" Q :");
p_mR(Q,S3,P4,C6);
printf(" R :");
p_mR(R,S10,P4,C6);
stop();
clrscrn();
transpose_mR(Q,Q_T);
printf(" Q_T :");
pE_mR(Q_T,S3,P4,C6);
inv_mR(R,invR);
printf(" invR :");
pE_mR(invR,S10,P4,C6);
stop();
clrscrn();
printf(" Solving this system yields a unique\n"
" least squares solution, namely \n\n");
mul_mR(invR,Q_T,invR_Q_T);
mul_mR(invR_Q_T,b,x);
printf(" x = invR * Q_T * b :");
p_mR(x,S10,P2,C6);
printf(" The Quadratic Curve to Data : \n\n"
" s = %+.2f %+.2f*t %+.2f*t**2\n\n"
,x[R1][C1],x[R2][C1],x[R3][C1]);
stop();
f_mR(A);
f_mR(b);
f_mR(Ab);
f_mR(Q);
f_mR(Q_T);
f_mR(R);
f_mR(invR);
f_mR(invR_Q_T);
f_mR(x);
}
/* ------------------------------------ */
int main(void)
{
fun();
return 0;
}
/* ------------------------------------ */
/* ------------------------------------ */
Trouver la meilleur équation du second degré, qui s'ajuste au mieux aux points donnés
On utilise la QR décomposition. Dans l'exemple suivant on utilise un autre algorithme sur le même exemple. Exemple de sortie écran :
-----------------------------------
Fitting a Quadratic Curve to Data :
A :
+1.00 +0.10 +0.01
+1.00 +0.20 +0.04
+1.00 +0.30 +0.09
+1.00 +0.40 +0.16
+1.00 +0.50 +0.25
b :
-0.19
+0.32
+1.04
+2.47
+3.74
Ab :
+1.00 +0.10 +0.01 -0.19
+1.00 +0.20 +0.04 +0.32
+1.00 +0.30 +0.09 +1.04
+1.00 +0.40 +0.16 +2.47
+1.00 +0.50 +0.25 +3.74
Press return to continue.
-----------------------------------
Q :
+0.4472 -0.6325 +0.5345
+0.4472 -0.3162 -0.2673
+0.4472 +0.0000 -0.5345
+0.4472 +0.3162 -0.2673
+0.4472 +0.6325 +0.5345
R :
+2.2361 +0.6708 +0.2460
+0.0000 +0.3162 +0.1897
-0.0000 -0.0000 +0.0374
Press return to continue.
-----------------------------------
Q_T :
+4.4721e-01 +4.4721e-01 +4.4721e-01 +4.4721e-01 +4.4721e-01
-6.3246e-01 -3.1623e-01 +0.0000e+00 +3.1623e-01 +6.3246e-01
+5.3452e-01 -2.6726e-01 -5.3452e-01 -2.6726e-01 +5.3452e-01
invR :
+4.4721e-01 -9.4868e-01 +1.8708e+00
+0.0000e+00 +3.1623e+00 -1.6036e+01
-0.0000e+00 -0.0000e+00 +2.6726e+01
Press return to continue.
-----------------------------------
Solving this system yields a unique
least squares solution, namely
x = invR * Q_T * b :
-0.41
+0.45
+15.93
The Quadratic Curve to Data :
s = -0.41 +0.45*t +15.93*t**2