Mathc matrices/01p
Valeurs propres. Vecteurs propres
Installer et compiler ces fichiers dans votre répertoire de travail.
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c00a.c |
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/* ------------------------------------ */
/* Save as : c00a.c */
/* ------------------------------------ */
#include "v_a.h"
/* ------------------------------------
* Create the files a_main.plt *
# a_main.plt
reset
set zeroaxis lt 8
set grid
set view 90.,0., 1., 1.
splot [-20:20] [-20:20] [-20:20]\
"g_u1" with linespoints lt 16 lw 3,\
"g_u2" with linespoints lt 15 lw 3,\
"g_u3" with linespoints lt 14 lw 3
reset
------------------------------------ */
#define RCA RC3
#define UC 20
/* ------------------------------------ */
int main(void)
{
time_t t;
srand(time(&t));
double a[RCA*RCA]={
+75,+4,+8,
+4,+5,+2,
+8,+2,+7
};
double **A = ca_A_mR(a,i_mR(RCA,RCA));
double **EigsVector = i_mR(RCA,RCA);
double **T_EigsVector = i_mR(RCA,RCA);
double **EigsValue = i_mR(RCA,RCA);
double **T1 = i_mR(RCA,RCA);
double **U = r_mR(i_mR(R3,UC),5);
double **AU = i_mR(R3,UC);
double **TR = i_mR(R1,RCA);
int i;
clrscrn();
printf(" Copy/Past into the octave windows \n\n");
p_Octave_mR(A,"a",P0);
printf(" [V, E] = eigs (a,%d) \n\n",RCA);
printf(" EignVector : Octave Form with smul_c1_mR(-1;);");
eigs_V_mR(A,EigsVector);
smul_c1_mR(-1.,C1,EigsVector);
p_mR(EigsVector,S14,P9,C5);
printf(" EigsValue : T_EigsVector * A * EigsVector");
transpose_mR(EigsVector,T_EigsVector);
mul_mR(T_EigsVector,A,T1);
mul_mR(T1,EigsVector,EigsValue);
p_mR(EigsValue,S14,P9,C5);
stop();
clrscrn();
printf(" EigsValue * EigsVector");
smul_c1_mR(EigsValue[R1][C1],C1,EigsVector);
smul_c1_mR(EigsValue[R2][C2],C2,EigsVector);
smul_c1_mR(EigsValue[R3][C3],C3,EigsVector);
p_mR(EigsVector,S14,P9,C5);
printf(" A * U ");
mul_mR(A,U,AU);
p_mR(AU,S6,P1,C5);
printf(" Create the files \"a_main.plt\", \"g_u1\", "
"\"g_u2\", \"g_u3\"\n\n"
" ... load \"a_main.plt\" ... with gnuplot. \n\n"
" See Below \n\n");
stop();
clrscrn();
printf(" Create the file\"g_u1\" \n\n"
" g_u1 : EigsValue * EigsVector \n\n");
for(i = C1; i <= RCA; i++)
{
printf("+0.0 +0.0 +0.0");
c_c_r_mR(EigsVector,i,TR,R1);
p_mR(TR,S0,P9,C5);
printf("\n");
}
stop();
clrscrn();
printf(" Create the file\"g_u2\" \n\n"
" g_u2 : U \n\n");
for(i = C1; i <= UC; i++)
{
printf("+0.0 +0.0 +0.0");
c_c_r_mR(U,i,TR,R1);
p_mR(TR,S0,P1,C5);
printf("\n");
}
stop();
clrscrn();
printf(" Create the file\"g_u3\" \n\n"
" g_u3 : AU = A * U \n\n");
for(i = C1; i <= UC; i++)
{
printf("+0.0 +0.0 +0.0");
c_c_r_mR(AU,i,TR,R1);
p_mR(TR,S0,P1,C5);
printf("\n");
}
stop();
f_mR(A);
f_mR(EigsVector);
f_mR(T_EigsVector);
f_mR(EigsValue);
f_mR(T1);
f_mR(U);
f_mR(AU);
f_mR(TR);
return 0;
}
/* ------------------------------------ */
/* ------------------------------------ */
Je vous propose de dessiner 20 vecteurs aléatoires multipliés par la matrice A. De comparer ce résultat avec les vecteurs propres multipliés par leurs valeurs propres.
Exemple de sortie écran :
Copy/Past into the octave windows
a=[
+75,+4,+8;
+4,+5,+2;
+8,+2,+7]
[V, E] = eigs (a,3)
EignVector : Octave Form with smul_c1_mR(-1;);
-0.991453599 -0.129595628 +0.014991154
-0.058987383 +0.547809280 +0.834521109
-0.116362580 +0.826504668 -0.550771989
EigsValue : T_EigsVector * A * EigsVector
+76.176908505 +0.000000000 +0.000000000
+0.000000000 +7.071207751 -0.000000000
-0.000000000 -0.000000000 +3.751883745
Press return to continue.
EigsValue * EigsVector
-75.525870084 -0.916397606 +0.056245066
-4.493476470 +3.873673224 +3.131026184
-8.864141617 +5.844386215 -2.066432471
A * U
-336.0 -407.0 +182.0 -363.0 +87.0
-21.0 -20.0 +24.0 -29.0 -13.0
-65.0 -71.0 +41.0 -25.0 +26.0
-237.0 +249.0 +304.0 +107.0 +47.0
-3.0 +10.0 -3.0 +28.0 -7.0
-39.0 +48.0 +43.0 +30.0 -15.0
+379.0 +83.0 +122.0 +126.0 -162.0
+33.0 -10.0 +5.0 -6.0 -31.0
+39.0 +21.0 -10.0 -2.0 -19.0
-260.0 +304.0 +91.0 +47.0 +43.0
+10.0 -3.0 +0.0 +9.0 -12.0
-3.0 +43.0 +25.0 -21.0 -17.0
Create the files "a_main.plt", "g_u1", "g_u2", "g_u3"
... load "a_main.plt" ... with gnuplot.
See Below