Mathc matrices/03i
Fonctions matricielles ex : cos(A)
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"
/* ------------------------------------ */
#define RCA RC5
/* ------------------------------------ */
/* ------------------------------------ */
double f(
double x)
{
return(sinh(x)*sinh(x));
}
char feq[] = "sinh(x)*sinh(x)";
/* ------------------------------------ */
double g(
double x)
{
return(cosh(x)*cosh(x));
}
char geq[] = "cosh(x)*cosh(x)";
/* ------------------------------------ */
/* ------------------------------------ */
void fun(void)
{
double **A = rEsymmetric_mR( i_mR(RCA,RCA),999,+1.E-4);
double **sinhA_P2 = i_mR(RCA,RCA);
double **coshA_P2 = i_mR(RCA,RCA);
double **EVector = i_mR(RCA,RCA);
double **EVectorT = i_mR(RCA,RCA);
double **EValue = i_mR(RCA,RCA);
double **sinhA_P2_EValue = i_mR(RCA,RCA);
double **coshA_P2_EValue = i_mR(RCA,RCA);
double **T1 = i_mR(RCA,RCA);
clrscrn();
printf(" A:");
p_mR(A, S10,P4,C6);
stop();
clrscrn();
eigs_V_mR(A,EVector);
transpose_mR(EVector,EVectorT);
/* EValue = EVectorT A EVector */
mul_mR(EVectorT,A,T1);
mul_mR(T1,EVector,EValue);
printf(" sinh(A)**2");
f_eigs_mR(f,EValue,sinhA_P2_EValue);
/* sinhA_P2 = EVector sinhA_P2_EValue EVectorT */
mul_mR(EVector,sinhA_P2_EValue,T1);
mul_mR(T1,EVectorT,sinhA_P2);
pE_mR(sinhA_P2, S10,P4,C6);
printf(" cosh(A)**2");
f_eigs_mR(g,EValue,coshA_P2_EValue);
/* coshA_P2 = EVector coshA_P2_EValue EVectorT */
mul_mR(EVector,coshA_P2_EValue,T1);
mul_mR(T1,EVectorT,coshA_P2);
pE_mR(coshA_P2, S10,P4,C6);
printf(" Ide = cosh(A)**2 - sinh(A)**2");
sub_mR(coshA_P2,sinhA_P2,T1);
p_mR(T1, S10,P4,C6);
f_mR(A);
f_mR(sinhA_P2);
f_mR(coshA_P2);
f_mR(EVector);
f_mR(EVectorT);
f_mR(EValue);
f_mR(coshA_P2_EValue);
f_mR(sinhA_P2_EValue);
f_mR(T1);
}
/* ------------------------------------ */
int main(void)
{
time_t t;
srand(time(&t));
do
{
fun();
} while(stop_w());
return 0;
}
/* ------------------------------------ */
/* ------------------------------------ */
Vérifions les égalités :
a) *******************
sinh(x) = (e**x - e**(-x)) / 2
cosh(x) = (e**x + e**(-x)) / 2
2 sinh(x) = e**x - e**(-x)
2 cosh(x) = e**x + e**(-x)
2 sinh(x) = e**x - e**(-x)
+ 2 cosh(x) = e**x + e**(-x)
----------------------------
2 sinh(x) + 2 cosh(x) = e**x - e**(-x) + e**x + e**(-x)
2 sinh(x) + 2 cosh(x) = e**x + e**x
2 sinh(x) + 2 cosh(x) = 2 e**x
sinh(x) + cosh(x) = e**x
e**x = sinh(x) + cosh(x)
b) *******************
sinh(x) = (e**x - e**(-x)) / 2
cosh(x) = (e**x + e**(-x)) / 2
2 sinh(x) = e**x - e**(-x)
2 cosh(x) = e**x + e**(-x)
2 sinh(x) = e**x - e**(-x)
- 2 cosh(x) = e**x + e**(-x)
----------------------------
2 sinh(x) - 2 cosh(x) = e**x - e**(-x) - (e**x + e**(-x))
2 sinh(x) - 2 cosh(x) = e**x - e**(-x) - e**x - e**(-x)
2 sinh(x) - 2 cosh(x) = - e**(-x) - e**(-x)
2 sinh(x) - 2 cosh(x) = - 2 e**(-x)
sinh(x) - cosh(x) = - e**(-x)
- e**(-x) = sinh(x) - cosh(x)
e**(-x) = cosh(x) - sinh(x)
c) *******************
e**x = sinh(x) + cosh(x) a)
e**(-x) = cosh(x) - sinh(x) b)
e**x = cosh(x) + sinh(x)
* e**(-x) = cosh(x) - sinh(x)
-----------------------------
e**x e**(-x) = (cosh(x) + sinh(x)) (cosh(x) - sinh(x))
(a+b)(a-b) = a**2-b**2
e**(x-x) = cosh(x)**2 - sinh(x)**2
e**(0) = cosh(x)**2 - sinh(x)**2
1 = cosh(x)**2 - sinh(x)**2
Exemple de sortie écran :
A:
+0.0300 +0.0058 +0.0155 +0.0039 +0.0210
+0.0058 +0.0218 +0.0037 +0.0102 +0.0102
+0.0155 +0.0037 +0.0249 -0.0028 +0.0103
+0.0039 +0.0102 -0.0028 +0.0076 +0.0065
+0.0210 +0.0102 +0.0103 +0.0065 +0.0176
Press return to continue.
sinh(A)**2
+1.6338e-03 +6.1375e-04 +1.0786e-03 +2.9843e-04 +1.2458e-03
+6.1375e-04 +7.3024e-04 +3.3764e-04 +3.7717e-04 +6.2842e-04
+1.0786e-03 +3.3764e-04 +9.8826e-04 +7.2425e-05 +7.8267e-04
+2.9843e-04 +3.7717e-04 +7.2425e-05 +2.2587e-04 +3.1965e-04
+1.2458e-03 +6.2842e-04 +7.8267e-04 +3.1965e-04 +1.0037e-03
cosh(A)**2
+1.0016e+00 +6.1375e-04 +1.0786e-03 +2.9843e-04 +1.2458e-03
+6.1375e-04 +1.0007e+00 +3.3764e-04 +3.7717e-04 +6.2842e-04
+1.0786e-03 +3.3764e-04 +1.0010e+00 +7.2425e-05 +7.8267e-04
+2.9843e-04 +3.7717e-04 +7.2425e-05 +1.0002e+00 +3.1965e-04
+1.2458e-03 +6.2842e-04 +7.8267e-04 +3.1965e-04 +1.0010e+00
Ide = cosh(A)**2 - sinh(A)**2
+1.0000 -0.0000 -0.0000 -0.0000 -0.0000
-0.0000 +1.0000 +0.0000 +0.0000 -0.0000
-0.0000 -0.0000 +1.0000 -0.0000 -0.0000
-0.0000 +0.0000 -0.0000 +1.0000 +0.0000
-0.0000 -0.0000 -0.0000 -0.0000 +1.0000
Press return to continue
Press X return to stop