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#include "headcpp.h"
#include "math.h"
#include "tbl.h"
#include "matrice.h"
#include "cheby.h"
double f_source (double x) {
static double constante = -4*exp(1.)/(1+exp(1.)*exp(1.)) ;
return constante + exp(x) ;
}
double f_solution (double x) {
static double constante = -4*exp(1.)/(1+exp(1.)*exp(1.)) ;
return exp(x) - sinh(1.)/sinh(2.)*exp(2*x) + constante/4 ;
}
//*******************
// First derivative
//*******************
Tbl ope_der (const Tbl& so) {
// Verification of size :
assert (so.get_ndim() == 1) ;
int size = so.get_dim(0) ;
// The result is set to zero
Tbl res (size) ;
res.annule_hard() ;
// the computation
for (int n=0 ; n<size ; n++)
for (int p=n+1 ; p<size ; p++)
if ((p+n)%2 == 1)
res.set(n) += p*so(p)*2 ;
// Normalisation of the first coef :
res.set(0) /= 2. ;
return res ;
}
//*******************
// Second derivative
//*******************
Tbl ope_der_sec (const Tbl& so) {
// Verification of size :
assert (so.get_ndim() == 1) ;
int size = so.get_dim(0) ;
// The result is set to zero
Tbl res (size) ;
res.annule_hard() ;
// the computation
for (int n=0 ; n<size ; n++)
for (int p=n+2 ; p<size ; p++)
if ((p+n)%2 == 0)
res.set(n) += p*(p*p-n*n)*so(p) ;
// Normalisation of the first coef :
res.set(0) /= 2. ;
return res ;
}
//*******************
// The operator
//*******************
Matrice matrix_ope (int n) {
// The result :
Matrice res(n,n) ;
res.set_etat_qcq() ;
// Work arrays :
Tbl so (n) ;
Tbl d_so (n) ;
Tbl dd_so (n) ;
// Column by column :
for (int col=0 ; col<n ; col++) {
so.annule_hard() ;
so.set(col) = 1 ;
// The derivatives
d_so = ope_der(so) ;
dd_so = ope_der_sec(so) ;
// Put in the matrix :
for (int line=0 ; line<n ; line++)
res.set(line, col) = dd_so(line) -4*d_so(line) + 4*so(line) ;
}
return res ;
}
int main () {
int nr ;
cout << "Please enter the number of coefficients :" << endl ;
cin >> nr ;
cout << "The operator matrix is " << endl ;
Matrice operator_mat (matrix_ope(nr)) ;
cout << operator_mat << endl ;
// Resolution with a collocation method :
Matrice systeme(nr, nr) ;
systeme.annule_hard() ;
// Boundary conditions are inforced by additional equations:
// Left boundary condition :
for (int i=0 ; i<nr ; i++)
systeme.set(0, i) = (i%2==0) ? 1 : -1 ;
// Right boundary condition :
for (int i=0 ; i<nr ; i++)
systeme.set(nr-1, i) = 1 ;
// The rest of the equations :
Tbl colocation(coloc_cheb(nr)) ;
for (int n=1 ; n<nr-1 ; n++)
for (int k=0 ; k<nr ; k++)
for (int j=0 ; j<nr ; j++)
systeme.set(n,k) += operator_mat(j, k) * cheby(j, colocation(n)) ;
systeme.set_lu() ;
cout << "The colocation matrix is" << endl ;
cout << systeme << endl ;
// Second member of the system :
Tbl sec_member (nr) ;
sec_member.set_etat_qcq() ;
// Boundary conditions :
sec_member.set(0) = 0 ;
sec_member.set(nr-1) = 0 ;
// Coefficients of the source :
for (int i=1 ; i<nr-1 ; i++)
sec_member.set(i) = f_source(colocation(i)) ;
cout << "Second member for the colocation system" << endl ;
cout << sec_member << endl ;
// The system is inverted :
Tbl coef_sol (systeme.inverse(sec_member)) ;
cout << "Coefficients of the solution : " << endl ;
cout << coef_sol << endl ;
// Output in a file for plotting purposes :
char name_out[30] ;
sprintf(name_out, "plot_tau_%i.dat", nr) ;
ofstream fiche (name_out) ;
int resolution = 200 ;
double x=-1 ;
double step = 2./resolution ;
// We will also compute the maximum difference between the solution and its numerical value :
double error_max = 0 ;
for (int i=0 ; i<resolution+1 ; i++) {
// One computes the solution at the current point
double val_func = 0 ;
for (int j=0 ; j<nr ; j++)
val_func += coef_sol(j)*cheby(j, x) ;
fiche << x << " " << val_func << " " << f_solution(x) << endl ;
double error = fabs(val_func-f_solution(x)) ;
if (error > error_max)
error_max = error ;
x += step ;
}
cout << "Error max : " << endl ;
cout << error_max << endl ;
return 0 ;
}
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