void TestLeastSquares() {
MatrixXf A = MatrixXf::Random(10, 2);
VectorXf b = VectorXf::Random(10);
Vector2f x;
std::cout << "=============================" << std::endl;
std::cout << "Testing least squares solvers" << std::endl;
std::cout << "=============================" << std::endl;
x = A.jacobiSvd(ComputeThinU | ComputeThinV).solve(b);
std::cout << "Solution using Jacobi SVD = " << x.transpose() << std::endl;
x = A.colPivHouseholderQr().solve(b);
std::cout << "Solution using column pivoting Householder QR = " << x.transpose() << std::endl;
// If the matrix A is ill-conditioned, then this is not a good method
x = (A.transpose() * A).ldlt().solve(A.transpose() * b);
std::cout << "Solution using normal equation = " << x.transpose() << std::endl;
}
int main(int argc, char **argv){
double t = atof(argv[1]);
float time = 0;
float x_values [x_spaces+1];
//float u_values [x_spaces+1];
float u_old[x_spaces+1];
float correction_factor;
for (int i=0; i<=x_spaces;i++){
x_values[i] = x_lower + (x_higher-x_lower)/x_spaces * i;
u_old[i] = func(x_values[i]);
}
MatrixXf A = MatrixXf::Zero(x_spaces+1,x_spaces+1);
VectorXf b(x_spaces+1,1);
VectorXf u_values(x_spaces+1,1);
// The next section deals with the euler explicit method
while (time <= t) {
for (int i=0;i<=x_spaces;i++){
if (i-1 >= 0){
A(i,i-1) = -delta_t/(4*delta_x)*(u_old[i-1]);
}
if (i+1 <= x_spaces){
A(i,i+1) = delta_t/(4*delta_x)*(u_old[i+1]);
}
A(i,i) = 1;
if (i-2<0){
correction_factor = -omega/8*(u_old[i+2] -4*u_old[i+1]+6*u_old[i]);
}else if (i-1<0){
correction_factor = -omega/8*(u_old[i+2] -4*u_old[i+1]+6*u_old[i]-4*u_old[i-1]);
}else if (i+1>x_spaces){
correction_factor = -omega/8*(-4*u_old[i+1]+6*u_old[i]-4*u_old[i-1]+u_old[i-2]);
}else if (i+2>x_spaces){
correction_factor = -omega/8*(6*u_old[i]-4*u_old[i-1]+u_old[i-2]);
}else{
correction_factor = -omega/8*(u_old[i+2] -4*u_old[i+1]+6*u_old[i]-4*u_old[i-1]+u_old[i-2]);
}
if (i-1 < 0){
b(i) = -delta_t/(2*delta_x)*(pow(u_old[i+1],2)/2-pow(ustart,2)/2)-delta_t/(4*delta_x)*pow(ustart,2) + u_old[i]+delta_t/(4*delta_x)*pow(u_old[i+1],2)+correction_factor;
}else if (i+1>x_spaces){
b(i) = -delta_t/(2*delta_x)*(pow(uend,2)/2-pow(u_old[i-1],2)/2)-delta_t/(4*delta_x)*pow(u_old[i-1],2) + u_old[i]+delta_t/(4*delta_x)*pow(uend,2)+correction_factor;
}else{
b(i) = -delta_t/(2*delta_x)*(pow(u_old[i+1],2)/2-pow(u_old[i-1],2)/2)-delta_t/(4*delta_x)*pow(u_old[i-1],2) + u_old[i]+delta_t/(4*delta_x)*pow(u_old[i+1],2)+correction_factor;
}
}
//b(x_spaces) = u_old[x_spaces] - v_Neumann /2*u_old[x_spaces];
u_values = A.colPivHouseholderQr().solve(b);
time = time + delta_t;
if (time >= t){
break;
}
for (int i=0;i<=x_spaces;i++){
u_old[i] = u_values(i);
}
u_old[0]=ustart;
u_values[0] = ustart;
u_old[x_spaces]=uend;
u_old[x_spaces]=uend;
}
for (int j=0; j<=x_spaces;j++){
if (j==0){
cout << "X Implicit Value Start" << "," << x_values[j] << ",";
}else if (j==x_spaces){
cout << x_values[j] << "," << "X Implicit Value End" << ",";
}else{
cout << x_values[j] << ",";
}
}
for (int k=0; k<=x_spaces;k++){
if (k==0){
cout << "Y Implicit Value Start" << "," << u_old[k] << ",";
}else if (k==x_spaces){
cout << u_old[k] << "," << "Y Implicit Value End"<<","<<endl;
}else{
cout << u_old[k] << ",";
}
}
/*
cout << "Here is the matrix A:\n" << A << endl;
cout << "Here is the vector b:\n" << b << endl;
cout << "A(0,1)"<<endl;
cout << A(0,1)<< endl;
*/
return 0;
}
int main(int argc, char **argv){
int n = atof(argv[1]);
string method = argv[2];
double h = 1.0/(n+1.0);
double max_error = pow(10,-6);
double error =1;
time_t t_start,t_end;
double dif_iterative, dif_matrix;
VectorXf x (n,1);
VectorXf xk (n,1);
VectorXf xkplus1 (n,1);
VectorXf f (n,1);
MatrixXf A = MatrixXf::Zero(n,n);
MatrixXf I = MatrixXf::Zero(n,n);
VectorXf b (n,1);
VectorXf xc (n,1);
for (int i=0; i<n;i++){
x(i) = h*(i+1);
f(i) = func(x(i));
}
for (int i=0;i<n;i++){
A(i,i) = 2;
if (i!=n-1){
A(i,i+1) = -1;
A(i+1,i) = -1;
}
}
b = f*pow(h,2);
//Solving the Equation
t_start = time(0);
xc = A.colPivHouseholderQr().solve(b);
t_end=time(0);
dif_matrix = difftime (t_end,t_start);
/*
cout << "Here is the matrix A:\n" << A << endl;
cout << "Here is the vector x:\n" << x << endl;
cout << "Here is the vector b:\n" << b << endl;
cout << "Here is the vector uc:\n" << xc << endl;
*/
//Jacobin part
MatrixXf P = MatrixXf::Zero(n,n);
MatrixXf Pinverse = MatrixXf::Zero(n,n);
if (method == "Jacobian"){
for (int i=0; i<n;i++){
P(i,i) = A(i,i);
}
} else if (method=="Gauss-Seidel"){
for (int i=0; i<n;i++){
P(i,i) = A(i,i);
for (int j=0; j<i;j++){
P(i,j) = A(i,j);
}
}
} else if (method=="SOR"){
double omega = atof(argv[3]);
for (int i=0; i<n;i++){
P(i,i) = A(i,i);
for (int j=0; j<i;j++){
P(i,j) = omega*A(i,j);
}
}
} else {
return 1;
}
Pinverse = P.inverse();
for (int i=0; i<n;i++){
xk(i) = 0;
I(i,i) = 1;
}
t_start = time(0);
double repeat_error=100*n;
int loop=1;
while(error >=max_error){
xkplus1 = (I-Pinverse*A)*xk+Pinverse*b;
xk = xkplus1;
error = maximum_array(xc,xkplus1,n);
//cout << loop<<endl;
//cout << error<<endl;
if (repeat_error == error){
break;
}
repeat_error = error;
loop +=1;
}
t_end=time(0);
dif_iterative = difftime (t_end,t_start);
cout << dif_iterative <<"," <<loop<<"," << dif_matrix <<","<<maximum_array(xc,xkplus1,n);
return 0;
}