/* Solve Ax = b using the conjugate gradient method. */
Matrix conjugateGradient(Matrix& A, Matrix& b)
{
double error_tol = .5; // error tolerance
int max_iter = 200; // max # of iterations
ColumnVector x(A.rows()); // the solution we will iteratively arrive at
int i = 0;
ColumnVector r = static_cast<ColumnVector>(b - A*x);
ColumnVector d = r;
double sigma_old = 0; // will be used later on, in the loop
double sigma_new = (r.transpose() * r)(0,0);
double sigma_0 = sigma_new;
while (i < max_iter && sigma_new > error_tol * error_tol * sigma_0)
{
ColumnVector q = A * d;
double alpha = sigma_new / (d.transpose() * q)(0,0);
x = x + alpha * d;
if (i % 50 == 0)
{
r = static_cast<ColumnVector>(b - A*x);
}else{
r = r - alpha * q;
}
sigma_old = sigma_new;
sigma_new = (r.transpose() * r)(0,0);
double beta = sigma_new / sigma_old;
d = r + beta * d;
i++;
}
shared_ptr<Matrix> final_x(new Matrix(static_cast<Matrix>(x)));
return *final_x;
}
/* solve Ax=b using the Method of Steepest Descent. */
Matrix steepestDescent(Matrix& A, Matrix& b)
{
// the Method of Steepest Descent *requires* a symmetric matrix.
if (isSymmetric(A)==false)
{
shared_ptr<Matrix> nullMat(new Matrix(0,0));
return *nullMat;
}
/* STEP 1: Start with a guess. Our guess is all ones. */
ColumnVector x(A.cols());
fill(x.begin(),x.end(),1);
/* This is NOT an infinite loop. There's a break statement inside. */
while(true)
{
/* STEP 2: Calculate the residual r_0 = b - Ax_0 */
ColumnVector r = static_cast<ColumnVector> (b - A*x);
if (r.length() < .01) break;
/* STEP 3: Calculate alpha */
double alpha = (r.transpose() * r)(0,0) / (r.transpose() * A * r)(0,0);
/* STEP 4: Calculate new X_1 where X_1 = X_0 + alpha*r_0 */
x = x + alpha * r;
}
shared_ptr<Matrix> final_x(new Matrix(static_cast<Matrix>(x)));
return *final_x;
}