void BackwardEulerUpdater::update( const std::vector<double> & uOld,
std::vector<double>* uNew,
const std::vector<double> & prem ) const
{
int N = uOld.size()-1;
double c = getAlpha();
// Construct b-vector
Eigen::VectorXd b = Eigen::VectorXd::Zero(N-1);
for (int i=0; i<N-1; i++) { b(i) = uOld[i+1]; }
b(0) += c*(*uNew)[0];
b(N-2) += c*(*uNew)[N];
// Construct early exercise premium vector
Eigen::VectorXd p = Eigen::VectorXd::Zero(0);
if ( prem.size()>0 ) {
p = Eigen::VectorXd::Zero(N-1);
for (int i=0; i<N-1; i++) { p(i) = prem[i+1]; }
}
// update
Eigen::VectorXd y = forward_subst(d_L, b);
Eigen::VectorXd x = backward_subst(d_U, y, p);
for (int i=0; i<N-1; i++) { (*uNew)[i+1] = x(i); }
}
void CrankNicolsonUpdater::update( const std::vector<double> & uOld,
std::vector<double>* uNew,
const std::vector<double> & prem ) const
{
int N = uOld.size()-1;
double c = getAlpha();
// uOld restricted in the interior region
Eigen::VectorXd u = Eigen::VectorXd::Zero(N-1);
for (int i=0; i<N-1; i++) { u(i) = uOld[i+1]; }
// Construct b-vector
Eigen::VectorXd b = d_B*u;
b(0) += 0.5*c*((*uNew)[0]+uOld[0]);
b(N-2) += 0.5*c*((*uNew)[N]+uOld[N]);
// Construct early exercise premium vector
Eigen::VectorXd p = Eigen::VectorXd::Zero(0);
if ( prem.size()>0 ) {
p = Eigen::VectorXd::Zero(N-1);
for (int i=0; i<N-1; i++) { p(i) = prem[i+1]; }
}
// update
Eigen::VectorXd y = forward_subst(d_L, b);
Eigen::VectorXd x = backward_subst(d_U, y, p);
for (int i=0; i<N-1; i++) { (*uNew)[i+1] = x(i); }
}
TEST_F(TriangularSolveTest, BackwardSubstMatchEigenComputedResults)
{
Eigen::MatrixXd U = Eigen::MatrixXd::Random(2,2);
Eigen::VectorXd b = Eigen::VectorXd::Random(2);
U.triangularView<Eigen::StrictlyLower>().setZero();
Eigen::VectorXd x = backward_subst(U, b);
Eigen::VectorXd y = U.triangularView<Eigen::Upper>().solve(b);
double tol = 1e-16;
EXPECT_NEAR((x-y).norm(), 0, tol);
}
void solve_qr_decomp(double* qr, int m, int n, int ldim, double* b, double *x)
{
double *z;
int j, i;
qr_transform(qr, m, n, ldim, b);
printf("\nz:\n");
print_matrix(b, n, 1);
backward_subst(qr, n, ldim, b, x);
}
Eigen::VectorXd spd_solve(const Eigen::MatrixXd & A,
const Eigen::VectorXd & b)
{
int n = b.size();
assert(A.rows() == n);
assert(A.cols() == n);
Eigen::MatrixXd U = cholesky(A);
Eigen::VectorXd y = forward_subst(U.transpose(), b);
Eigen::VectorXd x = backward_subst(U, y);
return x;
}
int main(int argc, char* argv[])
{
// Output precision
int p=9;
// Matrix dimension
int N=9;
// Matrix specification
Eigen::MatrixXd A = Eigen::MatrixXd::Zero(N,N);
Eigen::MatrixXd A1 = Eigen::MatrixXd::Zero(N,N);
for (int i=0; i<N; i++) {
A1(i,i) = 4.0;
if (i>=1) A1(i,i-1) = -1.0;
if (i>=2) A1(i,i-2) = 3.0;
if (i<=6) A1(i,i+2) = -2.0;
}
// Could be AT+A or AT*A
A = A1.transpose()+A1;
// Vector specification
Eigen::VectorXd b = Eigen::VectorXd::Zero(N);
for (int i=0; i<N; i++) {
b(i) = (i*i-9.0)/(i+5.0);
}
// Cholesky decomposition
Eigen::MatrixXd U = cholesky(A);
Eigen::MatrixXd L = U.transpose();
// Perform forward substitution and backward substitution
// Using Eigen's library method: Eigen::VectorXd x = A.llt().solve(b);
Eigen::VectorXd y = forward_subst(L, b);
Eigen::VectorXd x = backward_subst(U, y);
// Compute residual error
double R = (A*x-b).norm();
std::cout << "Residual error =, "
<< std::setprecision(p)
<< R
<< std::endl;
// Compute L-inverse, U-inverse, and A-inverse
Eigen::MatrixXd L_inv = Eigen::MatrixXd::Zero(N,N);
Eigen::MatrixXd U_inv = Eigen::MatrixXd::Zero(N,N);
Eigen::MatrixXd A_inv = Eigen::MatrixXd::Zero(N,N);
// L-inverse
for (int i=0; i<N; i++) {
Eigen::VectorXd ei = Eigen::VectorXd::Zero(N);
ei(i) = 1;
Eigen::VectorXd x = forward_subst(L, ei);
L_inv.col(i) = x;
}
// U-inverse
for (int i=0; i<N; i++) {
Eigen::VectorXd ei = Eigen::VectorXd::Zero(N);
ei(i) = 1;
Eigen::VectorXd x = backward_subst(U, ei);
U_inv.col(i) = x;
}
// A-inverse
for (int i=0; i<N; i++) {
Eigen::VectorXd ei = Eigen::VectorXd::Zero(N);
ei(i) = 1;
Eigen::VectorXd y = forward_subst(L, ei);
Eigen::VectorXd x = backward_subst(U, y);
A_inv.col(i) = x;
}
// Compute residual error by matrix inverse
Eigen::VectorXd v = A_inv*b;
R = (b-A*v).norm();
std::cout << "Residual error by inverse =, "
<< std::setprecision(p)
<< R
<< std::endl;
std::cout << "Cholesky decomposition" << std::endl;
for (int i=0; i<N; i++) {
std::cout << " |, ";
for (int j=0; j<N; j++) {
std::cout << std::right
<< std::setw(p+3)
<< std::fixed
<< std::setprecision(p)
<< A(i,j)
<< ", ";
}
std::cout << " |, =, |, ";
for (int j=0; j<N; j++) {
std::cout << std::right
<< std::setw(p+3)
<< std::fixed
<< std::setprecision(p)
<< L(i,j)
<< ", ";
}
std::cout << " |, ";
for (int j=0; j<N; j++) {
std::cout << std::right
<< std::setw(p+3)
<< std::fixed
<< std::setprecision(p)
<< U(i,j)
<< ", ";
}
std::cout << " |, " << std::endl;
}
std::cout << "Linear solve - x:" << std::endl;
for (int i=0; i<N; i++) {
std::cout << std::right
<< std::setw(p+3)
<< std::fixed
<< std::setprecision(p)
<< x(i)
<< std::endl;
}
std::cout << "Inverse relations:" << std::endl;
for (int i=0; i<N; i++) {
std::cout << " |, ";
for (int j=0; j<N; j++) {
std::cout << std::right
<< std::setw(p+3)
<< std::fixed
<< std::setprecision(p)
<< A_inv(i,j)
<< ", ";
}
std::cout << " |, =, |, ";
for (int j=0; j<N; j++) {
std::cout << std::right
<< std::setw(p+3)
<< std::fixed
<< std::setprecision(p)
<< U_inv(i,j)
<< ", ";
}
std::cout << " |, ";
for (int j=0; j<N; j++) {
std::cout << std::right
<< std::setw(p+3)
<< std::fixed
<< std::setprecision(p)
<< L_inv(i,j)
<< ", ";
}
std::cout << " |, " << std::endl;
}
std::cout << "Linear solve by inversion- x:" << std::endl;
for (int i=0; i<N; i++) {
std::cout << std::right
<< std::setw(p+3)
<< std::fixed
<< std::setprecision(p)
<< v(i)
<< std::endl;
}
// Verify accuracy of matrix inverse
std::cout << "L * L-inverse residual error =, "
<< std::scientific
<< std::setprecision(p)
<< (L*L_inv-Eigen::MatrixXd::Identity(N,N)).norm()
<< std::endl;
std::cout << "U * U-inverse residual error =, "
<< std::scientific
<< std::setprecision(p)
<< (U*U_inv-Eigen::MatrixXd::Identity(N,N)).norm()
<< std::endl;
std::cout << "A * A-inverse residual error =, "
<< std::scientific
<< std::setprecision(p)
<< (A*A_inv-Eigen::MatrixXd::Identity(N,N)).norm()
<< std::endl;
return 0;
}