bool QRDecomposition<T>::set(const MatrixT& A)
{
QR.copy(A);
tau.resize(Min(A.m,A.n));
for (int i=0;i<Min(A.m,A.n);i++) {
/* Compute the Householder transformation to reduce the j-th
column of the matrix to a multiple of the j-th unit vector */
VectorT c_full,c;
QR.getColRef(i,c_full);
c.setRef(c_full,i);
T tau_i = HouseholderTransform (c);
tau(i)=tau_i;
/* Apply the transformation to the remaining columns and
update the norms */
if (i+1 < A.n) {
MatrixT m;
m.setRef(QR,i,i+1);
HouseholderPreMultiply (tau_i, c, m);
}
}
return true;
}
void QRDecomposition<T>::getQ(MatrixT& Q) const
{
Assert(tau.n == Min(QR.m,QR.n));
Q.resize(QR.m,QR.m);
int i;
Q.setIdentity();
for (i=Min(QR.m,QR.n);i>0 && i--;) {
VectorT c,h;
QR.getColRef(i,c);
h.setRef(c,i);
MatrixT m;
m.setRef(Q,i,i);
HouseholderPreMultiply(tau(i),h,m);
}
}
void QRDecomposition<T>::leastSquares(const VectorT& b, VectorT& x, VectorT& residual) const
{
if(x.n == 0) x.resize(QR.n);
Assert(QR.m >= QR.n);
Assert(QR.m == b.n);
Assert(QR.n == x.n);
Assert(QR.m == residual.n);
MatrixT R;
R.setRef(QR,0,0,1,1,QR.n,QR.n);
VectorT c;
c.setRef(residual,0,1,QR.n);
/* compute rhs = Q^T b */
QtMul(b,residual);
/* Solve R x = rhs */
UBackSubstitute(R,c,x);
/* Compute residual = b - A x = Q (Q^T b - R x) */
c.setZero();
QMul(residual,residual);
}