/**
* Returns the Q matrix.
*/
public DMatrixRMaj getQ()
{
DMatrixRMaj Q = CommonOps_DDRM.identity(QR.numRows);
DMatrixRMaj Q_k = new DMatrixRMaj(QR.numRows, QR.numRows);
DMatrixRMaj u = new DMatrixRMaj(QR.numRows, 1);
DMatrixRMaj temp = new DMatrixRMaj(QR.numRows, QR.numRows);
int N = Math.Min(QR.numCols, QR.numRows);
// compute Q by first extracting the householder vectors from the columns of QR and then applying it to Q
for (int j = N - 1; j >= 0; j--)
{
CommonOps_DDRM.extract(QR, j, QR.numRows, j, j + 1, u, j, 0);
u.set(j, 1.0);
// A = (I - γ*u*u<sup>T</sup>)*A<br>
CommonOps_DDRM.setIdentity(Q_k);
CommonOps_DDRM.multAddTransB(-gammas[j], u, u, Q_k);
CommonOps_DDRM.mult(Q_k, Q, temp);
Q.set(temp);
}
return(Q);
}
/**
* <p>
* Creates a reflector from the provided vector and gamma.<br>
* <br>
* Q = I - γ u u<sup>T</sup><br>
* </p>
*
* <p>
* In practice {@link VectorVectorMult_DDRM#householder(double, DMatrixD1, DMatrixD1, DMatrixD1)} multHouseholder}
* should be used for performance reasons since there is no need to calculate Q explicitly.
* </p>
*
* @param u A vector. Not modified.
* @param gamma To produce a reflector gamma needs to be equal to 2/||u||.
* @return An orthogonal reflector.
*/
public static DMatrixRMaj createReflector(DMatrixRMaj u, double gamma)
{
if (!MatrixFeatures_DDRM.isVector(u))
{
throw new ArgumentException("u must be a vector");
}
DMatrixRMaj Q = CommonOps_DDRM.identity(u.getNumElements());
CommonOps_DDRM.multAddTransB(-gamma, u, u, Q);
return(Q);
}
/**
* <p>
* Creates a reflector from the provided vector.<br>
* <br>
* Q = I - γ u u<sup>T</sup><br>
* γ = 2/||u||<sup>2</sup>
* </p>
*
* <p>
* In practice {@link VectorVectorMult_DDRM#householder(double, DMatrixD1, DMatrixD1, DMatrixD1)} multHouseholder}
* should be used for performance reasons since there is no need to calculate Q explicitly.
* </p>
*
* @param u A vector. Not modified.
* @return An orthogonal reflector.
*/
public static DMatrixRMaj createReflector(DMatrix1Row u)
{
if (!MatrixFeatures_DDRM.isVector(u))
{
throw new ArgumentException("u must be a vector");
}
double norm = NormOps_DDRM.fastNormF(u);
double gamma = -2.0 / (norm * norm);
DMatrixRMaj Q = CommonOps_DDRM.identity(u.getNumElements());
CommonOps_DDRM.multAddTransB(gamma, u, u, Q);
return(Q);
}
/**
* Computes the QR decomposition of the provided matrix.
*
* @param A Matrix which is to be decomposed. Not modified.
*/
public void decompose(DMatrixRMaj A)
{
this.QR = (DMatrixRMaj)A.copy();
int N = Math.Min(A.numCols, A.numRows);
gammas = new double[A.numCols];
DMatrixRMaj A_small = new DMatrixRMaj(A.numRows, A.numCols);
DMatrixRMaj A_mod = new DMatrixRMaj(A.numRows, A.numCols);
DMatrixRMaj v = new DMatrixRMaj(A.numRows, 1);
DMatrixRMaj Q_k = new DMatrixRMaj(A.numRows, A.numRows);
for (int i = 0; i < N; i++)
{
// reshape temporary variables
A_small.reshape(QR.numRows - i, QR.numCols - i, false);
A_mod.reshape(A_small.numRows, A_small.numCols, false);
v.reshape(A_small.numRows, 1, false);
Q_k.reshape(v.getNumElements(), v.getNumElements(), false);
// use extract matrix to get the column that is to be zeroed
CommonOps_DDRM.extract(QR, i, QR.numRows, i, i + 1, v, 0, 0);
double max = CommonOps_DDRM.elementMaxAbs(v);
if (max > 0 && v.getNumElements() > 1)
{
// normalize to reduce overflow issues
CommonOps_DDRM.divide(v, max);
// compute the magnitude of the vector
double tau = NormOps_DDRM.normF(v);
if (v.get(0) < 0)
{
tau *= -1.0;
}
double u_0 = v.get(0) + tau;
double gamma = u_0 / tau;
CommonOps_DDRM.divide(v, u_0);
v.set(0, 1.0);
// extract the submatrix of A which is being operated on
CommonOps_DDRM.extract(QR, i, QR.numRows, i, QR.numCols, A_small, 0, 0);
// A = (I - γ*u*u<sup>T</sup>)A
CommonOps_DDRM.setIdentity(Q_k);
CommonOps_DDRM.multAddTransB(-gamma, v, v, Q_k);
CommonOps_DDRM.mult(Q_k, A_small, A_mod);
// save the results
CommonOps_DDRM.insert(A_mod, QR, i, i);
CommonOps_DDRM.insert(v, QR, i, i);
QR.unsafe_set(i, i, -tau * max);
// save gamma for recomputing Q later on
gammas[i] = gamma;
}
}
}