/**
* <p>
* Computes the householder vector "u" for the first column of submatrix j. Note this is
* a specialized householder for this problem. There is some protection against
* overflow and underflow.
* </p>
* <p>
* Q = I - γuu<sup>H</sup>
* </p>
* <p>
* This function finds the values of 'u' and 'γ'.
* </p>
*
* @param j Which submatrix to work off of.
*/
protected void householder(int j)
{
// find the element with the largest absolute value in the column and make a copy
double max = QrHelperFunctions_ZDRM.extractColumnAndMax(QR, j, numRows, j, u, 0);
if (max <= 0.0)
{
gammas[j] = 0;
error = true;
}
else
{
double gamma = QrHelperFunctions_ZDRM.computeTauGammaAndDivide(j, numRows, u, max, tau);
gammas[j] = gamma;
// divide u by u_0
double real_u_0 = u[j * 2] + tau.real;
double imag_u_0 = u[j * 2 + 1] + tau.imaginary;
QrHelperFunctions_ZDRM.divideElements(j + 1, numRows, u, 0, real_u_0, imag_u_0);
// write the reflector into the lower left column of the matrix
for (int i = j + 1; i < numRows; i++)
{
dataQR[(i * numCols + j) * 2] = u[i * 2];
dataQR[(i * numCols + j) * 2 + 1] = u[i * 2 + 1];
}
u[j * 2] = 1;
u[j * 2 + 1] = 0;
QrHelperFunctions_ZDRM.rank1UpdateMultR(QR, u, 0, gamma, j + 1, j, numRows, v);
// since the first element in the householder vector is known to be 1
// store the full upper hessenberg
if (j < numCols)
{
dataQR[(j * numCols + j) * 2] = -tau.real * max;
dataQR[(j * numCols + j) * 2 + 1] = -tau.imaginary * max;
}
}
}
