/**
* Tridiagonalize a real symmetric matrix A, i.e. A = Q * T * Q' such that Q is an orthogonal
* matrix and T is a tridiagonal matrix.
*
* <p>A = QTQ' <=> Q'AQ = T
*
* @param A a real symmetric matrix
* @param computeV if V is to be computed
* @return a {@code Matrix} array [Q, T]
*/
private static Matrix[] tridiagonalize(Matrix A, boolean computeV) {
A = full(A).copy();
int m = A.getRowDimension();
int n = A.getColumnDimension();
Matrix[] QT = new Matrix[2];
double[] a = ArrayOperator.allocateVector(n, 0);
double[] b = ArrayOperator.allocateVector(n, 0);
double[][] AData = ((DenseMatrix) A).getData();
double c = 0;
double s = 0;
double r = 0;
for (int j = 0; j < n - 2; j++) {
a[j] = AData[j][j];
// Householder transformation on columns of A(j+1:m, j+1:n)
// Compute the norm of A(j+1:m, j)
c = 0;
for (int i = j + 1; i < m; i++) {
c += Math.pow(AData[i][j], 2);
}
if (c == 0) continue;
s = Math.sqrt(c);
b[j] = AData[j + 1][j] > 0 ? -s : s;
r = Math.sqrt(s * (s + abs(AData[j + 1][j])));
/*double[] u1 = ArrayOperation.allocate1DArray(n, 0);
for (int k = j + 1; k < m; k++) {
u1[k] = AData[k][j];
}
u1[j + 1] -= b[j];
for (int k = j + 1; k < m; k++) {
u1[k] /= r;
}
Matrix H = eye(n).minus(new DenseMatrix(u1, 1).mtimes(new DenseMatrix(u1, 2)));
disp(new DenseMatrix(u1, 1));
disp(eye(n));
disp(H);
disp(A);
disp(H.mtimes(A).mtimes(H));
disp(A);*/
AData[j + 1][j] -= b[j];
for (int k = j + 1; k < m; k++) {
AData[k][j] /= r;
}
double[] w = new double[n - j - 1];
double[] u = new double[n - j - 1];
double[] v = new double[n - j - 1];
for (int i = j + 1, t = 0; i < m; i++, t++) {
u[t] = AData[i][j];
}
// v = B33 * u
for (int i = j + 1, t = 0; i < m; i++, t++) {
double[] ARow_i = AData[i];
s = 0;
for (int k = j + 1, l = 0; k < n; k++, l++) {
s += ARow_i[k] * u[l];
}
v[t] = s;
}
c = ArrayOperator.innerProduct(u, v) / 2;
for (int i = j + 1, t = 0; i < m; i++, t++) {
w[t] = v[t] - c * u[t];
}
/*disp(w);
Matrix B33 = new DenseMatrix(n - j - 1, n - j - 1, 0);
for (int i = j + 1, t = 0; i < m; i++, t++) {
double[] ARow_i = AData[i];
for (int k = j + 1, l = 0; k < n; k++, l++) {
B33.setEntry(t, l, ARow_i[k]);
}
}
disp(B33);
Matrix U = new DenseMatrix(u, 1);
disp(U.transpose().mtimes(U));
disp(U);
Matrix W = B33.mtimes(U).minus(U);
disp(W);
disp(B33.minus(plus(U.mtimes(W.transpose()), W.mtimes(U.transpose()))));
Matrix Hk = eye(n - j - 1).minus(U.mtimes(U.transpose()));
disp(Hk);
disp(Hk.mtimes(B33).mtimes(Hk));*/
for (int i = j + 1, t = 0; i < m; i++, t++) {
double[] ARow_i = AData[i];
for (int k = j + 1, l = 0; k < n; k++, l++) {
ARow_i[k] = ARow_i[k] - (u[t] * w[l] + w[t] * u[l]);
}
}
// disp(A);
/*fprintf("Householder transformation on n - 1 columns:\n");
disp(A);*/
// disp(A);
// Householder transformation on rows of A(j:m, j+1:n)
}
a[n - 2] = AData[n - 2][n - 2];
a[n - 1] = AData[n - 1][n - 1];
b[n - 2] = AData[n - 1][n - 2];
QT = unpack(A, a, b, computeV);
return QT;
}
