/**
* Only eigenvalues of a symmetric real matrix are computed.
*
* @param A a symmetric real matrix
* @return a 1D {@code double} array containing the eigenvalues in decreasing order (absolute
* value)
*/
public static double[] computeEigenvalues(Matrix A) {
SparseMatrix S = (SparseMatrix) decompose(A, false)[1];
int m = S.getRowDimension();
int n = S.getColumnDimension();
int len = m >= n ? n : m;
double[] s = ArrayOperator.allocateVector(len, 0);
for (int i = 0; i < len; i++) {
s[i] = S.getEntry(i, i);
}
return s;
}
/**
* Do eigenvalue decomposition for a real symmetric tridiagonal matrix, i.e. T = VDV'.
*
* @param T a real symmetric tridiagonal matrix
* @param computeV if V is to be computed
* @return a {@code Matrix} array [V, D]
*/
private static Matrix[] diagonalizeTD(Matrix T, boolean computeV) {
int m = T.getRowDimension();
int n = T.getColumnDimension();
int len = m >= n ? n : m;
int idx = 0;
/*
* The tridiagonal matrix T is
* s[0] e[0]
* e[0] s[1] e[1]
* e[1] ...
* s[len - 2] e[len - 2]
* e[len - 2] s[len - 1]
*/
double[] s = ArrayOperator.allocateVector(len, 0);
double[] e = ArrayOperator.allocateVector(len, 0);
for (int i = 0; i < len - 1; i++) {
s[i] = T.getEntry(i, i);
e[i] = T.getEntry(i, i + 1);
}
s[len - 1] = T.getEntry(len - 1, len - 1);
// V': each row of V' is a right singular vector
double[][] Vt = null;
if (computeV) Vt = eye(n, n).getData();
double[] mu = ArrayOperator.allocate1DArray(len, 0);
/*
* T0 = ITI', V = I
* T = IT0I' = VT0G1G1'Vt = VT0G1(G1'Vt)
* = VG1G1'T0G1(G1'Vt) = (VG1)(G1'T0G1)(G1'Vt)
* = (VG1)T1(G1'Vt)
* ...
* = (Gn-1...G2G1)D(G1G2...Gn-1)'
*
* where G = |cs sn|
* |-sn cs|
* G' = |cs -sn|
* |sn cs|
*/
/*
* Find B_hat, i.e. the bottommost unreduced submatrix of B.
* Index starts from 0.
*/
// *********************************************************
int i_start = 0;
int i_end = len - 1;
// int cnt_shifted = 0;
int ind = 1;
while (true) {
idx = len - 2;
while (idx >= 0) {
if (e[idx] == 0) {
idx--;
} else {
break;
}
}
i_end = idx + 1;
// Now idx = -1 or e[idx] != 0
// If idx = -1, then e[0] = 0, i_start = i_end = 0, e = 0
// Else if e[idx] != 0, then e[i] = 0 for i_end = idx + 1 <= i <= len - 1
while (idx >= 0) {
if (e[idx] != 0) {
idx--;
} else {
break;
}
}
i_start = idx + 1;
// Now idx = -1 or e[idx] = 0
// If idx = -1 i_start = 0
// Else if e[idx] = 0, then e[idx + 1] != 0, e[i_end - 1] != 0
// i.e. e[i] != 0 for i_start <= i <= i_end - 1
if (i_start == i_end) {
break;
}
// Apply the stopping criterion to B_hat
// If any e[i] is set to zero, return to loop
boolean set2Zero = false;
mu[i_start] = abs(s[i_start]);
for (int j = i_start; j < i_end; j++) {
mu[j + 1] = abs(s[j + 1]) * mu[j] / (mu[j] + abs(e[j]));
if (abs(e[j]) <= mu[j] * tol) {
e[j] = 0;
set2Zero = true;
}
}
if (set2Zero) {
continue;
}
implicitSymmetricShiftedQR(s, e, Vt, i_start, i_end, computeV);
// cnt_shifted++;
if (ind == maxIter) {
break;
}
ind++;
}
// fprintf("cnt_shifted: %d\n", cnt_shifted);
// *********************************************************
// Quick sort eigenvalues and eigenvectors
quickSort(s, Vt, 0, len - 1, "descend", computeV);
Matrix[] VD = new Matrix[2];
VD[0] = computeV ? new DenseMatrix(Vt).transpose() : null;
VD[1] = buildD(s, m, n);
/*disp("T:");
printMatrix(T);
disp("VDV':");
Matrix V = VD[0];
Matrix D = VD[1];
disp(V.mtimes(D).mtimes(V.transpose()));*/
return VD;
}
/**
* 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;
}