in modules/core/src/lda.cpp [316:752]
void hqr2() {
// This is derived from the Algol procedure hqr2,
// by Martin and Wilkinson, Handbook for Auto. Comp.,
// Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
// Initialize
int nn = this->n;
int n1 = nn - 1;
int low = 0;
int high = nn - 1;
double eps = std::pow(2.0, -52.0);
double exshift = 0.0;
double p = 0, q = 0, r = 0, s = 0, z = 0, t, w, x, y;
// Store roots isolated by balanc and compute matrix norm
double norm = 0.0;
for (int i = 0; i < nn; i++) {
if (i < low || i > high) {
d[i] = H[i][i];
e[i] = 0.0;
}
for (int j = std::max(i - 1, 0); j < nn; j++) {
norm = norm + std::abs(H[i][j]);
}
}
// Outer loop over eigenvalue index
int iter = 0;
while (n1 >= low) {
// Look for single small sub-diagonal element
int l = n1;
while (l > low) {
s = std::abs(H[l - 1][l - 1]) + std::abs(H[l][l]);
if (s == 0.0) {
s = norm;
}
if (std::abs(H[l][l - 1]) < eps * s) {
break;
}
l--;
}
// Check for convergence
// One root found
if (l == n1) {
H[n1][n1] = H[n1][n1] + exshift;
d[n1] = H[n1][n1];
e[n1] = 0.0;
n1--;
iter = 0;
// Two roots found
} else if (l == n1 - 1) {
w = H[n1][n1 - 1] * H[n1 - 1][n1];
p = (H[n1 - 1][n1 - 1] - H[n1][n1]) / 2.0;
q = p * p + w;
z = std::sqrt(std::abs(q));
H[n1][n1] = H[n1][n1] + exshift;
H[n1 - 1][n1 - 1] = H[n1 - 1][n1 - 1] + exshift;
x = H[n1][n1];
// Real pair
if (q >= 0) {
if (p >= 0) {
z = p + z;
} else {
z = p - z;
}
d[n1 - 1] = x + z;
d[n1] = d[n1 - 1];
if (z != 0.0) {
d[n1] = x - w / z;
}
e[n1 - 1] = 0.0;
e[n1] = 0.0;
x = H[n1][n1 - 1];
s = std::abs(x) + std::abs(z);
p = x / s;
q = z / s;
r = std::sqrt(p * p + q * q);
p = p / r;
q = q / r;
// Row modification
for (int j = n1 - 1; j < nn; j++) {
z = H[n1 - 1][j];
H[n1 - 1][j] = q * z + p * H[n1][j];
H[n1][j] = q * H[n1][j] - p * z;
}
// Column modification
for (int i = 0; i <= n1; i++) {
z = H[i][n1 - 1];
H[i][n1 - 1] = q * z + p * H[i][n1];
H[i][n1] = q * H[i][n1] - p * z;
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
z = V[i][n1 - 1];
V[i][n1 - 1] = q * z + p * V[i][n1];
V[i][n1] = q * V[i][n1] - p * z;
}
// Complex pair
} else {
d[n1 - 1] = x + p;
d[n1] = x + p;
e[n1 - 1] = z;
e[n1] = -z;
}
n1 = n1 - 2;
iter = 0;
// No convergence yet
} else {
// Form shift
x = H[n1][n1];
y = 0.0;
w = 0.0;
if (l < n1) {
y = H[n1 - 1][n1 - 1];
w = H[n1][n1 - 1] * H[n1 - 1][n1];
}
// Wilkinson's original ad hoc shift
if (iter == 10) {
exshift += x;
for (int i = low; i <= n1; i++) {
H[i][i] -= x;
}
s = std::abs(H[n1][n1 - 1]) + std::abs(H[n1 - 1][n1 - 2]);
x = y = 0.75 * s;
w = -0.4375 * s * s;
}
// MATLAB's new ad hoc shift
if (iter == 30) {
s = (y - x) / 2.0;
s = s * s + w;
if (s > 0) {
s = std::sqrt(s);
if (y < x) {
s = -s;
}
s = x - w / ((y - x) / 2.0 + s);
for (int i = low; i <= n1; i++) {
H[i][i] -= s;
}
exshift += s;
x = y = w = 0.964;
}
}
iter = iter + 1; // (Could check iteration count here.)
// Look for two consecutive small sub-diagonal elements
int m = n1 - 2;
while (m >= l) {
z = H[m][m];
r = x - z;
s = y - z;
p = (r * s - w) / H[m + 1][m] + H[m][m + 1];
q = H[m + 1][m + 1] - z - r - s;
r = H[m + 2][m + 1];
s = std::abs(p) + std::abs(q) + std::abs(r);
p = p / s;
q = q / s;
r = r / s;
if (m == l) {
break;
}
if (std::abs(H[m][m - 1]) * (std::abs(q) + std::abs(r)) < eps * (std::abs(p)
* (std::abs(H[m - 1][m - 1]) + std::abs(z) + std::abs(
H[m + 1][m + 1])))) {
break;
}
m--;
}
for (int i = m + 2; i <= n1; i++) {
H[i][i - 2] = 0.0;
if (i > m + 2) {
H[i][i - 3] = 0.0;
}
}
// Double QR step involving rows l:n and columns m:n
for (int k = m; k <= n1 - 1; k++) {
bool notlast = (k != n1 - 1);
if (k != m) {
p = H[k][k - 1];
q = H[k + 1][k - 1];
r = (notlast ? H[k + 2][k - 1] : 0.0);
x = std::abs(p) + std::abs(q) + std::abs(r);
if (x != 0.0) {
p = p / x;
q = q / x;
r = r / x;
}
}
if (x == 0.0) {
break;
}
s = std::sqrt(p * p + q * q + r * r);
if (p < 0) {
s = -s;
}
if (s != 0) {
if (k != m) {
H[k][k - 1] = -s * x;
} else if (l != m) {
H[k][k - 1] = -H[k][k - 1];
}
p = p + s;
x = p / s;
y = q / s;
z = r / s;
q = q / p;
r = r / p;
// Row modification
for (int j = k; j < nn; j++) {
p = H[k][j] + q * H[k + 1][j];
if (notlast) {
p = p + r * H[k + 2][j];
H[k + 2][j] = H[k + 2][j] - p * z;
}
H[k][j] = H[k][j] - p * x;
H[k + 1][j] = H[k + 1][j] - p * y;
}
// Column modification
for (int i = 0; i <= std::min(n1, k + 3); i++) {
p = x * H[i][k] + y * H[i][k + 1];
if (notlast) {
p = p + z * H[i][k + 2];
H[i][k + 2] = H[i][k + 2] - p * r;
}
H[i][k] = H[i][k] - p;
H[i][k + 1] = H[i][k + 1] - p * q;
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
p = x * V[i][k] + y * V[i][k + 1];
if (notlast) {
p = p + z * V[i][k + 2];
V[i][k + 2] = V[i][k + 2] - p * r;
}
V[i][k] = V[i][k] - p;
V[i][k + 1] = V[i][k + 1] - p * q;
}
} // (s != 0)
} // k loop
} // check convergence
} // while (n1 >= low)
// Backsubstitute to find vectors of upper triangular form
if (norm == 0.0) {
return;
}
for (n1 = nn - 1; n1 >= 0; n1--) {
p = d[n1];
q = e[n1];
// Real vector
if (q == 0) {
int l = n1;
H[n1][n1] = 1.0;
for (int i = n1 - 1; i >= 0; i--) {
w = H[i][i] - p;
r = 0.0;
for (int j = l; j <= n1; j++) {
r = r + H[i][j] * H[j][n1];
}
if (e[i] < 0.0) {
z = w;
s = r;
} else {
l = i;
if (e[i] == 0.0) {
if (w != 0.0) {
H[i][n1] = -r / w;
} else {
H[i][n1] = -r / (eps * norm);
}
// Solve real equations
} else {
x = H[i][i + 1];
y = H[i + 1][i];
q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
t = (x * s - z * r) / q;
H[i][n1] = t;
if (std::abs(x) > std::abs(z)) {
H[i + 1][n1] = (-r - w * t) / x;
} else {
H[i + 1][n1] = (-s - y * t) / z;
}
}
// Overflow control
t = std::abs(H[i][n1]);
if ((eps * t) * t > 1) {
for (int j = i; j <= n1; j++) {
H[j][n1] = H[j][n1] / t;
}
}
}
}
// Complex vector
} else if (q < 0) {
int l = n1 - 1;
// Last vector component imaginary so matrix is triangular
if (std::abs(H[n1][n1 - 1]) > std::abs(H[n1 - 1][n1])) {
H[n1 - 1][n1 - 1] = q / H[n1][n1 - 1];
H[n1 - 1][n1] = -(H[n1][n1] - p) / H[n1][n1 - 1];
} else {
cdiv(0.0, -H[n1 - 1][n1], H[n1 - 1][n1 - 1] - p, q);
H[n1 - 1][n1 - 1] = cdivr;
H[n1 - 1][n1] = cdivi;
}
H[n1][n1 - 1] = 0.0;
H[n1][n1] = 1.0;
for (int i = n1 - 2; i >= 0; i--) {
double ra, sa, vr, vi;
ra = 0.0;
sa = 0.0;
for (int j = l; j <= n1; j++) {
ra = ra + H[i][j] * H[j][n1 - 1];
sa = sa + H[i][j] * H[j][n1];
}
w = H[i][i] - p;
if (e[i] < 0.0) {
z = w;
r = ra;
s = sa;
} else {
l = i;
if (e[i] == 0) {
cdiv(-ra, -sa, w, q);
H[i][n1 - 1] = cdivr;
H[i][n1] = cdivi;
} else {
// Solve complex equations
x = H[i][i + 1];
y = H[i + 1][i];
vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
vi = (d[i] - p) * 2.0 * q;
if (vr == 0.0 && vi == 0.0) {
vr = eps * norm * (std::abs(w) + std::abs(q) + std::abs(x)
+ std::abs(y) + std::abs(z));
}
cdiv(x * r - z * ra + q * sa,
x * s - z * sa - q * ra, vr, vi);
H[i][n1 - 1] = cdivr;
H[i][n1] = cdivi;
if (std::abs(x) > (std::abs(z) + std::abs(q))) {
H[i + 1][n1 - 1] = (-ra - w * H[i][n1 - 1] + q
* H[i][n1]) / x;
H[i + 1][n1] = (-sa - w * H[i][n1] - q * H[i][n1
- 1]) / x;
} else {
cdiv(-r - y * H[i][n1 - 1], -s - y * H[i][n1], z,
q);
H[i + 1][n1 - 1] = cdivr;
H[i + 1][n1] = cdivi;
}
}
// Overflow control
t = std::max(std::abs(H[i][n1 - 1]), std::abs(H[i][n1]));
if ((eps * t) * t > 1) {
for (int j = i; j <= n1; j++) {
H[j][n1 - 1] = H[j][n1 - 1] / t;
H[j][n1] = H[j][n1] / t;
}
}
}
}
}
}
// Vectors of isolated roots
for (int i = 0; i < nn; i++) {
if (i < low || i > high) {
for (int j = i; j < nn; j++) {
V[i][j] = H[i][j];
}
}
}
// Back transformation to get eigenvectors of original matrix
for (int j = nn - 1; j >= low; j--) {
for (int i = low; i <= high; i++) {
z = 0.0;
for (int k = low; k <= std::min(j, high); k++) {
z = z + V[i][k] * H[k][j];
}
V[i][j] = z;
}
}
}