docs: private void hqr2 (double[][] V, double[][] H) {
     
        //  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 n = nn-1;
        int low = 0;
        int high = nn-1;
        double eps = Math.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 = Math.max(i-1,0); j < nn; j++) {
              norm = norm + Math.abs(H[i][j]);
           }
        }
     
        // Outer loop over eigenvalue index
     
        int iter = 0;
        while (n >= low) {
     
           // Look for single small sub-diagonal element
     
           int l = n;
           while (l > low) {
              s = Math.abs(H[l-1][l-1]) + Math.abs(H[l][l]);
              if (s == 0.0) {
                 s = norm;
              }
              if (Math.abs(H[l][l-1]) < eps * s) {
                 break;
              }
              l--;
           }
         
           // Check for convergence
           // One root found
     
           if (l == n) {
              H[n][n] = H[n][n] + exshift;
              d[n] = H[n][n];
              e[n] = 0.0;
              n--;
              iter = 0;
     
           // Two roots found
     
           } else if (l == n-1) {
              w = H[n][n-1] * H[n-1][n];
              p = (H[n-1][n-1] - H[n][n]) / 2.0;
              q = p * p + w;
              z = Math.sqrt(Math.abs(q));
              H[n][n] = H[n][n] + exshift;
              H[n-1][n-1] = H[n-1][n-1] + exshift;
              x = H[n][n];
     
              // Real pair
     
              if (q >= 0) {
                 if (p >= 0) {
                    z = p + z;
                 } else {
                    z = p - z;
                 }
                 d[n-1] = x + z;
                 d[n] = d[n-1];
                 if (z != 0.0) {
                    d[n] = x - w / z;
                 }
                 e[n-1] = 0.0;
                 e[n] = 0.0;
                 x = H[n][n-1];
                 s = Math.abs(x) + Math.abs(z);
                 p = x / s;
                 q = z / s;
                 r = Math.sqrt(p * p+q * q);
                 p = p / r;
                 q = q / r;
     
                 // Row modification
     
                 for (int j = n-1; j < nn; j++) {
                    z = H[n-1][j];
                    H[n-1][j] = q * z + p * H[n][j];
                    H[n][j] = q * H[n][j] - p * z;
                 }
     
                 // Column modification
     
                 for (int i = 0; i <= n; i++) {
                    z = H[i][n-1];
                    H[i][n-1] = q * z + p * H[i][n];
                    H[i][n] = q * H[i][n] - p * z;
                 }
     
                 // Accumulate transformations
     
                 for (int i = low; i <= high; i++) {
                    z = V[i][n-1];
                    V[i][n-1] = q * z + p * V[i][n];
                    V[i][n] = q * V[i][n] - p * z;
                 }
     
              // Complex pair
     
              } else {
                 d[n-1] = x + p;
                 d[n] = x + p;
                 e[n-1] = z;
                 e[n] = -z;
              }
              n = n - 2;
              iter = 0;
     
           // No convergence yet
     
           } else {
     
              // Form shift
     
              x = H[n][n];
              y = 0.0;
              w = 0.0;
              if (l < n) {
                 y = H[n-1][n-1];
                 w = H[n][n-1] * H[n-1][n];
              }
     
              // Wilkinson's original ad hoc shift
     
              if (iter == 10) {
                 exshift += x;
                 for (int i = low; i <= n; i++) {
                    H[i][i] -= x;
                 }
                 s = Math.abs(H[n][n-1]) + Math.abs(H[n-1][n-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 = Math.sqrt(s);
                      if (y < x) {
                         s = -s;
                      }
                      s = x - w / ((y - x) / 2.0 + s);
                      for (int i = low; i <= n; 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 = n-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 = Math.abs(p) + Math.abs(q) + Math.abs(r);
                 p = p / s;
                 q = q / s;
                 r = r / s;
                 if (m == l) {
                    break;
                 }
                 if (Math.abs(H[m][m-1]) * (Math.abs(q) + Math.abs(r)) <
                    eps * (Math.abs(p) * (Math.abs(H[m-1][m-1]) + Math.abs(z) +
                    Math.abs(H[m+1][m+1])))) {
                       break;
                 }
                 m--;
              }
     
              for (int i = m+2; i <= n; 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 <= n-1; k++) {
                 boolean notlast = (k != n-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 = Math.abs(p) + Math.abs(q) + Math.abs(r);
                    if (x != 0.0) {
                       p = p / x;
                       q = q / x;
                       r = r / x;
                    }
                 }
                 if (x == 0.0) {
                    break;
                 }
                 s = Math.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 <= Math.min(n,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 (n >= low)
        
        // Backsubstitute to find vectors of upper triangular form
  
        if (norm == 0.0) {
           return;
        }
     
        for (n = nn-1; n >= 0; n--) {
           p = d[n];
           q = e[n];
     
           // Real vector
     
           if (q == 0) {
              int l = n;
              H[n][n] = 1.0;
              for (int i = n-1; i >= 0; i--) {
                 w = H[i][i] - p;
                 r = 0.0;
                 for (int j = l; j <= n; j++) {
                    r = r + H[i][j] * H[j][n];
                 }
                 if (e[i] < 0.0) {
                    z = w;
                    s = r;
                 } else {
                    l = i;
                    if (e[i] == 0.0) {
                       if (w != 0.0) {
                          H[i][n] = -r / w;
                       } else {
                          H[i][n] = -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][n] = t;
                       if (Math.abs(x) > Math.abs(z)) {
                          H[i+1][n] = (-r - w * t) / x;
                       } else {
                          H[i+1][n] = (-s - y * t) / z;
                       }
                    }
     
                    // Overflow control
     
                    t = Math.abs(H[i][n]);
                    if ((eps * t) * t > 1) {
                       for (int j = i; j <= n; j++) {
                          H[j][n] = H[j][n] / t;
                       }
                    }
                 }
              }
     
           // Complex vector
     
           } else if (q < 0) {
              int l = n-1;
  
              // Last vector component imaginary so matrix is triangular
     
              if (Math.abs(H[n][n-1]) > Math.abs(H[n-1][n])) {
                 H[n-1][n-1] = q / H[n][n-1];
                 H[n-1][n] = -(H[n][n] - p) / H[n][n-1];
              } else {
                 cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);
                 H[n-1][n-1] = cdivr;
                 H[n-1][n] = cdivi;
              }
              H[n][n-1] = 0.0;
              H[n][n] = 1.0;
              for (int i = n-2; i >= 0; i--) {
                 double ra,sa,vr,vi;
                 ra = 0.0;
                 sa = 0.0;
                 for (int j = l; j <= n; j++) {
                    ra = ra + H[i][j] * H[j][n-1];
                    sa = sa + H[i][j] * H[j][n];
                 }
                 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][n-1] = cdivr;
                       H[i][n] = 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 * (Math.abs(w) + Math.abs(q) +
                          Math.abs(x) + Math.abs(y) + Math.abs(z));
                       }
                       cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
                       H[i][n-1] = cdivr;
                       H[i][n] = cdivi;
                       if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {
                          H[i+1][n-1] = (-ra - w * H[i][n-1] + q * H[i][n]) / x;
                          H[i+1][n] = (-sa - w * H[i][n] - q * H[i][n-1]) / x;
                       } else {
                          cdiv(-r-y*H[i][n-1],-s-y*H[i][n],z,q);
                          H[i+1][n-1] = cdivr;
                          H[i+1][n] = cdivi;
                       }
                    }
     
                    // Overflow control
  
                    t = Math.max(Math.abs(H[i][n-1]),Math.abs(H[i][n]));
                    if ((eps * t) * t > 1) {
                       for (int j = i; j <= n; j++) {
                          H[j][n-1] = H[j][n-1] / t;
                          H[j][n] = H[j][n] / 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 <= Math.min(j,high); k++) {
                 z = z + V[i][k] * H[k][j];
              }
              V[i][j] = z;
           }
        }
     }