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Edit File : eigenvalue_decomposition.rb \| Size : 21.62 KB
# frozen_string_literal: false class Matrix # Adapted from JAMA: http://math.nist.gov/javanumerics/jama/ # Eigenvalues and eigenvectors of a real matrix. # # Computes the eigenvalues and eigenvectors of a matrix A. # # If A is diagonalizable, this provides matrices V and D # such that A = VDV.inv, where D is the diagonal matrix with entries # equal to the eigenvalues and V is formed by the eigenvectors. # # If A is symmetric, then V is orthogonal and thus A = VDV.t class EigenvalueDecomposition # Constructs the eigenvalue decomposition for a square matrix +A+ # def initialize(a) # @d, @e: Arrays for internal storage of eigenvalues. # @v: Array for internal storage of eigenvectors. # @h: Array for internal storage of nonsymmetric Hessenberg form. raise TypeError, "Expected Matrix but got #{a.class}" unless a.is_a?(Matrix) @size = a.row_count @d = Array.new(@size, 0) @e = Array.new(@size, 0) if (@symmetric = a.symmetric?) @v = a.to_a tridiagonalize diagonalize else @v = Array.new(@size) { Array.new(@size, 0) } @h = a.to_a @ort = Array.new(@size, 0) reduce_to_hessenberg hessenberg_to_real_schur end end # Returns the eigenvector matrix +V+ # def eigenvector_matrix Matrix.send(:new, build_eigenvectors.transpose) end alias_method :v, :eigenvector_matrix # Returns the inverse of the eigenvector matrix +V+ # def eigenvector_matrix_inv r = Matrix.send(:new, build_eigenvectors) r = r.transpose.inverse unless @symmetric r end alias_method :v_inv, :eigenvector_matrix_inv # Returns the eigenvalues in an array # def eigenvalues values = @d.dup @e.each_with_index{\|imag, i\| values[i] = Complex(values[i], imag) unless imag == 0} values end # Returns an array of the eigenvectors # def eigenvectors build_eigenvectors.map{\|ev\| Vector.send(:new, ev)} end # Returns the block diagonal eigenvalue matrix +D+ # def eigenvalue_matrix Matrix.diagonal(eigenvalues) end alias_method :d, :eigenvalue_matrix # Returns [eigenvector_matrix, eigenvalue_matrix, eigenvector_matrix_inv] # def to_ary [v, d, v_inv] end alias_method :to_a, :to_ary private def build_eigenvectors # JAMA stores complex eigenvectors in a strange way # See http://web.archive.org/web/20111016032731/http://cio.nist.gov/esd/emaildir/lists/jama/msg01021.html @e.each_with_index.map do \|imag, i\| if imag == 0 Array.new(@size){\|j\| @v[j][i]} elsif imag > 0 Array.new(@size){\|j\| Complex(@v[j][i], @v[j][i+1])} else Array.new(@size){\|j\| Complex(@v[j][i-1], -@v[j][i])} end end end # Complex scalar division. private def cdiv(xr, xi, yr, yi) if (yr.abs > yi.abs) r = yi/yr d = yr + ryi [(xr + rxi)/d, (xi - rxr)/d] else r = yr/yi d = yi + ryr [(rxr + xi)/d, (rxi - xr)/d] end end # Symmetric Householder reduction to tridiagonal form. private def tridiagonalize # This is derived from the Algol procedures tred2 by # Bowdler, Martin, Reinsch, and Wilkinson, Handbook for # Auto. Comp., Vol.ii-Linear Algebra, and the corresponding # Fortran subroutine in EISPACK. @size.times do \|j\| @d[j] = @v[@size-1][j] end # Householder reduction to tridiagonal form. (@size-1).downto(0+1) do \|i\| # Scale to avoid under/overflow. scale = 0.0 h = 0.0 i.times do \|k\| scale = scale + @d[k].abs end if (scale == 0.0) @e[i] = @d[i-1] i.times do \|j\| @d[j] = @v[i-1][j] @v[i][j] = 0.0 @v[j][i] = 0.0 end else # Generate Householder vector. i.times do \|k\| @d[k] /= scale h += @d[k] @d[k] end f = @d[i-1] g = Math.sqrt(h) if (f > 0) g = -g end @e[i] = scale * g h -= f * g @d[i-1] = f - g i.times do \|j\| @e[j] = 0.0 end # Apply similarity transformation to remaining columns. i.times do \|j\| f = @d[j] @v[j][i] = f g = @e[j] + @v[j][j] * f (j+1).upto(i-1) do \|k\| g += @v[k][j] * @d[k] @e[k] += @v[k][j] * f end @e[j] = g end f = 0.0 i.times do \|j\| @e[j] /= h f += @e[j] * @d[j] end hh = f / (h + h) i.times do \|j\| @e[j] -= hh * @d[j] end i.times do \|j\| f = @d[j] g = @e[j] j.upto(i-1) do \|k\| @v[k][j] -= (f * @e[k] + g * @d[k]) end @d[j] = @v[i-1][j] @v[i][j] = 0.0 end end @d[i] = h end # Accumulate transformations. 0.upto(@size-1-1) do \|i\| @v[@size-1][i] = @v[i][i] @v[i][i] = 1.0 h = @d[i+1] if (h != 0.0) 0.upto(i) do \|k\| @d[k] = @v[k][i+1] / h end 0.upto(i) do \|j\| g = 0.0 0.upto(i) do \|k\| g += @v[k][i+1] * @v[k][j] end 0.upto(i) do \|k\| @v[k][j] -= g * @d[k] end end end 0.upto(i) do \|k\| @v[k][i+1] = 0.0 end end @size.times do \|j\| @d[j] = @v[@size-1][j] @v[@size-1][j] = 0.0 end @v[@size-1][@size-1] = 1.0 @e[0] = 0.0 end # Symmetric tridiagonal QL algorithm. private def diagonalize # This is derived from the Algol procedures tql2, by # Bowdler, Martin, Reinsch, and Wilkinson, Handbook for # Auto. Comp., Vol.ii-Linear Algebra, and the corresponding # Fortran subroutine in EISPACK. 1.upto(@size-1) do \|i\| @e[i-1] = @e[i] end @e[@size-1] = 0.0 f = 0.0 tst1 = 0.0 eps = Float::EPSILON @size.times do \|l\| # Find small subdiagonal element tst1 = [tst1, @d[l].abs + @e[l].abs].max m = l while (m < @size) do if (@e[m].abs <= epstst1) break end m+=1 end # If m == l, @d[l] is an eigenvalue, # otherwise, iterate. if (m > l) iter = 0 begin iter = iter + 1 # (Could check iteration count here.) # Compute implicit shift g = @d[l] p = (@d[l+1] - g) / (2.0 @e[l]) r = Math.hypot(p, 1.0) if (p < 0) r = -r end @d[l] = @e[l] / (p + r) @d[l+1] = @e[l] * (p + r) dl1 = @d[l+1] h = g - @d[l] (l+2).upto(@size-1) do \|i\| @d[i] -= h end f += h # Implicit QL transformation. p = @d[m] c = 1.0 c2 = c c3 = c el1 = @e[l+1] s = 0.0 s2 = 0.0 (m-1).downto(l) do \|i\| c3 = c2 c2 = c s2 = s g = c * @e[i] h = c * p r = Math.hypot(p, @e[i]) @e[i+1] = s * r s = @e[i] / r c = p / r p = c * @d[i] - s * g @d[i+1] = h + s * (c * g + s * @d[i]) # Accumulate transformation. @size.times do \|k\| h = @v[k][i+1] @v[k][i+1] = s * @v[k][i] + c * h @v[k][i] = c * @v[k][i] - s * h end end p = -s * s2 * c3 * el1 * @e[l] / dl1 @e[l] = s * p @d[l] = c * p # Check for convergence. end while (@e[l].abs > epstst1) end @d[l] = @d[l] + f @e[l] = 0.0 end # Sort eigenvalues and corresponding vectors. 0.upto(@size-2) do \|i\| k = i p = @d[i] (i+1).upto(@size-1) do \|j\| if (@d[j] < p) k = j p = @d[j] end end if (k != i) @d[k] = @d[i] @d[i] = p @size.times do \|j\| p = @v[j][i] @v[j][i] = @v[j][k] @v[j][k] = p end end end end # Nonsymmetric reduction to Hessenberg form. private def reduce_to_hessenberg # This is derived from the Algol procedures orthes and ortran, # by Martin and Wilkinson, Handbook for Auto. Comp., # Vol.ii-Linear Algebra, and the corresponding # Fortran subroutines in EISPACK. low = 0 high = @size-1 (low+1).upto(high-1) do \|m\| # Scale column. scale = 0.0 m.upto(high) do \|i\| scale = scale + @h[i][m-1].abs end if (scale != 0.0) # Compute Householder transformation. h = 0.0 high.downto(m) do \|i\| @ort[i] = @h[i][m-1]/scale h += @ort[i] @ort[i] end g = Math.sqrt(h) if (@ort[m] > 0) g = -g end h -= @ort[m] * g @ort[m] = @ort[m] - g # Apply Householder similarity transformation # @h = (I-uu'/h)@h(I-uu')/h) m.upto(@size-1) do \|j\| f = 0.0 high.downto(m) do \|i\| f += @ort[i]@h[i][j] end f = f/h m.upto(high) do \|i\| @h[i][j] -= f@ort[i] end end 0.upto(high) do \|i\| f = 0.0 high.downto(m) do \|j\| f += @ort[j]@h[i][j] end f = f/h m.upto(high) do \|j\| @h[i][j] -= f@ort[j] end end @ort[m] = scale@ort[m] @h[m][m-1] = scaleg end end # Accumulate transformations (Algol's ortran). @size.times do \|i\| @size.times do \|j\| @v[i][j] = (i == j ? 1.0 : 0.0) end end (high-1).downto(low+1) do \|m\| if (@h[m][m-1] != 0.0) (m+1).upto(high) do \|i\| @ort[i] = @h[i][m-1] end m.upto(high) do \|j\| g = 0.0 m.upto(high) do \|i\| g += @ort[i] * @v[i][j] end # Double division avoids possible underflow g = (g / @ort[m]) / @h[m][m-1] m.upto(high) do \|i\| @v[i][j] += g * @ort[i] end end end end end # Nonsymmetric reduction from Hessenberg to real Schur form. private def hessenberg_to_real_schur # 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 nn = @size n = nn-1 low = 0 high = nn-1 eps = Float::EPSILON exshift = 0.0 p = q = r = s = z = 0 # Store roots isolated by balanc and compute matrix norm norm = 0.0 nn.times do \|i\| if (i < low \|\| i > high) @d[i] = @h[i][i] @e[i] = 0.0 end ([i-1, 0].max).upto(nn-1) do \|j\| norm = norm + @h[i][j].abs end end # Outer loop over eigenvalue index iter = 0 while (n >= low) do # Look for single small sub-diagonal element l = n while (l > low) do s = @h[l-1][l-1].abs + @h[l][l].abs if (s == 0.0) s = norm end if (@h[l][l-1].abs < eps * s) break end l-=1 end # 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-=1 iter = 0 # Two roots found elsif (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(q.abs) @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 end @d[n-1] = x + z @d[n] = @d[n-1] if (z != 0.0) @d[n] = x - w / z end @e[n-1] = 0.0 @e[n] = 0.0 x = @h[n][n-1] s = x.abs + z.abs p = x / s q = z / s r = Math.sqrt(p * p+q * q) p /= r q /= r # Row modification (n-1).upto(nn-1) do \|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 end # Column modification 0.upto(n) do \|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 end # Accumulate transformations low.upto(high) do \|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 end # Complex pair else @d[n-1] = x + p @d[n] = x + p @e[n-1] = z @e[n] = -z end 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] end # Wilkinson's original ad hoc shift if (iter == 10) exshift += x low.upto(n) do \|i\| @h[i][i] -= x end s = @h[n][n-1].abs + @h[n-1][n-2].abs x = y = 0.75 * s w = -0.4375 * s * s end # MATLAB's new ad hoc shift if (iter == 30) s = (y - x) / 2.0 s = s + w if (s > 0) s = Math.sqrt(s) if (y < x) s = -s end s = x - w / ((y - x) / 2.0 + s) low.upto(n) do \|i\| @h[i][i] -= s end exshift += s x = y = w = 0.964 end end iter = iter + 1 # (Could check iteration count here.) # Look for two consecutive small sub-diagonal elements m = n-2 while (m >= l) do 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 = p.abs + q.abs + r.abs p /= s q /= s r /= s if (m == l) break end if (@h[m][m-1].abs * (q.abs + r.abs) < eps * (p.abs * (@h[m-1][m-1].abs + z.abs + @h[m+1][m+1].abs))) break end m-=1 end (m+2).upto(n) do \|i\| @h[i][i-2] = 0.0 if (i > m+2) @h[i][i-3] = 0.0 end end # Double QR step involving rows l:n and columns m:n m.upto(n-1) do \|k\| 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 = p.abs + q.abs + r.abs next if x == 0 p /= x q /= x r /= x end s = Math.sqrt(p * p + q * q + r * r) if (p < 0) s = -s end if (s != 0) if (k != m) @h[k][k-1] = -s * x elsif (l != m) @h[k][k-1] = -@h[k][k-1] end p += s x = p / s y = q / s z = r / s q /= p r /= p # Row modification k.upto(nn-1) do \|j\| p = @h[k][j] + q * @h[k+1][j] if (notlast) p += r * @h[k+2][j] @h[k+2][j] = @h[k+2][j] - p * z end @h[k][j] = @h[k][j] - p * x @h[k+1][j] = @h[k+1][j] - p * y end # Column modification 0.upto([n, k+3].min) do \|i\| p = x * @h[i][k] + y * @h[i][k+1] if (notlast) p += z * @h[i][k+2] @h[i][k+2] = @h[i][k+2] - p * r end @h[i][k] = @h[i][k] - p @h[i][k+1] = @h[i][k+1] - p * q end # Accumulate transformations low.upto(high) do \|i\| p = x * @v[i][k] + y * @v[i][k+1] if (notlast) p += z * @v[i][k+2] @v[i][k+2] = @v[i][k+2] - p * r end @v[i][k] = @v[i][k] - p @v[i][k+1] = @v[i][k+1] - p * q end end # (s != 0) end # k loop end # check convergence end # while (n >= low) # Backsubstitute to find vectors of upper triangular form if (norm == 0.0) return end (nn-1).downto(0) do \|k\| p = @d[k] q = @e[k] # Real vector if (q == 0) l = k @h[k][k] = 1.0 (k-1).downto(0) do \|i\| w = @h[i][i] - p r = 0.0 l.upto(k) do \|j\| r += @h[i][j] * @h[j][k] end if (@e[i] < 0.0) z = w s = r else l = i if (@e[i] == 0.0) if (w != 0.0) @h[i][k] = -r / w else @h[i][k] = -r / (eps * norm) end # 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][k] = t if (x.abs > z.abs) @h[i+1][k] = (-r - w * t) / x else @h[i+1][k] = (-s - y * t) / z end end # Overflow control t = @h[i][k].abs if ((eps * t) * t > 1) i.upto(k) do \|j\| @h[j][k] = @h[j][k] / t end end end end # Complex vector elsif (q < 0) l = n-1 # Last vector component imaginary so matrix is triangular if (@h[n][n-1].abs > @h[n-1][n].abs) @h[n-1][n-1] = q / @h[n][n-1] @h[n-1][n] = -(@h[n][n] - p) / @h[n][n-1] else cdivr, cdivi = 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 end @h[n][n-1] = 0.0 @h[n][n] = 1.0 (n-2).downto(0) do \|i\| ra = 0.0 sa = 0.0 l.upto(n) do \|j\| ra = ra + @h[i][j] * @h[j][n-1] sa = sa + @h[i][j] * @h[j][n] end w = @h[i][i] - p if (@e[i] < 0.0) z = w r = ra s = sa else l = i if (@e[i] == 0) cdivr, cdivi = 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 * (w.abs + q.abs + x.abs + y.abs + z.abs) end cdivr, cdivi = cdiv(xr-zra+qsa, xs-zsa-qra, vr, vi) @h[i][n-1] = cdivr @h[i][n] = cdivi if (x.abs > (z.abs + q.abs)) @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 cdivr, cdivi = 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 end end # Overflow control t = [@h[i][n-1].abs, @h[i][n].abs].max if ((eps * t) * t > 1) i.upto(n) do \|j\| @h[j][n-1] = @h[j][n-1] / t @h[j][n] = @h[j][n] / t end end end end end end # Vectors of isolated roots nn.times do \|i\| if (i < low \|\| i > high) i.upto(nn-1) do \|j\| @v[i][j] = @h[i][j] end end end # Back transformation to get eigenvectors of original matrix (nn-1).downto(low) do \|j\| low.upto(high) do \|i\| z = 0.0 low.upto([j, high].min) do \|k\| z += @v[i][k] * @h[k][j] end @v[i][j] = z end end end end end Back

Edit File : eigenvalue_decomposition.rb | Size : 21.62 KB

# frozen_string_literal: false
class Matrix
  # Adapted from JAMA: http://math.nist.gov/javanumerics/jama/

# Eigenvalues and eigenvectors of a real matrix.
  #
  # Computes the eigenvalues and eigenvectors of a matrix A.
  #
  # If A is diagonalizable, this provides matrices V and D
  # such that A = V*D*V.inv, where D is the diagonal matrix with entries
  # equal to the eigenvalues and V is formed by the eigenvectors.
  #
  # If A is symmetric, then V is orthogonal and thus A = V*D*V.t

class EigenvalueDecomposition

# Constructs the eigenvalue decomposition for a square matrix +A+
    #
    def initialize(a)
      # @d, @e: Arrays for internal storage of eigenvalues.
      # @v: Array for internal storage of eigenvectors.
      # @h: Array for internal storage of nonsymmetric Hessenberg form.
      raise TypeError, "Expected Matrix but got #{a.class}" unless a.is_a?(Matrix)
      @size = a.row_count
      @d = Array.new(@size, 0)
      @e = Array.new(@size, 0)

if (@symmetric = a.symmetric?)
        @v = a.to_a
        tridiagonalize
        diagonalize
      else
        @v = Array.new(@size) { Array.new(@size, 0) }
        @h = a.to_a
        @ort = Array.new(@size, 0)
        reduce_to_hessenberg
        hessenberg_to_real_schur
      end
    end

# Returns the eigenvector matrix +V+
    #
    def eigenvector_matrix
      Matrix.send(:new, build_eigenvectors.transpose)
    end
    alias_method :v, :eigenvector_matrix

# Returns the inverse of the eigenvector matrix +V+
    #
    def eigenvector_matrix_inv
      r = Matrix.send(:new, build_eigenvectors)
      r = r.transpose.inverse unless @symmetric
      r
    end
    alias_method :v_inv, :eigenvector_matrix_inv

# Returns the eigenvalues in an array
    #
    def eigenvalues
      values = @d.dup
      @e.each_with_index{|imag, i| values[i] = Complex(values[i], imag) unless imag == 0}
      values
    end

# Returns an array of the eigenvectors
    #
    def eigenvectors
      build_eigenvectors.map{|ev| Vector.send(:new, ev)}
    end

# Returns the block diagonal eigenvalue matrix +D+
    #
    def eigenvalue_matrix
      Matrix.diagonal(*eigenvalues)
    end
    alias_method :d, :eigenvalue_matrix

# Returns [eigenvector_matrix, eigenvalue_matrix, eigenvector_matrix_inv]
    #
    def to_ary
      [v, d, v_inv]
    end
    alias_method :to_a, :to_ary

private def build_eigenvectors
      # JAMA stores complex eigenvectors in a strange way
      # See http://web.archive.org/web/20111016032731/http://cio.nist.gov/esd/emaildir/lists/jama/msg01021.html
      @e.each_with_index.map do |imag, i|
        if imag == 0
          Array.new(@size){|j| @v[j][i]}
        elsif imag > 0
          Array.new(@size){|j| Complex(@v[j][i], @v[j][i+1])}
        else
          Array.new(@size){|j| Complex(@v[j][i-1], -@v[j][i])}
        end
      end
    end

# Complex scalar division.

private def cdiv(xr, xi, yr, yi)
      if (yr.abs > yi.abs)
        r = yi/yr
        d = yr + r*yi
        [(xr + r*xi)/d, (xi - r*xr)/d]
      else
        r = yr/yi
        d = yi + r*yr
        [(r*xr + xi)/d, (r*xi - xr)/d]
      end
    end

# Symmetric Householder reduction to tridiagonal form.

private def tridiagonalize

#  This is derived from the Algol procedures tred2 by
      #  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
      #  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
      #  Fortran subroutine in EISPACK.

@size.times do |j|
        @d[j] = @v[@size-1][j]
      end

# Householder reduction to tridiagonal form.

(@size-1).downto(0+1) do |i|

# Scale to avoid under/overflow.

scale = 0.0
        h = 0.0
        i.times do |k|
          scale = scale + @d[k].abs
        end
        if (scale == 0.0)
          @e[i] = @d[i-1]
          i.times do |j|
            @d[j] = @v[i-1][j]
            @v[i][j] = 0.0
            @v[j][i] = 0.0
          end
        else

# Generate Householder vector.

i.times do |k|
            @d[k] /= scale
            h += @d[k] * @d[k]
          end
          f = @d[i-1]
          g = Math.sqrt(h)
          if (f > 0)
            g = -g
          end
          @e[i] = scale * g
          h -= f * g
          @d[i-1] = f - g
          i.times do |j|
            @e[j] = 0.0
          end

# Apply similarity transformation to remaining columns.

i.times do |j|
            f = @d[j]
            @v[j][i] = f
            g = @e[j] + @v[j][j] * f
            (j+1).upto(i-1) do |k|
              g += @v[k][j] * @d[k]
              @e[k] += @v[k][j] * f
            end
            @e[j] = g
          end
          f = 0.0
          i.times do |j|
            @e[j] /= h
            f += @e[j] * @d[j]
          end
          hh = f / (h + h)
          i.times do |j|
            @e[j] -= hh * @d[j]
          end
          i.times do |j|
            f = @d[j]
            g = @e[j]
            j.upto(i-1) do |k|
              @v[k][j] -= (f * @e[k] + g * @d[k])
            end
            @d[j] = @v[i-1][j]
            @v[i][j] = 0.0
          end
        end
        @d[i] = h
      end

# Accumulate transformations.

0.upto(@size-1-1) do |i|
        @v[@size-1][i] = @v[i][i]
        @v[i][i] = 1.0
        h = @d[i+1]
        if (h != 0.0)
          0.upto(i) do |k|
            @d[k] = @v[k][i+1] / h
          end
          0.upto(i) do |j|
            g = 0.0
            0.upto(i) do |k|
              g += @v[k][i+1] * @v[k][j]
            end
            0.upto(i) do |k|
              @v[k][j] -= g * @d[k]
            end
          end
        end
        0.upto(i) do |k|
          @v[k][i+1] = 0.0
        end
      end
      @size.times do |j|
        @d[j] = @v[@size-1][j]
        @v[@size-1][j] = 0.0
      end
      @v[@size-1][@size-1] = 1.0
      @e[0] = 0.0
    end

# Symmetric tridiagonal QL algorithm.

private def diagonalize
      #  This is derived from the Algol procedures tql2, by
      #  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
      #  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
      #  Fortran subroutine in EISPACK.

1.upto(@size-1) do |i|
        @e[i-1] = @e[i]
      end
      @e[@size-1] = 0.0

f = 0.0
      tst1 = 0.0
      eps = Float::EPSILON
      @size.times do |l|

# Find small subdiagonal element

tst1 = [tst1, @d[l].abs + @e[l].abs].max
        m = l
        while (m < @size) do
          if (@e[m].abs <= eps*tst1)
            break
          end
          m+=1
        end

# If m == l, @d[l] is an eigenvalue,
        # otherwise, iterate.

if (m > l)
          iter = 0
          begin
            iter = iter + 1  # (Could check iteration count here.)

# Compute implicit shift

g = @d[l]
            p = (@d[l+1] - g) / (2.0 * @e[l])
            r = Math.hypot(p, 1.0)
            if (p < 0)
              r = -r
            end
            @d[l] = @e[l] / (p + r)
            @d[l+1] = @e[l] * (p + r)
            dl1 = @d[l+1]
            h = g - @d[l]
            (l+2).upto(@size-1) do |i|
              @d[i] -= h
            end
            f += h

# Implicit QL transformation.

p = @d[m]
            c = 1.0
            c2 = c
            c3 = c
            el1 = @e[l+1]
            s = 0.0
            s2 = 0.0
            (m-1).downto(l) do |i|
              c3 = c2
              c2 = c
              s2 = s
              g = c * @e[i]
              h = c * p
              r = Math.hypot(p, @e[i])
              @e[i+1] = s * r
              s = @e[i] / r
              c = p / r
              p = c * @d[i] - s * g
              @d[i+1] = h + s * (c * g + s * @d[i])

# Accumulate transformation.

@size.times do |k|
                h = @v[k][i+1]
                @v[k][i+1] = s * @v[k][i] + c * h
                @v[k][i] = c * @v[k][i] - s * h
              end
            end
            p = -s * s2 * c3 * el1 * @e[l] / dl1
            @e[l] = s * p
            @d[l] = c * p

# Check for convergence.

end while (@e[l].abs > eps*tst1)
        end
        @d[l] = @d[l] + f
        @e[l] = 0.0
      end

# Sort eigenvalues and corresponding vectors.

0.upto(@size-2) do |i|
        k = i
        p = @d[i]
        (i+1).upto(@size-1) do |j|
          if (@d[j] < p)
            k = j
            p = @d[j]
          end
        end
        if (k != i)
          @d[k] = @d[i]
          @d[i] = p
          @size.times do |j|
            p = @v[j][i]
            @v[j][i] = @v[j][k]
            @v[j][k] = p
          end
        end
      end
    end

# Nonsymmetric reduction to Hessenberg form.

private def reduce_to_hessenberg
      #  This is derived from the Algol procedures orthes and ortran,
      #  by Martin and Wilkinson, Handbook for Auto. Comp.,
      #  Vol.ii-Linear Algebra, and the corresponding
      #  Fortran subroutines in EISPACK.

low = 0
      high = @size-1

(low+1).upto(high-1) do |m|

# Scale column.

scale = 0.0
        m.upto(high) do |i|
          scale = scale + @h[i][m-1].abs
        end
        if (scale != 0.0)

# Compute Householder transformation.

h = 0.0
          high.downto(m) do |i|
            @ort[i] = @h[i][m-1]/scale
            h += @ort[i] * @ort[i]
          end
          g = Math.sqrt(h)
          if (@ort[m] > 0)
            g = -g
          end
          h -= @ort[m] * g
          @ort[m] = @ort[m] - g

# Apply Householder similarity transformation
          # @h = (I-u*u'/h)*@h*(I-u*u')/h)

m.upto(@size-1) do |j|
            f = 0.0
            high.downto(m) do |i|
              f += @ort[i]*@h[i][j]
            end
            f = f/h
            m.upto(high) do |i|
              @h[i][j] -= f*@ort[i]
            end
          end

0.upto(high) do |i|
            f = 0.0
            high.downto(m) do |j|
              f += @ort[j]*@h[i][j]
            end
            f = f/h
            m.upto(high) do |j|
              @h[i][j] -= f*@ort[j]
            end
          end
          @ort[m] = scale*@ort[m]
          @h[m][m-1] = scale*g
        end
      end

# Accumulate transformations (Algol's ortran).

@size.times do |i|
        @size.times do |j|
          @v[i][j] = (i == j ? 1.0 : 0.0)
        end
      end

(high-1).downto(low+1) do |m|
        if (@h[m][m-1] != 0.0)
          (m+1).upto(high) do |i|
            @ort[i] = @h[i][m-1]
          end
          m.upto(high) do |j|
            g = 0.0
            m.upto(high) do |i|
              g += @ort[i] * @v[i][j]
            end
            # Double division avoids possible underflow
            g = (g / @ort[m]) / @h[m][m-1]
            m.upto(high) do |i|
              @v[i][j] += g * @ort[i]
            end
          end
        end
      end
    end

# Nonsymmetric reduction from Hessenberg to real Schur form.

private def hessenberg_to_real_schur

#  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

nn = @size
      n = nn-1
      low = 0
      high = nn-1
      eps = Float::EPSILON
      exshift = 0.0
      p = q = r = s = z = 0

# Store roots isolated by balanc and compute matrix norm

norm = 0.0
      nn.times do |i|
        if (i < low || i > high)
          @d[i] = @h[i][i]
          @e[i] = 0.0
        end
        ([i-1, 0].max).upto(nn-1) do |j|
          norm = norm + @h[i][j].abs
        end
      end

# Outer loop over eigenvalue index

iter = 0
      while (n >= low) do

# Look for single small sub-diagonal element

l = n
        while (l > low) do
          s = @h[l-1][l-1].abs + @h[l][l].abs
          if (s == 0.0)
            s = norm
          end
          if (@h[l][l-1].abs < eps * s)
            break
          end
          l-=1
        end

# 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-=1
          iter = 0

# Two roots found

elsif (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(q.abs)
          @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
            end
            @d[n-1] = x + z
            @d[n] = @d[n-1]
            if (z != 0.0)
              @d[n] = x - w / z
            end
            @e[n-1] = 0.0
            @e[n] = 0.0
            x = @h[n][n-1]
            s = x.abs + z.abs
            p = x / s
            q = z / s
            r = Math.sqrt(p * p+q * q)
            p /= r
            q /= r

# Row modification

(n-1).upto(nn-1) do |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
            end

# Column modification

0.upto(n) do |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
            end

# Accumulate transformations

low.upto(high) do |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
            end

# Complex pair

else
            @d[n-1] = x + p
            @d[n] = x + p
            @e[n-1] = z
            @e[n] = -z
          end
          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]
          end

# Wilkinson's original ad hoc shift

if (iter == 10)
            exshift += x
            low.upto(n) do |i|
              @h[i][i] -= x
            end
            s = @h[n][n-1].abs + @h[n-1][n-2].abs
            x = y = 0.75 * s
            w = -0.4375 * s * s
          end

# MATLAB's new ad hoc shift

if (iter == 30)
             s = (y - x) / 2.0
             s *= s + w
             if (s > 0)
                s = Math.sqrt(s)
                if (y < x)
                  s = -s
                end
                s = x - w / ((y - x) / 2.0 + s)
                low.upto(n) do |i|
                  @h[i][i] -= s
                end
                exshift += s
                x = y = w = 0.964
             end
          end

iter = iter + 1  # (Could check iteration count here.)

# Look for two consecutive small sub-diagonal elements

m = n-2
          while (m >= l) do
            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 = p.abs + q.abs + r.abs
            p /= s
            q /= s
            r /= s
            if (m == l)
              break
            end
            if (@h[m][m-1].abs * (q.abs + r.abs) <
              eps * (p.abs * (@h[m-1][m-1].abs + z.abs +
              @h[m+1][m+1].abs)))
                break
            end
            m-=1
          end

(m+2).upto(n) do |i|
            @h[i][i-2] = 0.0
            if (i > m+2)
              @h[i][i-3] = 0.0
            end
          end

# Double QR step involving rows l:n and columns m:n

m.upto(n-1) do |k|
            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 = p.abs + q.abs + r.abs
              next if x == 0
              p /= x
              q /= x
              r /= x
            end
            s = Math.sqrt(p * p + q * q + r * r)
            if (p < 0)
              s = -s
            end
            if (s != 0)
              if (k != m)
                @h[k][k-1] = -s * x
              elsif (l != m)
                @h[k][k-1] = -@h[k][k-1]
              end
              p += s
              x = p / s
              y = q / s
              z = r / s
              q /= p
              r /= p

# Row modification

k.upto(nn-1) do |j|
                p = @h[k][j] + q * @h[k+1][j]
                if (notlast)
                  p += r * @h[k+2][j]
                  @h[k+2][j] = @h[k+2][j] - p * z
                end
                @h[k][j] = @h[k][j] - p * x
                @h[k+1][j] = @h[k+1][j] - p * y
              end

# Column modification

0.upto([n, k+3].min) do |i|
                p = x * @h[i][k] + y * @h[i][k+1]
                if (notlast)
                  p += z * @h[i][k+2]
                  @h[i][k+2] = @h[i][k+2] - p * r
                end
                @h[i][k] = @h[i][k] - p
                @h[i][k+1] = @h[i][k+1] - p * q
              end

# Accumulate transformations

low.upto(high) do |i|
                p = x * @v[i][k] + y * @v[i][k+1]
                if (notlast)
                  p += z * @v[i][k+2]
                  @v[i][k+2] = @v[i][k+2] - p * r
                end
                @v[i][k] = @v[i][k] - p
                @v[i][k+1] = @v[i][k+1] - p * q
              end
            end  # (s != 0)
          end  # k loop
        end  # check convergence
      end  # while (n >= low)

# Backsubstitute to find vectors of upper triangular form

if (norm == 0.0)
        return
      end

(nn-1).downto(0) do |k|
        p = @d[k]
        q = @e[k]

# Real vector

if (q == 0)
          l = k
          @h[k][k] = 1.0
          (k-1).downto(0) do |i|
            w = @h[i][i] - p
            r = 0.0
            l.upto(k) do |j|
              r += @h[i][j] * @h[j][k]
            end
            if (@e[i] < 0.0)
              z = w
              s = r
            else
              l = i
              if (@e[i] == 0.0)
                if (w != 0.0)
                  @h[i][k] = -r / w
                else
                  @h[i][k] = -r / (eps * norm)
                end

# 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][k] = t
                if (x.abs > z.abs)
                  @h[i+1][k] = (-r - w * t) / x
                else
                  @h[i+1][k] = (-s - y * t) / z
                end
              end

# Overflow control

t = @h[i][k].abs
              if ((eps * t) * t > 1)
                i.upto(k) do |j|
                  @h[j][k] = @h[j][k] / t
                end
              end
            end
          end

# Complex vector

elsif (q < 0)
          l = n-1

# Last vector component imaginary so matrix is triangular

if (@h[n][n-1].abs > @h[n-1][n].abs)
            @h[n-1][n-1] = q / @h[n][n-1]
            @h[n-1][n] = -(@h[n][n] - p) / @h[n][n-1]
          else
            cdivr, cdivi = 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
          end
          @h[n][n-1] = 0.0
          @h[n][n] = 1.0
          (n-2).downto(0) do |i|
            ra = 0.0
            sa = 0.0
            l.upto(n) do |j|
              ra = ra + @h[i][j] * @h[j][n-1]
              sa = sa + @h[i][j] * @h[j][n]
            end
            w = @h[i][i] - p

if (@e[i] < 0.0)
              z = w
              r = ra
              s = sa
            else
              l = i
              if (@e[i] == 0)
                cdivr, cdivi = 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 * (w.abs + q.abs +
                  x.abs + y.abs + z.abs)
                end
                cdivr, cdivi = 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 (x.abs > (z.abs + q.abs))
                  @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
                  cdivr, cdivi = 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
                end
              end

# Overflow control

t = [@h[i][n-1].abs, @h[i][n].abs].max
              if ((eps * t) * t > 1)
                i.upto(n) do |j|
                  @h[j][n-1] = @h[j][n-1] / t
                  @h[j][n] = @h[j][n] / t
                end
              end
            end
          end
        end
      end

# Vectors of isolated roots

nn.times do |i|
        if (i < low || i > high)
          i.upto(nn-1) do |j|
            @v[i][j] = @h[i][j]
          end
        end
      end

# Back transformation to get eigenvectors of original matrix

(nn-1).downto(low) do |j|
        low.upto(high) do |i|
          z = 0.0
          low.upto([j, high].min) do |k|
            z += @v[i][k] * @h[k][j]
          end
          @v[i][j] = z
        end
      end
    end

end
end

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