@sgratzl/science/src/lin/decompose.js

Scientific and statistical computing in JavaScript.

import hypot from '../core/hypot';

export default function decompose() {

  function decompose(A) {
    var n = A.length, // column dimension
        V = [],
        d = [],
        e = [];

    for (var i = 0; i < n; i++) {
      V[i] = [];
      d[i] = [];
      e[i] = [];
    }

    var symmetric = true;
    for (var j = 0; j < n; j++) {
      for (var i = 0; i < n; i++) {
        if (A[i][j] !== A[j][i]) {
          symmetric = false;
          break;
        }
      }
    }

    if (symmetric) {
      for (var i = 0; i < n; i++) V[i] = A[i].slice();

      // Tridiagonalize.
      science_lin_decomposeTred2(d, e, V);

      // Diagonalize.
      science_lin_decomposeTql2(d, e, V);
    } else {
      var H = [];
      for (var i = 0; i < n; i++) H[i] = A[i].slice();

      // Reduce to Hessenberg form.
      science_lin_decomposeOrthes(H, V);

      // Reduce Hessenberg to real Schur form.
      science_lin_decomposeHqr2(d, e, H, V);
    }

    var D = [];
    for (var i = 0; i < n; i++) {
      var row = D[i] = [];
      for (var j = 0; j < n; j++) row[j] = i === j ? d[i] : 0;
      D[i][e[i] > 0 ? i + 1 : i - 1] = e[i];
    }
    return {D: D, V: V};
  }

  return decompose;
};

// Symmetric Householder reduction to tridiagonal form.
function science_lin_decomposeTred2(d, e, V) {
  // 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.

  var n = V.length;

  for (var j = 0; j < n; j++) d[j] = V[n - 1][j];

  // Householder reduction to tridiagonal form.
  for (var i = n - 1; i > 0; i--) {
    // Scale to avoid under/overflow.

    var scale = 0,
        h = 0;
    for (var k = 0; k < i; k++) scale += Math.abs(d[k]);
    if (scale === 0) {
      e[i] = d[i - 1];
      for (var j = 0; j < i; j++) {
        d[j] = V[i - 1][j];
        V[i][j] = 0;
        V[j][i] = 0;
      }
    } else {
      // Generate Householder vector.
      for (var k = 0; k < i; k++) {
        d[k] /= scale;
        h += d[k] * d[k];
      }
      var f = d[i - 1];
      var g = Math.sqrt(h);
      if (f > 0) g = -g;
      e[i] = scale * g;
      h = h - f * g;
      d[i - 1] = f - g;
      for (var j = 0; j < i; j++) e[j] = 0;

      // Apply similarity transformation to remaining columns.

      for (var j = 0; j < i; j++) {
        f = d[j];
        V[j][i] = f;
        g = e[j] + V[j][j] * f;
        for (var k = j+1; k <= i - 1; k++) {
          g += V[k][j] * d[k];
          e[k] += V[k][j] * f;
        }
        e[j] = g;
      }
      f = 0;
      for (var j = 0; j < i; j++) {
        e[j] /= h;
        f += e[j] * d[j];
      }
      var hh = f / (h + h);
      for (var j = 0; j < i; j++) e[j] -= hh * d[j];
      for (var j = 0; j < i; j++) {
        f = d[j];
        g = e[j];
        for (var k = j; k <= i - 1; k++) V[k][j] -= (f * e[k] + g * d[k]);
        d[j] = V[i - 1][j];
        V[i][j] = 0;
      }
    }
    d[i] = h;
  }

  // Accumulate transformations.
  for (var i = 0; i < n - 1; i++) {
    V[n - 1][i] = V[i][i];
    V[i][i] = 1.0;
    var h = d[i + 1];
    if (h != 0) {
      for (var k = 0; k <= i; k++) d[k] = V[k][i + 1] / h;
      for (var j = 0; j <= i; j++) {
        var g = 0;
        for (var k = 0; k <= i; k++) g += V[k][i + 1] * V[k][j];
        for (var k = 0; k <= i; k++) V[k][j] -= g * d[k];
      }
    }
    for (var k = 0; k <= i; k++) V[k][i + 1] = 0;
  }
  for (var j = 0; j < n; j++) {
    d[j] = V[n - 1][j];
    V[n - 1][j] = 0;
  }
  V[n - 1][n - 1] = 1;
  e[0] = 0;
}

// Symmetric tridiagonal QL algorithm.
function science_lin_decomposeTql2(d, e, V) {
  // 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.

  var n = V.length;

  for (var i = 1; i < n; i++) e[i - 1] = e[i];
  e[n - 1] = 0;

  var f = 0;
  var tst1 = 0;
  var eps = 1e-12;
  for (var l = 0; l < n; l++) {
    // Find small subdiagonal element
    tst1 = Math.max(tst1, Math.abs(d[l]) + Math.abs(e[l]));
    var m = l;
    while (m < n) {
      if (Math.abs(e[m]) <= eps*tst1) { break; }
      m++;
    }

    // If m == l, d[l] is an eigenvalue,
    // otherwise, iterate.
    if (m > l) {
      var iter = 0;
      do {
        iter++;  // (Could check iteration count here.)

        // Compute implicit shift
        var g = d[l];
        var p = (d[l + 1] - g) / (2 * e[l]);
        var r = hypot(p, 1);
        if (p < 0) r = -r;
        d[l] = e[l] / (p + r);
        d[l + 1] = e[l] * (p + r);
        var dl1 = d[l + 1];
        var h = g - d[l];
        for (var i = l+2; i < n; i++) d[i] -= h;
        f += h;

        // Implicit QL transformation.
        p = d[m];
        var c = 1;
        var c2 = c;
        var c3 = c;
        var el1 = e[l + 1];
        var s = 0;
        var s2 = 0;
        for (var i = m - 1; i >= l; i--) {
          c3 = c2;
          c2 = c;
          s2 = s;
          g = c * e[i];
          h = c * p;
          r = 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.
          for (var k = 0; k < n; 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;
          }
        }
        p = -s * s2 * c3 * el1 * e[l] / dl1;
        e[l] = s * p;
        d[l] = c * p;

        // Check for convergence.
      } while (Math.abs(e[l]) > eps*tst1);
    }
    d[l] = d[l] + f;
    e[l] = 0;
  }

  // Sort eigenvalues and corresponding vectors.
  for (var i = 0; i < n - 1; i++) {
    var k = i;
    var p = d[i];
    for (var j = i + 1; j < n; j++) {
      if (d[j] < p) {
        k = j;
        p = d[j];
      }
    }
    if (k != i) {
      d[k] = d[i];
      d[i] = p;
      for (var j = 0; j < n; j++) {
        p = V[j][i];
        V[j][i] = V[j][k];
        V[j][k] = p;
      }
    }
  }
}

// Nonsymmetric reduction to Hessenberg form.
function science_lin_decomposeOrthes(H, V) {
  // 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.

  var n = H.length;
  var ort = [];

  var low = 0;
  var high = n - 1;

  for (var m = low + 1; m < high; m++) {
    // Scale column.
    var scale = 0;
    for (var i = m; i <= high; i++) scale += Math.abs(H[i][m - 1]);

    if (scale !== 0) {
      // Compute Householder transformation.
      var h = 0;
      for (var i = high; i >= m; i--) {
        ort[i] = H[i][m - 1] / scale;
        h += ort[i] * ort[i];
      }
      var g = Math.sqrt(h);
      if (ort[m] > 0) g = -g;
      h = h - ort[m] * g;
      ort[m] = ort[m] - g;

      // Apply Householder similarity transformation
      // H = (I-u*u'/h)*H*(I-u*u')/h)
      for (var j = m; j < n; j++) {
        var f = 0;
        for (var i = high; i >= m; i--) f += ort[i] * H[i][j];
        f /= h;
        for (var i = m; i <= high; i++) H[i][j] -= f * ort[i];
      }

      for (var i = 0; i <= high; i++) {
        var f = 0;
        for (var j = high; j >= m; j--) f += ort[j] * H[i][j];
        f /= h;
        for (var j = m; j <= high; j++) H[i][j] -= f * ort[j];
      }
      ort[m] = scale * ort[m];
      H[m][m - 1] = scale * g;
    }
  }

  // Accumulate transformations (Algol's ortran).
  for (var i = 0; i < n; i++) {
    for (var j = 0; j < n; j++) V[i][j] = i === j ? 1 : 0;
  }

  for (var m = high-1; m >= low+1; m--) {
    if (H[m][m - 1] !== 0) {
      for (var i = m + 1; i <= high; i++) ort[i] = H[i][m - 1];
      for (var j = m; j <= high; j++) {
        var g = 0;
        for (var i = m; i <= high; i++) g += ort[i] * V[i][j];
        // Double division avoids possible underflow
        g = (g / ort[m]) / H[m][m - 1];
        for (var i = m; i <= high; i++) V[i][j] += g * ort[i];
      }
    }
  }
}

// Nonsymmetric reduction from Hessenberg to real Schur form.
function science_lin_decomposeHqr2(d, e, H, V) {
  // 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.

  var nn = H.length,
      n = nn - 1,
      low = 0,
      high = nn - 1,
      eps = 1e-12,
      exshift = 0,
      p = 0,
      q = 0,
      r = 0,
      s = 0,
      z = 0,
      t,
      w,
      x,
      y;

  // Store roots isolated by balanc and compute matrix norm
  var norm = 0;
  for (var i = 0; i < nn; i++) {
    if (i < low || i > high) {
      d[i] = H[i][i];
      e[i] = 0;
    }
    for (var j = Math.max(i - 1, 0); j < nn; j++) norm += Math.abs(H[i][j]);
  }

  // Outer loop over eigenvalue index
  var iter = 0;
  while (n >= low) {
    // Look for single small sub-diagonal element
    var l = n;
    while (l > low) {
      s = Math.abs(H[l - 1][l - 1]) + Math.abs(H[l][l]);
      if (s === 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;
      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;
      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) {
        z = p + (p >= 0 ? z : -z);
        d[n - 1] = x + z;
        d[n] = d[n - 1];
        if (z !== 0) d[n] = x - w / z;
        e[n - 1] = 0;
        e[n] = 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 /= r;
        q /= r;

        // Row modification
        for (var 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 (var 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 (var 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;
      w = 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 (var 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 (var i = low; i <= n; i++) {
            H[i][i] -= s;
          }
          exshift += s;
          x = y = w = 0.964;
        }
      }

      iter++;   // (Could check iteration count here.)

      // Look for two consecutive small sub-diagonal elements
      var 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 (var i = m+2; i <= n; i++) {
        H[i][i-2] = 0;
        if (i > m+2) H[i][i-3] = 0;
      }

      // Double QR step involving rows l:n and columns m:n
      for (var k = m; k <= n - 1; k++) {
        var 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);
          x = Math.abs(p) + Math.abs(q) + Math.abs(r);
          if (x != 0) {
            p /= x;
            q /= x;
            r /= x;
          }
        }
        if (x == 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 += s;
          x = p / s;
          y = q / s;
          z = r / s;
          q /= p;
          r /= p;

          // Row modification
          for (var 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 (var i = 0; i <= Math.min(n, k + 3); 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;
            }
            H[i][k] = H[i][k] - p;
            H[i][k + 1] = H[i][k + 1] - p * q;
          }

          // Accumulate transformations
          for (var 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) { return; }

  for (n = nn - 1; n >= 0; n--) {
    p = d[n];
    q = e[n];

    // Real vector
    if (q == 0) {
      var l = n;
      H[n][n] = 1.0;
      for (var i = n - 1; i >= 0; i--) {
        w = H[i][i] - p;
        r = 0;
        for (var j = l; j <= n; j++) { r = r + H[i][j] * H[j][n]; }
        if (e[i] < 0) {
          z = w;
          s = r;
        } else {
          l = i;
          if (e[i] === 0) {
            H[i][n] = -r / (w !== 0 ? w : eps * norm);
          } else {
            // Solve real equations
            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 (var j = i; j <= n; j++) H[j][n] = H[j][n] / t;
          }
        }
      }
    // Complex vector
    } else if (q < 0) {
      var 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 {
        var zz = science_lin_decomposeCdiv(0, -H[n - 1][n], H[n - 1][n - 1] - p, q);
        H[n - 1][n - 1] = zz[0];
        H[n - 1][n] = zz[1];
      }
      H[n][n - 1] = 0;
      H[n][n] = 1;
      for (var i = n-2; i >= 0; i--) {
        var ra = 0,
            sa = 0,
            vr,
            vi;
        for (var 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) {
          z = w;
          r = ra;
          s = sa;
        } else {
          l = i;
          if (e[i] == 0) {
            var zz = science_lin_decomposeCdiv(-ra,-sa,w,q);
            H[i][n - 1] = zz[0];
            H[i][n] = zz[1];
          } 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 & vi == 0) {
              vr = eps * norm * (Math.abs(w) + Math.abs(q) +
                Math.abs(x) + Math.abs(y) + Math.abs(z));
            }
            var zz = science_lin_decomposeCdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
            H[i][n - 1] = zz[0];
            H[i][n] = zz[1];
            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 {
              var zz = science_lin_decomposeCdiv(-r-y*H[i][n - 1],-s-y*H[i][n],z,q);
              H[i + 1][n - 1] = zz[0];
              H[i + 1][n] = zz[1];
            }
          }

          // Overflow control
          t = Math.max(Math.abs(H[i][n - 1]),Math.abs(H[i][n]));
          if ((eps * t) * t > 1) {
            for (var 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 (var i = 0; i < nn; i++) {
    if (i < low || i > high) {
      for (var j = i; j < nn; j++) V[i][j] = H[i][j];
    }
  }

  // Back transformation to get eigenvectors of original matrix
  for (var j = nn - 1; j >= low; j--) {
    for (var i = low; i <= high; i++) {
      z = 0;
      for (var k = low; k <= Math.min(j, high); k++) z += V[i][k] * H[k][j];
      V[i][j] = z;
    }
  }
}

// Complex scalar division.
function science_lin_decomposeCdiv(xr, xi, yr, yi) {
  if (Math.abs(yr) > Math.abs(yi)) {
    var r = yi / yr,
        d = yr + r * yi;
    return [(xr + r * xi) / d, (xi - r * xr) / d];
  } else {
    var r = yr / yi,
        d = yi + r * yr;
    return [(r * xr + xi) / d, (r * xi - xr) / d];
  }
}