jama_eig.h

一个矩阵计算的库 包括几乎所有需要的矩阵运算 对3d计算和图形处理有很大帮助

#ifndef JAMA_EIG_H
#define JAMA_EIG_H


#include "tnt_array1d.h"
#include "tnt_array2d.h"
#include "tnt_math_utils.h"

#include <algorithm>
// for min(), max() below

#include <cmath>
// for abs() below

using namespace TNT;
using namespace std;

namespace JAMA
{

/** 

    Computes eigenvalues and eigenvectors of a real (non-complex)
    matrix. 
<P>
    If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is
    diagonal and the eigenvector matrix V is orthogonal. That is,
	the diagonal values of D are the eigenvalues, and
    V*V' = I, where I is the identity matrix.  The columns of V 
    represent the eigenvectors in the sense that A*V = V*D.
    
<P>
    If A is not symmetric, then the eigenvalue matrix D is block diagonal
    with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
    a + i*b, in 2-by-2 blocks, [a, b; -b, a].  That is, if the complex
    eigenvalues look like
<pre>

          u + iv     .        .          .      .    .
            .      u - iv     .          .      .    .
            .        .      a + ib       .      .    .
            .        .        .        a - ib   .    .
            .        .        .          .      x    .
            .        .        .          .      .    y
</pre>
        then D looks like
<pre>

            u        v        .          .      .    .
           -v        u        .          .      .    . 
            .        .        a          b      .    .
            .        .       -b          a      .    .
            .        .        .          .      x    .
            .        .        .          .      .    y
</pre>
    This keeps V a real matrix in both symmetric and non-symmetric
    cases, and A*V = V*D.
    
    
    
    <p>
    The matrix V may be badly
    conditioned, or even singular, so the validity of the equation
    A = V*D*inverse(V) depends upon the condition number of V.

   <p>
	(Adapted from JAMA, a Java Matrix Library, developed by jointly 
	by the Mathworks and NIST; see  http://math.nist.gov/javanumerics/jama).
**/

template <class Real>
class Eigenvalue
{


   /** Row and column dimension (square matrix).  */
    int n;

   int issymmetric; /* boolean*/

   /** Arrays for internal storage of eigenvalues. */

   TNT::Array1D<Real> d;         /* real part */
   TNT::Array1D<Real> e;         /* img part */

   /** Array for internal storage of eigenvectors. */
    TNT::Array2D<Real> V;

   /** Array for internal storage of nonsymmetric Hessenberg form.
   @serial internal storage of nonsymmetric Hessenberg form.
   */
   TNT::Array2D<Real> H;
   

   /** Working storage for nonsymmetric algorithm.
   @serial working storage for nonsymmetric algorithm.
   */
   TNT::Array1D<Real> ort;


   // Symmetric Householder reduction to tridiagonal form.

   void tred2() {

   //  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.

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

      // Householder reduction to tridiagonal form.
   
      for (int i = n-1; i > 0; i--) {
   
         // Scale to avoid under/overflow.
   
         Real scale = 0.0;
         Real h = 0.0;
         for (int k = 0; k < i; k++) {
            scale = scale + abs(d[k]);
         }
         if (scale == 0.0) {
            e[i] = d[i-1];
            for (int j = 0; j < i; j++) {
               d[j] = V[i-1][j];
               V[i][j] = 0.0;
               V[j][i] = 0.0;
            }
         } else {
   
            // Generate Householder vector.
   
            for (int k = 0; k < i; k++) {
               d[k] /= scale;
               h += d[k] * d[k];
            }
            Real f = d[i-1];
            Real g = sqrt(h);
            if (f > 0) {
               g = -g;
            }
            e[i] = scale * g;
            h = h - f * g;
            d[i-1] = f - g;
            for (int j = 0; j < i; j++) {
               e[j] = 0.0;
            }
   
            // Apply similarity transformation to remaining columns.
   
            for (int j = 0; j < i; j++) {
               f = d[j];
               V[j][i] = f;
               g = e[j] + V[j][j] * f;
               for (int k = j+1; k <= i-1; k++) {
                  g += V[k][j] * d[k];
                  e[k] += V[k][j] * f;
               }
               e[j] = g;
            }
            f = 0.0;
            for (int j = 0; j < i; j++) {
               e[j] /= h;
               f += e[j] * d[j];
            }
            Real hh = f / (h + h);
            for (int j = 0; j < i; j++) {
               e[j] -= hh * d[j];
            }
            for (int j = 0; j < i; j++) {
               f = d[j];
               g = e[j];
               for (int k = j; k <= i-1; k++) {
                  V[k][j] -= (f * e[k] + g * d[k]);
               }
               d[j] = V[i-1][j];
               V[i][j] = 0.0;
            }
         }
         d[i] = h;
      }
   
      // Accumulate transformations.
   
      for (int i = 0; i < n-1; i++) {
         V[n-1][i] = V[i][i];
         V[i][i] = 1.0;
         Real h = d[i+1];
         if (h != 0.0) {
            for (int k = 0; k <= i; k++) {
               d[k] = V[k][i+1] / h;
            }
            for (int j = 0; j <= i; j++) {
               Real g = 0.0;
               for (int k = 0; k <= i; k++) {
                  g += V[k][i+1] * V[k][j];
               }
               for (int k = 0; k <= i; k++) {
                  V[k][j] -= g * d[k];
               }
            }
         }
         for (int k = 0; k <= i; k++) {
            V[k][i+1] = 0.0;
         }
      }
      for (int j = 0; j < n; j++) {
         d[j] = V[n-1][j];
         V[n-1][j] = 0.0;
      }
      V[n-1][n-1] = 1.0;
      e[0] = 0.0;
   } 

   // Symmetric tridiagonal QL algorithm.
   
   void tql2 () {

   //  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.
   
      for (int i = 1; i < n; i++) {
         e[i-1] = e[i];
      }
      e[n-1] = 0.0;
   
      Real f = 0.0;
      Real tst1 = 0.0;
      Real eps = pow(2.0,-52.0);
      for (int l = 0; l < n; l++) {

         // Find small subdiagonal element
   
         tst1 = max(tst1,abs(d[l]) + abs(e[l]));
         int m = l;

        // Original while-loop from Java code
         while (m < n) {
            if (abs(e[m]) <= eps*tst1) {
               break;
            }
            m++;
         }

   
         // If m == l, d[l] is an eigenvalue,
         // otherwise, iterate.
   
         if (m > l) {
            int iter = 0;
            do {
               iter = iter + 1;  // (Could check iteration count here.)
   
               // Compute implicit shift
   
               Real g = d[l];
               Real p = (d[l+1] - g) / (2.0 * e[l]);
               Real r = hypot(p,1.0);
               if (p < 0) {
                  r = -r;
               }
               d[l] = e[l] / (p + r);
               d[l+1] = e[l] * (p + r);
               Real dl1 = d[l+1];
               Real h = g - d[l];
               for (int i = l+2; i < n; i++) {
                  d[i] -= h;
               }
               f = f + h;
   
               // Implicit QL transformation.
   
               p = d[m];
               Real c = 1.0;
               Real c2 = c;
               Real c3 = c;
               Real el1 = e[l+1];
               Real s = 0.0;
               Real s2 = 0.0;
               for (int 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 (int 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 (abs(e[l]) > eps*tst1);
         }
         d[l] = d[l] + f;
         e[l] = 0.0;
      }
     
      // Sort eigenvalues and corresponding vectors.
   
      for (int i = 0; i < n-1; i++) {
         int k = i;
         Real p = d[i];
         for (int 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 (int j = 0; j < n; j++) {
               p = V[j][i];
               V[j][i] = V[j][k];
               V[j][k] = p;
            }
         }
      }
   }

   // Nonsymmetric reduction to Hessenberg form.

   void orthes () {
   
      //  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.
   
      int low = 0;
      int high = n-1;
   
      for (int m = low+1; m <= high-1; m++) {
   
         // Scale column.
   
         Real scale = 0.0;
         for (int i = m; i <= high; i++) {
            scale = scale + abs(H[i][m-1]);
         }
         if (scale != 0.0) {
   
            // Compute Householder transformation.
   
            Real h = 0.0;
            for (int i = high; i >= m; i--) {
               ort[i] = H[i][m-1]/scale;
               h += ort[i] * ort[i];
            }
            Real g = 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 (int j = m; j < n; j++) {
               Real f = 0.0;
               for (int i = high; i >= m; i--) {
                  f += ort[i]*H[i][j];
               }
               f = f/h;
               for (int i = m; i <= high; i++) {
                  H[i][j] -= f*ort[i];
               }
           }
   
           for (int i = 0; i <= high; i++) {
               Real f = 0.0;
               for (int j = high; j >= m; j--) {
                  f += ort[j]*H[i][j];
               }
               f = f/h;
               for (int 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 (int i = 0; i < n; i++) {
         for (int j = 0; j < n; j++) {
            V[i][j] = (i == j ? 1.0 : 0.0);
         }
      }

      for (int m = high-1; m >= low+1; m--) {
         if (H[m][m-1] != 0.0) {
            for (int i = m+1; i <= high; i++) {
               ort[i] = H[i][m-1];
            }
            for (int j = m; j <= high; j++) {
               Real g = 0.0;
               for (int 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 (int i = m; i <= high; i++) {
                  V[i][j] += g * ort[i];
               }
            }
         }
      }
   }


   // Complex scalar division.

   Real cdivr, cdivi;
   void cdiv(Real xr, Real xi, Real yr, Real yi) {
      Real r,d;
      if (abs(yr) > abs(yi)) {
         r = yi/yr;
         d = yr + r*yi;
         cdivr = (xr + r*xi)/d;
         cdivi = (xi - r*xr)/d;
      } else {
         r = yr/yi;
         d = yi + r*yr;
         cdivr = (r*xr + xi)/d;
         cdivi = (r*xi - xr)/d;
      }
   }


   // Nonsymmetric reduction from Hessenberg to real Schur form.

   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 n = nn-1;
      int low = 0;
      int high = nn-1;
      Real eps = pow(2.0,-52.0);
      Real exshift = 0.0;
      Real p=0,q=0,r=0,s=0,z=0,t,w,x,y;
   
      // Store roots isolated by balanc and compute matrix norm
   
      Real 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 = max(i-1,0); j < nn; j++) {
            norm = norm + 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 = abs(H[l-1][l-1]) + abs(H[l][l]);
            if (s == 0.0) {
               s = norm;
            }
            if (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