eigenvaluedecomposition.cs

C#下的矩阵计算方法,从Japack该过来的,已经试用过,很好用.

			// 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
				if (l == n) 
				{
					// One root found
					H[n,n] = H[n,n] + exshift;
					d[n] = H[n,n];
					e[n] = 0.0;
					n--;
					iter = 0;
				} 
				else if (l == n-1) 
				{
					// Two roots found
					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];
		 
					if (q >= 0) 
					{
						// Real pair
						z = (p >= 0) ? (p + 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;
						}
					}
					else 
					{
						// Complex pair
						d[n-1] = x + p;
						d[n] = x + p;
						e[n-1] = z;
						e[n] = -z;
					}
						
					n = n - 2;
					iter = 0;
				}
				else 
				{
					// No convergence yet	 
					
					// 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;
		 
					// 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++) 
					{
						bool 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;
							}
						}
					}
				}
			}
				
			// 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) 
							{
								H[i,n] = (w != 0.0) ? (-r / w) : (-r / (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;
								H[i+1,n] = (Math.Abs(x) > Math.Abs(z)) ? ((-r - w * t) / x) : ((-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;
						}
					}
				}
				else if (q < 0) 
				{
					// Complex vector
					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;
				}
		}

		/// <summary>Returns the real parts of the eigenvalues.</summary>
		public double[] RealEigenvalues
		{
			get 
			{ 
				return this.d; 
			}
		}
	
		/// <summary>Returns the imaginary parts of the eigenvalues.</summary>	
		public double[] ImaginaryEigenvalues
		{
			get 
			{ 
				return this.e; 
			}
		}

		/// <summary>Returns the eigenvector matrix.</summary>
		public Matrix EigenvectorMatrix
		{
			get 
			{ 
				return this.V; 
			}
		}
	
		/// <summary>Returns the block diagonal eigenvalue matrix.</summary>
		public Matrix DiagonalMatrix
		{
			get
			{
				Matrix X = new Matrix(n, n);
				double[][] x = X.Array;
	
				for (int i = 0; i < n; i++) 
				{
					for (int j = 0; j < n; j++)
						x[i][j] = 0.0;
	
					x[i][i] = d[i];
					if (e[i] > 0)
					{
						x[i][i+1] = e[i];
					}
					else if (e[i] < 0) 
					{
						x[i][i-1] = e[i];
					}
				}
				
				return X;
			}			
		}

		private static double Hypotenuse(double a, double b) 
		{
			if (Math.Abs(a) > Math.Abs(b))
			{
				double r = b / a;
				return Math.Abs(a) * Math.Sqrt(1 + r * r);
			}

			if (b != 0)
			{
				double r = a / b;
				return Math.Abs(b) * Math.Sqrt(1 + r * r);
			}

			return 0.0;
		}
	}
}