LUDecomposition.java

   /** LU Decomposition.
   <P>
   For an m-by-n matrix A with m >= n, the LU decomposition is an m-by-n
   unit lower triangular matrix L, an n-by-n upper triangular matrix U,
   and a permutation vector piv of length m so that A(piv,:) = L*U.
   If m < n, then L is m-by-m and U is m-by-n.
   <P>
   The LU decompostion with pivoting always exists, even if the matrix is
   singular, so the constructor will never fail.  The primary use of the
   LU decomposition is in the solution of square systems of simultaneous
   linear equations.  This will fail if isNonsingular() returns false.
   */

public class LUDecomposition implements java.io.Serializable {

/* ------------------------
   Class variables
 * ------------------------ */

   /** Array for internal storage of decomposition.
   @serial internal array storage.
   */
   private double[][] LU;

   /** Row and column dimensions, and pivot sign.
   @serial column dimension.
   @serial row dimension.
   @serial pivot sign.
   */
   private int m, n, pivsign; 

   /** Internal storage of pivot vector.
   @serial pivot vector.
   */
   private int[] piv;

/* ------------------------
   Constructor
 * ------------------------ */

   /** LU Decomposition
       Structure to access L, U and piv.
   @param  A Rectangular matrix
   */

   public LUDecomposition (Matrix A) {

   // Use a "left-looking", dot-product, Crout/Doolittle algorithm.

      LU = A.getArrayCopy();
      m = A.getRowDimension();
      n = A.getColumnDimension();
      piv = new int[m];
      for (int i = 0; i < m; i++) {
         piv[i] = i;
      }
      pivsign = 1;
      double[] LUrowi;
      double[] LUcolj = new double[m];

      // Outer loop.

      for (int j = 0; j < n; j++) {

         // Make a copy of the j-th column to localize references.

         for (int i = 0; i < m; i++) {
            LUcolj[i] = LU[i][j];
         }

         // Apply previous transformations.

         for (int i = 0; i < m; i++) {
            LUrowi = LU[i];

            // Most of the time is spent in the following dot product.

            int kmax = Math.min(i,j);
            double s = 0.0;
            for (int k = 0; k < kmax; k++) {
               s += LUrowi[k]*LUcolj[k];
            }

            LUrowi[j] = LUcolj[i] -= s;
         }
   
         // Find pivot and exchange if necessary.

         int p = j;
         for (int i = j+1; i < m; i++) {
            if (Math.abs(LUcolj[i]) > Math.abs(LUcolj[p])) {
               p = i;
            }
         }
         if (p != j) {
            for (int k = 0; k < n; k++) {
               double t = LU[p][k]; LU[p][k] = LU[j][k]; LU[j][k] = t;
            }
            int k = piv[p]; piv[p] = piv[j]; piv[j] = k;
            pivsign = -pivsign;
         }

         // Compute multipliers.
         
         if (j < m & LU[j][j] != 0.0) {
            for (int i = j+1; i < m; i++) {
               LU[i][j] /= LU[j][j];
            }
         }
      }
   }

/* ------------------------
   Temporary, experimental code.
   ------------------------ *\

   \** LU Decomposition, computed by Gaussian elimination.
   <P>
   This constructor computes L and U with the "daxpy"-based elimination
   algorithm used in LINPACK and MATLAB.  In Java, we suspect the dot-product,
   Crout algorithm will be faster.  We have temporarily included this
   constructor until timing experiments confirm this suspicion.
   <P>
   @param  A             Rectangular matrix
   @param  linpackflag   Use Gaussian elimination.  Actual value ignored.
   @return               Structure to access L, U and piv.
   *\

   public LUDecomposition (Matrix A, int linpackflag) {
      // Initialize.
      LU = A.getArrayCopy();
      m = A.getRowDimension();
      n = A.getColumnDimension();
      piv = new int[m];
      for (int i = 0; i < m; i++) {
         piv[i] = i;
      }
      pivsign = 1;
      // Main loop.
      for (int k = 0; k < n; k++) {
         // Find pivot.
         int p = k;
         for (int i = k+1; i < m; i++) {
            if (Math.abs(LU[i][k]) > Math.abs(LU[p][k])) {
               p = i;
            }
         }
         // Exchange if necessary.
         if (p != k) {
            for (int j = 0; j < n; j++) {
               double t = LU[p][j]; LU[p][j] = LU[k][j]; LU[k][j] = t;
            }
            int t = piv[p]; piv[p] = piv[k]; piv[k] = t;
            pivsign = -pivsign;
         }
         // Compute multipliers and eliminate k-th column.
         if (LU[k][k] != 0.0) {
            for (int i = k+1; i < m; i++) {
               LU[i][k] /= LU[k][k];
               for (int j = k+1; j < n; j++) {
                  LU[i][j] -= LU[i][k]*LU[k][j];
               }
            }
         }
      }
   }

\* ------------------------
   End of temporary code.
 * ------------------------ */

/* ------------------------
   Public Methods
 * ------------------------ */

   /** Is the matrix nonsingular?
   @return     true if U, and hence A, is nonsingular.
   */

   public boolean isNonsingular () {
      for (int j = 0; j < n; j++) {
         if (LU[j][j] == 0)
            return false;
      }
      return true;
   }

   /** Return lower triangular factor
   @return     L
   */

   public Matrix getL () {
      Matrix X = new Matrix(m,n);
      double[][] L = X.getArray();
      for (int i = 0; i < m; i++) {
         for (int j = 0; j < n; j++) {
            if (i > j) {
               L[i][j] = LU[i][j];
            } else if (i == j) {
               L[i][j] = 1.0;
            } else {
               L[i][j] = 0.0;
            }
         }
      }
      return X;
   }

   /** Return upper triangular factor
   @return     U
   */

   public Matrix getU () {
      Matrix X = new Matrix(n,n);
      double[][] U = X.getArray();
      for (int i = 0; i < n; i++) {
         for (int j = 0; j < n; j++) {
            if (i <= j) {
               U[i][j] = LU[i][j];
            } else {
               U[i][j] = 0.0;
            }
         }
      }
      return X;
   }

   /** Return pivot permutation vector
   @return     piv
   */

   public int[] getPivot () {
      int[] p = new int[m];
      for (int i = 0; i < m; i++) {
         p[i] = piv[i];
      }
      return p;
   }

   /** Return pivot permutation vector as a one-dimensional double array
   @return     (double) piv
   */

   public double[] getDoublePivot () {
      double[] vals = new double[m];
      for (int i = 0; i < m; i++) {
         vals[i] = (double) piv[i];
      }
      return vals;
   }

   /** Determinant
   @return     det(A)
   @exception  IllegalArgumentException  Matrix must be square
   */

   public double det () {
      if (m != n) {
         throw new IllegalArgumentException("Matrix must be square.");
      }
      double d = (double) pivsign;
      for (int j = 0; j < n; j++) {
         d *= LU[j][j];
      }
      return d;
   }

   /** Solve A*X = B
   @param  B   A Matrix with as many rows as A and any number of columns.
   @return     X so that L*U*X = B(piv,:)
   @exception  IllegalArgumentException Matrix row dimensions must agree.
   @exception  RuntimeException  Matrix is singular.
   */

   public Matrix solve (Matrix B) {
      if (B.getRowDimension() != m) {
         throw new IllegalArgumentException("Matrix row dimensions must agree.");
      }
      if (!this.isNonsingular()) {
         throw new RuntimeException("Matrix is singular.");
      }

      // Copy right hand side with pivoting
      int nx = B.getColumnDimension();
      Matrix Xmat = B.getMatrix(piv,0,nx-1);
      double[][] X = Xmat.getArray();

      // Solve L*Y = B(piv,:)
      for (int k = 0; k < n; k++) {
         for (int i = k+1; i < n; i++) {
            for (int j = 0; j < nx; j++) {
               X[i][j] -= X[k][j]*LU[i][k];
            }
         }
      }
      // Solve U*X = Y;
      for (int k = n-1; k >= 0; k--) {
         for (int j = 0; j < nx; j++) {
            X[k][j] /= LU[k][k];
         }
         for (int i = 0; i < k; i++) {
            for (int j = 0; j < nx; j++) {
               X[i][j] -= X[k][j]*LU[i][k];
            }
         }
      }
      return Xmat;
   }
  private static final long serialVersionUID = 1;
}