Esempio n. 1
0
        /* ------------------------
           Constructor
         * ------------------------ */
        ///<summary>QR Decomposition, computed by Householder reflections.</summary>
        ///<param name="A">Rectangular matrix</param>
        ///<returns>Structure to access R and the Householder vectors and compute Q.</returns>
        public QRDecomposition(Matrix A)
        {
            // Initialize.
              QR = A.ArrayCopy();
              m = A.RowDimension;
              n = A.ColumnDimension;
              Rdiag = new double[n];

              // Main loop.
              for (int k = 0; k < n; k++) {
             // Compute 2-norm of k-th column without under/overflow.
             double nrm = 0;
             for (int i = k; i < m; i++) {
            nrm = Maths.Hypot(nrm,QR[i][k]);
             }

             if (nrm != 0.0) {
            // Form k-th Householder vector.
            if (QR[k][k] < 0) {
               nrm = -nrm;
            }
            for (int i = k; i < m; i++) {
               QR[i][k] /= nrm;
            }
            QR[k][k] += 1.0;

            // Apply transformation to remaining columns.
            for (int j = k+1; j < n; j++) {
               double s = 0.0;
               for (int i = k; i < m; i++) {
                  s += QR[i][k]*QR[i][j];
               }
               s = -s/QR[k][k];
               for (int i = k; i < m; i++) {
                  QR[i][j] += s*QR[i][k];
               }
            }
             }
             Rdiag[k] = -nrm;
              }
        }
Esempio n. 2
0
        /* ------------------------
           Constructor
         * ------------------------ */
        /** LU Decomposition
           @param  A   Rectangular matrix
           @return     Structure to access L, U and piv.
           */
        public LUDecomposition(Matrix A)
        {
            // Use a "left-looking", dot-product, Crout/Doolittle algorithm.

              LU = A.ArrayCopy();
              m = A.RowDimension;
              n = A.ColumnDimension;
              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 k2 = piv[p]; piv[p] = piv[j]; piv[j] = k2;
            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];
            }
             }
              }
        }
        /* ------------------------
           Constructor
         * ------------------------ */
        ///<summary>Construct the singular value decomposition</summary>
        ///<param name="Arg">Rectangular matrix</param>
        ///<returns>Structure to access U, S and V.</returns>
        public SingularValueDecomposition(Matrix Arg)
        {
            // Derived from LINPACK code.
              // Initialize.
              double[][] A = Arg.ArrayCopy();
              m = Arg.RowDimension;
              n = Arg.ColumnDimension;

              /* Apparently the failing cases are only a proper subset of (m<n),
             so let's not throw error.  Correct fix to come later?
              if (m<n) {
              throw new System.ArgumentException("Jama SVD only works for m >= n"); }
              */
              int nu = Math.Min(m,n);
              s = new double [Math.Min(m+1,n)];
              U = new double [m][];
              for (int i = 0; i < m; i++)//Added by KJ
              {
              U[i] = new double[m];
              }
              V = new double [n][];
              for (int i = 0; i < n; i++)//Added by KJ
              {
              V[i] = new double[n];
              }
              double[] e = new double [n];
              double[] work = new double [m];
              bool wantu = true;
              bool wantv = true;

              // Reduce A to bidiagonal form, storing the diagonal elements
              // in s and the super-diagonal elements in e.

              int nct = Math.Min(m-1,n);
              int nrt = Math.Max(0,Math.Min(n-2,m));
              for (int k = 0; k < Math.Max(nct,nrt); k++) {
             if (k < nct) {

            // Compute the transformation for the k-th column and
            // place the k-th diagonal in s[k].
            // Compute 2-norm of k-th column without under/overflow.
            s[k] = 0;
            for (int i = k; i < m; i++) {
               s[k] = Maths.Hypot(s[k],A[i][k]);
            }
            if (s[k] != 0.0) {
               if (A[k][k] < 0.0) {
                  s[k] = -s[k];
               }
               for (int i = k; i < m; i++) {
                  A[i][k] /= s[k];
               }
               A[k][k] += 1.0;
            }
            s[k] = -s[k];
             }
             for (int j = k+1; j < n; j++) {
            if ((k < nct) & (s[k] != 0.0))  {

            // Apply the transformation.

               double t = 0;
               for (int i = k; i < m; i++) {
                  t += A[i][k]*A[i][j];
               }
               t = -t/A[k][k];
               for (int i = k; i < m; i++) {
                  A[i][j] += t*A[i][k];
               }
            }

            // Place the k-th row of A into e for the
            // subsequent calculation of the row transformation.

            e[j] = A[k][j];
             }
             if (wantu & (k < nct)) {

            // Place the transformation in U for subsequent back
            // multiplication.

            for (int i = k; i < m; i++) {
               U[i][k] = A[i][k];
            }
             }
             if (k < nrt) {

            // Compute the k-th row transformation and place the
            // k-th super-diagonal in e[k].
            // Compute 2-norm without under/overflow.
            e[k] = 0;
            for (int i = k+1; i < n; i++) {
               e[k] = Maths.Hypot(e[k],e[i]);
            }
            if (e[k] != 0.0) {
               if (e[k+1] < 0.0) {
                  e[k] = -e[k];
               }
               for (int i = k+1; i < n; i++) {
                  e[i] /= e[k];
               }
               e[k+1] += 1.0;
            }
            e[k] = -e[k];
            if ((k+1 < m) & (e[k] != 0.0)) {

            // Apply the transformation.

               for (int i = k+1; i < m; i++) {
                  work[i] = 0.0;
               }
               for (int j = k+1; j < n; j++) {
                  for (int i = k+1; i < m; i++) {
                     work[i] += e[j]*A[i][j];
                  }
               }
               for (int j = k+1; j < n; j++) {
                  double t = -e[j]/e[k+1];
                  for (int i = k+1; i < m; i++) {
                     A[i][j] += t*work[i];
                  }
               }
            }
            if (wantv) {

            // Place the transformation in V for subsequent
            // back multiplication.

               for (int i = k+1; i < n; i++) {
                  V[i][k] = e[i];
               }
            }
             }
              }

              // Set up the final bidiagonal matrix or order p.

              int p = Math.Min(n,m+1);
              if (nct < n) {
             s[nct] = A[nct][nct];
              }
              if (m < p) {
             s[p-1] = 0.0;
              }
              if (nrt+1 < p) {
             e[nrt] = A[nrt][p-1];
              }
              e[p-1] = 0.0;

              // If required, generate U.

              if (wantu) {
             for (int j = nct; j < nu; j++) {
            for (int i = 0; i < m; i++) {
               U[i][j] = 0.0;
            }
            U[j][j] = 1.0;
             }
             for (int k = nct-1; k >= 0; k--) {
            if (s[k] != 0.0) {
               for (int j = k+1; j < nu; j++) {
                  double t = 0;
                  for (int i = k; i < m; i++) {
                     t += U[i][k]*U[i][j];
                  }
                  t = -t/U[k][k];
                  for (int i = k; i < m; i++) {
                     U[i][j] += t*U[i][k];
                  }
               }
               for (int i = k; i < m; i++ ) {
                  U[i][k] = -U[i][k];
               }
               U[k][k] = 1.0 + U[k][k];
               for (int i = 0; i < k-1; i++) {
                  U[i][k] = 0.0;
               }
            } else {
               for (int i = 0; i < m; i++) {
                  U[i][k] = 0.0;
               }
               U[k][k] = 1.0;
            }
             }
              }

              // If required, generate V.

              if (wantv) {
             for (int k = n-1; k >= 0; k--) {
            if ((k < nrt) & (e[k] != 0.0)) {
               for (int j = k+1; j < nu; j++) {
                  double t = 0;
                  for (int i = k+1; i < n; i++) {
                     t += V[i][k]*V[i][j];
                  }
                  t = -t/V[k+1][k];
                  for (int i = k+1; i < n; i++) {
                     V[i][j] += t*V[i][k];
                  }
               }
            }
            for (int i = 0; i < n; i++) {
               V[i][k] = 0.0;
            }
            V[k][k] = 1.0;
             }
              }

              // Main iteration loop for the singular values.

              int pp = p-1;
              int iter = 0;
              double eps = Math.Pow(2.0,-52.0);
              double tiny = Math.Pow(2.0,-966.0);
              while (p > 0) {
             int k,kase;

             // Here is where a test for too many iterations would go.

             // This section of the program inspects for
             // negligible elements in the s and e arrays.  On
             // completion the variables kase and k are set as follows.

             // kase = 1     if s(p) and e[k-1] are negligible and k<p
             // kase = 2     if s(k) is negligible and k<p
             // kase = 3     if e[k-1] is negligible, k<p, and
             //              s(k), ..., s(p) are not negligible (qr step).
             // kase = 4     if e(p-1) is negligible (convergence).

             for (k = p-2; k >= -1; k--) {
            if (k == -1) {
               break;
            }
            if (Math.Abs(e[k]) <=
                  tiny + eps*(Math.Abs(s[k]) + Math.Abs(s[k+1]))) {
               e[k] = 0.0;
               break;
            }
             }
             if (k == p-2) {
            kase = 4;
             } else {
            int ks;
            for (ks = p-1; ks >= k; ks--) {
               if (ks == k) {
                  break;
               }
               double t = (ks != p ? Math.Abs(e[ks]) : 0.0) +
                          (ks != k+1 ? Math.Abs(e[ks-1]) : 0.0);
               if (Math.Abs(s[ks]) <= tiny + eps*t)  {
                  s[ks] = 0.0;
                  break;
               }
            }
            if (ks == k) {
               kase = 3;
            } else if (ks == p-1) {
               kase = 1;
            } else {
               kase = 2;
               k = ks;
            }
             }
             k++;

             // Perform the task indicated by kase.

             switch (kase) {

            // Deflate negligible s(p).

            case 1: {
               double f = e[p-2];
               e[p-2] = 0.0;
               for (int j = p-2; j >= k; j--) {
                  double t = Maths.Hypot(s[j],f);
                  double cs = s[j]/t;
                  double sn = f/t;
                  s[j] = t;
                  if (j != k) {
                     f = -sn*e[j-1];
                     e[j-1] = cs*e[j-1];
                  }
                  if (wantv) {
                     for (int i = 0; i < n; i++) {
                        t = cs*V[i][j] + sn*V[i][p-1];
                        V[i][p-1] = -sn*V[i][j] + cs*V[i][p-1];
                        V[i][j] = t;
                     }
                  }
               }
            }
            break;

            // Split at negligible s(k).

            case 2: {
               double f = e[k-1];
               e[k-1] = 0.0;
               for (int j = k; j < p; j++) {
                  double t = Maths.Hypot(s[j],f);
                  double cs = s[j]/t;
                  double sn = f/t;
                  s[j] = t;
                  f = -sn*e[j];
                  e[j] = cs*e[j];
                  if (wantu) {
                     for (int i = 0; i < m; i++) {
                        t = cs*U[i][j] + sn*U[i][k-1];
                        U[i][k-1] = -sn*U[i][j] + cs*U[i][k-1];
                        U[i][j] = t;
                     }
                  }
               }
            }
            break;

            // Perform one qr step.

            case 3: {

               // Calculate the shift.

               double scale = Math.Max(Math.Max(Math.Max(Math.Max(
                       Math.Abs(s[p-1]),Math.Abs(s[p-2])),Math.Abs(e[p-2])),
                       Math.Abs(s[k])),Math.Abs(e[k]));
               double sp = s[p-1]/scale;
               double spm1 = s[p-2]/scale;
               double epm1 = e[p-2]/scale;
               double sk = s[k]/scale;
               double ek = e[k]/scale;
               double b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;
               double c = (sp*epm1)*(sp*epm1);
               double shift = 0.0;
               if ((b != 0.0) | (c != 0.0)) {
                  shift = Math.Sqrt(b*b + c);
                  if (b < 0.0) {
                     shift = -shift;
                  }
                  shift = c/(b + shift);
               }
               double f = (sk + sp)*(sk - sp) + shift;
               double g = sk*ek;

               // Chase zeros.

               for (int j = k; j < p-1; j++) {
                  double t = Maths.Hypot(f,g);
                  double cs = f/t;
                  double sn = g/t;
                  if (j != k) {
                     e[j-1] = t;
                  }
                  f = cs*s[j] + sn*e[j];
                  e[j] = cs*e[j] - sn*s[j];
                  g = sn*s[j+1];
                  s[j+1] = cs*s[j+1];
                  if (wantv) {
                     for (int i = 0; i < n; i++) {
                        t = cs*V[i][j] + sn*V[i][j+1];
                        V[i][j+1] = -sn*V[i][j] + cs*V[i][j+1];
                        V[i][j] = t;
                     }
                  }
                  t = Maths.Hypot(f,g);
                  cs = f/t;
                  sn = g/t;
                  s[j] = t;
                  f = cs*e[j] + sn*s[j+1];
                  s[j+1] = -sn*e[j] + cs*s[j+1];
                  g = sn*e[j+1];
                  e[j+1] = cs*e[j+1];
                  if (wantu && (j < m-1)) {
                     for (int i = 0; i < m; i++) {
                        t = cs*U[i][j] + sn*U[i][j+1];
                        U[i][j+1] = -sn*U[i][j] + cs*U[i][j+1];
                        U[i][j] = t;
                     }
                  }
               }
               e[p-2] = f;
               iter = iter + 1;
            }
            break;

            // Convergence.

            case 4: {

               // Make the singular values positive.

               if (s[k] <= 0.0) {
                  s[k] = (s[k] < 0.0 ? -s[k] : 0.0);
                  if (wantv) {
                     for (int i = 0; i <= pp; i++) {
                        V[i][k] = -V[i][k];
                     }
                  }
               }

               // Order the singular values.

               while (k < pp) {
                  if (s[k] >= s[k+1]) {
                     break;
                  }
                  double t = s[k];
                  s[k] = s[k+1];
                  s[k+1] = t;
                  if (wantv && (k < n-1)) {
                     for (int i = 0; i < n; i++) {
                        t = V[i][k+1]; V[i][k+1] = V[i][k]; V[i][k] = t;
                     }
                  }
                  if (wantu && (k < m-1)) {
                     for (int i = 0; i < m; i++) {
                        t = U[i][k+1]; U[i][k+1] = U[i][k]; U[i][k] = t;
                     }
                  }
                  k++;
               }
               iter = 0;
               p--;
            }
            break;
             }
              }
        }
Esempio n. 4
0
        ///<Sumary>Least squares solution of A*X = B</Sumary>
        ///<param name="B">A Matrix with as many rows as A and any number of columns.</param>
        ///<returns> X that minimizes the two norm of Q*R*X-B.</returns>
        public Matrix Solve(Matrix B)
        {
            if (B.RowDimension != m) {
             throw new ArgumentException("Matrix row dimensions must agree.");
              }
              if (!IsFullRank()) {
             throw new Exception("Matrix is rank deficient.");
              }

              // Copy right hand side
              int nx = B.ColumnDimension;
              double[][] X = B.ArrayCopy();

              // Compute Y = transpose(Q)*B
              for (int k = 0; k < n; k++) {
             for (int j = 0; j < nx; j++) {
            double s = 0.0;
            for (int i = k; i < m; i++) {
               s += QR[i][k]*X[i][j];
            }
            s = -s/QR[k][k];
            for (int i = k; i < m; i++) {
               X[i][j] += s*QR[i][k];
            }
             }
              }
              // Solve R*X = Y;
              for (int k = n-1; k >= 0; k--) {
             for (int j = 0; j < nx; j++) {
            X[k][j] /= Rdiag[k];
             }
             for (int i = 0; i < k; i++) {
            for (int j = 0; j < nx; j++) {
               X[i][j] -= X[k][j]*QR[i][k];
            }
             }
              }
              return (new Matrix(X,n,nx).GetMatrix(0,n-1,0,nx-1));
        }
        ///<summary>Solve A*X = B</summary>
        ///<param name="B">A Matrix with as many rows as A and any number of columns.</param>
        ///<returns>X so that L*L'*X = B</returns>
        ///<exception cref="ArgumentException">System.ArgumentException  Matrix row dimensions must agree.</exception>
        ///<exception cref="Exception">System.Exception  Matrix is not symmetric positive definite.</exception>
        public Matrix Solve(Matrix B)
        {
            if (B.RowDimension != n) {
             throw new ArgumentException("Matrix row dimensions must agree.");
              }
              if (!isspd) {
             throw new Exception("Matrix is not symmetric positive definite.");
              }

              // Copy right hand side.
              double[][] X = B.ArrayCopy();
              int nx = B.ColumnDimension;

              // Solve L*Y = B;
              for (int k = 0; k < n; k++) {
            for (int j = 0; j < nx; j++) {
               for (int i = 0; i < k ; i++) {
                   X[k][j] -= X[i][j]*L[k][i];
               }
               X[k][j] /= L[k][k];
            }
              }

              // Solve L'*X = Y;
              for (int k = n-1; k >= 0; k--) {
            for (int j = 0; j < nx; j++) {
               for (int i = k+1; i < n ; i++) {
                   X[k][j] -= X[i][j]*L[i][k];
               }
               X[k][j] /= L[k][k];
            }
              }

              return new Matrix(X,n,nx);
        }