public static void RayleighQuotient(MatrixR A, double tolerance, int flag, out VectorR x, out double lambda) { int n = A.GetCols(); double delta = 0.0; Random random = new Random(); x = new VectorR(n); if (flag != 2) { for (int i = 0; i < n; i++) { x[i] = random.NextDouble(); } x.Normalize(); lambda = VectorR.DotProduct(x, MatrixR.Transform(A, x)); } else { lambda = 0.0; Rayleigh(A, 1e-2, out x, out lambda); } double temp = lambda; MatrixR identity = new MatrixR(n, n); LinearSystem ls = new LinearSystem(); do { temp = lambda; double d = ls.LUCrout(A - lambda * identity.Identity(), x); x.Normalize(); lambda = VectorR.DotProduct(x, MatrixR.Transform(A, x)); delta = Math.Abs((temp - lambda) / lambda); }while (delta > tolerance); }
public static void Inverse(MatrixR A, double s, double tolerance, out VectorR x, out double lambda) { int n = A.GetCols(); x = new VectorR(n); lambda = 0.0; double delta = 0.0; MatrixR identity = new MatrixR(n, n); A = A - s * (identity.Identity()); LinearSystem ls = new LinearSystem(); A = ls.LUInverse(A); Random random = new Random(); for (int i = 0; i < n; i++) { x[i] = random.NextDouble(); } do { VectorR temp = x; x = MatrixR.Transform(A, x); x.Normalize(); if (VectorR.DotProduct(temp, x) < 0) { x = -x; } VectorR dx = temp - x; delta = dx.GetNorm(); }while (delta > tolerance); lambda = s + 1.0 / (VectorR.DotProduct(x, MatrixR.Transform(A, x))); }
public static void Rayleigh(MatrixR A, double tolerance, out VectorR x, out double lambda) { int n = A.GetCols(); double delta = 0.0; Random random = new Random(); x = new VectorR(n); for (int i = 0; i < n; i++) { x[i] = random.NextDouble(); } x.Normalize(); VectorR x0 = MatrixR.Transform(A, x); x0.Normalize(); lambda = VectorR.DotProduct(x, x0); double temp = lambda; do { temp = lambda; x0 = x; x0.Normalize(); x = MatrixR.Transform(A, x0); lambda = VectorR.DotProduct(x, x0); delta = Math.Abs((temp - lambda) / lambda); }while (delta > tolerance); x.Normalize(); }
public static void Power(MatrixR A, double tolerance, out VectorR x, out double lambda) { int n = A.GetCols(); x = new VectorR(n); lambda = 0.0; double delta = 0.0; Random random = new Random(); for (int i = 0; i < n; i++) { x[i] = random.NextDouble(); } do { VectorR temp = x; x = MatrixR.Transform(A, x); x.Normalize(); if (VectorR.DotProduct(temp, x) < 0) { x = -x; } VectorR dx = temp - x; delta = dx.GetNorm(); }while (delta > tolerance); lambda = VectorR.DotProduct(x, MatrixR.Transform(A, x)); }
public VectorR GaussJordan(MatrixR A, VectorR b) { Triangulate(A, b); int n = b.GetSize(); VectorR x = new VectorR(n); for (int i = n - 1; i >= 0; i--) { double d = A[i, i]; if (Math.Abs(d) < epsilon) { throw new ArgumentException("Diagonal element is too small!"); } x[i] = (b[i] - VectorR.DotProduct(A.GetRowVector(i), x)) / d; } return(x); }
public static VectorR NewtonMultiEquations(MFunction f, VectorR x0, double tolerance) { LinearSystem ls = new LinearSystem(); VectorR dx = new VectorR(x0.GetSize()); do { MatrixR A = Jacobian(f, x0); if (Math.Sqrt(VectorR.DotProduct(f(x0), f(x0)) / x0.GetSize()) < tolerance) { return(x0); } dx = ls.GaussJordan(A, -f(x0)); x0 = x0 + dx; }while (Math.Sqrt(VectorR.DotProduct(dx, dx)) > tolerance); return(x0); }
public static VectorR TriVectorProduct(VectorR v1, VectorR v2, VectorR v3) { if (v1.size != 3) { throw new ArgumentOutOfRangeException( "v1", v1, "Vector v1 must be 3 dimensional!"); } if (v1.size != 3) { throw new ArgumentOutOfRangeException( "v2", v2, "Vector v2 must be 3 dimensional!"); } if (v1.size != 3) { throw new ArgumentOutOfRangeException( "v3", v3, "Vector v3 must be 3 dimensional!"); } return(v2 * VectorR.DotProduct(v1, v3) - v3 * VectorR.DotProduct(v1, v2)); }
public static VectorR TridiagonalEigenvector(double s, double tolerance, out double lambda) { int n = Alpha.GetLength(0); double[] gamma = (double[])Beta.Clone(); double[] beta = (double[])Beta.Clone(); double[] alpha = new double[n]; for (int i = 0; i < n; i++) { alpha[i] = Alpha[i] - s; } double[] gamma1, alpha1, beta1; LUDecomposition(gamma, alpha, beta, out gamma1, out alpha1, out beta1); VectorR x = new VectorR(n); Random random = new Random(); for (int i = 0; i < n; i++) { x[i] = random.NextDouble(); } x.Normalize(); VectorR x1 = new VectorR(n);; double sign; do { x1 = x.Clone(); LUSolver(gamma1, alpha1, beta1, x); x.Normalize(); if (VectorR.DotProduct(x1, x) < 0.0) { sign = -1.0; x = -x; } else { sign = 1.0; } }while ((x - x1).GetNorm() > tolerance); lambda = s + sign / x.GetNorm(); return(x); }
public static MatrixR operator *(MatrixR m1, MatrixR m2) { if (m1.GetCols() != m2.GetRows()) { throw new ArgumentOutOfRangeException( "Columns", m1, "The numbers of columns of the first matrix must be equal to" + " the number of rows of the second matrix!"); } MatrixR result = new MatrixR(m1.GetRows(), m2.GetCols()); VectorR v1 = new VectorR(m1.GetCols()); VectorR v2 = new VectorR(m2.GetRows()); for (int i = 0; i < m1.GetRows(); i++) { v1 = m1.GetRowVector(i); for (int j = 0; j < m2.GetCols(); j++) { v2 = m2.GetColVector(j); result[i, j] = VectorR.DotProduct(v1, v2); } } return(result); }
public static MatrixR Tridiagonalize(MatrixR A) { int n = A.GetCols(); MatrixR A1 = new MatrixR(n, n); A1 = A.Clone(); double h, g, unorm; for (int i = 0; i < n - 2; i++) { VectorR u = new VectorR(n - i - 1); for (int j = i + 1; j < n; j++) { u[j - i - 1] = A[i, j]; } unorm = u.GetNorm(); if (u[0] < 0.0) { unorm = -unorm; } u[0] += unorm; for (int j = i + 1; j < n; j++) { A[j, i] = u[j - i - 1]; } h = VectorR.DotProduct(u, u) * 0.5; VectorR v = new VectorR(n - i - 1); MatrixR a1 = new MatrixR(n - i - 1, n - i - 1); for (int j = i + 1; j < n; j++) { for (int k = i + 1; k < n; k++) { a1[j - i - 1, k - i - 1] = A[j, k]; } } v = MatrixR.Transform(a1, u) / h; g = VectorR.DotProduct(u, v) / (2.0 * h); v -= g * u; for (int j = i + 1; j < n; j++) { for (int k = i + 1; k < n; k++) { A[j, k] = A[j, k] - v[j - i - 1] * u[k - i - 1] - u[j - i - 1] * v[k - i - 1]; } } A[i, i + 1] = -unorm; } Alpha = new double[n]; Beta = new double[n - 1]; Alpha[0] = A[0, 0]; for (int i = 1; i < n; i++) { Alpha[i] = A[i, i]; Beta[i - 1] = A[i - 1, i]; } MatrixR V = new MatrixR(n, n); V = V.Identity(); for (int i = 0; i < n - 2; i++) { VectorR u = new VectorR(n - i - 1); for (int j = i + 1; j < n; j++) { u[j - i - 1] = A.GetColVector(i)[j]; } h = VectorR.DotProduct(u, u) * 0.5; VectorR v = new VectorR(n - 1); MatrixR v1 = new MatrixR(n - 1, n - i - 1); for (int j = 1; j < n; j++) { for (int k = i + 1; k < n; k++) { v1[j - 1, k - i - 1] = V[j, k]; } } v = MatrixR.Transform(v1, u) / h; for (int j = 1; j < n; j++) { for (int k = i + 1; k < n; k++) { V[j, k] -= v[j - 1] * u[k - i - 1]; } } } return(V); }