//u'' = f; u(a) = u0 u(b) = u1 public static void SolveSecondDerevetivWhithBounderyCondition(int SIZE, double a_border, double b_border, ICardinalStratagy calculate, double u0, double u1, FunctionLib.Function function) { int degree = 4; int N = SIZE; int M = N + degree - 2; int m = degree - 2; Matrix U = new Matrix(m, M); double h = MyMath.Basic.GetStep(N, a_border, b_border); CardinalSpline spline = new CardinalSpline(calculate); Matrix D = (1d / Math.Pow(h, m)) * CardinalOperators.DerevetiveMatrix(degree, SIZE); //условие 1: (c,ksi(a)) = u0 Vector ksi_a = spline.SplineVector(a_border, a_border, b_border, N, degree); //условие 1: (c,ksi(b)) = u1 Vector ksi_b = spline.SplineVector(b_border, a_border, b_border, N, degree); for (int i = 0; i < M; i++) { U[0, i] = ksi_a[i]; U[1, i] = ksi_b[i]; } //------------- //Составляем линейную систему Matrix A = ConstructMatrix(D, U); //Составляем правую часть Vector b = MyMath.Basic.GetVectorFunction(N, a_border, b_border, function); Vector y = Matrix.Transpose(D) * b; Vector y_prime = new Vector(y.Length + m); y_prime[y.Length] = u0; y_prime[y.Length + 1] = u1; for (int i = 0; i < y.Length; i++) { y_prime[i] = y[i]; } Console.WriteLine("right part = " + y_prime); // Console.WriteLine(A); Vector x = Solver.BCG(A, y_prime, 0.0001d); Console.WriteLine("x = " + x); Vector c = x.SubVector(0, M); Vector f = spline.GetVectorFunction(N, a_border, b_border, c, h, degree); Console.WriteLine("f = " + f); f = spline.GetVectorFunction(10 * N, a_border, b_border, c, h, degree); b = MyMath.Basic.GetVectorFunction(10 * N, a_border, b_border, function); Console.WriteLine("||err|| = " + (f - b).Norm); }
public void Test() { int N = 10; int degree = 2; double h = 1; double a_border = 0, b_border = 1; double u0 = 1; Matrix D = CardinalDifferentialEquation.FirstDirevetive(N); Matrix KSI = CardinalDifferentialEquation.CreateKSIMatrix(degree, h, N + 1); Matrix DD = D * Matrix.Transpose(D); //KSI[0, 0] -= KSI[0, 1]; //KSI[N, N] -= KSI[0, 1]; //DD[0, 0] = DD[N, N] = 2d; Console.WriteLine(DD); Console.WriteLine(KSI); Matrix A = KSI * DD; //KSI[0, 0] += KSI[0, 1]; //KSI[N, N] += KSI[0, 1]; for (int i = 0; i < N + 1; i++) { A[i, i] *= 1d; } A = 2d * A; //A[0, 0] = 1.5; // A[N, N] = 1.5; Console.WriteLine(A); Vector y = Vector.GetConstVector(1d, N); for (int i = 0; i < N; i++) { y[i] = i * h; } Console.WriteLine(D * y); Vector Dy = (KSI * D) * y; Vector b = new Vector(N + 2); for (int i = 0; i < N + 1; i++) { b[i] = 2d * Dy[i]; } b[N + 1] = u0; Console.WriteLine("b = " + b); Matrix B = new Matrix(N + 2); for (int i = 0; i < N + 1; i++) { for (int j = 0; j < N + 1; j++) { B[i, j] = A[i, j]; } } for (int i = 0; i < N; i++) { B[N + 1, i] = CardinalSpline.Cardinal(degree + 1, a_border, a_border + (i - (degree + 1) + 1) * h, h); B[i, N + 1] = -B[N + 1, i]; } Console.WriteLine(B); Vector x = Vector.GetConstVector(2d, N + 2); x[N + 1] = 0; for (int i = 0; i < N + 1; i++) { x[i] = i + 1; } Console.WriteLine("x = " + x); Console.WriteLine("B*x = " + B * x); Console.WriteLine("b = " + b); Vector c = Solver.BCG(B, b, 0.00000001d); Console.WriteLine("anser = " + c); Console.WriteLine(B * c); }
public void SecondDerevitiveSinTest() { int N = 50; int degree = 2; double a_border = 0, b_border = 6.29; double h = MyMath.Basic.GetStep(N, a_border, b_border); double u0 = FunctionLib.cos(a_border), u1 = FunctionLib.cos(b_border); Vector f = MyMath.Basic.GetVectorFunction(N, a_border, b_border, FunctionLib.cos); Matrix D = CardinalDifferentialEquation.SecondDirevetive(N); Matrix DD = D * Matrix.Transpose(D); // Console.WriteLine(DD); Matrix A = (1d / (h * h)) * DD; Console.WriteLine(D * f); Vector Dy = D * f; Vector b = new Vector(N + 4); for (int i = 0; i < N + 2; i++) { b[i] = Dy[i]; } b[N + 2] = u0; b[N + 3] = u1; Console.WriteLine("b = " + b); Matrix B = new Matrix(N + 4); for (int i = 0; i < N + 2; i++) { for (int j = 0; j < N + 2; j++) { B[i, j] = A[i, j]; } } for (int i = 0; i < N + 2; i++) { B[N + 2, i] = CardinalSpline.Cardinal(degree + 2, a_border, a_border + (i - (degree + 2) + 1) * h, h); B[N + 3, i] = CardinalSpline.Cardinal(degree + 2, b_border, a_border + (i - (degree + 2) + 1) * h, h); B[i, N + 2] = -B[N + 2, i]; B[i, N + 3] = -B[N + 3, i]; } // Console.WriteLine(B); //Console.WriteLine("b = " + b); Vector c = Solver.BCG(B, b, 0.00000001d); Console.WriteLine("anser = " + c); Console.WriteLine("Ax = " + B * c); Vector d = c.SubVector(0, N + 2); CardinalSpline spline = new CardinalSpline(); Vector anser = spline.GetVectorFunction(10, a_border, b_border, d, h, degree + 2); Vector expect = MyMath.Basic.GetVectorFunction(10, a_border, b_border, FunctionLib.cos); Console.WriteLine(anser); Console.WriteLine(expect); Console.WriteLine("||u - u_spl|| = " + (expect - anser).Norm); }
public void SinTest() { int N = 50; int degree = 2; double a_border = 0, b_border = 6.29; double h = MyMath.Basic.GetStep(N, a_border, b_border); double u0 = 0; Vector f = MyMath.Basic.GetVectorFunction(N, a_border, b_border, FunctionLib.cos); Matrix D = CardinalDifferentialEquation.FirstDirevetive(N); //Matrix KSI = CardinalDifferentialEquation.CreateKSIMatrix(degree, h, N + 1); Matrix DD = D * Matrix.Transpose(D); //KSI[0, 0] -= KSI[0, 1]; //KSI[N, N] -= KSI[0, 1]; //DD[0, 0] = DD[N, N] = 2d; Console.WriteLine(DD); //Console.WriteLine(KSI); Matrix A = (1d / h) * DD; // Console.WriteLine(A); Console.WriteLine(D * f); Vector Dy = D * f; Vector b = new Vector(N + 2); for (int i = 0; i < N + 1; i++) { b[i] = Dy[i]; } b[N + 1] = u0; Console.WriteLine("b = " + b); Matrix B = new Matrix(N + 2); for (int i = 0; i < N + 1; i++) { for (int j = 0; j < N + 1; j++) { B[i, j] = A[i, j]; } } for (int i = 0; i < N; i++) { B[N + 1, i] = CardinalSpline.Cardinal(degree + 1, a_border, a_border + (i - (degree + 1) + 1) * h, h); B[i, N + 1] = -B[N + 1, i]; } //Console.WriteLine(B); Console.WriteLine("b = " + b); Vector c = Solver.BCG(B, b, 0.00000001d); Console.WriteLine("anser = " + c); Console.WriteLine("Ax = " + B * c); Vector d = c.SubVector(0, N + 1); CardinalSpline spline = new CardinalSpline(); Vector anser = spline.GetVectorFunction(100, a_border, b_border, d, h, degree + 1); Vector expect = MyMath.Basic.GetVectorFunction(100, a_border, b_border, FunctionLib.sin); Console.WriteLine("||u - u_spl|| = " + (expect - anser).Norm); }