diff --git a/examples/demo.py b/examples/demo.py index a57f143..c3596aa 100644 --- a/examples/demo.py +++ b/examples/demo.py @@ -1,5 +1,6 @@ import sys -sys.path.append('../') + +sys.path.append("../") from numethods import * @@ -13,6 +14,7 @@ def demo_linear_solvers(): print("Jacobi:", Jacobi(A, b, tol=1e-12).solve()) print("Gauss-Seidel:", GaussSeidel(A, b, tol=1e-12).solve()) + def demo_qr(): A = Matrix([[2, -1], [1, 2], [1, 1]]) @@ -38,11 +40,7 @@ def demo_qr(): print("Least squares solution:", x_ls) - def demo_roots(): - f = lambda x: x**2 - 2 - df = lambda x: 2 * x - # Newton steps = NewtonRoot(f, df, x0=1.0).trace() print("Newton Method Trace (x^2 - 2):") @@ -74,6 +72,7 @@ def demo_interpolation(): print("Newton interpolation at", t, "=", newt.evaluate(t)) print("Lagrange interpolation at", t, "=", lagr.evaluate(t)) + def demo_differentiation(): f = lambda x: x**3 # f'(x) = 3x^2, f''(x) = 6x x0 = 2.0 @@ -86,7 +85,6 @@ def demo_differentiation(): print("Second derivative:", SecondDerivative(f, x0)) - if __name__ == "__main__": demo_linear_solvers() demo_roots() diff --git a/tutorials/tutorial4_root_finding.ipynb b/tutorials/tutorial4_root_finding.ipynb index 33c9d07..1dba70e 100644 --- a/tutorials/tutorial4_root_finding.ipynb +++ b/tutorials/tutorial4_root_finding.ipynb @@ -161,7 +161,7 @@ } ], "source": [ - "from numethods import Bisection, FixedPoint, NewtonRoot, Secant\n", + "from numethods import Bisection, FixedPoint, NewtonRoot, Secant, print_trace\n", "import math\n", "\n", "# Example function: f(x) = x^2 - 2\n", @@ -193,6 +193,105 @@ " print('Fixed-point failed:', e)\n" ] }, + { + "cell_type": "code", + "execution_count": 2, + "id": "7f8fe13e", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Newton Method Trace (x^2 - 2):\n", + " iter | x | f(x) | df(x) | x_new | error\n", + "------------------------------------------------------------------------------\n", + " 0 | 1 | -1 | 2 | 1.5 | 0.5\n", + " 1 | 1.5 | 0.25 | 3 | 1.41667 | 0.0833333\n", + " 2 | 1.41667 | 0.00694444 | 2.83333 | 1.41422 | 0.00245098\n", + " 3 | 1.41422 | 6.0073e-06 | 2.82843 | 1.41421 | 2.1239e-06\n", + " 4 | 1.41421 | 4.51061e-12 | 2.82843 | 1.41421 | 1.59472e-12\n", + "\n", + "Secant Method Trace (x^2 - 2):\n", + " iter | x0 | x1 | x2 | f(x0) | f(x1) | error\n", + "-------------------------------------------------------------------------------------------\n", + " 0 | 0 | 2 | 1 | -2 | 2 | 1\n", + " 1 | 2 | 1 | 1.33333 | 2 | -1 | 0.333333\n", + " 2 | 1 | 1.33333 | 1.42857 | -1 | -0.222222 | 0.0952381\n", + " 3 | 1.33333 | 1.42857 | 1.41379 | -0.222222 | 0.0408163 | 0.0147783\n", + " 4 | 1.42857 | 1.41379 | 1.41421 | 0.0408163 | -0.00118906 | 0.000418335\n", + " 5 | 1.41379 | 1.41421 | 1.41421 | -0.00118906 | -6.00729e-06 | 2.12421e-06\n", + " 6 | 1.41421 | 1.41421 | 1.41421 | -6.00729e-06 | 8.93146e-10 | 3.15775e-10\n", + " 7 | 1.41421 | 1.41421 | 1.41421 | 8.93146e-10 | -8.88178e-16 | 2.22045e-16\n", + "\n", + "Bisection Method Trace (x^2 - 2):\n", + " iter | a | b | c | f(a) | f(b) | f(c) | interval\n", + "--------------------------------------------------------------------------------------------------------\n", + " 0 | 0 | 2 | 1 | -2 | 2 | -1 | 2\n", + " 1 | 1 | 2 | 1.5 | -1 | 2 | 0.25 | 1\n", + " 2 | 1 | 1.5 | 1.25 | -1 | 0.25 | -0.4375 | 0.5\n", + " 3 | 1.25 | 1.5 | 1.375 | -0.4375 | 0.25 | -0.109375 | 0.25\n", + " 4 | 1.375 | 1.5 | 1.4375 | -0.109375 | 0.25 | 0.0664062 | 0.125\n", + " 5 | 1.375 | 1.4375 | 1.40625 | -0.109375 | 0.0664062 | -0.0224609 | 0.0625\n", + " 6 | 1.40625 | 1.4375 | 1.42188 | -0.0224609 | 0.0664062 | 0.0217285 | 0.03125\n", + " 7 | 1.40625 | 1.42188 | 1.41406 | -0.0224609 | 0.0217285 | -0.000427246 | 0.015625\n", + " 8 | 1.41406 | 1.42188 | 1.41797 | -0.000427246 | 0.0217285 | 0.0106354 | 0.0078125\n", + " 9 | 1.41406 | 1.41797 | 1.41602 | -0.000427246 | 0.0106354 | 0.00510025 | 0.00390625\n", + " 10 | 1.41406 | 1.41602 | 1.41504 | -0.000427246 | 0.00510025 | 0.00233555 | 0.00195312\n", + " 11 | 1.41406 | 1.41504 | 1.41455 | -0.000427246 | 0.00233555 | 0.000953913 | 0.000976562\n", + " 12 | 1.41406 | 1.41455 | 1.41431 | -0.000427246 | 0.000953913 | 0.000263274 | 0.000488281\n", + " 13 | 1.41406 | 1.41431 | 1.41418 | -0.000427246 | 0.000263274 | -8.20011e-05 | 0.000244141\n", + " 14 | 1.41418 | 1.41431 | 1.41425 | -8.20011e-05 | 0.000263274 | 9.06326e-05 | 0.00012207\n", + " 15 | 1.41418 | 1.41425 | 1.41422 | -8.20011e-05 | 9.06326e-05 | 4.31482e-06 | 6.10352e-05\n", + " 16 | 1.41418 | 1.41422 | 1.4142 | -8.20011e-05 | 4.31482e-06 | -3.88434e-05 | 3.05176e-05\n", + " 17 | 1.4142 | 1.41422 | 1.41421 | -3.88434e-05 | 4.31482e-06 | -1.72643e-05 | 1.52588e-05\n", + " 18 | 1.41421 | 1.41422 | 1.41421 | -1.72643e-05 | 4.31482e-06 | -6.47477e-06 | 7.62939e-06\n", + " 19 | 1.41421 | 1.41422 | 1.41421 | -6.47477e-06 | 4.31482e-06 | -1.07998e-06 | 3.8147e-06\n", + " 20 | 1.41421 | 1.41422 | 1.41421 | -1.07998e-06 | 4.31482e-06 | 1.61742e-06 | 1.90735e-06\n", + " 21 | 1.41421 | 1.41421 | 1.41421 | -1.07998e-06 | 1.61742e-06 | 2.68718e-07 | 9.53674e-07\n", + " 22 | 1.41421 | 1.41421 | 1.41421 | -1.07998e-06 | 2.68718e-07 | -4.05632e-07 | 4.76837e-07\n", + " 23 | 1.41421 | 1.41421 | 1.41421 | -4.05632e-07 | 2.68718e-07 | -6.84571e-08 | 2.38419e-07\n", + " 24 | 1.41421 | 1.41421 | 1.41421 | -6.84571e-08 | 2.68718e-07 | 1.0013e-07 | 1.19209e-07\n", + " 25 | 1.41421 | 1.41421 | 1.41421 | -6.84571e-08 | 1.0013e-07 | 1.58366e-08 | 5.96046e-08\n", + " 26 | 1.41421 | 1.41421 | 1.41421 | -6.84571e-08 | 1.58366e-08 | -2.63102e-08 | 2.98023e-08\n", + " 27 | 1.41421 | 1.41421 | 1.41421 | -2.63102e-08 | 1.58366e-08 | -5.23681e-09 | 1.49012e-08\n", + " 28 | 1.41421 | 1.41421 | 1.41421 | -5.23681e-09 | 1.58366e-08 | 5.2999e-09 | 7.45058e-09\n", + " 29 | 1.41421 | 1.41421 | 1.41421 | -5.23681e-09 | 5.2999e-09 | 3.15445e-11 | 3.72529e-09\n", + "\n", + "Fixed-Point Iteration Trace (x^2 - 2):\n", + " iter | x | x_new | error\n", + "----------------------------------------------------\n", + " 0 | 1 | 1.5 | 0.5\n", + " 1 | 1.5 | 1.41667 | 0.0833333\n", + " 2 | 1.41667 | 1.41422 | 0.00245098\n", + " 3 | 1.41422 | 1.41421 | 2.1239e-06\n", + " 4 | 1.41421 | 1.41421 | 1.59495e-12\n" + ] + } + ], + "source": [ + "# Newton\n", + "steps = NewtonRoot(f, df, x0=1.0).trace()\n", + "print(\"Newton Method Trace (x^2 - 2):\")\n", + "print_trace(steps)\n", + "\n", + "# Secant\n", + "steps = Secant(f, 0, 2).trace()\n", + "print(\"\\nSecant Method Trace (x^2 - 2):\")\n", + "print_trace(steps)\n", + "\n", + "# Bisection\n", + "steps = Bisection(f, 0, 2).trace()\n", + "print(\"\\nBisection Method Trace (x^2 - 2):\")\n", + "print_trace(steps)\n", + "\n", + "# Fixed-point: solve\n", + "g = lambda x: 0.5 * (x + 2 / x)\n", + "steps = FixedPoint(g, 1.0).trace()\n", + "print(\"\\nFixed-Point Iteration Trace (x^2 - 2):\")\n", + "print_trace(steps)" + ] + }, { "cell_type": "markdown", "id": "833b538c",