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