denizdonmez/numethods
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new examples

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Deniz Donmez
2025-09-15 10:40:52 +03:00
parent 6addbce56a
commit 591495159e
1 changed files with 62 additions and 29 deletions
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examples/demo.py
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@@ -2,27 +2,6 @@ import sys
sys.path.append('../') sys.path.append('../')
from numethods import * from numethods import *
A = Matrix([[2, -1], [1, 2], [1, 1]])
b = Vector([1, 2, 3])
# Factorization
qr = QRHouseholder(A)
Q, R = qr.Q, qr.R
print("Q =", Q)
print("R =", R)
qrm = QRModifiedGramSchmidt(A)
Qm, Rm = qrm.Q, qrm.R
print("Qm =", Qm)
print("Rm =", Rm)
print("Q^T Q =", Q.T @ Q)
print("Qm^T Qm =", Qm.T @ Qm)
print("A=Qm Rm =", Qm @ Rm)
# Solve Ax = b (least squares, since A is tall)
x_ls = LeastSquaresSolver(A, b).solve()
print("Least squares solution:", x_ls)
def demo_linear_solvers(): def demo_linear_solvers():
A = Matrix([[4, -1, 0], [-1, 4, -1], [0, -1, 3]]) A = Matrix([[4, -1, 0], [-1, 4, -1], [0, -1, 3]])
@@ -34,16 +13,56 @@ def demo_linear_solvers():
print("Jacobi:", Jacobi(A, b, tol=1e-12).solve()) print("Jacobi:", Jacobi(A, b, tol=1e-12).solve())
print("Gauss-Seidel:", GaussSeidel(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]])
b = Vector([1, 2, 3])
# Factorization
qr = QRHouseholder(A)
Q, R = qr.Q, qr.R
print("Q =", Q)
print("R =", R)
qrm = QRModifiedGramSchmidt(A)
Qm, Rm = qrm.Q, qrm.R
print("Qm =", Qm)
print("Rm =", Rm)
print("Q^T Q =", Q.T @ Q)
print("Qm^T Qm =", Qm.T @ Qm)
print("A=Qm Rm =", Qm @ Rm)
print("A=Q R =", Q @ R)
# Solve Ax = b (least squares, since A is tall)
x_ls = LeastSquaresSolver(A, b).solve()
print("Least squares solution:", x_ls)
def demo_roots(): def demo_roots():
f = lambda x: x**3 - x - 2 f = lambda x: x**2 - 2
df = lambda x: 3 * x**2 - 1 df = lambda x: 2 * x
print("Bisection:", Bisection(f, 1, 2).solve())
print("Secant:", Secant(f, 1.0, 2.0).solve()) # Newton
print("Newton root:", NewtonRoot(f, df, 1.5).solve()) steps = NewtonRoot(f, df, x0=1.0).trace()
# A simple contraction for demonstration; not general-purpose print("Newton Method Trace (x^2 - 2):")
g = lambda x: (x + 2 / x**2) / 2 print_trace(steps)
print("Fixed point (demo):", FixedPoint(g, 1.5).solve())
# Secant
steps = Secant(f, 0, 2).trace()
print("\nSecant Method Trace (x^2 - 2):")
print_trace(steps)
# Bisection
steps = Bisection(f, 0, 2).trace()
print("\nBisection Method Trace (x^2 - 2):")
print_trace(steps)
# Fixed-point: solve
g = lambda x: 0.5 * (x + 2 / x)
steps = FixedPoint(g, 1.0).trace()
print("\nFixed-Point Iteration Trace (x^2 - 2):")
print_trace(steps)
def demo_interpolation(): def demo_interpolation():
@@ -55,8 +74,22 @@ def demo_interpolation():
print("Newton interpolation at", t, "=", newt.evaluate(t)) print("Newton interpolation at", t, "=", newt.evaluate(t))
print("Lagrange interpolation at", t, "=", lagr.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
print("Forward :", ForwardDiff(f, x0))
print("Backward :", BackwardDiff(f, x0))
print("Central :", CentralDiff(f, x0))
print("4th order:", CentralDiff4th(f, x0))
print("Richardson:", RichardsonExtrap(f, x0))
print("Second derivative:", SecondDerivative(f, x0))
if __name__ == "__main__": if __name__ == "__main__":
demo_linear_solvers() demo_linear_solvers()
demo_roots() demo_roots()
demo_interpolation() demo_interpolation()
demo_qr()
demo_differentiation()