From 591495159ef29a3edac10aaf7e3d7ea28a4a3013 Mon Sep 17 00:00:00 2001
From: Deniz Donmez <deniz.donmez_03@metu.edu.tr>
Date: Mon, 15 Sep 2025 10:40:52 +0300
Subject: [PATCH] new examples

---
 examples/demo.py | 91 +++++++++++++++++++++++++++++++++---------------
 1 file changed, 62 insertions(+), 29 deletions(-)

diff --git a/examples/demo.py b/examples/demo.py
index 5a19eff..a57f143 100644
--- a/examples/demo.py
+++ b/examples/demo.py
@@ -2,27 +2,6 @@ import sys
 sys.path.append('../')
 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():
     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("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():
-    f = lambda x: x**3 - x - 2
-    df = lambda x: 3 * x**2 - 1
-    print("Bisection:", Bisection(f, 1, 2).solve())
-    print("Secant:", Secant(f, 1.0, 2.0).solve())
-    print("Newton root:", NewtonRoot(f, df, 1.5).solve())
-    # A simple contraction for demonstration; not general-purpose
-    g = lambda x: (x + 2 / x**2) / 2
-    print("Fixed point (demo):", FixedPoint(g, 1.5).solve())
+    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):")
+    print_trace(steps)
+
+    # 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():
@@ -55,8 +74,22 @@ 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
+
+    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__":
     demo_linear_solvers()
     demo_roots()
     demo_interpolation()
+    demo_qr()
+    demo_differentiation()