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# Systems of Non-linear equations
So far we've solved system of linear equations where the equations come in the form:
$$
f(x) = a_1x_1 + a_2x_2 + \ldots + a_nx_n-b = 0
$$
where the $a$'s and $b$ are constants. Equations that do not fit this format are called *non linear* equations. For example:
$$
x^2 + xy = 10
$$
$$
y+3xy^2=57
$$
these are two nonlinear equations with two unknowns. To solve this system we can
This roots of single equations and solved system of equations, but what if the system of equations are dependent on non-linear.
$$
\begin{align*}
f_{1}(x_{1},x_{2},\ldots ,x_{n}) &= 0 \\
f_{2}(x_{1},x_{2},\ldots ,x_{n}) &= 0 \\
&\ \vdots \\
f_{n}(x_{1},x_{2},\ldots ,x_{n}) &= 0
\end{align*}
$$
We've applied the Newton-Raphson to
## Problem 1: Newton-Raphson for a Nonlinear system
Use the multiple-equation Newton-Raphson method to determine the roots of the following system of non-linear equations
$$
\begin{cases}
u(x,y) = x^2 + xy - 10 = 0 \\
v(x,y) = y + 3xy^2 - 57 = 0
\end{cases}
$$
after
a) 1 iteration
b) Write a python function that iterates until the tolerance is within 0.002
Note the solution to $x$ =2 and $y$ = 3. Initiate the computation with guesses of $x$ = 1.5 and $y$ = 3.5.
Solution:
a)
```python
u = lambda x, y: x**2 + x*y - 10
v = lambda x, y: y + 3*x*y**2 - 57
# Initial guesses
x_0 = 1.5
y_0 = 3.5
# Evaluate partial derivatives at initial guesses
dudx = 2*x_0+y_0
dudy = x_0
dvdx = 3*y_0**2
dvdy = 1 + 6*x_0*y_0
# Find determinant of the Jacobian
det = dudx * dvdy - dudy * dvdx
# Values of functions valuated at initial guess
u_0 = u(x_0,y_0)
v_0 = v(x_0,y_0)
# Substitute into newton-raphson equation
x_1 = x_0 - (u_0*dvdy-v_0*dudy) / det
y_1 = y_0 - (v_0*dudx-u_0*dvdx) / det
print(x_1)
print(y_1)
```
b)
```python
import numpy as np
def newton_raphson_system(f, df, g0, tol=1e-8, max_iter=50, verbose=False):
"""
Newton-Raphson solver for a 2x2 system of nonlinear equations.
Parameters
----------
f : array of callables
f[i](x, y) should evaluate the i-th equation.
df : 2D array of callables
df[i][j](x, y) is the partial derivative of f[i] wrt variable j.
g0 : array-like
Initial guess [x0, y0].
tol : float
Convergence tolerance.
max_iter : int
Maximum iterations.
verbose : bool
If True, prints iteration details.
Returns
-------
(x, y) : solution vector
iters : number of iterations
"""
x, y = g0
for k in range(1, max_iter + 1):
# Evaluate system of equations
F1 = f[0](x, y)
F2 = f[1](x, y)
# Evaluate Jacobian entries
J11 = df[0][0](x, y)
J12 = df[0][1](x, y)
J21 = df[1][0](x, y)
J22 = df[1][1](x, y)
# Determinant
det = J11 * J22 - J12 * J21
if det == 0:
raise ZeroDivisionError("Jacobian is singular")
# Solve for updates using 2x2 inverse
dx = ( F1*J22 - F2*J12) / det
dy = ( F2*J11 - F1*J21) / det3
# Update variables
x_new = x - dx
y_new = y - dy
if verbose:
print(f"iter {k}: x={x_new:.6f}, y={y_new:.6f}, "
f"|dx|={abs(dx):.2e}, |dy|={abs(dy):.2e}")
# Convergence check
if max(abs(dx), abs(dy)) < tol:
return np.array([x_new, y_new]), k
x, y = x_new, y_new
raise ValueError("Newton-Raphson did not converge")
# -------------------
# Example usage
# -------------------
# Define system: u(x,y), v(x,y)
u = lambda x, y: x**2 + x*y - 10
v = lambda x, y: y + 3*x*y**2 - 57
# Partial derivatives
dudx = lambda x, y: 2*x + y
dudy = lambda x, y: x
dvdx = lambda x, y: 3*y**2
dvdy = lambda x, y: 1 + 6*x*y
f = np.array([u, v])
df = np.array([[dudx, dudy],
[dvdx, dvdy]])
g0 = np.array([1.5, 3.5])
sol, iters = newton_raphson_system(f, df, g0, tol=1e-10, verbose=True)
print("Solution:", sol, "in", iters, "iterations")
```
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