PyTorch Newton's Method Implementations

This repository contains implementations of Newton's method using PyTorch for both single-variable and multi-variable optimization problems.

Files

1. newton.py - Implementation of Newton's method for single-variable functions
2. newton_multivariable.py - Implementation of Newton's method for multi-variable systems of equations
3. loss_backword.py - Example using PyTorch's autograd for gradient computation

Newton's Method for Multi-variable Systems

The newton_multivariable.py file implements Newton's method to find the intersection of:

• A circle: x₁² + x₂² = 4
• A parabola: x₁ = x₂²

The implementation provides two solution approaches:

1. Using torch.linalg.solve to solve the linear system J * dx = y
2. Using matrix inversion with torch.linalg.inv to compute x_new = x - J⁻¹y

Key features:

• Automatic differentiation with torch.autograd.functional.jacobian
• Custom implementation of the Jacobian matrix
• Convergence checking with tolerance
• Maximum iteration limit

Requirements

• Python 3.x
• PyTorch

Usage

To run the multi-variable Newton's method:

python newton_multivariable.py

The implementation will find the intersection points of the circle and parabola, demonstrating the convergence of Newton's method for systems of nonlinear equations.