6.1 N Dimensional Search Procedures

The speaker introduces the concept of trial and error searching in the context of complex problem-solving, like constructing a bridge. They set the stage for a demonstration involving a model bridge with weights and strings to illustrate the principles of tension and angles in a real-world scenario. The emphasis is on understanding the scientific method behind solving such problems, even when they appear complicated.

This paragraph delves into a practical demonstration of the physics involved in bridge construction. The setup includes a model with a river, canyon, and a bridge with weights and strings. The focus is on the angles formed by the strings and the tension within them as weights are adjusted. The speaker explains how changing the mass affects the angles and tension, highlighting the importance of positive tension for structural integrity.

The speaker transitions to the mathematical formulation of the bridge problem, identifying unknown tensions and angles that need to be solved. They explain that six unknowns require at least six equations, derived from geometric constraints and Newton's laws for static equilibrium. The paragraph outlines the process of writing down these equations, emphasizing the importance of understanding each component for successful problem-solving.

In this section, the speaker breaks down the complexity of the equations involved in the bridge problem. They discuss the challenge of solving non-linear equations with transcendental functions and the impracticality of finding an analytical solution by hand. The speaker introduces the concept of using numerical methods and computers to find solutions to such complex systems of equations.

The speaker introduces the extension of the Newton-Raphson method to multi-dimensional problems, which is essential for solving complex equations like those in the bridge model. They explain the process of turning the non-linear equations into a form that can be solved using numerical methods, emphasizing the role of computers in handling the complexity and the iterative nature of the search for solutions.

This paragraph explains how to apply the Newton-Raphson method to the multi-variable equations derived from the bridge problem. The speaker details the process of making an initial guess, using a Taylor expansion to approximate the corrections needed for the guess, and then solving for these corrections using matrix equations. The emphasis is on the iterative process and the use of partial derivatives to update the guess.

The speaker further elaborates on the matrix formulation of the problem, explaining how to express the system of equations in a compact form using matrices and vectors. They discuss the process of evaluating the partial derivatives and using numerical methods to approximate them, leading to a solvable matrix equation for the unknown corrections. The paragraph also touches on the practical aspects of solving such matrix equations using computer libraries.

The speaker concludes by encouraging the audience to engage with the equations, experiment with different values, and understand the impact on the solution. They suggest trying to 'break' the code to see how it fails with poor initial guesses and to explore enhancements like back-tracking. For those interested, they propose an extension of the problem to include three masses, offering a more complex challenge.

In the final paragraph, the speaker reflects on the power of mathematical abstraction, showing how the one-dimensional Newton-Raphson method is fundamentally the same as the multi-dimensional matrix equation. They discuss the formal solution involving the inverse of the Jacobian matrix and highlight the importance of understanding the underlying principles rather than just the computational techniques.