Lecture 10: Systems of equations; optimization in one variable (part I)

Midterm is Monday; one 8.5x11 sheet allowed. Focus is problem-solving, not algorithm regurgitation. Multivariable root-finding discussed: Newton's method extended using Jacobians; Broyden's method (Secant method extension) approximates Jacobians efficiently to avoid repeated computations. Linear algebra and notation are important. The instructor details the midterm exam format, allowed materials (one 8.5x11 sheet of notes), acceptable note-taking methods, and emphasizes problem-solving over rote memorization of algorithms. The focus is on understanding when and how to apply algorithms rather than their intricate implementation details. This segment contrasts single-variable and multivariable root-finding problems. It explains that while solving linear systems is straightforward, nonlinear equations involving functions like cosine, sine, or exponents require more sophisticated methods because Gauss elimination no longer applies. The instructor explains how Newton's method, successful in single-variable root finding, extends to multiple variables using the Jacobian matrix. The discussion includes the size and calculation of the Jacobian matrix, serving as a foundational concept for subsequent multivariable root-finding techniques. This segment highlights the computational challenges of multivariable root finding, particularly when dealing with complex functions requiring extensive simulations or when the Jacobian matrix needs recalculation at each iteration, limiting the effectiveness of strategies like LU factorization. The need for a more efficient solver is established. This segment details the process of solving an optimization problem using Lagrange multipliers. The speaker clearly explains the substitution of variables, the formulation of the Lagrange multiplier function, and the resulting simplification of the problem, making it accessible for viewers to understand the mathematical steps involved in this optimization technique.This segment focuses on calculating the derivative of the Lagrange multiplier function and solving for the unknown matrix, delta J. The speaker breaks down the complex calculations into manageable steps, explaining the logic behind each step and simplifying the expressions to arrive at a concise solution. This detailed explanation makes the complex mathematical process easier to follow.This segment shows the derivation of the formula for updating the Jacobian matrix (delta J) in Broyden's method. The speaker meticulously substitutes values, simplifies expressions, and ultimately arrives at a closed-form solution for delta J. This step-by-step approach makes the derivation clear and understandable, highlighting the elegance and efficiency of the method.