MS-A0211 - Differential and Integral Calculus 2, Lecture, 10.1.2023-20.2.2023
This course space end date is set to 20.02.2023 Search Courses: MS-A0211
Lecture 8 - 2.2.2023
Completion requirements
In this lecture we studied:
- The Jacobian matrix. We introduced vector - valued functions of several variables F and the notion of Jacobian matrix J_F.
- Newton's method. Newton's method is a numerical method for finding approximate solutions of (systems of) equations. We first considered one equation in one variable f(x) = 0. That is, we want to find where the graph crosses the x-axis. The idea of the method is to guess an initial point x_0. Then, approximate f(x) at x_0 by its tangent line y = f(x_0) + f'(x_0)(x-x_0). We can find where this tangent line crosses the x-axis: x_1 = x_0 -f(x_0)/f'(x_0). Finally, iterate this procedure with the general expression: x_{n+1} = x_n-f(x_n)/f'(x_n). For many examples this algorithm works efficiently. We showed an example of implementing this on Maple. Then, we considered the case of m-equations in m unknowns (we assumed first m=2). That is, we want to solve F(x)=0, where now F(x) and x are vectors of m variables. We considered also an initial guess x_0, and we obtained the iterative expression: x_{n+1} = x_n -J_F^(-1)(x_n) F(x_n). Here (-1) means matrix inverse. We showed an example of implementing this on Maple.
The definition of the Jacobian matrix can be found in section 12.6 (functions from n-space to m-space) and the Newton's method in section 13.6 of Adam and Essex’s Calculus book (seventh edition).
Maple codes of some examples seen during the lecture: