Basic Examples of Hermite Interpolation & Cubic Spline Interpolation (also Free vs Clamped Boundary)

https://www.youtube.com/watch?v=bIb8wnd7qhU. Numerical Methods course (Numerical Analysis course) Lecture 22 at Bethel University, St. Paul, MN, Spring 2020. This is a calculus-based advanced undergraduate numerical analysis course. Numerical Analysis Textbook, by Burden, Faires, and Burden: https://amzn.to/2V2f5oI. Amazon Prime Student 6-Month Trial: https://amzn.to/3iUKwdP. In this video I do two basic examples with Hermite polynomial interpolation and cubic splines. These can be related to Lagrange interpolating polynomials as well. For cubic splines, I also consider a free boundary (natural boundary) problem and a clamped boundary problem. Lecture Documents: https://drive.google.com/drive/folders/1sYHvAmZB_lWHL9nIcnbSY8TIEA_7fYjG?usp=sharing Check out my math blog: https://infinityisreallybig.com/ Bethel University is a Christian liberal arts university in St. Paul, Minnesota with strong science, engineering, mathematics and computer science departments: https://www.bethel.edu/ (0:00) Introduction (0:34) Lecture plan (2:33) Example 1: two data points and two slopes (4:04) The system of 4 linear equations and 4 unknowns (5:00) Solution using elementary row operations on an augmented matrix to reduced row echelon form (10:24) Use a TI calculator to solve the system using RREF (12:34) The answer and double-checking it symbolically (13:12) The graph of the answer (13:51) Relationship to Lagrange polynomials (15:45) Hermite basis polynomials (18:09) Graphs of the Hermite basis polynomials (20:01) Confirm this works with Mathematica (21:02) Use divided differences to confirm the answer (25:00) Example 2: three data points and three slopes (26:02) Will use both Hermite interpolation and a cubic spline (27:06) Hermite interpolation with divided differences (29:18) Cubic splines as just a piecewise interpolation (30:48) System of 8 linear equations in 8 unknowns (33:09) The answer (using technology) (33:58) Compare the graphs (34:38) The cubic spline does not have a continuous 2nd derivative (35:23) Free boundary and clamped boundary conditions (37:20) Free boundary conditions and solution (38:51) Clamped boundary conditions and solution (39:51) Compare the graphs (41:24) Spreadsheet implementation of Hermite interpolation (44:12) Use Mathematica to animate how the graph changes as the 2nd slope changes (2 nodes) (45:17) Mathematica for the 3 node case (45:59) Solve in Mathematica AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.