Description

Lecture 4
Numerical Interpolation / TDMA
Jung-Il Choi
0. Interpolation
• Problem statement
 For given data
𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑤𝑤 = 1, … , 𝑛𝑛
determine function 𝑓𝑓: ℝ → ℝ𝑠𝑠𝑠𝑠𝑠𝑠𝑤 𝑤𝑤𝑤𝑡𝑡𝑤𝑤
𝑓𝑓 𝑥𝑥𝑖𝑖 = 𝑦𝑦𝑖𝑖 𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑤𝑤 = 1, … , 𝑛𝑛
 Given a new 𝑥𝑥∗, we can interpolate its function value 𝑦𝑦(𝑥𝑥∗). 𝑦𝑦(𝑥𝑥) is interpolating function.
 Example
• Linear interpolation
The estimated point is assumed to lie on the line joining the nearest points to the left and right. Linear interpolation at 𝑥𝑥 is

𝑥𝑥 2 8
𝑦𝑦 10 2

2 − 10
→ 𝑦𝑦 5 =10+ 5 −2 =6
8 − 2

• Lagrange interpolation
 Lagrange polynomial interpolation finds a single polynomial, 𝑃𝑃(𝑥𝑥).
 As an interpolation function, it should have the property 𝑃𝑃 𝑥𝑥𝑖𝑖 = 𝑓𝑓(𝑥𝑥𝑖𝑖) for every point in the dataset. It is useful to write them as a linear combination of Lagrange basis polynomials, 𝑙𝑙𝑖𝑖(𝑥𝑥)
 And
• Lagrange interpolation
Interpolation result(line-by-line code)
• Lagrange interpolation
Interpolation result(Scipy)

• Bivariate functions

• Linear algebra

Back substitution (forward sweep)

• Example
6 3 9 0
0 2 5 2
0 0 4 3
11 / 17
• Cubic Spline

• Cubic Spline
• Cubic Spline
→ 𝑟𝑟𝑟𝑟𝑟𝑟𝑠𝑠𝑤𝑤𝑟𝑟𝑟𝑟𝑟𝑟 2𝑚𝑚𝑚𝑚𝑟𝑟𝑟𝑟 𝑠𝑠𝑚𝑚𝑛𝑛𝑠𝑠𝑤𝑤𝑟𝑟𝑡𝑡𝑤𝑤𝑛𝑛𝑤𝑤𝑠𝑠

• Cubic Spline
 Free run-out (natural spline):

 Parabolic run-out:

 Periodic:

• Cubic Spline
Interpolation result(line-by-line code)
• Cubic Spline
Interpolation result(line-by-line code)

Q&A Thanks for listening