Higher Degree Polynomial Interpolation

• Suppose we had a data set of 102 data points.

• A polynomial interpolation to this data would have degree 101.

• Can have as much as 100 oscillations, typically of large magnitude.

• These are found by taking the derivative (deg 100 polynomial) and setting equal to zero, and then solving for critical points.

Draftsman’s Spline

• Woodworking, ship hull design, etc.
• Smooth fit determined by a natural balance of internal and external forces (tension, weights).
• Applications in other areas help create naturally smooth curves.

Clamped Spline

title

• Fit flexible strip of wood through pegs
• Spline is clamped at endpoints
• Specific derivative (shape) condition at endpoints

title

Two Jokes

## 
## 
## Sometimes I tuck my knees into my chest and lean forward.
## 
## That's just how I roll.

## 
## 
## Who invented the round table?
## 
## Sir Cumference

Linear Interpolation with R

• Program from our book.
• Fit equation of line to two data points \((x_1,y_1), \, (x_2,y_2)\).
• For \(y = mx + b\), need \(m\) and \(b\).

\[ \small{ \begin{aligned} m & = \frac{y_2 - y_1}{x_2-x_1} \\ b & = y_2 - m x_2 \end{aligned} } \]

Linear Spline with R

• Program from our book.
• Returns coefficients for each linear spline.

Data Plot Linear Function

• Graphs the linear spline with data

Example 1: Linear Spline

## $m
## [1] -1  1  3
## 
## $b
## [1] -3 -1 -1

## [1] 1.4

Cubic Spline Plot using R

## Warning: package 'pracma' was built under R version 4.2.2

Example 2: Cubic Spline Function & Plot

## [1] 1.3232