Trigonometric interpolation

### 17.1.4 Trigonometric interpolation

The triginterp command computes a trigonometric polynomial which interpolates given data.

- triginterp takes four arguments:
  - L, a list of numbers.
  - a, a number (the beginning of an interval).
  - b, a number (the end of the interval).
  - x, the name of a variable.
  The last three arguments can also be given as x=a..b.
- triginterp(L,a,b,x) or triginterp(L,x=a..b) returns the trigonometric polynomial that interpolates data given in the list L. It is assumed that the list L contains ordinate components of the points with equidistant abscissa components between a and b such that the first element of L corresponds to a and the last element to b.

L may be a list of experimental measurements of some quantity taken at regular intervals, with the first observation at time t=a and the last observation at time t=b. The resulting trigonometric polynomial has period T=n/n−1(b−a), where n=size(L) is the number of observations.

As an example, assume that the following data is obtained by measuring the temperature every three hours during one day:

Hour of the day | 0 | 3 | 6 | 9 | 12 | 15 | 18 | 21
Temperature (∘C) | 11 | 10 | 17 | 24 | 32 | 26 | 23 | 19

Suppose that an estimate of the temperature at 13:45 is required. To obtain a trigonometric interpolation of the data:

tp:=triginterp([11,10,17,24,32,26,23,19],x=0..21)

81
4
| + |
1
8
|  | ⎛ ⎜ ⎝ | −21 | √ |
2
| −42 | ⎞ ⎟ ⎠ | cos | ⎛ ⎜ ⎜ ⎝ |
1
12
| π x | ⎞ ⎟ ⎟ ⎠ | + |
1
8
|  | ⎛ ⎜ ⎝ | −11 | √ |
2
| −12 | ⎞ ⎟ ⎠ | sin | ⎛ ⎜ ⎜ ⎝ |
1
12
| π x | ⎞ ⎟ ⎟ ⎠ | + |
3
4
| cos | ⎛ ⎜ ⎜ ⎝ |
1
6
| π x | ⎞ ⎟ ⎟ ⎠ | −
7
4
| sin | ⎛ ⎜ ⎜ ⎝ |
1
6
| π x | ⎞ ⎟ ⎟ ⎠ | + |
1
8
|  | ⎛ ⎜ ⎝ | 21 | √ |
2
| −42 | ⎞ ⎟ ⎠ | cos | ⎛ ⎜ ⎜ ⎝ |
1
4
| π x | ⎞ ⎟ ⎟ ⎠ | + |
1
8
|  | ⎛ ⎜ ⎝ | −11 | √ |
2
| +12 | ⎞ ⎟ ⎠ | sin | ⎛ ⎜ ⎜ ⎝ |
1
4
| π x | ⎞ ⎟ ⎟ ⎠ | + |
cos | ⎛ ⎜ ⎜ ⎝ |
1
3
| π x | ⎞ ⎟ ⎟ ⎠
2

To plot the interpolant, enter:

labels=["time","temperature"]; legend=["hours","°C"]; plot(tp,x=0..21)

The interpolant is smooth, without oscillations, and appears realistic in the given context. Now a temperature at 13:45 can be approximated by the value of tp for x=13.75:

tp | x=13.75

29.4863181684 |  |  |  |  |  |  |  |  |  |