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10.2.6  Trigonometric interpolation

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

Example

The data 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 (ba)
n−1
,

where n=size(L) is the number of observations. For a concrete example, assume that the following data is obtained by measuring the temperature every three hours:

Hour of the day036912151821
Temperature (C)1110172432262319

Furthermore, assume 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
         

Now a temperature at 13:45 hrs can be approximated with the value of tp for x=13.75.

tp | x=13.75
     
29.4863181684           

If one of the input parameters is inexact, the result will also be inexact. For example:

Digits:=3:; triginterp([11,10,17,24,32,26,23,19],x=0..21.0)
     
 
20.2−8.96 cos
0.262 x
−3.44 sin
0.262 x
+ 0.75 cos
0.524 x
         
 
1.75 sin
0.524 x
−1.54 cos
0.785 x
− 0.445 sin
0.785 x
+ 0.5 cos
1.05 x
         

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