Question: Polynomial Interpolation

Thanks for the help on my previous question.

I set up my dataset as follows:

points := [[2.612499, 16.24096], [3.437499, 14.21687], [3.437498, 11.22891], [3.437497, 8.192769], [4.262499, 6.361444], [5.224999, 7.662649], [7.974999, 7.51807], [10.175, 8.192769], [10.725, 9.59036], [12.375, 8.289155], [14.1625, 8.626504], [16.0875, 8.578312], [17.7375, 7.469878], [19.9375, 7.469878], [21.5875, 7.469878], [22.6875, 7.180721], [23.65, 8.289155], [24.0625, 9.638553], [24.75, 11.03614], [25.7125, 8.819275], [26.125, 6.024094], [27.225, 8.192769], [27.775, 5.349395], [28.875, 6.024094], [28.7375, 7.903613], [29.2875, 9.253011], [31.625, 8.578312], [35.75, 8.915661], [37.2625, 9.349396], [38.3625, 9.542167], [40.0125, 8.144576], [41.1125, 7.807227], [42.4875, 6.698793], [43.8625, 6.409636], [45.5125, 10.12048], [47.575, 12.43373], [50.1875, 14.0241], [52.3875, 14.45783], [54.45, 9.493975], [55, 11.75904], [57.6125, 9.831324], [58.7125, 9.59036], [61.05, 9.59036], [62.425, 9.638553]]

> with(CurveFitting);
> PolynomialInterpolation(points, x);
 

I got the following equation, which obviously not the right one.

  1.366320976*10^25*x-3.324247252*10^(-8)*x^32+4.137100219*10^(-10)*x^33-4.344114895*10^24-4.494388947*10^(-12)*x^34+4.224221225*10^(-14)*x^35-3.396220559*10^(-16)*x^36+2.301551012*10^(-18)*x^37-1.289093607*10^(-20)*x^38+5.806976582*10^(-23)*x^39-2.021464086*10^(-25)*x^40+5.102260529*10^(-28)*x^41-8.306717170*10^(-31)*x^42+6.547175495*10^(-34)*x^43-2.024150984*10^25*x^2+1.883573820*10^25*x^3+6.147560142*10^24*x^5-2.398075636*10^24*x^6+7.568813465*10^23*x^7-1.974267448*10^23*x^8+4.325909499*10^22*x^9-8.065873369*10^21*x^10+1.293212669*10^21*x^11-1.798329683*10^20*x^12+5.410457272*10^12*x^19-2.332032412*10^18*x^14+2.198819368*10^17*x^15-1.838902957*10^16*x^16+1.368923639*10^15*x^17-9.097553817*10^13*x^18+2.184523675*10^19*x^13-2.884909536*10^11*x^20+1.381164550*10^10*x^21-5.943072632*10^8*x^22+2.299764834*10^7*x^23-8.004522973*10^5*x^24-1.238974514*10^25*x^4+0.2348239451e-5*x^31+25052.76841*x^25-704.5979381*x^26+17.78630688*x^27-.4023095853*x^28+0.8135588709e-2*x^29-0.1466603550e-3*x^30

I tried increasing the digits to 12 and form = La Grange and it did chnege the outcome.

Cubic Spline works but I'm interested if possible in an equation describing this data set. It happens to be real data of measured temperature over time.

Thanks ahead for your assistance.