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Optimal distribution of interpolation points

Thermodynamic Formulae

saturation steam pressure for humid air above water or ice

hPa 100 Pa * :

T k E.symb k 1 el e k 2 el T * k 3 el T + /^* :

k.w 614.0726 17.62 243.12 31 mat :

k.i 614.0726 22.46 272.62 31 mat :

coefficients or water and ice

ϑ E ϑ k.w E.symb Pa * ϑ 0.01 ≥ (ϑ 60 ≤ (& ϑ k.i E.symb Pa * ϑ 40 - (≥ (ϑ 0.01 < (& not defined cases :

put together
for water and ice

Formula to be interpolated:
Saturation steam pressure in humid air

ϑ E

Test

20 E Pa 2344

Temperature range from

T.min 25 - :

°C

T.max 60 :

°C

number of points

n 20 :

set to 20 for better illustration later needed 120

wanted: list of temperature points that produce the best fit for

E

Parameters of the method

T k κ.symb T k E.symb T 2 diff :

Curvature

x κ x k.i κ.symb x 0.01 < x k.w κ.symb cases :

Function to interpolate

x f x E.l :

todo: calculate initial step size

Initial step

h.0 14 :

Sampling point of the reference curvature in the first interval

f 0.15 :

Exponent in the step scaling formula

p 0.45 :

Centering shift

s 0.6 - :

X 1 el T.min :

X 2 el X 1 el h.0 + :

κ.0 X 1 el f h.0 * + κ : 0.693

1 κ h.0 * 42.3

j 2 n 1 - (range X j 1 + el X j el h.0 κ.0 X j el κ / (p^* + : for

X n el 62.8

Adjust initial step such that the range is covered

F X f vectorize eval :

X F s + x linterp x f - X 0 + augment 2 1 sys

X 25 - 11 - 1.7661 - 5.1079 11.6412 17.2598 22.2297 26.7113 30.8094 34.5972 38.1276 41.4407 44.5674 47.5321 50.3545 53.0508 55.6345 58.1168 60.5074 62.8144 201 mat