6bf51e0a-6b43-476b-bfeb-38a3b9e363a8 127 4 5 false false 0 0 decimal

&[DATE] &[TIME] - &[FILENAME]

Optimal distribution of interpolation points

Thermodynamic Formulae

saturation steam pressure above water or ice

T k E.symb e k 1 el T / k 2 el + k 3 el T * + k 4 el T 2^* + k 5 el T ln * +^:

k.w 6096.9385 - 16.635794 2.711193 102 -^* - 1.673952 105 -^* 2.433502 51 mat :

k.i 6024.5282 - 24.7219 1.0613868 102 -^* 1.3198825 105 -^* - 0.49382577 - 51 mat :

coefficients or water and ice

T k E.simp k 2 el T k 5 el^* e k 1 el T /^* e k 3 el T *^* e k 4 el T 2^*^* :

let us simplify so we can calculate derivatives

k.ws 6096.9385 - 1678160661 2.711193 102 -^* - 1.673952 105 -^* 2.433502 51 mat :

k.is 6024.5282 - 5452362876616 1.0613868 102 -^* 1.3198825 105 -^* - 0.49382577 - 51 mat :

ϑ E ϑ 273.15 + k.ws E.simp ϑ 0.01 ≥ (ϑ 100 ≤ (& ϑ 273.15 + k.is E.simp cases :

put together
for water and ice

caution: temperatures not below or above +-100 °C

Enhancement factor

t f 1.0047 :

valid for -50 °C to 60 °C

Formula to be interpolated:
Saturation steam pressure in humid air

ϑ E.l ϑ E ϑ f * :

Test

20 E 2339

20 E.l 2350

20 - (E.l 104

Temperature range from

T.min 20 - :

°C

T.max 60 :

°C

number of points

n 20 :

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

E.l

Parameters of the method

Sampling point of the reference curvature in the first interval

f 0.1 :

Exponent in the step scaling formula

p 0.46 :

Centering shift

s 0.516736 - :

Initial step

h.0 14 :

Function to interpolate, unitless

x f x E.l :

T k E.simp T 2 Diff k 2 el e T nthroot T 2^k 4 el T * k 3 el + (* k 1 el +^* k 5 el T * 1 - k 5 el + (T * 2 T 2^2 k 4 el T * k 3 el + (* - k 3 el + (* k 1 el + (* - (* T 2^2 k 4 el T * k 3 el + (* - k 3 el + (* k 1 el + (2^+ 2 T * T 3^k 4 el * k 1 el + (* + (* T 1 k 5 el - 2 + 1 +^/

Curvature

simplify, we get, unitless:

T k κ.symb k 2 el e T 2^k 4 el T * k 3 el + (* k 1 el + T /^* T k 5 el 1 - k 5 el + (T * 2 T 2^2 k 4 el T * k 3 el + (* - k 3 el + (* k 1 el + (* - (* 2 T 3^k 4 el * k 1 el + (* + (* T 2^2 k 4 el T * k 3 el + (* - k 3 el + (* k 1 el + (2^+ (* T 4 k 5 el -^/ :

x κ x 273.15 + k.is κ.symb x 0.01 ≥ (x 100 ≤ (& x 273.15 + k.ws κ.symb cases :

X 1 el T.min 1 round :

X 2 el X 1 el h.0 + 1 round :

1 κ h.0 * 56.4

κ.0 X 1 el f h.0 * + κ : 0.919

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

X n el 57.4

Adjust initial step such that the range is covered

F X f vectorize eval s + :

X T.max ≤ vectorize (sum 20

Points in range

F 103.2 369.9 799.7 12501781239430723823464855316486749985569690108901217013500148801630017750201 mat

absolute Error

x err X F x linterp x f - :

x err.% X F x linterp x f - x f / :

relative Error

X E P X X 0.5 0.5 21 mat al_convr1d 2 n range el stack 1 csort : j 1 P T.max ≤ vectorize (sum range : E j el P j el err : E min E max 21 sys 41 line :

X E res X 1 el err 11 mat : j 2 n range X j el T.max ≤ res j el X j el err : if 0.5 X j el X j 1 - el + (* T.max ≤ res j n + el 0.5 X j el X j 1 - el + (* err : if 21 line for res min res max 21 sys 31 line :

E X E : 0.516736 - 37.1413 21 sys

x err X 0 + augment X 0.5 0.5 2 1 mat al_convr1d 2 n range el 0 o 5 red augment E 4 1 sys

E 2 el E 1 el + 2 / 18.3123

E 2 el E 1 el - 2 / 18.829

Martin Kraska

X 20 - 6 - 3.7 10.2 41 mat

X 0.5 0.5 21 mat al_convr1d 2 n range el 13 - 1.15 - 6.95 12.9 41 mat

P X X 0.5 0.5 21 mat al_convr1d 2 n range el stack 1 csort : 20 - 13 - 6 - 1.15 - 41 mat

j.max P T.max ≤ vectorize (sum : 39

j 1 j.max range :

E 011 mat :

E j el P j el err :

E 0.517 - 37.1 0.517 - 26.5 41 mat

E min E max 21 sys 0.517 - 37.1 21 sys

X 20 - 6 - 3.7 10.2 15.6 20.3 24.4 28.1 31.5 34.6 37.5 40.2 42.7 45.1 47.4 49.6 51.7 53.7 55.6 57.4 201 mat

A B Hadamard r A rows : c A cols : A length B length ≡ (A rows B rows ≡ (& Hp r c matrix : j 1 r range k 1 c range Hp j k el A j k el B j k el * : for for Hp 31 line input matrices must have same sizes error if 31 line :

list of interpolating points

E.sq E E Hadamard j el j 1 E length sum E length / : 168.6425

mean absolute square error

to be minimised!