72b25d9c-070b-486d-a5b4-32799769ef38 26 4 5 decimal

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

Non-linear integration

time 8 sec ≡

Watch how you thread wrt the limits of the integrand. For instance, if the integrand is "Divide by 0" , take this fact into account in the low limit.

t0 1 time :

n 2.5 :

n t f t n^t exp 1 - / :

t q π 2 / t 2^* cos :

x lin t 3^t 0 x int :

u FC t q t 0 u int :

1 lin 0.25

1 FC 0.7799

x Nlin n t f t 0 x int :

Division by 0

2 Nlin #

ε 1012 -^:

x lin n t f t ε x int :

2 lin 1.0475

x Nlin n t f t ε x sin int :

2 Nlin 0.2249

x Nlin n t f t ε 1 0.5 x + / int :

ε Nlin 1.0475

Fnct x ε 3 π * ε 0.05 + (range : i 1 x rows range u i el x i el : Fnct i el n u i el f : 21 line for u Fnct augment 31 line :

NONlin x ε 3 π * ε 0.1 + (range : i 1 x rows range u i el x i el : F i el u i el Nlin : 21 line for u F augment 31 line :

NON linear integral

Fnct NONlin 2 1 sys

Two ways model fit: In this example, the path of integration may make non sense for the unaware reader, NOT so ! Data sets that have shape of arc of hyperbola are difficult to fit by single model functions. In association with "Gaussians" which are very rich in modeling shape, we have considerably enriched the fitting session. Assume we have "cleaned" a data set with this NON-linear integral, we may have chance to get a Thiele fit, which would generate an entire analytic function, representing the data set.

1 time t0 - 9 sec *

XY ε ε Nlin 0.25 0.25 Nlin 0.5 0.5 Nlin 0.75 0.75 Nlin 11 Nlin 1.5 1.5 Nlin 22 Nlin 33 Nlin 44 Nlin 66 Nlin 102 mat :

XY 11012 -^* 1.0475 0.25 0.4963 0.5 0.2755 0.75 0.1703 1 0.1136 1.5 0.0589 2 0.035 3 0.0157 4 0.0086 6 0.0035 102 mat

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

XY XY 1 col XY 2 col . 10 black plotG :

NON linear integral

Fnct NONlin XY 3 1 sys

Didactic

Assume we were given a noisy [not clean] data set and that after the fitting session we have conserved NONlin as the fit. For the Thiele suite, we obtain some support points XY that we pass to Thiele procedure. Thiele(X,Y,Δ)calculates the triangular table, it contains the "K" coefficients of the continued fraction to produce f(x) entire function that can be in turn derived and integrated.

X Y Δ Thiele ε 1018 -^: n X rows : i 1 n range M i 1 el Y i el : for j 2 n range i j n range dM M i j 1 - el M j 1 - j 1 - el - : dM dM abs ε < ε M i j el X i el X j 1 - el - dM / : if : 21 line for for i 1 n range c i 1 el X i el : c i 2 el M i i el : 21 line for Δ 0 ≡ c M if 61 line :

XY 11012 -^* 1.0475 0.25 0.4963 0.5 0.2755 0.75 0.1703 1 0.1136 1.5 0.0589 2 0.035 3 0.0157 4 0.0086 6 0.0035 102 mat :

vx XY 1 col :

vy XY 2 col :

Thiele 11012 -^* 1.0475 0.25 0.4536 - 0.5 1.2879 - 0.75 5.8979 1 0.2575 1.5 108.4294 - 2 0.0102 - 3 431.1187 - 4 0.0041 - 6 1343.1271 - 102 mat :

K Thiele 2 col :

vx vy 0 Thiele

x Cf K 1 el x vx 1 el - K 2 el x vx 2 el - K 3 el x vx 3 el - K 4 el x vx 4 el - K 5 el x vx 5 el - K 6 el x vx 6 el - K 7 el x vx 7 el - K 8 el x vx 8 el - K 9 el x vx 9 el - K 10 el / + / + / + / + / + / + / + / + / + :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 21 line 31 line :

L ε :

H 8 :

N 99 :

U L H N xd :

data i 1 U rows range vx i el U i el : vy i el U i el Cf : 21 line for vx vy augment 31 line :

data Fnct 2 1 sys

vx vy 1 Thiele Activate to see Thiele table ≡

vx vy 0 Thiele Activate to see Thiele K ≡