<p> Utilities </p>

9a4dc1fe-edd7-4d76-892f-11d875b22c81 14 6 12 decimal

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

A fitting session by the reduced technique . . .

Essentially, the fitting is based on Data Reduction Technique [DRT],

one of very many DRT techniques. The original data is sourced from the

Mathsoft Collaboratory, as is ... un-explained wrt what is what in there.

In this Smath document, we refit from simpler reduced .

1. Smath complains about the integral representation of Γ(b,x).

We integrate on upper limit [length of data] @ low accuracy [100].

2. From the reduced fit, approximate the correction factor by a 4rth polyfit.

3. Finish the fit by applying the λ correction to the original reduced fit.

Data Reduction is a transform technique from the original domain to

a more manageable one. Parent to DRT fitting, in Process Control we

use an algorithm called LEAD/LAG. In short: is consists in un-shapping

the correction for re-shapping otherwise with better controlling properties.

. . . instead of approximating "ratio", it may be tempting to work from the

reduced directly ... not offered

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XY 0 127.42 0.1 123.469 0.2 118.008 0.3 111.187 0.4 102.811 0.5 93.554 0.6 84.168 0.7 74.818 0.8 66.373 0.9 58.587 1 51.678 1.1 45.739 1.2 40.675 1.3 36.404 1.4 32.74 1.5 29.581 1.6 26.848 1.7 24.472 1.8 22.395 1.9 20.573 2 18.97 2.1 17.555 2.2 16.3 2.3 15.184 2.4 14.188 2.5 13.295 2.6 12.492 2.7 11.768 2.8 11.12 2.9 10.547 3 10.028 312 mat :

data char size clr plot k 1 data rows range r3 k el char : r3 k el char : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Smath reduced fit from 'n' only yo 10.028 : a 127.42 : b 1 : n 1.815 : 14 mat

data XY o 5 black plot : x 03 0.1 range : 21 sys

Reduced Γ(b,x) adapted from Carlos i 1 x length range f i el t b 1 -^t - exp * t x i el x x length el int : ff i el yo a f i el n^* + : 21 line for

x f ff :

Ω x z f vectorize augment :

Y XY 2 col :

ratio x Y ff / vectorize augment :

Reduced Γ(a,x)on the range of the data set

data Ω 2 1 sys ratio

CWD C:\SmathFile CurrentDirectory :

ratio CWD Γ(a,x) wfile 1

ratio CWD Γ(a,x) rfile :

<p> Utilities </p>

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

Error stem plot for QuickPlot unicolor data Error stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

Bar plot unicolor style, bar width user. data dt BarSize v data 2 col : t v bar t dt - 0 t dt - v t dt + v t dt + 042 mat : j 1 data rows range j 1 ≡ B data j 1 el data j 2 el bar : B B data j 1 el data j 2 el bar stack : if 11 line for B 41 line :

XY ratio : X XY 1 col : Y XY 2 col : pts XY o 4 black plot : n ratio rows : 51 line

x φ .25 x x 2^x 3^41 mat :

Cholesky companion solver j 1 x φ length range i 1 X length range M i j el X i el φ j el : for for Φ M transpose M * : d M transpose Y * : β Φ 1 -^eval d * : 41 line

β 3.936116 2.041613 1.205603 - 0.175581 41 mat

recast the polyfit x β fit β x φ * :

U u 03 0.1 range : i 1 u rows range y i el u i el β fit eval : for u y augment 31 line :

Deviation i 1 XY rows range Δ i el Y i el X i el β fit - eval : for X Δ augment 21 line :

pts U 2 1 sys Deviation Error Deviation 1 row 0.01 BarSize 2 1 sys

1. Recast the 4rth order polyfit. 2. Discretize on the data set range. 3. Extract the correction vector " λ "

U u 03 0.1 range : i 1 u rows range y i el u i el β fit : for u y augment 31 line :

'λ' correction to reduced Γ(b,x)
.......
The corrected reduced fit λ U 2 col : Ωnew x λ z f vectorize * vectorize augment : 21 sys

data Ωnew 2 1 sys

From this superb Γ(b,x) fit on the long range, it is easy to conclude the 1rst point is freaked. The incomplete Γ(1,x) is 1 @ x = 0 This data set may represent some statistics, some physics ... was published in Mathsoft Collaboratory. It may just be an invention from some "Smart Quizer"

vx Ωnew 1 col : vy Ωnew 2 col : x s vx vy x cinterp : 31 line

spline x 03 0.01 range : i 1 x rows range y i el x i el s eval : for x y augment 31 line :

data spline 2 1 sys