3dc8b046-08af-4058-ba87-e7db7925d9f2 48 4 5 decimal

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

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 :

X Y . 8 red plotG

The "Super Smoothlet" ...

... outputs an extraordinary perfect fit to "Carbon α 12". Smath spits the project in ~ 5 sec. Notice the two sets of eval(,): eval[cinterp(vx,vy,t)] & eval[Sup(j,X,Y)]. A better fit at both ends is obtained with the pspline [Mathcad 11]. The 1rst derivative reveals this missing feature in Smath. This project is public, suggestive of fitting techniques as applicable "per case". Note: Sup(j,X,Y) is constructed via the local definition. Smath does not take an index in the definition of a function. Remark that we manipulate 3 points in the procedure. Rarely a fitting session can be completely automated [a urban myth]. There are lot more hours in that project than you can guess.

t0 1 time :

Parameters

n 4 :

λ 1 :

xc 0 :

σ 0.375 :

μ 0 :

bw 1 :

n λ xc x P 1 - (k^λ * 2 x xc - (* (n 2 k * -^* k! (n 2 k * - (! (* / (k 0 n 2 / abs sum :

1. 'λ' zoom cosmetic to sup(x) almost NO influence to the fit. 2. 'bw' [bandwidth], fixed = 1 3. 'σ' governs the fit, i.e: besides 'bw', 'σ' sets amplitude between min/max of sup(x) operates on the weigthed moving average of the Sup(j,X,Y) kernel.

x μ σ dnorm 1 σ 2 π * sqrt * / e 12 / x μ - σ / (2^* -^* :

x sup n λ xc x P x μ σ dnorm * : 434 x 2^* 3 - x 2^+ (* + (* 92 sqrt * π sqrt * e 32 x 2^* 9 /^* /

span 0.7 0 sup 0.7 0.7 sup 22 mat :

span x sup x sup x diff 3 1 sys

span 0.7 0.531923 0.7 0.0596075 - 22 mat

0 sup 0.7 sup abs / 8.9238

x sup x 2 - 2 solve 1.6507 - 0.5246 - 0.5246 1.6507 41 mat

Carbon12 0.175 317.303 0.35 195.232 0.525 124.387 0.7 43.733 0.875 40.463 1.05 55.722 1.225 90.599 1.4 115.668 1.575 184.332 1.75 140.736 1.925 100.409 2.1 57.902 2.275 42.643 2.45 58.992 2.625 98.229 2.8 206.131 2.975 316.212 3.15 345.64 3.325 284.605 3.5 235.559 3.675 90.599 3.85 35.014 4.025 36.104 4.2 11.035 242 mat :

Xo Carbon12 1 col :

X Carbon12 1 col :

Yo Carbon12 2 col :

Y Carbon12 2 col :

n X rows :

n 24

j X Y Sup m j 11 line : X i el X m el - bw / sup Y i el * (i 1 X rows sum X i el X m el - bw / sup (i 1 X rows sum / 21 line :

X 2 el 0.319 :

Y 9 el 230 :

W j 1 X rows range s j el j X Y Sup eval : constrain 1rst node s 1 el 301.5 : 31 line for s 21 line :

XW X W augment :

vx Xo :

vy XW 2 col :

t 0.175 4.2 0.180 range :

XW 0.175 301.5 0.319 212.419 0.525 132.948 0.7 70.127 0.875 37.642 1.05 48.892 1.225 93.052 1.4 143.028 1.575 168.217 1.75 152.09 1.925 104.526 2.1 54.35 2.275 31.11 2.45 53.392 2.625 123.308 2.8 221.034 2.975 307.124 3.15 340.85 3.325 304.367 3.5 215.563 3.675 118.199 3.85 48.228 4.025 10.832 4.2 7.717 - 242 mat

t rows 805

fit i 1 t rows range u i el t i el : v i el vx vy t i el cinterp eval : 21 line for u v augment 21 line :

C12 Xo Yo + 7 blue plotG :

XW XW 1 col XW 2 col o 7 blak plotG :

Title 1360 Carbon α 12 12 black 15 mat :

0 fit C12 XW Title 5 1 sys

1 time t0 - 5.7 sec *

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - range deriv i el 0.5 Y i 1 - el Y i el - X i 1 - el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv n 1 - el : X deriv augment 101 line :

fit Deriv 0 2 1 sys

Erratum: pspline in Mathcad 11

Δ Yo W - :

Δ transpose 1617 - 9 - 26 - 372 - 27 - 1611 - 4 - 412625 - 15 - 9520 - 2028 - 13 - 2519124 mat

iii 1 Δ rows range P iii el 3 : P Δ rows el 2 : 21 line for

ii 1 Δ rows range v ii el ii : for

i 1 v length range YΔ 3 i * el Δ i el : for

Core component of the stem style P v replicate Mathcad 8 Pro. Treasury k 1 : i 1 v length range P i el 1 ≡ continue j 1 P i el range T k j + el v i el : for k k P i el + : 21 line if for T 41 line :

datum 0 :

u P v replicate :

V u YΔ augment :

Δx P Xo replicate :

Residuals Δx V 2 col augment :

Title 145 Cα12 SuperSmoothlet 12 black 15 mat :

Residuals on SuperSmoothlet... extensible at will.

Residuals datum Title 3 1 sys

Δ Max Δ Min 21 mat 25.3 27.6 - 21 mat

jmGiraud: 2015/10/06

Padé rational converted J_Fraction

x fit 1403.4 2933.1 x 6.8679 - 33.952 x 3.4766 + 0.32404 x 4.6785 - 4.1246 x 0.10541 + 0.50485 x 5.9138 - 14.081 x 1.0965 + / + / - / + / - / + / - :

x f x fit (x diff :

x q 0.175 x ≤ (x 4.2 ≤ (& x fit out of range if :

x q 0.5 x f * 2 1 sys

1. The derivative can't be constrained ! the plot is 0.5f(x) 2. 'Carbon12' data can't be plotted, i.e: f(x) does NOT discretise.