<p> Derivative, Cumul Integration, Ksmmoth </p>

9809058a-20d4-472e-96e5-00990c3c12bb 48 3 5 decimal

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

Iterate the GivenFind solve bloc

Inspired from Mathcad 8 Pro. It reproduces the solver 1/1. *****

Predefine the solver invariants

t0 1 time :

a b GivenFind a b FindRoot :

range 100 :

a 0.65 : b 12 : c 5 : d 4 : 14 mat

init Y 1 ≡ :

Note:

GivenFind outputs same values as Mathcad 8 Pro.

This iterative bloc works well but painful to initialise.

Unlike Mathcad that initialises from μ = 0, Smath

must start much lower. No description was given

from the originator. Once the data set in hand, you can

apply finite differences [derivative, cumulative integration].

In this example, the GivenFind solve bloc is noisy.

We apply the Gaussian "Ksmooth" and deliver the project.

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n 0.5 range * :

i 1 range range idx i el i : μ i el 0.4999 1 n / + (- i n / + : 21 line for

Declare the system to solve for μ. j system Y a b 2^* c d * Y * - (μ j el^- 0 ≡ :

Generalise an iterative collection of the solution j Iterate Y Find j system init GivenFind : μ sol Y Find : 21 line above => define the indexed formation μ sol 31 line :

Collect the solver j 1 range range : Solved j Iterate vectorize : 21 line

User 35 :

sol μ Solved augment :

X μ User el : Y User Iterate : 12 mat

data j μ sol 13 mat idx μ sol 2 col augment stack :

1 time t0 - min 0.3

data j μ sol 1 0.5 - 0.1 2 0.48 - 0.11 3 0.46 - 0.13 4 0.44 - 0.14 5 0.42 - 0.15 6 0.4 - 0.17 7 0.38 - 0.18 8 0.36 - 0.2 93 mat

sol X Y + 15 black 1 5 mat 2 1 sys

X Y 12 mat 0.18 2.043 12 mat

Collect the raw project XY μ sol 2 col augment :

<p> Derivative, Cumul Integration, Ksmmoth </p>

Finite differences "FD" derivative of data set data D 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 deriv rows el : X deriv augment 91 line :

Finite differences cumulative integration data Σ vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 51 line :

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 :

σ 0.3708158988058244 :

The Gaussian ksmooth

d K 12 π * sqrt σ * / d 2^2 σ 2^* / - exp * :

Smath recognises the silent 'bw', something
like the Mathcad "global definition". W j X Y Ksmooth μ j 11 line : X i el X μ el - bw / K Y i el * (i 1 X rows sum X i el X μ el - bw / K (i 1 X rows sum / eval 21 line : j 1 X rows range s j el j X Y Ksmooth eval : for s 31 line :

Gaussian Ksmooth Kernel

x K

X Y bw 13 mat

W 11 mat

X XY 1 col : Y XY 2 col : bw 0.0625 : (bandwidth ≡ XW X W augment : 41 line

Genius Smath:

augment(X,W) needs 'W' be calculated from stand alone Ksmooth(j,X,Y)

which Ksmooth(j,X,Y) renews itself from 'bw' and by same token

executes itself from [X, Y]

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Sanity smoothed derivative

XY D XW D 2 1 sys

Export the smoothed 'sol' [XW]

sol XW 2 1 sys

Cumulative needs NO smoothing

XY Σ

Export j μ sol 13 mat idx μ W augment stack :

Export j μ sol 1 0.5 - 0.11 2 0.48 - 0.12 3 0.46 - 0.13 4 0.44 - 0.14 5 0.42 - 0.15 6 0.4 - 0.17 7 0.38 - 0.18 8 0.36 - 0.2 9 0.34 - 0.22 10 0.32 - 0.24 113 mat