<p> [F, D, Cumul, Interp Triplet </p>

966acb61-5f0d-4af3-8c32-fe2bb7eac2a8 37 6 9 decimal

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

Smath OFFicial 5346

Cumulative Area, Integral Curve, Cumulative Integration Curve

In-situ Cumulative Area
between  user [a,b] limts 11 x 2^+ / x 020 int 1.520838

u Φ 11 x 2^+ / :

t0 1 time :

If you can't get Maple symbolic expansion, you are restricted to "Finite Differences"

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x cumul u Φ u x - x int maple : 2 x * 1 x 2^+ /

Integral curve x Int u Φ u x - x int :

Transit numerical u C x cumul :

Cumulative integration curve z Cumul u C u x - x int :

x Φ x Int x Cumul 3 1 sys <p> [F, D, Cumul, Interp Triplet </p>

f L H N Project Discretise the function dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for i 1 U rows range vx i el U i el : vy i el U i el f : 21 line for Derivative backward substitution deriv 1 el 0 : deriv N 1 + el 0 : i 2 N 1 - range deriv i el 0.5 vy i 1 - el vy i el - vx i 1 - el vx i el - / vy i el vy i 1 - el - vx i el vx i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv N el deriv deriv rows el : Cumulative Integration 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 Nest the export vx vy augment vx deriv augment vx Y augment 31 mat 151 line :

Collect the interpolation triplet data x linterp data 1 col data 2 col x linterp : data x ainterp data 1 col data 2 col x ainterp : data x cinterp data 1 col data 2 col x cinterp : 31 sys

x f 11 x 2^+ / :

range & populate the triplet L 5 - : H 5 : N 200 : h H L - N / : 14 mat

Sampled data are noisy. Applying FD cumulative integration, result will be impecable [smmoothed]. FD => Finite Differences. The FD derivative is a drastic noise amplifier. You can apply "Ksmooth". There is an alternative to smooth data: "Savitsky-Golay", applied either globally or piecewise. The result has several components: 1. polynomial based smoothed data set. 2. 1rst, 2nd and more smoothed derivatives. RemToDo for Smath ToolBox.

Collect/Export project triplet F f L H N Project 11 el : D f L H N Project 21 el : Σ f L H N Project 31 el : 31 line

Discrete triplet for export

F D Σ 3 1 sys

In-situ interpolation F 1.23456789 linterp D 1.23456789 ainterp Σ 1.23456789 cinterp 31 mat 0.396291 0.3988 - 2.263304 31 mat

r 101 : row search ≡

Search the project triplet F r row D r row Σ r row 31 mat 0112 mat 0 0.049875 12 mat 0 1.373398 12 mat 31 mat

x Fnct D x linterp :

scalar "Fnite Differences" derivative. This case   
will apply on instance NO symbolic derivative
could be found ... try function "Blanc-mange" x h f D1 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* :

linterp, ainterp, cinterp will NOT extrapolate the data set. here > ± 5 <

x Fnct x h f D1

1 time t0 - 23.414 sec *

Algo style scalar incremental analysis

Application:

software of low maths level

x f 11 x 2^+ / :

1. a function can be represented in term of "increment h"

x h φ h x h + φ x φ - (* 2 x φ * + :

2. φ(x,h) has integral rpresentation in term of "increment h"

Integral Curve => AntiDerivative [courbe intégrale] hh x hh + φ x φ - (* 2 x φ * + u int maple hh x hh + φ x φ - (* 2 x φ * + (u *

Cumulative Integration 4 hh x hh + φ x φ - (* 2 x φ * + (u * u int (* maple 2 hh x hh + φ x φ - (* 2 x φ * + (* u 2^*

x f 11 x 2^+ / : User function ≡

Incremental analysis procedure

Algo style Memo formula of incremental analysis of this demo example h 0.001 : increment ≡ x h f D1 x 2 h * + f - x 2 h * - f + 4 h * / - : 1rst Derivative ≡ x h f D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / : 2nd Derivative ≡ x h f h_int h x h + f x f - (* 2 x f * + (x * : Integral Curve [antiderivative] ≡ x h f h_cumul 2 h x h + f x f - (* 2 x f * + (x 2^* (* : Cumulative Integration ≡ 51 line

x f x h f D1 x h f D2 x h f h_int 0.5 x h f h_cumul * x f x diff 6 1 sys

0.5 h 2^(* 0.0000005

u Φ x f : x Int u Φ u x - x int : 21 line

cumul integration scaled ½ to enable view plot

x Int x h f h_int - 0.0000005 2 1 sys

D2(x,h,f) starts generate noise @~ 1e-08