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

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

1. Discrete algo style function [f,Lo,Hi,N] 2. function => Derivative =>Cumulative Integration

t0 1 time :

f L H N Project Discretise the function dx H L - N / : i 1 N 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 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 x x β 3 el x * β 2 el + β 4 el x 2^* + (* β 1 el + (* 1 x x β 7 el x * β 6 el + β 8 el x 2^* + (* β 5 el + (* + / :

x Deriv x f (x diff :

β 4.69292878185 2.485645088668 0.005049502793 - 0.008399478307 - 5.19337552684 0.698811411628 0.043875966805 - 0.000673223801 - 81 mat :

[L,H] range Npts L 0 : H 15 : N 500 : h H L - N / : 14 mat 11 sys

"if" rejects d/dx(f(x)) 0 x ≤ (x 20 ≤ (& x f NOT d/dx(f(x)) if

Collect the project components 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 sys

F 1.23456789 linterp 1.138291

D 1.23456789 ainterp 0.372191

"F" is the function data set XY "D" is the 1rst derivative data set XD "Σ" is the cumulative integral data set XΣ

Σ 1.23456789 cinterp 0.978738

The integral operator is numeric, thus accepts the bounded function. The derivative operator is not symbolic, therefore does NOT parse any and all "if" construction: whether f(x) or else user algo based programmed module(s).

F D 2 1 sys

u Q x f :

z Int u Q u 0 x int :

x Int z Int :

0 x ≤ ( x 15 ≤ ( & x Int not sanity fit if Σ

1 time t0 - 11 sec *

Recast the FD deriv in scalar form to support
the  internal "if"  ... success.
FD = Finite Differences. Unfortunately the 
derivative operator does not over plot. x h deriv 0 x ≤ (x 15 ≤ (& 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* OUT of fitted range if :

Most important remark

@ infinite 'N' both propositions FD & deriv operator are = accurate. Check 'N' = 1000

xx 1 :

Point sanity check

xx f xx diff 0.418

xx h deriv 0.421

x h deriv 0 4.693 . 15 red 1 5 mat 2 1 sys

x βDeriv 0 x ≤ (x 15 ≤ (& x Deriv OUT of fitted range if :

0 βDeriv 4.692929

x βDeriv 0 4.693 . 15 red 1 5 mat 2 1 sys x βDeriv x h deriv -

u #

u 0 :

...diff(#,#) NOT defined u erf u diff #

Alternately to this case or as applicable: visit Convolution "Finite Differences Smath Algo.sm" erf(x) is discontinuous by "Signum"

x Erf x erf :

Data set from foreign source

data 5 - 0.03846 4.95 - 0.03921 4.9 - 0.03998 4.85 - 0.04078 4.8 - 0.0416 4.75 - 0.04244 4.7 - 0.04331 4.65 - 0.0442 4.6 - 0.04513 4.55 - 0.04608 4.5 - 0.04706 4.45 - 0.04807 4.4 - 0.04912 4.35 - 0.05019 4.3 - 0.05131 4.25 - 0.05246 4.2 - 0.05365 4.15 - 0.05488 4.1 - 0.05615 4.05 - 0.05746 4 - 0.05882 3.95 - 0.06023 3.9 - 0.06169 3.85 - 0.0632 3.8 - 0.06477 3.75 - 0.06639 3.7 - 0.06807 3.65 - 0.06982 3.6 - 0.07163 3.55 - 0.07352 3.5 - 0.07547 3.45 - 0.0775 3.4 - 0.07962 3.35 - 0.08182 3.3 - 0.0841 3.25 - 0.08649 3.2 - 0.08897 3.15 - 0.09155 3.1 - 0.09425 3.05 - 0.09706 3 - 0.1 2.95 - 0.10307 2.9 - 0.10627 2.85 - 0.10962 2.8 - 0.11312 2.75 - 0.11679 2.7 - 0.12063 2.65 - 0.12465 2.6 - 0.12887 2.55 - 0.13329 2.5 - 0.13793 2.45 - 0.14281 2.4 - 0.14793 2.35 - 0.15332 2.3 - 0.15898 2.25 - 0.16495 2.2 - 0.17123 2.15 - 0.17786 2.1 - 0.18484 2.05 - 0.19222 2 - 0.2 1.95 - 0.20822 1.9 - 0.21692 1.85 - 0.22612 1.8 - 0.23585 1.75 - 0.24615 1.7 - 0.25707 1.65 - 0.26864 1.6 - 0.2809 1.55 - 0.2939 1.5 - 0.30769 1.45 - 0.32232 1.4 - 0.33784 1.35 - 0.3543 1.3 - 0.37175 1.25 - 0.39024 1.2 - 0.40984 1.15 - 0.43057 1.1 - 0.45249 1.05 - 0.47562 1 - 0.5 0.95 - 0.52562 0.9 - 0.55249 0.85 - 0.58055 0.8 - 0.60976 0.75 - 0.64 0.7 - 0.67114 0.65 - 0.70299 0.6 - 0.73529 0.55 - 0.76775 0.5 - 0.8 0.45 - 0.8316 0.4 - 0.86207 0.35 - 0.89087 0.3 - 0.91743 0.25 - 0.94118 0.2 - 0.96154 0.15 - 0.978 0.1 - 0.9901 0.05 - 0.99751 1.8 1014 -^* - 1 0.05 0.99751 0.1 0.9901 0.15 0.978 0.2 0.96154 0.25 0.94118 0.3 0.91743 0.35 0.89087 0.4 0.86207 0.45 0.8316 0.5 0.8 0.55 0.76775 0.6 0.73529 0.65 0.70299 0.7 0.67114 0.75 0.64 0.8 0.60976 0.85 0.58055 0.9 0.55249 0.95 0.52562 1 0.5 1.05 0.47562 1.1 0.45249 1.15 0.43057 1.2 0.40984 1.25 0.39024 1.3 0.37175 1.35 0.3543 1.4 0.33784 1.45 0.32232 1.5 0.30769 1.55 0.2939 1.6 0.2809 1.65 0.26864 1.7 0.25707 1.75 0.24615 1.8 0.23585 1.85 0.22612 1.9 0.21692 1.95 0.20822 2 0.2 2.05 0.19222 2.1 0.18484 2.15 0.17786 2.2 0.17123 2.25 0.16495 2.3 0.15898 2.35 0.15332 2.4 0.14793 2.45 0.14281 2.5 0.13793 2.55 0.13329 2.6 0.12887 2.65 0.12465 2.7 0.12063 2.75 0.11679 2.8 0.11312 2.85 0.10962 2.9 0.10627 2.95 0.10307 3 0.1 3.05 0.09706 3.1 0.09425 3.15 0.09155 3.2 0.08897 3.25 0.08649 3.3 0.0841 3.35 0.08182 3.4 0.07962 3.45 0.0775 3.5 0.07547 3.55 0.07352 3.6 0.07163 3.65 0.06982 3.7 0.06807 3.75 0.06639 3.8 0.06477 3.85 0.0632 3.9 0.06169 3.95 0.06023 4 0.05882 4.05 0.05746 4.1 0.05615 4.15 0.05488 4.2 0.05365 4.25 0.05246 4.3 0.05131 4.35 0.05019 4.4 0.04912 4.45 0.04807 4.5 0.04706 4.55 0.04608 4.6 0.04513 4.65 0.0442 4.7 0.04331 4.75 0.04244 4.8 0.0416 4.85 0.04078 4.9 0.03998 4.95 0.03921 5 0.03846 2012 mat :

data rows 201

data Σ Finite differences Cumul Integration 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 61 line :

Finite differences Cumulative Integral

data Σ data 2 1 sys

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 deriv rows el : X deriv augment 101 line :

Finite Differences Derivative

data Deriv 0 2 1 sys

About finite Diff's derivative of data sets You may want to try the centered recursive [Hildebrand] but you will be greatly deceived, though the recursive !may! show a slight smoothing effect on lightly noisy data sets. Recursive or this one proposed amplifies noise or simply not enough decimal places ... try this data set with 3 decimal places to make your own proof.

These two tools will be most precious if you have: interpolated data [linterp, cinterp, ainterp]. These 3 aren't scalar in Smath, thus excluding further scalar processing down the WS.