dad98d66-3bca-4edf-b428-e04317c82362 31 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 :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 21 line 31 line :

X Data 1 col :

Data 1 2.61173 2 8.48228 3 16.15875 4 26.15419 5 39.09752 6 55.73891 7 76.9396 8 103.63542 9 136.7611 10 177.12511 11 225.23407 12 281.08655 13 343.98411 14 412.43203 15 484.20056 16 556.57528 17 626.74962 18 692.24603 19 751.23737 20 802.68943 21 846.32288 22 882.45309 23 911.78366 24 935.21381 25 953.69087 26 968.11438 27 979.28442 28 987.8817 29 994.46746 30 999.494 302 mat :

Y Data 2 col :

L 0 :

H 30 :

N 100 :

U L H N xd :

x F X Y x ainterp :

pts X Y . 12 blue plotG :

pts x F 2 1 sys

x D x F x diff :

M i 1 U rows range u i el i : F i el U i el F : 21 line for U F augment 21 line :

x D

In Smath: linterp, ainterp, cinterp are NOT scalar functions. So, they can't be sub-used analytically. They are just useful gadgets to fill-in the gap between clean data. Or create zombie plot between zombie data points. In this example, 'Data' is a "clean" data set so that the red trace does not wiggle "apparently" but sufficiently to render derivative totally useless. Cubic splines are graphing tool, that's it ! Parametric splines is another story of 100's pages. In short, Hermite, Bézier parametric splines are the core component of drawing software [Autocad, LibreCad, FreeCad ... etc.

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 4 : 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 :

The finite differences catches the wiggling of "ainterp". Data smoothing is another myth. From inspection, the 2nd derivative could be rebuilt around Fourier polynomial fit. Smath is not much geared in terms of coding this proposal.

"ainterp" noisy 1rst Derivative

M Deriv 0 2 1 sys

"ainterp" useless 2nd Derivative

M Deriv Deriv 0 2 1 sys

Finite Differences Derivative of the clean original data

Data Deriv 0 2 1 sys

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 of "ainterp"

M Deriv Σ 1 1 sys

Extending the use of scalar function

β 1031.1298 4.2853 0.2817 15.8683 - 41 mat :

x β f β 1 el 1 β 2 el β 3 el x * - exp + / β 4 el + :

L 0 :

H 30 :

N 200 :

U L H N xd :

data i 1 U rows range u i el i : F i el U i el β f : 21 line for U F augment 21 line :

The finite differences derivative & cumulative integral are just perfect tools to populate data sets from scalar functions or w/o the scalar function if the dataset is clean and supplied well populated.

Finite Differences Derivative

data Deriv data Deriv Deriv data Deriv Deriv Deriv 3 1 sys data Deriv Deriv Deriv

x β D x β f (x diff :

In CAS [Computer Algebra System], a "scalar function" is scalar wrt "x". In there, 'x' is a vanishing iterator. In short: the function runs over itself for immediate use or for future sub-analytical use. The other "scalar" is a number. A scalar function is a "book style assignment".

x β Int x β D x int maple :

x β D x β Int

x D x β f (x diff :

x Int x D x int maple :

x D x Int

About "cubic splines" : There is not proprietary rights as they date back to Hermite and well published in the Mathematician Community. Linear, parabolic, natural are just variantes. And there are many more, Bézier/Bernstein, B-splines, De Boor ... etc. The traditional l-p-c Mathcad splines are the same in other CAS . They are built around the same tridiagonal matrix, solved for the spline coefficients. The Smath "Akima ainterp" looks linear end [lspline] for whatever sub-use in RK DE solvers.