plotG

dad98d66-3bca-4edf-b428-e04317c82362 56 3 5 decimal

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

plotG

data char size clr plotG k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

1. The crappy Akima

t0 1 time :

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 :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 41 line :

X Data 1 col : Y Data 2 col : 21 line

L 0 : H 30 : N 500 : 13 mat

U L H N xd :

x F X Y x ainterp :

pts Data . 12 blue plotG :

pts x F 2 1 sys

x D x F x diff :

In Smath: linterp, ainterp, cinterp are NOT scalar functions. So, they can't be sub-used analytically. The finite differences catches the wiggling of "ainterp", thus not smooth. The 2nd deriv' out of use.

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

Deriv(data)

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 - / + (* eval : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows el : X deriv augment 101 line :

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 eval for vx Y augment 61 line :

Logistic

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 + :

Logistic i 1 U rows range F i el U i el β f eval : for U F augment 21 line :

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

D M Deriv :

"ainterp" useless ... "cinterp " acceptable in reduced range from visual inspection. alternate: cinterp(X,Y,x) ... ainterp(X,Y,x)

D Deriv Logistic Deriv Deriv 2 1 sys

"ainterp" noisy 1rst Derivative

D 0 2 1 sys

Finite Differences Derivative of the clean original data

Data Deriv 0 2 1 sys

cumul D Σ :

Finite differences Cumulative Integral of "ainterp"

cumul

Finite Differences Derivative

Logistic Deriv Logistic Deriv Deriv Logistic Deriv Deriv Deriv 3 1 sys Logistic Deriv Deriv Deriv

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. The scalar 'x' is the "analytical iterator".

x β D x β f (x diff :

x β Int x β D x int maple :

x β D x β Int

2. The acceptable Akima

XY 0 0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 1 0.6 0.167 0.7 0.048 0.8 0.022 0.9 0.012 1 0.00794 112 mat :

X XY 1 col : Y XY 2 col : support XY + 10 black plotG : 31 line

L 0 : H 1 : Npts 200 : 31 line

U L H Npts xd :

x Akima X Y x ainterp :

x Cinterp X Y x cinterp :

Akim i 1 U rows range vy i el U i el Akima eval : for U vy augment 21 line :

Cin i 1 U rows range vy i el U i el Cinterp eval : for U vy augment 21 line :

1 time t0 - s 38.3

Akim Cin support 3 1 sys

Resume ainterp(X,Y,x) Akima spline

1. Absolute crap for the Logistic example

2. Adequate for the Gaussian.

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There are other techniques to spline this particular Gaussian

[not presented here]. Though Gaussians may look simple,

they are the most difficult data sets to fit model functions.

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