XY data

29abd409-ae77-43b2-a2a2-d25cb63a47c9 77 4 5 decimal

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

XY data

XY 0 0.0385 0.1 0.04 0.2 0.0416 0.3 0.0433 0.4 0.0451 0.5 0.0471 0.6 0.0491 0.7 0.0513 0.8 0.0536 0.9 0.0561 1 0.0588 1.1 0.0617 1.2 0.0648 1.3 0.0681 1.4 0.0716 1.5 0.0755 1.6 0.0796 1.7 0.0841 1.8 0.089 1.9 0.0943 2 0.1 2.1 0.1063 2.2 0.1131 2.3 0.1206 2.4 0.1289 2.5 0.1379 2.6 0.1479 2.7 0.159 2.8 0.1712 2.9 0.1848 3 0.2 3.1 0.2169 3.2 0.2358 3.3 0.2571 3.4 0.2809 3.5 0.3077 3.6 0.3378 3.7 0.3717 3.8 0.4098 3.9 0.4525 4 0.5 4.1 0.5525 4.2 0.6098 4.3 0.6711 4.4 0.7353 4.5 0.8 4.6 0.8621 4.7 0.9174 4.8 0.9615 4.9 0.9901 51 5.1 0.9901 5.2 0.9615 5.3 0.9174 5.4 0.8621 5.5 0.8 5.6 0.7353 5.7 0.6711 5.8 0.6098 5.9 0.5525 6 0.5 6.1 0.4525 6.2 0.4098 6.3 0.3717 6.4 0.3378 6.5 0.3077 6.6 0.2809 6.7 0.2571 6.8 0.2358 6.9 0.2169 7 0.2 7.1 0.1848 7.2 0.1712 7.3 0.159 7.4 0.1479 7.5 0.1379 7.6 0.1289 7.7 0.1206 7.8 0.1131 7.9 0.1063 8 0.1 8.1 0.0943 8.2 0.089 8.3 0.0841 8.4 0.0796 8.5 0.0755 8.6 0.0716 8.7 0.0681 8.8 0.0648 8.9 0.0617 9 0.0588 9.1 0.0561 9.2 0.0536 9.3 0.0513 9.4 0.0491 9.5 0.0471 9.6 0.0451 9.7 0.0433 9.8 0.0416 9.9 0.04 10 0.0385 1012 mat :

Example 0 . . . scalar Ridder RemToDo

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XY rows 101

XY data

XY 0 0.0385 0.1 0.04 0.2 0.0416 0.3 0.0433 0.4 0.0451 0.5 0.0471 0.6 0.0491 0.7 0.0513 0.8 0.0536 0.9 0.0561 1 0.0588 1.1 0.0617 1.2 0.0648 1.3 0.0681 1.4 0.0716 1.5 0.0755 1.6 0.0796 1.7 0.0841 1.8 0.089 1.9 0.0943 2 0.1 2.1 0.1063 2.2 0.1131 2.3 0.1206 2.4 0.1289 2.5 0.1379 2.6 0.1479 2.7 0.159 2.8 0.1712 2.9 0.1848 3 0.2 3.1 0.2169 3.2 0.2358 3.3 0.2571 3.4 0.2809 3.5 0.3077 3.6 0.3378 3.7 0.3717 3.8 0.4098 3.9 0.4525 4 0.5 4.1 0.5525 4.2 0.6098 4.3 0.6711 4.4 0.7353 4.5 0.8 4.6 0.8621 4.7 0.9174 4.8 0.9615 4.9 0.9901 51 5.1 0.9901 5.2 0.9615 5.3 0.9174 5.4 0.8621 5.5 0.8 5.6 0.7353 5.7 0.6711 5.8 0.6098 5.9 0.5525 6 0.5 6.1 0.4525 6.2 0.4098 6.3 0.3717 6.4 0.3378 6.5 0.3077 6.6 0.2809 6.7 0.2571 6.8 0.2358 6.9 0.2169 7 0.2 7.1 0.1848 7.2 0.1712 7.3 0.159 7.4 0.1479 7.5 0.1379 7.6 0.1289 7.7 0.1206 7.8 0.1131 7.9 0.1063 8 0.1 8.1 0.0943 8.2 0.089 8.3 0.0841 8.4 0.0796 8.5 0.0755 8.6 0.0716 8.7 0.0681 8.8 0.0648 8.9 0.0617 9 0.0588 9.1 0.0561 9.2 0.0536 9.3 0.0513 9.4 0.0491 9.5 0.0471 9.6 0.0451 9.7 0.0433 9.8 0.0416 9.9 0.04 10 0.0385 1012 mat :

4 appVersion 0.98.6179.21440 ≡

X XY 1 col : Y XY 2 col : 12 mat x f X Y x ainterp : x λ 1 X 1 el x ≤ (x X X rows el ≤ (& cases : 31 sys

◄ 'n' derivative order

x n d iter 2 : TOL 0.0000000001 : h 1 : 13 mat j iter 29 range D j el n 2 h 1.4 j^/ * / (n^1 - (n k -^n! k! (n k - (! (* / * x h 1.4 j^/ - k 2 h 1.4 j^/ * n / (* + f * (k 0 n sum * eval : D j el 1.4 2 j *^1.4 2 j *^1 - / D j el * 1 1.4 2 j *^1 - / D j 1 - el * + eval : 21 line for Next ▼  ... construct the scalar derivative iter 3 < D iter el D iter 1 - el - abs TOL ≤ break iter iter 1 + : d D iter el : 21 line if 11 line while d 51 line :

t0 1 time :

x f x λ * x 2 d x λ * 2 1 sys

1 time t0 - min 3.2

iter 2 : TOL 0.0000000001 : h 1 : n 1 : 14 mat

29 2.4 min * # 13 mat

iter 2 : TOL 0.0000000001 : h 1 : n 2 : 14 mat

29 2.9 min * # 13 mat

By default, Mathsoft derivative is Ridder but derivative of scalar functions. Thus, does not apply to splined functions. If f(x) symbolic, will evaluate symbolic.

Example 1 . . . scalar infinitesimal-derivatives

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X XY 1 col : Y XY 2 col : 12 mat x f X Y x ainterp : x λ 1 X 1 el x ≤ (x X X rows el ≤ (& cases : 31 sys

x fnct x h + f x h - f + 2 / ≡

1rst derivative  applied on interpolating functions x h D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / :

2nd derivative applied on interpolating functions x h D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / :

x h D3 x 3 h * - f 8 x 2 h * - f * - 13 x h - f * + 13 x h + f * - 8 x 2 h * + f * + x 3 h * + f - 8 h 3^* / :

x h D4 x 3 h * - f - 12 x 2 h * - f * + 39 x h - f * - 56 x f * + 39 x h + f * - 12 x 2 h * + f * + x 3 h * + f - 6 h 4^* / :

x 0.25 D3 x λ * 0.5 x 0.5 D4 ( * x λ * 2 1 sys x 0.1 D1 x λ * x 0.1 D2 x λ * 2 1 sys

Data XY :

Example 2 . . . successive discrete convolution-derivatives

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Discrete derivative by convolution ... [-1 point at each end]. data scale convol vx data 1 col : vy data 2 col : 12 mat f 101 - 13 mat transpose : Dy vy f al_convr1d eval : Dy Dy 2 Dy rows 1 - 11 submatrix : Deriv vx scale Dy * augment : Deriv Deriv 2 Deriv rows 1 - 12 submatrix : 61 line :

Sanity vs Symbolic.
Support ainterp/cinterp
... not offered. Derivative_1 Data 1 convol : Derivative_2 Derivative_1 1 convol : Derivative_3 Derivative_2 1 convol : 31 sys

Derivative_1 Derivative_2 Derivative_3

Example 3 . . . Mathsoft . . . dspline(s,X,Y,t)

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X XY 1 col : Y XY 2 col : 12 mat N Y rows 1 - : r 2 N range : 12 mat 21 sys

Prepare the Tridiagonal Matrices

Prep.0 Install the constant vector 'br'

i r br i el 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 : br N 1 + el 0 : 21 line for

Prep.1 Install the tridiagonal matrix

M 11 el 1 :

M N 1 + N 1 + el 1 :

l_p_c cubic splines

0. Construct the tridiagonal linear spline matrix [lspline]

i r M i i 1 - el X i el X i 1 - el - 6 / : for

i r M i i el X i 1 + el X i 1 - el - 3 / : for

i r M i i 1 + el X i 1 + el X i el - 6 / : for

slin M 1 -^br * eval :

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

sp M 1 -^br * eval :

2. Construct the tridiagonal cubic spline matrix [cspline]

M 12 el X 1 el X 3 el - (X 3 el X 2 el - / (:

M N 1 + N el X N el X N 2 - el - (X N 2 - el X N 1 - el - / :

M 13 el X 2 el X 1 el - (X 3 el X 2 el - (/ :

M N 1 + N 1 - el X N 1 - el X N el - (X N 1 - el X N el - (/ :

sc M 1 -^br * eval :

In 's' , plug the the selected type of spline [slin, sp,sc]
'slin & sp' join cubic 2 points entrance and along and departutre.
'sc' joins cubic 3 points entrance and along and departure. s sp :

Mathcad 8 Pro. code conversion InterpSpline(s,X,Y,t)

s X Y t InterpSpline Mathcad 8 Pro. code conversion t X 1 el ≤ idx 1221 mat : t X N el ≥ idx N N 1 + 21 mat : i 1 : X i el t < i i 1 + : while idx i 1 - i 21 mat : 31 line if if hx X idx 2 el el X idx 1 el el - (: hy Y idx 2 el el Y idx 1 el el - (: hs s idx 2 el el s idx 1 el el - (: 31 line a hs 6 hx * / : b s idx 1 el el 2 / : c hy hx / hx 6 / s idx 2 el el 2 s idx 1 el el * + (* - (: d Y idx 1 el el : 41 line a t X idx 1 el el - (3^* b t X idx 1 el el - (2^* + c t X idx 1 el el - (* + d + 51 line :

◄ 's' from your collection above

recast the 'interpolation' code x cubic s X Y x InterpSpline :

interpolate within 'X range' 3.5 cubic 0.3077

The complimentary scalar dspline

s X Y t dspline t X 2 el ≤ k 1 : t X X rows 1 - el ≥ k X rows 1 - : k 1 : t X k 1 + el > k k 1 + : while 21 line if if c Y k 1 + el Y k el - X k 1 + el X k el - / s k 1 + el 2 s k el * + (X k 1 + el X k el - 6 / (* - : s k 1 + el s k el - X k 1 + el X k el - / (t X k el - (2^2 / * s k el t X k el - (* + c + 31 line eval :

x D1 s X Y x dspline : x D2 X s x ainterp : x c 1 X 1 el x ≤ (x X X rows el ≤ (& cases : 31 sys

The complimentary fintegrate

Prepare the pseudo-integrand ... 'N' from initial  spline setup s X Y fintegrate i 1 N range h i el X i 1 + el X i el - : f i 1 + el h i el 2 / Y i el Y i 1 + el + (* h i el (3^24 / s i el s i 1 + el + (* - : 21 line for f 21 line :

x D x diff NOT supported ≡

Σ s X Y fintegrate :

discrete, not ideally smooth D2 X s augment :

Export the discrete cumulative integration Σcumulative n X rows 1 - : Sum 1 el 0 : Sum n 1 + el 0 : i 1 n range Sum i 1 + el Sum i el Σ i el + eval : for X Sum augment 51 line :

2nd derivative discrete [lpc]

D2 0 2 1 sys x cubic x c * x D1 x c * x D2 x c * 3 1 sys Σcumulative

s X Y fintegrate sum 2.7468

j 51 :

Σcumulative j row 5 1.2737 12 mat

D2 data

XY 0 0.0097 0.1 0.0097 0.2 0.0116 0.3 0.0038 0.4 0.0333 0.5 0.0169 - 0.6 0.0342 0.7 0.0001 0.8 0.0253 0.9 0.0185 1 0.0206 1.1 0.0192 1.2 0.0227 1.3 0.0102 1.4 0.0566 1.5 0.0034 1.6 0.05 1.7 0.0367 1.8 0.043 1.9 0.0311 2 0.0726 2.1 0.0385 2.2 0.0735 2.3 0.0876 2.4 0.0562 2.5 0.1077 2.6 0.1129 2.7 0.1007 2.8 0.1442 2.9 0.1626 3 0.1655 3.1 0.1955 3.2 0.2526 3.3 0.234 3.4 0.3113 3.5 0.3208 3.6 0.3856 3.7 0.4169 3.8 0.4666 3.9 0.4765 4 0.5073 4.1 0.4943 4.2 0.3957 4.3 0.323 4.4 0.0524 4.5 0.2324 - 4.6 0.6826 - 4.7 1.1172 - 4.8 1.5684 - 4.9 1.909 - 5 2.0155 - 5.1 1.909 - 5.2 1.5684 - 5.3 1.1172 - 5.4 0.6826 - 5.5 0.2324 - 5.6 0.0524 5.7 0.323 5.8 0.3957 5.9 0.4943 6 0.5073 6.1 0.4765 6.2 0.4666 6.3 0.4169 6.4 0.3856 6.5 0.3208 6.6 0.3113 6.7 0.234 6.8 0.2526 6.9 0.1955 7 0.1655 7.1 0.1626 7.2 0.1442 7.3 0.1007 7.4 0.1129 7.5 0.1077 7.6 0.0562 7.7 0.0876 7.8 0.0735 7.9 0.0385 8 0.0726 8.1 0.0311 8.2 0.043 8.3 0.0367 8.4 0.05 8.5 0.0034 8.6 0.0566 8.7 0.0102 8.8 0.0227 8.9 0.0192 9 0.0206 9.1 0.0185 9.2 0.0253 9.3 0.0001 9.4 0.0342 9.5 0.0169 - 9.6 0.0333 9.7 0.0038 9.8 0.0116 9.9 0.0097 10 0.0097 1012 mat :

ksmooth kernel

σ 0.3708158988058244 :

The Gaussian ksmooth

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

Smath recognises the silent 'bw', something
like the Mathcad "global definition". W j X Y Ksmooth X i el X j el - bw / K Y i el * (i 1 X rows sum X i el X j el - bw / K (i 1 X rows sum / : 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

from below

NOT smooth for sub-use ►

0. 'XW' discrete export
1. adjust 'bw' smoothing.
2. adjust plot/export scale.
3. Akima(x) for in-situ X XY 1 col : Y XY 2 col : 12 mat bw 0.375 : (bandwidth ≡ XW X W augment : 31 sys

discrete XW ► smoothed 2nd derivative ... adjust 'bw'.

1.0625 XW * 2 1 sys

Example 4 . . . symbolic derivatives

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x erf 2 π sqrt / t 2^- exp t 0 x int * ≡

1rst derivative ...
is simply the integrand x D 2 π sqrt / x 2^- exp * :

n 2 :

'n' derivative x D x n diff 4 x 2^- exp * 12 x 2^* - (* π sqrt / -

constrain the plot ["] x u 12 - x ≤ (x 2 ≤ (& cases :

x erf x u * x D x D x 1 diff 0.5 x D x 3 diff * 4 1 sys

x y f x 9^y 8^* 2 x * + y 3^+ :

x y f x diff x y f y diff 21 mat 214 x 8^* y 8^* + (* x 8^y 8^* + y 2^38 y 5^* x 9^* + (* 21 mat

Mathsoft reference

Example 5 . . . derivative-convolution single vector

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[L,H] range Npts L 5 - : H 5 : m 200 : 14 mat 11 sys

set N = even/odd to escape divide by 0 maybe other singularities.

λ 99 : start plot point to center ≡

dx H L - m 1 - / :

i λ - m 1 - range y i λ 1 + (+ el dx i * erf : for

y: Ausgangsdaten, mit m Punkten abgetastete Simusfunktion

y: Fonction originelle, fonction sinus, échantillon de points de m

<= y original function, sampled at 'm' points

f 101 - 31 mat :

Die zugehörige x-Differenz ist 2 dx

Le pas correspondant à x est 2 dx

the corresponding x step is 2 dx dy approximates the first derivative of y using divided central differences

y f i el * y i 1 + el y i 1 - / - ≡

dy 12 dx * / y f al_convr1d * :

fnct dx λ - m 1 - range (* y augment :

t0 1 time :

fnct fnct 1 m 2 - (12 submatrix :

n f length :

Deriv dx λ 1 + (- m range (* dy augment :

Deriv Deriv n m 1 - 12 submatrix :

fnct Deriv 2 1 sys

1 time t0 - 0.22 s *

Example 6 . . . derivative finite differences dual vector

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data Deriv 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 1 deriv * augment 91 line :

fnct fnct Deriv 2 1 sys

The derivative is a noise amplfier. If it looks noisy, the data set is though impercivable by the eye.

Example 7 . . . push-down derivative

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There are many different styles of derivatives from accumulated data Read more literature [F.B. Hildebrand]. The most common style: T-bar, it has smoothing effect. T-bar is what we use in Process Control and Instrumentation. This technique applies to plant processing that need not too brutal rate of increase, vice versa rate of decrease from ramped SP [Set Point].

Example 8 . . . working a real project

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t0 1 time :

dn_GearsBDF is 1/1 MCD The MCD solver "Radau" introduced Mathcad 2001i

Y Clear 1

http://oai.cwi.nl/oai/asset/4619/04619D.pdf

t Y f 7100171 Y 1 el * - 43 Y 2 el * + 832 Y 3 el * + (* + 10000 / 171 Y 1 el * 875 Y 2 el * - 100 / 21003 Y 3 el * - 43 Y 4 el * + (* 7 Y 5 el * + 200 / 1652 Y 2 el * 7 Y 4 el * - (* 171 Y 3 el * + 100 / 349 Y 5 el * - 86 Y 6 el Y 7 el + (* + 200 / Y 6 el 4328000 Y 8 el * + (* - 323 Y 4 el * 57 Y 5 el * + 23 Y 7 el * + (* + 100 / 28000 Y 6 el * Y 8 el * 181 Y 7 el * - 100 / 28000 Y 6 el * Y 8 el * - 181 Y 7 el * + 100 / 81 mat :

IC 1000000 0.0057 81 mat :

5 possible robust solvers. # # # # # mwode113 2.5 min ≡ # # # # # mwode15s 40 sec ≡ # # # # # mwode23 2.5 min ≡ # # # # # mwode23s 50sec ≡ # # # # # mwode45 3 min ≡ 51 line

3 possible robust solvers. # # # # # dn_AdamsMoulton 3 min ≡ # # # # # dn_ExplicitRK45 2 min ≡ # # # # # dn_GearsBDF 10 sec ≡ 31 line

sol IC 03221000 t Y f dn_GearsBDF :

X sol 1 col : C sol 6 col : D sol 7 col : 31 line

X C augment X D augment 2 1 sys

Rkadapt: adequate for these 3 pairs reactions sol IC 0101000 t Y f dn_GearsBDF :

X sol 1 col : U sol 2 col : V sol 5 col : W sol 3 col : Z sol 4 col : A sol 8 col : B sol 9 col : 71 line

X U augment X V augment 2 1 sys X W augment X Z augment 2 1 sys X A augment X B augment 2 1 sys

1 time t0 - 9.9 s *

Radau Symbolic

n x L 1 - (k^2 n * 2 k * - (! (x n 2 k * -^* (* 2 n^k! (* n k - (! (n 2 k * - (! (* (* / (k 0 n 2 / Floor sum 11 line :

n x Radau n 1 - x L n x L + x 1 + / :

x y axes x y / - :

ε 1012 -^:

5 x L 6 x L axes 3 1 sys 5 x ε - Radau axes 2 1 sys

k 16 range Legendre k el k x L : for

k 15 range Radau k el k x Radau : for

Legendre x 1 - 3 x 2^* + 2 / x 3 - 5 x 2^* + (* 2 / 35 x 2^* 6 - 7 x 2^* + (* + 8 / x 157 x 2^* 10 - 9 x 2^* + (* + (* 8 / 5 - 21 x 2^* 5 x 2^15 - 11 x 2^* + (* + (* + 16 / 61 mat

Radau 11 - x 23 x * + (* + 21 x + (* / 1 x 2^35 x * + (* - 3 x * + 21 x + (* / - 3 x 223 - 5 x 2^* + (* 15 x * - (* 35 x 3^* + (* + 81 x + (* / 3121 x 5^* + (* 5 x 2^* 67 x 2^* - 14 x * + (* - 15 x * + 81 x + (* / 51 mat

x U X U x ainterp : x V X V x ainterp : 21 sys

user 1 :

Interpolate in-situ user U user V 21 mat 0.2555 0.4585 21 mat

X U augment X V augment user 0.75 user 0 2 2 mat 3 1 sys

Mathcad/Mathsoft 11 and earlier does not have built-in cubic interpolation. They transit via l_p_c splines. Smath has two built-in cubic interpolation [Akima, cinterp]. From the onset, Smath will interpolate in-situ w/o the freak-trick PTC [l_p_c] interpolation. Sub-derivatives, immediate from infinitesimal analysis illustrated below, scalar plot style. Smath ODE solvers are native from code, intentionally producing discrete solution(s) for exporting to Engineers/Scientists...

Sub-derivatives

x fnct x h + f x h - f + 2 / ≡

x λ 10 x ≤ (x 5 ≤ (& cases :

x f x V :

Typical demo h 0.1 ≡

x h D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / : x h D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / : x h D3 x 3 h * - f 8 x 2 h * - f * - 13 x h - f * + 13 x h + f * - 8 x 2 h * + f * + x 3 h * + f - 8 h 3^* / : x h D4 x 3 h * - f - 12 x 2 h * - f * + 39 x h - f * - 56 x f * + 39 x h + f * - 12 x 2 h * + f * + x 3 h * + f - 6 h 4^* / : 41 sys

x 0.1 D1 x λ * x 0.1 D2 x λ * 2 1 sys

whatever it may mean
wrt HIRES project. x 0.1 D3 x λ * x 0.1 D4 x λ * 21 sys

x f x V :

Golden-Ratio search [min/max]

TOL 1012 -^:

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 71 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

X f 0.5 1.5 y1 y2 ≥ TOL Extrema 1 el :

x f x λ * X X f + 40 red 1 5 mat 2 1 sys

X 0.986597