5c835387-107c-4a7f-b47f-010482502ecf 11 4 5 decimal

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Generally, experimental projects will collect measured values ... call them tabular ordinates. Collect plenty to be representative. Associate with time 't' ... 1rst derivative 'speed' 2nd 'acceleration'. Hildebrand has smoothing effect, not exclusive 'ksmooth' raw collection. This demo document is not exclusive of other methods.

Companionship with F.B. Hildebrand

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x f 11 x 2^+ / :

f L H N n Tabular 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 n round eval : 21 line for vx vy augment 41 line :

data f 5 - 5509 Tabular :

t data 1 col :

Y data 2 col :

◄  Hildebrand f data 2 col :

h t t length el t 1 el - data rows 1 - / : 15 /

Lo 5 - 15 - 1 - 22 mat :

Hi 5151 - 22 mat :

H F.B. HILDEBRAND (3.11.9) ... derivative @ tabular ordinates i 5 data rows 5 - (range D i el 1840 h * / (3 f i 4 - el * 32 f i 3 - el * - 168 f i 2 - el * + 672 f i 1 - el * - 672 f i 1 + el * + 168 f i 2 + el * - 32 f i 3 + el * + 3 f i 4 + el * - (* : for t 1 D rows 11 submatrix D augment 31 line :

Derivative at tabular points . For the plot, X data are added. If you don't have equally spaced points, interpolate/discretise [L,H,N]

H data 2 1 sys

x q 11 x 2^+ / : points generatrix ≡

H rows 45

x q x q x diff Lo Hi 6 1 sys

h 15 /

x f t Y x cinterp :

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^* / :

5 - x ≤ ( x 5 ≤ ( & x h D1 OUT of data set range if 5 - x ≤ ( x 5 ≤ ( & x h D2 OUT of data set range if 5 - x ≤ ( x 5 ≤ ( & x h D3 OUT of data set range if

clean Export the clean D3 x t : h t t length el t 1 el - t rows 1 - / : i 1 x rows range Deriv i el x i el h D3 eval : for x Deriv augment 51 line :

clean Lo Hi 3 1 sys