3aa40909-6ba0-409b-9b13-bc672b60c2bc 10 4 5 decimal

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Infinitesimal Derivatives ... for functions w/o symbolic

Algo style Memo formula of incremental analysis of this demo example x deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : x D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / : (1rst Derivative ≡ x D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / : (2nd Derivative ≡ x 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 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^* / : 51 line

h 0.01 :

x f x sqrt :

x Symbolic x f x diff x f x 2 diff x f x 3 diff x f x 4 diff 41 sys :

2 D1 0.353553390544667

sanity 0.353553390593275

This example of infinitesimal analysis will be useful for scalar derivative(s) of functions that don't have symbolic expansion, i.e: scalar spline functions. This demo demonstrates insensibility to size 'h'. This is not a rule from experimented applications.

x λ 1 0.1 x ≤ (x 3 ≤ (& cases :

x Symbolic 4 el x λ * x D4 2 1 sys x Symbolic 3 el x λ * x D3 2 1 sys x Symbolic 2 el x λ * x D2 2 1 sys x Symbolic 1 el x λ * x D1 2 1 sys

x f x diff x f x 2 diff x f x 3 diff x f x 4 diff 41 sys 12 x sqrt * / 14 x sqrt 2^* x sqrt * / - 38 x sqrt 4^* x sqrt * / 3 x 4 x sqrt 2^* + (* 16 x sqrt 9^* / - 41 sys

Sanity check

10 D
10 D
8½ D
5½ D 2 D1 2 D2 2 D3 2 D4 41 mat 0.35355339054 0.0883883475 - 0.0662912665 0.08286795803 - 41 mat

x 2 : 2

12 x sqrt * / 14 x sqrt (2^* x sqrt * / - 38 x sqrt (4^* x sqrt * / 3 x 4 x sqrt (2^* + (* 16 x sqrt (9^* / - 41 sys 0.3535533906 0.0883883476 - 0.0662912607 0.0828640759 - 41 sys