F.B. Hildebrand [3.11.9]

96fd91e8-b34f-4bbc-8936-1a337695e1bb 50 4 5 decimal

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

Ridder scalar derivatives [D1, D2, D3] . . . DEMO

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Generally, experimental projects will collect noisy measured values ... call them tabular values. Collect plenty to be representative. This document is a guide line to projects that produce clean discrete. For clean D3 data set must have 6 decimals at least. 1. F.B.Hildebrand is kind of 'yardstic' ... D3 has dropped from 101/78. 2. Discrete convolution is unscaled ... NOT offered. 3. Ridder agrees closely with Hlidebrand. If you have hyper noisy data, low populated [Corn Thermies] smooth wisely ... possibly fit, populate ... suite.

Companionship with F.B. Hildebrand [3.11.9]

]]> F.B. Hildebrand [3.11.9] XY data

XY 0 0.038462 0.1 0.039984 0.2 0.041597 0.3 0.043309 0.4 0.045126 0.5 0.047059 0.6 0.049116 0.7 0.051308 0.8 0.053648 0.9 0.056148 1 0.058824 1.1 0.06169 1.2 0.064767 1.3 0.068074 1.4 0.071633 1.5 0.075472 1.6 0.079618 1.7 0.084104 1.8 0.088968 1.9 0.094251 2 0.1 2.1 0.10627 2.2 0.113122 2.3 0.120627 2.4 0.128866 2.5 0.137931 2.6 0.147929 2.7 0.158983 2.8 0.171233 2.9 0.184843 3 0.2 3.1 0.21692 3.2 0.235849 3.3 0.257069 3.4 0.280899 3.5 0.307692 3.6 0.337838 3.7 0.371747 3.8 0.409836 3.9 0.452489 4 0.5 4.1 0.552486 4.2 0.609756 4.3 0.671141 4.4 0.735294 4.5 0.8 4.6 0.862069 4.7 0.917431 4.8 0.961538 4.9 0.990099 51 5.1 0.990099 5.2 0.961538 5.3 0.917431 5.4 0.862069 5.5 0.8 5.6 0.735294 5.7 0.671141 5.8 0.609756 5.9 0.552486 6 0.5 6.1 0.452489 6.2 0.409836 6.3 0.371747 6.4 0.337838 6.5 0.307692 6.6 0.280899 6.7 0.257069 6.8 0.235849 6.9 0.21692 7 0.2 7.1 0.184843 7.2 0.171233 7.3 0.158983 7.4 0.147929 7.5 0.137931 7.6 0.128866 7.7 0.120627 7.8 0.113122 7.9 0.10627 8 0.1 8.1 0.094251 8.2 0.088968 8.3 0.084104 8.4 0.079618 8.5 0.075472 8.6 0.071633 8.7 0.068074 8.8 0.064767 8.9 0.06169 9 0.058824 9.1 0.056148 9.2 0.053648 9.3 0.051308 9.4 0.049116 9.5 0.047059 9.6 0.045126 9.7 0.043309 9.8 0.041597 9.9 0.039984 10 0.038462 1012 mat :

t XY 1 col : Y XY 2 col : 12 mat f XY 2 col : 21 mat

h t t length el t 1 el - XY rows 1 - / : 0.1

H F.B. HILDEBRAND (3.11.9) ... derivative @ tabular ordinates i 5 XY 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 * - (* eval : for t 1 D rows 11 submatrix D augment 31 line :

Derivative at tabular points .

H XY 2 1 sys

H H eval :

H 2 col Min 0.6488 -

XY H 5 H rows range 12 range el :

t XY 1 col : Y XY 2 col : 12 mat f XY 2 col : 21 mat

h t t length el t 1 el - XY rows 1 - / : 0.1

H F.B. HILDEBRAND (3.11.9) ... derivative @ tabular ordinates i 5 XY 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 * - (* eval : for t 1 D rows 11 submatrix D augment 31 line :

Derivative at tabular points .

H XY 2 1 sys

H H eval :

H 2 col Min 2 -

XY H 5 H rows range 12 range el :

t XY 1 col : Y XY 2 col : 12 mat f XY 2 col : 21 mat

h t t length el t 1 el - XY rows 1 - / : 0.1

H F.B. HILDEBRAND (3.11.9) ... derivative @ tabular ordinates i 5 XY 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 * - (* eval : for t 1 D rows 11 submatrix D augment 31 line :

Derivative at tabular points .

H XY 2 1 sys

H 2 col Min 4.6409 -

H rows 78

XY data

XY 0 0.038462 0.1 0.039984 0.2 0.041597 0.3 0.043309 0.4 0.045126 0.5 0.047059 0.6 0.049116 0.7 0.051308 0.8 0.053648 0.9 0.056148 1 0.058824 1.1 0.06169 1.2 0.064767 1.3 0.068074 1.4 0.071633 1.5 0.075472 1.6 0.079618 1.7 0.084104 1.8 0.088968 1.9 0.094251 2 0.1 2.1 0.10627 2.2 0.113122 2.3 0.120627 2.4 0.128866 2.5 0.137931 2.6 0.147929 2.7 0.158983 2.8 0.171233 2.9 0.184843 3 0.2 3.1 0.21692 3.2 0.235849 3.3 0.257069 3.4 0.280899 3.5 0.307692 3.6 0.337838 3.7 0.371747 3.8 0.409836 3.9 0.452489 4 0.5 4.1 0.552486 4.2 0.609756 4.3 0.671141 4.4 0.735294 4.5 0.8 4.6 0.862069 4.7 0.917431 4.8 0.961538 4.9 0.990099 51 5.1 0.990099 5.2 0.961538 5.3 0.917431 5.4 0.862069 5.5 0.8 5.6 0.735294 5.7 0.671141 5.8 0.609756 5.9 0.552486 6 0.5 6.1 0.452489 6.2 0.409836 6.3 0.371747 6.4 0.337838 6.5 0.307692 6.6 0.280899 6.7 0.257069 6.8 0.235849 6.9 0.21692 7 0.2 7.1 0.184843 7.2 0.171233 7.3 0.158983 7.4 0.147929 7.5 0.137931 7.6 0.128866 7.7 0.120627 7.8 0.113122 7.9 0.10627 8 0.1 8.1 0.094251 8.2 0.088968 8.3 0.084104 8.4 0.079618 8.5 0.075472 8.6 0.071633 8.7 0.068074 8.8 0.064767 8.9 0.06169 9 0.058824 9.1 0.056148 9.2 0.053648 9.3 0.051308 9.4 0.049116 9.5 0.047059 9.6 0.045126 9.7 0.043309 9.8 0.041597 9.9 0.039984 10 0.038462 1012 mat :

Part 0 . . . scalar piecewise Ridder D1, D2, D3

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X XY 1 col : Y XY 2 col : 12 mat u f X Y u ainterp : 21 sys

1srtDeriv n 1 :

stepsize h 0.000001 :

u D1 n 2 h * / (n^1 - (n k -^n! k! (n k - (! (* / * u h - k 2 h * n / (* + f * (k 0 n sum * :

D1 u 010 0.1 range : j 1 u rows range d1 j el u j el D1 : for u d1 augment 31 line :

D1 D1 eval :

X D1 1 col : Y D1 2 col : 12 mat u q X Y u ainterp : 21 sys

stepsize h 0.1 :

NOT sanity checked !

u D2 n 2 h * / (n^1 - (n k -^n! k! (n k - (! (* / * u h - k 2 h * n / (* + q * (k 0 n sum * :

u D1 u 04 int 0.4615

D2 u 010 0.1 range : j 1 u rows range d2 j el u j el D2 : for u d2 augment 31 line :

D2 D2 eval :

u D2 u 05 int 0.0148 -

Export your source project derivatives XY D1 2 col D2 2 col augment :

Convenient Ridder first & 2nd derivatives ...

D1 D2 XY 3 1 sys

derivatives 0 0.038 0.015 0.009 0.1 0.04 0.016 0.009 0.2 0.042 0.017 0.01 0.3 0.043 0.018 0.011 0.4 0.045 0.019 0.011 0.5 0.047 0.02 0.012 0.6 0.049 0.021 0.013 0.7 0.051 0.023 0.015 0.8 0.054 0.024 0.016 0.9 0.056 0.026 0.017 1 0.059 0.028 0.019 1.1 0.062 0.03 0.021 1.2 0.065 0.032 0.023 1.3 0.068 0.034 0.025 1.4 0.072 0.037 0.028 154 mat

X D2 1 col : Y D2 2 col : 12 mat u ω X Y u ainterp : 21 sys

reduced stepsize affects magnitude h 0.25 :

u D3 n 2 h * / (n^1 - (n k -^n! k! (n k - (! (* / * u h - k 2 h * n / (* + ω * (k 0 n sum * :

D3 u 010 0.1 range : j 1 u rows range d3 j el u j el D3 : for u d3 augment 31 line :

D3 D3 eval :

Ridder third derivative NOT clean ... needs smoothing

D1 D2 XY D3 4 1 sys

D1 2 col Min D2 2 col Min D3 2 col Min 31 mat 0.6451 - 2.0354 - 3.8874 - 31 mat

Hildebrand 0.6488 - 2 - 4.6409 - 31 mat

Data XY :

Part 1 . . . 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 :

Hildebrand 0.6488 - 2 - 4.6409 - 31 mat

convol 0.1289 - 0.077 - 0.0335 - 31 mat

raw evaluation
i.e: not scaled 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

Part 2 . . . scalar algo Ridder D1, D2

]]> XY data

XY 0 0.038462 0.1 0.039984 0.2 0.041597 0.3 0.043309 0.4 0.045126 0.5 0.047059 0.6 0.049116 0.7 0.051308 0.8 0.053648 0.9 0.056148 1 0.058824 1.1 0.06169 1.2 0.064767 1.3 0.068074 1.4 0.071633 1.5 0.075472 1.6 0.079618 1.7 0.084104 1.8 0.088968 1.9 0.094251 2 0.1 2.1 0.10627 2.2 0.113122 2.3 0.120627 2.4 0.128866 2.5 0.137931 2.6 0.147929 2.7 0.158983 2.8 0.171233 2.9 0.184843 3 0.2 3.1 0.21692 3.2 0.235849 3.3 0.257069 3.4 0.280899 3.5 0.307692 3.6 0.337838 3.7 0.371747 3.8 0.409836 3.9 0.452489 4 0.5 4.1 0.552486 4.2 0.609756 4.3 0.671141 4.4 0.735294 4.5 0.8 4.6 0.862069 4.7 0.917431 4.8 0.961538 4.9 0.990099 51 5.1 0.990099 5.2 0.961538 5.3 0.917431 5.4 0.862069 5.5 0.8 5.6 0.735294 5.7 0.671141 5.8 0.609756 5.9 0.552486 6 0.5 6.1 0.452489 6.2 0.409836 6.3 0.371747 6.4 0.337838 6.5 0.307692 6.6 0.280899 6.7 0.257069 6.8 0.235849 6.9 0.21692 7 0.2 7.1 0.184843 7.2 0.171233 7.3 0.158983 7.4 0.147929 7.5 0.137931 7.6 0.128866 7.7 0.120627 7.8 0.113122 7.9 0.10627 8 0.1 8.1 0.094251 8.2 0.088968 8.3 0.084104 8.4 0.079618 8.5 0.075472 8.6 0.071633 8.7 0.068074 8.8 0.064767 8.9 0.06169 9 0.058824 9.1 0.056148 9.2 0.053648 9.3 0.051308 9.4 0.049116 9.5 0.047059 9.6 0.045126 9.7 0.043309 9.8 0.041597 9.9 0.039984 10 0.038462 1012 mat :

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

4 appVersion 0.98.6179.21440 ≡

XY rows 101

▼ 'n' derivative order

x n d iter 5 : N iter 1 + : TOL 0.00000001 : h 1 : 14 mat j 219 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 N < 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 1 d x λ * x 2 d x λ * 2 1 sys

1 time t0 - min 2.6

Part 3 . . . scalar algo Ridder . . . D3

]]> XY data

XY 0 0.038462 0.1 0.039984 0.2 0.041597 0.3 0.043309 0.4 0.045126 0.5 0.047059 0.6 0.049116 0.7 0.051308 0.8 0.053648 0.9 0.056148 1 0.058824 1.1 0.06169 1.2 0.064767 1.3 0.068074 1.4 0.071633 1.5 0.075472 1.6 0.079618 1.7 0.084104 1.8 0.088968 1.9 0.094251 2 0.1 2.1 0.10627 2.2 0.113122 2.3 0.120627 2.4 0.128866 2.5 0.137931 2.6 0.147929 2.7 0.158983 2.8 0.171233 2.9 0.184843 3 0.2 3.1 0.21692 3.2 0.235849 3.3 0.257069 3.4 0.280899 3.5 0.307692 3.6 0.337838 3.7 0.371747 3.8 0.409836 3.9 0.452489 4 0.5 4.1 0.552486 4.2 0.609756 4.3 0.671141 4.4 0.735294 4.5 0.8 4.6 0.862069 4.7 0.917431 4.8 0.961538 4.9 0.990099 51 5.1 0.990099 5.2 0.961538 5.3 0.917431 5.4 0.862069 5.5 0.8 5.6 0.735294 5.7 0.671141 5.8 0.609756 5.9 0.552486 6 0.5 6.1 0.452489 6.2 0.409836 6.3 0.371747 6.4 0.337838 6.5 0.307692 6.6 0.280899 6.7 0.257069 6.8 0.235849 6.9 0.21692 7 0.2 7.1 0.184843 7.2 0.171233 7.3 0.158983 7.4 0.147929 7.5 0.137931 7.6 0.128866 7.7 0.120627 7.8 0.113122 7.9 0.10627 8 0.1 8.1 0.094251 8.2 0.088968 8.3 0.084104 8.4 0.079618 8.5 0.075472 8.6 0.071633 8.7 0.068074 8.8 0.064767 8.9 0.06169 9 0.058824 9.1 0.056148 9.2 0.053648 9.3 0.051308 9.4 0.049116 9.5 0.047059 9.6 0.045126 9.7 0.043309 9.8 0.041597 9.9 0.039984 10 0.038462 1012 mat :

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=3' 3rd order derivative

x n d iter 2 : N iter 1 + : TOL 0.00001 : h 1 : 14 mat j 219 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 N < 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 :

can't evaluate ... thus escaped 53 d #

t0 1 time :

x 0 4.95 0.05 range : d3.left x x 3 d vectorize augment : 21 sys

x 5.05 10 5.1 range : d3.right x x 3 d vectorize augment : 21 sys

1 time t0 - min 1.3

d3.left d3.right 2 1 sys

Unexplained overscale ? d3.left 2 col Min 5.474 -

Sanity noise d(x,3) noise d3.left 160 range 12 range el :

noise 0 0.0021 0.05 0.0034 0.1 0.0052 0.15 0.0068 0.2 0.0086 0.25 0.0099 0.3 0.0112 0.35 0.0117 82 mat

noise

0.6488 - 2 - 4.6409 - 31 mat

► document Demo data set creator

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L H N xd range discretise U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 51 line :

L 0 : H 10 : N 100 : 13 mat U L H N xd : 21 mat

x φ 11 x 5 - (2^+ / :

Data i 1 N 1 + (range vy i el U i el φ eval : for U vy augment 21 line :

Data

jmG: 20230531