sub-spaces

811a93bb-c4a0-4c94-b1cc-aa524fab516b 43 15 12 decimal

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

RKfixed integration ... discontinuous

t0 1 time :

XY 004 0.75 8 0.75 12 0.2 42 mat :

X XY 1 col : Y XY 2 col : x f X Y x linterp : x t D x f : sol 00121000 x t D rkfixed : 51 sys

3 appVersion 0.98.6179 ≡

Native project cumulative Σ, XY

XY sol 2 1 sys

Cumulative Σ over magnified XY

X 4 Y * augment sol 2 1 sys

sol sol rows row 12 6.399 12 mat

============================================================

RKfixed integration ... f(x)

α 1523043253 - 27354.7 - 43256.75 25388.3 - 40061 mat :

Fréchet: extreme value-II x f α 1 el α 2 el x α 4 el - (/ (α 5 el^* α 2 el x α 4 el - (/ (α 3 el^- exp * α 6 el + :

end 12 :

x c 10 x ≤ (x end ≤ (& cases :

0 f 403

x Y D x f :

init 0 :

sol init 0 end 100 x Y D rkfixed :

x f x c * sol 2 1 sys

RKfixed integration ... f(x)

x f x ln :

end 6 :

ε 1012 -^:

x Y D x ln :

x c 10 x ≤ (x end ≤ (& cases :

sol 0 ε end 1000 x Y D dn_GearsBDF :

x f x c * sol e 2 e 1 - 2 2 mat 3.858 1.35 + 100 black 1 5 mat 4 1 sys

1 time t0 - 5.5 s *

X sol 1 col : Y sol 2 col : 12 mat x q X Y x linterp : 21 sys

jnct x ln x q - 0 ≡ x 3.5 4 solve : 3.8582

jnct ln 1.3502

FD integration ...noisy data set

data_1 1 0.132 - 2 0.11 - 3 0.045 4 0.0045 5 0.138 - 6 0.467 7 0.5 8 0.78 9 1.342 10 0.993 11 1.105 12 1.119 13 1.18 14 1.143 15 0.655 16 1.008 17 0.771 18 1.208 19 0.938 20 0.783 21 0.734 22 0.763 23 1.044 24 0.754 25 0.797 252 mat :

data_2 26 1.086 27 0.426 28 0.563 29 0.814 30 0.665 31 0.561 - 32 0.429 - 33 0.187 34 0.153 - 35 0.422 - 36 0.127 - 37 0.018 - 38 0.247 - 39 0.422 - 40 0.738 - 41 0.711 - 42 0.863 - 43 0.769 - 44 0.485 - 45 0.79 - 46 0.737 - 47 0.708 - 48 0.965 - 49 0.899 - 50 0.947 - 252 mat :

data_3 51 0.871 - 52 1.138 - 53 1.191 - 54 0.046 - 55 1.016 - 56 1.275 - 57 0.617 - 58 0.908 - 59 1.09 - 60 0.972 - 61 0.594 - 62 0.5 - 63 0.346 - 64 0.191 - 65 0.177 - 66 0.362 - 67 0.547 - 68 0.455 69 0.417 70 0.129 - 71 0.352 72 0.906 73 0.386 74 0.047 - 75 0.214 252 mat :

data_4 76 0.309 77 0.615 78 0.846 79 0.698 80 0.819 81 1.113 82 1.042 83 1.114 84 0.653 85 0.667 86 0.657 87 0.846 88 0.547 89 0.053 90 1.244 91 1.046 92 0.793 93 0.383 94 0.335 95 0.272 96 0.537 97 0.248 98 0.67 99 0.226 - 100 0.208 252 mat :

Data data_1 data_2 data_3 data_4 stack :

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 0 : Y 1 el 0 : 12 mat 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 for vx Y augment 61 line :

Data Σ Data 2 1 sys Data

FD integration ... discontinuous exact Riemann rectangle

sub-spaces

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

data sub subspace j 1 data rows 1 - (range Vsub j el i 1 sub range vx i el data j el i sub / data 1 j + el data j el - (* + : for vx 21 line eval : 11 line for v Vsub 1 el : j 2 Vsub rows range v v Vsub j el stack : for v 31 line 21 line :

XY 004 0.75 8 0.75 12 0.2 42 mat :

In this example, independent of the number of points. RKfixed depends upon number of points [see above].

Xo XY 1 col : Yo XY 2 col : 21 sys

X Xo : Y Yo : 21 sys

Equally sub-divide  the control polygon sub 10 :

XY X Y augment : origin X Y + 15 blue plotG : 21 line

New Xo sub subspace Yo sub subspace augment : New Xo 1 el Yo 1 el 12 mat New stack : 21 line

isolate New New :

X New 1 col : Y New 2 col : 12 mat SUB X Y . 12 black plotG : 21 line

Hermite Parametric Polyline CAD style ... adjust fillet

origin XY SUB 3 1 sys

New 00 0.4 0.075 0.8 0.15 1.2 0.225 1.6 0.3 2 0.375 2.4 0.45 2.8 0.525 3.2 0.6 3.6 0.675 4 0.75 4.4 0.75 4.8 0.75 5.2 0.75 5.6 0.75 6 0.75 6.4 0.75 172 mat

data Σ FD Cumulative Integration vx data 1 col : vy data 2 col : 12 mat Y 0 : Y 1 el 0 : 12 mat 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 for vx Y augment 51 line :

New Σ New 2 1 sys

4 0.75 * 2 / 4 0.75 * + (0.75 0.2 - (4 * 2 / (+ 0.2 4 * (+ 6.400

New Σ New Σ rows row 12.000 6.400 12 mat

XY 004 0.75 8 0.75 12 0.2 42 mat :

X XY 1 col : Y XY 2 col : x f X Y x linterp : 31 sys

exact ... acc [50] x f x 012 int 6.400

user original polygon Xo 0.5 - 123456718 mat transpose : Yo 6 7.5 7.5 1.25 1.25 64718 mat transpose : 21 sys

X Xo : Y Yo : 21 sys

Equally sub-divide  the control polygon sub 10 :

XY X Y augment : origin X Y + 15 blue plotG : 21 line

New Xo sub subspace Yo sub subspace augment : New Xo 1 el Yo 1 el 12 mat New stack : 21 line

isolate New New :

X New 1 col : Y New 2 col : 12 mat SUB X Y . 12 black plotG : 21 line

Hermite Parametric Polyline CAD style ... adjust fillet

origin XY SUB 3 1 sys

New 0.5 - 6 0.35 - 6.15 0.2 - 6.3 0.05 - 6.45 0.1 6.6 0.25 6.75 0.4 6.9 0.55 7.05 0.7 7.2 0.85 7.35 1 7.5 1.1 7.5 1.2 7.5 1.3 7.5 1.4 7.5 1.5 7.5 1.6 7.5 172 mat

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 0 : Y 1 el 0 : 12 mat 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 for vx Y augment 61 line :

New Σ New 2 1 sys

More direct Linterp(data,x) ... sub-range quick

sub XY # # range # # range el ≡

Linear interpolation/extrapolation data x Linterp vx data 1 col : vy data 2 col : x vx 1 el ≤ k 1 : x vx vx rows el ≥ k vx rows 1 - : k 1 : x vx k 1 + el ≥ k k 1 + : while 21 line if if vy k el vy k 1 + el vy k el - vx k 1 + el vx k el - / x vx k el - (* + 41 line :

XY 0.175 301.5 0.319 212.419 0.525 132.948 0.7 70.127 0.875 37.642 1.05 48.892 1.225 93.052 1.4 143.028 1.575 168.217 1.75 152.09 1.925 104.526 2.1 54.35 2.275 31.11 2.45 53.392 2.625 123.308 2.8 221.034 2.975 307.124 3.15 340.85 3.325 304.367 3.5 215.563 3.675 118.199 3.85 48.228 4.025 10.832 4.2 7.717 - 242 mat :

check extrapolation XY 0.125 Linterp XY 4.5 Linterp 21 mat 33240 - 21 mat

Smath built-in X Y x linterp

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

data point(s) sanity check XY 0.175 Linterp XY 1.4 Linterp XY 3.5 Linterp XY 4.2 Linterp 41 mat 301.5 143.028 215.563 7.717 - 41 mat

a b c spec o 5 black 13 mat : pts XY a b c spec pts : x f XY 1 col XY 2 col x ainterp : x c 1 0.175 x ≤ (x 4.2 ≤ (& cases : 41 sys

Carbon-12 α from smoothlet especially designed for this project *.sm document

pts x f x c * 2 1 sys

x f x 0.175 4.2 int 545

RKfixed ... Simpson incompatible f(x) ... X^Y not scalar wrt 'x'

α 1523043253 - 27354.7 - 43256.75 25388.3 - 40061 mat :

Fréchet: extreme value-II x f α 1 el α 2 el x α 4 el - (/ (α 5 el^* α 2 el x α 4 el - (/ (α 3 el^- exp * α 6 el + :

end 12 :

x c 10 x ≤ (x end ≤ (& cases :

0 f 403

x Y D x f :

init 0 :

sol init 0 end 1200 x Y D rkfixed :

x f x c * sol 2 1 sys

sol sol rows row 12 29806.99 12 mat

'N' ... 100 12 29800.48 12 mat

Advanced cumulative integration on canvas

∫Adaptive(f,a,b,N,ε) Carlos

f a b N ε ∫Adaptive sum 0 : i 1 : tol i el ε : aa i el a : h i el b a - (2 / : 51 line .... fa i el a f : fc i el a h i el + f : fb i el b f : s i el h i el fa i el 4 fc i el * + fb i el + (3 / * : L i el 1 : 51 line 13 mat above => initialise the while loop i 0 > fd aa i el h i el 2 / + f : fe aa i el 3 h i el * 2 / + f : 12 mat s1 h i el fa i el 4 fd * + fc i el + 6 / (* : s2 h i el fc i el 4 fe * + fb i el + 6 / (* : 12 mat v1 aa i el : v2 fa i el : v3 fc i el : v4 fb i el : v5 h i el : v6 ε : v7 s i el : v8 L i el : 18 mat i i 1 - : s1 s2 + v7 - abs v6 < sum sum s1 s2 + (+ : v8 N ≥ Solution Insufficient Number of Intervals : i i 1 + : aa i el v1 v5 + : fa i el v3 : fc i el fe : fb i el v4 : 14 mat h i el v5 2 / : tol i el v6 2 / : s i el s2 : L i el v8 1 + : 14 mat i i 1 + : aa i el v1 : fa i el v2 : fc i el fd : fb i el v3 : h i el h i 1 - el : s i el s1 : 16 mat L i el L i 1 - el : 61 line if if 51 line while sum 41 line eval :

x 0.5 1 - ^ x - exp * π sqrt x sqrt erf * 2 1 sys

eval(,) gains 1 min
.....
N 100 plenty
'ε' canvas accuracy
.....
'a' infinite limit
'b' user end canvas plot x f x 0.5 1 -^x - exp * eval : N 100 : ε 0.001 : a 1015 -^: b e : 51 sys

x Canvas f a x N ε ∫Adaptive :

x c 1 a x ≤ (x e ≤ (& cases :

t0 1 time :

Canvas(x) is scalar wrt pixels runs 126 *∫Adaptive(f,a,x,N,ε)

x f x Canvas x c * 2 1 sys

1 time t0 - min 2.8

x 0.5 1 -^x - exp * π sqrt x sqrt erf vectorize * - 0 ≡ x 0.25 0.5 solve 0.384

Int(f,a,b,n) ... Simpson/Romberg Kernel [Jean]

Basic Romberg-Simpson ... integration acc from options ... NUMERIC f a b Romberg w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 01 int * 31 line eval :

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from  l_p_c_spline 
Integration accuracy from user command line 'n' ... NUMERIC f a b n Int w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : ... α 0 : β 1 : h β α - n / : 15 mat Simpson 13 / h 2 / α G β G + (* h α k h * + G k 1 n 1 - sum * + (* 23 / h * α h 2 / - k h * + G k 1 n sum * + eval : 32 / b a - (* Simpson * 41 line :

Romberg kernel for rescueing Simpson ...

α 1523043253 - 27354.7 - 43256.75 25388.3 - 40061 mat :

Fréchet: extreme value-II
X^Y [Power Function] is NOT
scalar wrt native Simpson 
thus fails to integrate. x f α 1 el α 2 el x α 4 el - (/ (α 5 el^* α 2 el x α 4 el - (/ (α 3 el^- exp * α 6 el + :

x Y D x f :

init 0 :

end 12 :

x c 10 x ≤ (x end ≤ (& cases :

sol init 0 end 1000 x Y D rkfixed :

x f x c * sol 2 1 sys

sol sol rows row 122980712 mat

x Frechet f 0 x 10 Int :

t0 1 time :

x f x c * x Frechet x c * 2 1 sys

12 Frechet 29807

1 time t0 - min 0.3

x f 1 x / :

fixed ....
sanity ...
asymptotic f 12 Romberg 2 ln f 122000 Int 31 mat 0.693147180841258 0.693147180559945 0.693147180559946 31 mat

Observe . . . two cumulative integrators

1. ODE solvers vary in timing and accuracy wrt N

2. Int(f,0,x,n) integrates Simpson in the Romberg range [0 .. 1]

3. rkfixed exports a data set solution.

4. Int(f,0,x,n) calculates @ point i.e: scalar for quick plot.

. . . algo style in program .

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a 8 π 3 - (* 3 π * 4 π - (* / : 0.140012

Approximation erf(x)

x Erf x sign 1 x 2^4 π / a x 2^* + 1 a x 2^* + / (* - exp - sqrt * :

x D1 2 π sqrt / e x 2^-^* :

x erf x diff 2 π sqrt / e x 2^-^* ≡

n x Dn x D1 x n diff :

x 1 :

exact 15 D
useless x erf x Erf 21 mat 0.842700792949715 0.842925571821284 21 mat

x erf x D1 2 x Dn 0.5 3 x Dn ( * 4 1 sys

Vectorize these TWO x erf vectorize x 0 π int x Gamma vectorize x 1 π int 21 mat 2.577 2.611 21 mat

Infinitesimal Derivative 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (*

Finite differences "FD" derivative of data set 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 deriv augment 91 line :

The finite differences derivative applies to data sets, not too noisy data sets. The "FD" is comparable to more advanced numarical methods. Be cautious if you apply "smoothing". Some productive "smoothers": 1. Savitsky-Golay [SG] 2. Supsmooth 3. Window smoothing [Fourier ...] 4. Time series 5. Ksmooth. Alternately, you may attempt to model the data set [globally/piecewise]. Modeling is an art [F.B. Hildebrand].

More Essentials . . . copy/paste in new sheet

]]> Infinitesimal Stuff [7 min] ... disabled

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

u b 0 u 2 - ≤ 16 / 2 u + (3^* 2 - u ≤ (u 1 - ≤ (& 16 / 2 u + (3^* 23 / 1 u + (3^* - 1 - u ≤ (u 0 ≤ (& 23 / 1 u - (3^* - 16 / 2 u - (3^* + 0 u ≤ (u 1 ≤ (& 16 / 2 u - (3^* 1 u ≤ (u 2 ≤ (& 0 cases :

Gaussian basis 'b-spline'

2 - x ≤ ( x 2 ≤ ( & x b Undefined if

0 b 0.6667

u v Bsp v m el u m - b * (m 1 v rows sum :

X 1.875 1.875 1.95 2 2.05 2.125 2.125 2.125 18 mat : Y 11101 - 1 - 1 - 1 - 18 mat : 21 line

XY X transpose Y transpose augment : Points X transpose Y transpose . 14 black plotG : 21 line

u oy u Y transpose Bsp : u ox u X transpose Bsp : 21 sys

infinitesimal setup
f(x)≡ox(x), oy(x) h 0.001 : x f x ox : 12 mat x f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : 21 sys

x f deriv

infinitesimal setup h 0.001 : x f x b : 12 mat x f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : 21 sys

x f deriv

data t 1.75 6.75 1.8 range : i 1 t rows range vx i el t i el ox : vy i el t i el oy : 21 line for vx vy augment 31 line :

XY data Points 3 1 sys

x Σ u b u 2 - x int :

x Σ

x f x b :

Algo style Memo formula of incremental analysis of this demo example h 0.0001 : (increment ≡ x h f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : x h f D1 x 2 h * - f x 2 h * + f - 4 h * / - : (1rst Derivative ≡ x h f 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 h f h_int h x h + f x f - (* 2 x f * + (x * : (Integral Curve [antiderivative] ≡ 51 line

constrain the canvas x c 12 - x ≤ (x 2 ≤ (& cases :

cumul u b u 2 - x int x c * u b u 0 x int x c * 21 sys :

x Maths x f x c * x h f D1 x c * x h f D2 x c * x h f h_int x c * x Σ x c * 51 sys :

x Maths cumul 1 el x c * cumul 2 el x c * 2 1 sys

CWD C:\SmathFile CurrentDirectory :

data CWD b_spline wfile 1

data CWD b_spline rfile :

X data 1 col : Y data 2 col : 12 mat x s X Y x ainterp : x c 1 1.875 x ≤ (x 2.125 ≤ (& cases : 31 sys

h 0.001 : x f x s : x h f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : x h f h_int h x h + f x f - (* 2 x f * + (x * : 41 sys

x s x c * x h f deriv x c * 5 x h f h_int * x c * 2 1 sys

scalar plot, interpolate x splineΣ u s u 1.875 x int :

2 splineΣ 0.0938

x 1.875 2.125 1.88 range : i 1 x rows range Σ i el u s u x 1 el x i el int : for cumul x Σ augment : 31 line

cumul x splineΣ x c * 2 2 splineΣ + 20 red 1 5 mat 2 1 sys

https://www.geometrictools.com/Documentation/FiniteDifferences.pdf

https://www.geometrictools.com/Documentation/DocUpdateHistory.pdf

https://www.geometrictools.com/Documentation/ReconstructHeightFromNormals.pdf

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^* / :

u b 0 u 2 - ≤ 16 / 2 u + (3^* 2 - u ≤ (u 1 - ≤ (& 16 / 2 u + (3^* 23 / 1 u + (3^* - 1 - u ≤ (u 0 ≤ (& 23 / 1 u - (3^* - 16 / 2 u - (3^* + 0 u ≤ (u 1 ≤ (& 16 / 2 u - (3^* 1 u ≤ (u 2 ≤ (& 0 cases :

x c 12 - x ≤ (x 2 ≤ (& cases :

known exact u b u 2 - 2 int 1.000000000000000

NO derivative from rules u b u diff

h 106 -^:

x f x b :

x f deriv 8 x h + f x h - f - (* x 2 h * + f - x 2 h * - f + 12 h * / :

0 b 0.6667

2 b 0.000000000000000

x b x f deriv 2 1 sys

2 - f deriv 0.000000000000000

The hyperfast Golden-Ratio search [min/max]

x f x f deriv : a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : 21 line

TOL 1012 -^:

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema 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 61 line :

t0 1 time :

X f 01 y1 y2 ≤ TOL Extrema 1 el f 1 - 0 y1 y2 ≥ TOL Extrema 1 el 21 mat :

1 time t0 - 1.3 s *

x f x c * X 1 el X 1 el f + 15 red X 2 el X 2 el f + 15 red 2 5 mat 2 1 sys

X 0.666662 0.666662 - 21 mat

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"