sub-spaces

7deeeaee-d386-4b42-af21-29d3fee7c730 89 4 5 decimal

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

ODE Spline Integration ... discontinuous

XY 004 0.75 8 0.75 12 0.2 42 mat :

X XY 1 col : Y XY 2 col : 12 mat x f X Y x linterp : x t D x f : sol 0012120 x t D dn_GearsBDF : 41 sys

Cumulative Σ over magnified XY

X 4 Y * augment sol 2 1 sys

Σ x 012 0.1 range : i 1 x rows range Σ i el u f u x 1 el x i el int eval : for x Σ augment 31 line :

@ acc 10000 didn't stop 20 min 10000 NOT needed here but needed for next Simpson below.

sol sol rows row Σ Σ rows row 21 mat 12 6.3989 12 mat 12 6.4 12 mat 21 mat

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

Missleading error message
"above max allowed"
Two terms f(x) are X^Y
NOT recognized by Smath x f x 012 int #

About ODE integrators ... 1. They range from exact or ~ exact. 2. They range from few decimals only. 3. They may be pure crap.

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 :

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

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

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

Infinitesimal Analysis ... 'h' governs !

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^* / :

ε   ... otherwise "Division by zero" ε 1031 -^:

x Φ x 2 x - (* sqrt :

arc length integrand x arc 1 x Φ x diff (2^+ sqrt :

short of simplification x arcLength x arc x int maple : x 2 - x + (* 1 - x + x 2 - x + (* sqrt + ln * x 2 - x + (* - sqrt x 2 - x + (* sqrt * /

x MCD 1 - x + asin 1 - x 2 - x + (* / sqrt * x 2 - x + (* - sqrt * π 2 / - :

Two equivalent forms

assumed -π C ε arcLength : 3.1415926535898 -

ABOUT this plot ... The red arc(x) exhibits two mathematical singularities 1. infinite limit @ [0] 2. singulsr end point @ [2]

x Φ x arc x arcLength C - 3 1 sys

Difference integrator

hatch

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und 
g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*n vertical lines (hatching) between functions f(x) and g(x)
in x-range from x1 to x2 f g x.1 x.2 N hatch dx x.2 x.1 - N / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

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 :

x f x Gamma :

Difference  ∫ algo style utilities a b c d n ∫ Δ f c d n Int f a b n Int at :

a 2.5 : b 4 : c 0.5 : d 1 : 14 mat

a b c d 10 ∫ 3.72

at(2) # # at

at(3) # # # at

at(3) # c d ∫ a b ∫ at

x Gamma vectorize x a b int x Gamma vectorize x c d int - 3.71963817

x y1 x Gamma vectorize : x y2 0 : x g 0 : x f x Gamma : 41 sys

Filing the differential Integrator

'N' random in critical region N 1000 :

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 :

size/define critical region.
Plot first before filling xmin d : xmax a : ymin 0 : ymax 2 : 22 mat space_x xmax xmin - (: space_y ymax ymin - (: 21 line 21 sys

A function that tests an arbitrary point 
for its membership in the critical region. x y Test 1 y x y1 ≤ (* y x y2 ≥ (* :

A routine that finds an array of N random 
pairs in the critical region. Pts i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 21 line :

a b c spec . 6 #808080 13 mat : fill Pts a b c spec pts : 21 line

Filling a range of a critial domain

x y1 x y2 fill f g a b 20 hatch f g c d 11 hatch 5 1 sys

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 4.0

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

an easy test ...

infinite limit x f 1 x / eval :

'N' iterations
_ precision _iter 100 : ε 1015 -^: 21 sys

x f x int maple x ln

lower limit [any]
_ upper limit _a 104 -^: b 1 : 21 sys

t0 1 time :

f a b iter ε ∫Adaptive 9.21034037197532

b ln a ln - 9.21034037197532 - 8.6 1013 -^*

1 time t0 - min 1.9

a damned freak from up above Σ
does not support acc 10000  ► x f x a b int 22.73

Premise algo companion . . . ∫kernel

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x a b Clear 1

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

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 ∫kernel 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 :

x f x ln :

lo 1015 -^:

t x Ω x ln :

x ∫ t x Ω x int maple : x 1 - x ln + (*

lo t part t u Ω u lo t int :

acc 100 lo 4 part 4 ∫ - 0.1873 -

Native ∫ vs kernel

testing  convergence ►  100000 n 10000 :

x Δ x ∫ f lo x n ∫kernel - :

lo 1012 -^: hi 4 : n 100 : 13 mat

4 Δ 0.000000009242 -

Premise ∫ 0 : t lo hi lo 0.1 + range : i 1 t rows range ∫ i el f lo t i el n ∫kernel : for t ∫ augment 41 line :

Premise x ∫ 2 1 sys

NON-linear integration ... acc 50

μ 0 :

σ 1 :

x λ 14 - x ≤ (x 4 ≤ (& cases :

u F u μ σ pdf.Normal :

lo 4 - : hi 4 : N 199 : 13 mat

shape range NON-linear path s 5 :

x q s 1 x 2^+ / :

ω u lo hi lo hi lo - (N / + range : i 1 u rows range vy i el u i el q eval : for u vy augment 31 line :

x g ω 1 col ω 2 col x ainterp :

ω x g 2 1 sys

non-linear path of integration f x g :

'x' is the running  canvas ...
as well the running discrete x Int u F u 0 f int :

user 0.75 :

0 Int 0.5

lo 4 - : hi 4 : N 99 : 13 mat

symbol below ʃ

interpolate «» user Int : 0.4993

INT x lo hi lo hi lo - (N / + range : i 1 x rows range ʃ i el x i el Int eval : for x ʃ augment 31 line :

Scalar/Interpolate NON-linear

x F x Int x λ * 0.125 x g * x λ * user «» + 20 black 1 5 mat 4 1 sys

Export discrete NON-linear

INT user «» + 20 black 1 5 mat 3 1 sys

Σ u lo hi lo hi lo - (N / + range : i 1 u rows range ʃ i el x q x u 1 el u i el int eval : for u ʃ augment 31 line :

PDF style NOT normalized

x q Σ 2 1 sys

Observe . . .

1. 's' has NO influence on the plateau magnitude

however, sets the shape over the running range .

2. Do not extrapolate Int(x).

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Some freak ODE integrators ...

1.73750207472565

x f x 0.5 1 -^x - exp * eval : a 1018 -^: b e : 12 mat 21 sys

# # # # # Rkadapt # # # # # rkfixed # # # # # rkm9mka # # # # # rkm9mkn # # # # # rkm9st 51 line

N 1000 :

x Y D x f :

sol 0 a b N x Y D Rkadapt :

freak ! sol sol rows row 2.718282 469.379272 12 mat

sol 0 a b N x Y D rkfixed :

freak ! sol sol rows row 2.718282 2.718283 106^* 12 mat

sol 0 a b N x Y D rkm9mka :

3½ D! sol sol rows row 2.718282 1.737671 12 mat

sol 0 a b N x Y D rkm9mkn :

3½ D ! sol sol rows row 2.718282 1.737671 12 mat

sol 0 a b N x Y D rkm9st :

3½ D ! sol sol rows row 2.718282 1.737671 12 mat

plotG(vx,vy,char,size,color) ... Hermite Polyline modules

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 :

z n RepEnd m z rows : i 1 n range w i el z 1 el : for i n 1 + (m n + (range w i el z i n - el : for i m n + 1 + (m 2 n * + (range w i el z m el : for w 51 line :

data Strip nest stripped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 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 :

Round OFF Matrix/Vector M n Round i 1 M rows range m i el M i el n round : for m 21 line :

Xo 1.875 1.95 2 2.05 2.125 15 mat transpose : Yo 1101 - 1 - 15 mat transpose : 21 sys

t0 1 time :

X Xo : Y Yo : 21 sys

Equally sub-divide  the control polygon sub 2 :

<=  populate  Hermite wisely t 01 0.075 range :

Collect QuickPlot

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

Polyline Construct

r 2 : X X r RepEnd : Y Y r RepEnd : 31 sys

x c 10 x ≤ (x 1 ≤ (& cases :

Hermite PolyLine t P 16 / t 3^- 3 t 2^* + 3 t * - 1 + (3 t 3^* 6 t 2^* - 4 + (3 t 3^* - 3 t 2^* + 3 t * + 1 + (t 3^41 mat * :

Hermite basis cubics 0...1

x P 1 el x c * x P 2 el x c * x P 3 el x c * x P 4 el x c * 4 1 sys

t j XP μ j 11 line : t P 1 el X μ el * t P 2 el X μ 1 + el * + t P 3 el X μ 2 + el * + t P 4 el X μ 3 + el * + 21 line :

t j YP μ j 11 line : t P 1 el Y μ el * t P 2 el Y μ 1 + el * + t P 3 el Y μ 2 + el * + t P 4 el Y μ 3 + el * + 21 line :

U j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hx i ct el t i el j XP eval : for ct ct 1 + : 21 line for Hx 41 line :

V j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hy i ct el t i el j YP eval : for ct ct 1 + : 21 line for Hy 41 line :

X rows r 1 + (- 10

collect the project xy U Unwrap V Unwrap augment : 11 sys

Hermite Parametric Polyline CAD style ... adjust fillet

origin xy XY SUB 4 1 sys

data xy :

Discrete Construction

1. Finite differences derivative X data 1 col : Y data 2 col : 12 mat 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 1 - el : D X deriv augment : 2. Arc integrand ................. xx D 1 col : δx D 2 col : 12 mat i 1 δx rows range yy i el 1 δx i el (2^+ sqrt eval : 11 line for Arc xx yy augment : 3. Arc length .................... vx Arc 1 col : vy Arc 2 col : 12 mat S 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : S i el S i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for Σ vx S augment : 181 line

Derivative of the function

Arc integrand

Arc

Cumulative arc length

1 time t0 - 8.3 s *

SUB New :

origin Xo Yo augment :

origin SUB xy 3 1 sys D Arc Σ 0 2 1 sys

Total arc length Σ Σ rows row 2.125 2.108 12 mat

The complimentary l_p_c spline ...

all spline stuff

Prepare the pseudo-integrand s X Y fintegrate i 1 N range h i el X i 1 + el X i el - : f i 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 :

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 :

The while loop form of the spline interpolation function s X Y t interpspline Find the knots that bracket 't' i0 1 : j X rows : j i0 - 1 > i i0 j + 2 / Floor : X i el t > j i : i0 i : if 21 line while Evaluate the spline based on those knots h X j el X i0 el - : a X j el t - h / : b t X i0 el - h / : a Y i0 el * b Y j el * + h 2^6 / a a 2^1 - (* s i0 el * b b 2^1 - (* s j el * + (* + 91 line :

The long cubic spline [lpc], scalar interpolation. s X Y t InterpSpline 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 + 41 line :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

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 :

data

data 03 - 0.2 2.951056 - 0.4 2.589484 - 0.6 1.581057 - 0.8 0.05423 1 1.5 1.2 2.112192 1.4 2.110927 1.6 1.8785 1.8 1.607987 2 1.363636 2.2 1.15897 2.4 0.991538 2.6 0.855127 2.8 0.743554 3 0.651639 3.2 0.575282 3.4 0.511303 3.6 0.457245 3.8 0.411207 4 0.371707 212 mat :

data

Part 1. get the 's' spline coefficients

t0 1 time :

l_p_cspline sanity Mathcad

X data 1 col : Y data 2 col : n data rows : 31 line

Initialise the blank diagonal matrix

M 11 el 1 :

M n n el 1 :

N n 1 - : r 2 N range : 21 line

Prepare the cubic splines

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

Fill-in the respective l_p_c splines tridiagonal matrices

Y i 1 + el X i 1 + el * Y i 1 + el X i el * - 2 Y i 1 + el * - Y i el X i 1 + el * + Y i el X i el * - 2 Y i el * + X i 1 + el X i el - (3^/

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 - / (-

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

lspline M :

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

pspline M :

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 - (/ :

cspline M :

Discrete as needed L X 1 el : H X n el : Num 100 : 13 mat U L H Num xd : 21 mat

s lspline 1 -^br * :

select spline type

assign/isolate s s :

s transpose 0 7.41249 17.24424 20.638799 5.770435 - 25.984607 - 16 mat

Part 2. The complimentary cumulative integrate

data

Σ s X Y fintegrate :

Σ Σ :

v X 1 Σ rows range 11 range el :

pointΣ v Σ augment o 5 black plot :

view the point-integrand

pointΣ

Export the cumulative integral Σcumul 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 :

Σcumul Σcumul :

Σ Σcumul o 5 black plot :

l_p_c spline cumulative integrate

Σ Σcumul 2 1 sys

x f X Y x cinterp :

cumulative i 1 X rows range y i el x f x X 1 el X i el int eval : for X y augment 21 line :

cumulative

Part 3. The complimentary scalar dspline

x D s X Y x dspline :

3 D 0.417663 -

D2 X s augment :

x plotD L x ≤ (x H ≤ (& (x D out spline if :

x D2 D2 1 col D2 2 col x linterp :

The scalar dspline

x plotD 0 2 1 sys

2nd derivative directly from [X,s]

x D2 0 2 1 sys

Part 4. The [lpc] spline interpolation

x Spline s X Y x interpspline :

Data data o 5 black plot :

x λ 1 L x ≤ (x H ≤ (& cases :

"Proofing" the cubic spline project

x Spline x λ * 0 3 - o 5 black 0.2 2.9511 - o 5 black 0.4 2.5895 - o 5 black 0.6 1.5811 - o 5 black 0.8 0.0542 o 5 black 1 1.5 o 5 black 1.2 2.1122 o 5 black 1.4 2.1109 o 5 black 1.6 1.8785 o 5 black 1.8 1.608 o 5 black 2 1.3636 o 5 black 2.2 1.159 o 5 black 2.4 0.9915 o 5 black 2.6 0.8551 o 5 black 2.8 0.7436 o 5 black 3 0.6516 o 5 black 3.2 0.5753 o 5 black 3.4 0.5113 o 5 black 3.6 0.4572 o 5 black 3.8 0.4112 o 5 black 4 0.3717 o 5 black 21 5 mat 2 1 sys

1 time t0 - 10.9 s *

Sources and credits: 1. Mathsoft for the tridiagonal systems. ...dspline(s,X,Y,t) 2. Robert Adair for the algos: ...InterpSpline(s,X,Y,t) ...fintegrate(s,vx,vy) 3. Smath adapted: Jean Giraud

Stem(data)

Stem construction over a data set data Stem stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

Part_1: Integral Convolution

W1 1 : W2 0.75 : m 0.75 : α 2 : 14 mat

t U α t 21 mat min :

t L α - t α - 21 mat max :

t X 1.5 2 ln sqrt * W1 π sqrt * / 42 ln * t W1 / (2^* - exp * : t H 1.5 2 ln sqrt * W2 π sqrt * / 42 ln * t W2 / (2^* - exp * : t Convol u X t u - H * u t L t U int t m ≤ (* : acc 50 for this symmetric Gaussian 41 sys

x U x L 2 1 sys

t0 1 time :

0.225 1 min t Convol m t - H - 0 ≡ t 0 0.25 solve

Cvl t 2 - m 1.95 - range : i 1 t rows range y i el t i el Convol eval : for t y augment Stem 31 line :

x X m x - H Cvl m 1 m 0.5 - 2 2 mat m 0.75 m 14 green 1 5 mat x Convol 6 1 sys

0.225 Convol 0.241408

1 time t0 - min 3.5

Part_2: Convolution Laser 560 TOPO/HDA NC'

A B W xo x Convol 12 / A * 114 / 8 x xo - (* 2 ln * B W 2^* - (W 2 ln sqrt * / * erf + (* 1 - 16 / B * 16 x xo - (* 2 ln * B W 2^* - (2 ln / * exp * :

x Φ x Heaviside :

A B xo x g A x xo - (Φ * B x xo - (* - exp * :

x c 11 - x ≤ (x 5 ≤ (& cases :

'd' is a constant wrt 'W'
Observe influence on plot
[W,1 ...W,0.5]wrt 'd' h 1 : W 0.5 : xo 0 : 13 mat d 22 ln sqrt * W π sqrt * / : 21 sys

play 'w' 0, 0.125, 0.25 w 0.0625 : xo 0 : 12 mat

red is strictly Φ(x) blue is pregnant of Φ(x) ... play 'w' At this point, on the account [h,d,w,xo], we can expect Convol(h,d,w,xo,x) to fit a data set similar to Laser 560 TOPO/HDA NC'

h d w xo x Convol x c * h d xo x g x c * 2 1 sys

1D simplex Emeritus

plot(data,char,size,clr)

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Romberg Simplex 1D kernel

k gencm1 N 2 k matrix : i 1 k range N 1 i el i 1 - : N 2 i el k i - : 21 line for N 31 line :

nr nc v genm M nr nc matrix : i 1 nr range j 1 nc range M i j el v : for for M 31 line :

11 gencm1 012345678910109876543210211 mat

330 genm 00000000033 mat

Carlos added,
enables recursive N 2 k matrix ≡ M nr nc matrix ≡ 21 sys

M idx M transpose 1 col rows :

5 gencm1 idx 5

Carlos  'submat' M 1 n 1 + (range 2 w range el M 1 n 1 + (2 w submatrix ≡

330 genm idx 3

without submatrix(...) n k gencm n 1 ≡ k gencm1 M n 1 + 10 genm : i 0 k range N n 1 - k 1 + (i - gencm : w N idx 1 - : O 1 w 1 + i genm : N O N stack : M M N augment : 51 line for w M idx 0 - : M 1 n 1 + (range 2 w range el 41 line if 11 line :

15 gencm 012344321025 mat

Only true for n=2 and any 'k'

24 gencm 11 mat 000001111222334012340123012010432103210210100315 mat 11 mat

weights(n,s)is universal for any dimension n s weights d 2 s * 1 + : i 0 s range w i 1 + el 1 - (i^22 s * -^* d n + 2 i * - (d^i! (d n + i - (! (* / * : for w 31 line :

Simplex integration algorithm f n s ∫ v 0 : Pt n s pts : w n s 1 - weights : i 1 w rows range P Pt i el : v v w i el P j col f j 1 P idx sum * + : 21 line for v 51 line :

n s pts d 2 s * 1 + : i 0 s range M n s 1 + (i - gencm : M M 2 n 1 + 1 M idx submatrix : Pt i el 2 M * 1 + d n + 2 i * - / : 31 line for Pt 31 line :

Specific for 1D

Recast the simplex domain

Simplex 1D integrator  ... domain [a,b] ... 's' convergence f a b s ∫ u b a - (34 / 2 x 1 el * 1 - (* 14 / 2 x 1 el * 1 - (3^* - 12 / + (* a + : x f u f 12 x 1 el * 1 - (2^- (* : simplex 32 / b a - (* f 1 s ∫ * : simplex 41 line :

x f x - exp x sqrt / :

106 -^f 1000

a 1012 -^: b 0.1 : 12 mat

x c 1 a x ≤ (x b ≤ (& cases :

The merit of simplex is NOT arguable though it needs be explored ... below

x f x c * b 50 b 50 - 2 2 mat 2 1 sys

whatever [acc] ... can't integrate
Adaptive MCD ... 1.49364826562485
[Wolfram α]    ... 1.49364826562485 x f x a b int 167.251

sanity 12 D f 0111 ∫ 1.493648265624

π sqrt 1 erf * 1.49364826562485

Explore f 018 ∫ f 019 ∫ f 0110 ∫ f 0111 ∫ f 0112 ∫ f 0113 ∫ f 0114 ∫ 71 mat : 1.49364827281499 1.49364826611909 1.49364826565873 1.49364826562379 1.49364826563812 1.49364826566673 1.49364826557079 71 mat

Simplex reverses convergence as 's' increases. Once you have determined the direction wrt f(x) use the reported best 's' ... here 's=11'

Search: 's' reverse convergence s k 6 : s k 15 range f 01 s ∫ f 01 s 1 - ∫ ≥ index k el s : k k 1 + : 21 line k if for s k 1 + : 31 line :

s 11

sanity ...  1.49364826562485 ... 12 D f 0111 ∫ 1.49364826562379

π sqrt 1 erf * f 0111 ∫ - 11012 -^*

1D Legendre ...

Legendre wr 63 ... 64

wr63 0.0494723666239310 0.0000000000000000 0.0494118330399182 0.0494521871161596 - 0.0494118330399182 0.0494521871161596 0.0492303804237476 0.0987833564469453 - 0.0492303804237476 0.0987833564469453 0.0489284528205120 0.1478727863578720 - 0.0489284528205120 0.1478727863578720 0.0485067890978838 0.1966003467915067 - 0.0485067890978838 0.1966003467915067 0.0479664211379951 0.2448467932459534 - 0.0479664211379951 0.2448467932459534 0.0473086713122689 0.2924940585862514 - 0.0473086713122689 0.2924940585862514 0.0465351492453837 0.3394255419745844 - 0.0465351492453837 0.3394255419745844 0.0456477478762926 0.3855263942122479 - 0.0456477478762926 0.3855263942122479 0.0446486388259414 0.4306837987951116 - 0.0446486388259414 0.4306837987951116 0.0435402670830276 0.4747872479948044 - 0.0435402670830276 0.4747872479948044 0.0423253450208158 0.5177288132900333 - 0.0423253450208158 0.5177288132900333 0.0410068457596664 0.5594034094862850 - 0.0410068457596664 0.5594034094862850 0.0395879958915441 0.5997090518776252 - 0.0395879958915441 0.5997090518776252 0.0380722675843496 0.6385471058213654 - 0.0380722675843496 0.6385471058213654 0.0364633700854573 0.6758225281149861 - 0.0364633700854573 0.6758225281149861 0.0347652406453559 0.7114440995848458 - 0.0347652406453559 0.7114440995848458 0.0329820348837793 0.7453246483178474 - 0.0329820348837793 0.7453246483178474 0.0311181166222198 0.7773812629903724 - 0.0311181166222198 0.7773812629903724 0.0291780472082805 0.8075354957734567 - 0.0291780472082805 0.8075354957734567 0.0271665743590979 0.8357135543195029 - 0.0271665743590979 0.8357135543195029 0.0250886205533450 0.8618464823641238 - 0.0250886205533450 0.8618464823641238 0.0229492710048899 0.8858703285078534 - 0.0229492710048899 0.8858703285078534 0.0207537612580391 0.9077263027785316 - 0.0207537612580391 0.9077263027785316 0.0185074644601613 0.9273609206218432 - 0.0185074644601613 0.9273609206218432 0.0162158784103383 0.9447261340410098 - 0.0162158784103383 0.9447261340410098 0.0138846126161156 0.9597794497589419 - 0.0138846126161156 0.9597794497589419 0.0115193760768800 0.9724840346975701 - 0.0115193760768800 0.9724840346975701 0.0091259686763267 0.9828088105937273 - 0.0091259686763267 0.9828088105937273 0.0067102917659601 0.9907285468921895 - 0.0067102917659601 0.9907285468921895 0.0042785083468638 0.9962240127779701 - 0.0042785083468638 0.9962240127779701 0.0018398745955771 0.9992829840291237 - 0.0018398745955771 0.9992829840291237 632 mat :

wr64 0.0486909570091397 0.0243502926634244 - 0.0486909570091397 0.0243502926634244 0.0485754674415034 0.0729931217877990 - 0.0485754674415034 0.0729931217877990 0.0483447622348030 0.1214628192961206 - 0.0483447622348030 0.1214628192961206 0.0479993885964583 0.1696444204239928 - 0.0479993885964583 0.1696444204239928 0.0475401657148303 0.2174236437400071 - 0.0475401657148303 0.2174236437400071 0.0469681828162100 0.2646871622087674 - 0.0469681828162100 0.2646871622087674 0.0462847965813144 0.3113228719902110 - 0.0462847965813144 0.3113228719902110 0.0454916279274181 0.3572201583376681 - 0.0454916279274181 0.3572201583376681 0.0445905581637566 0.4022701579639916 - 0.0445905581637566 0.4022701579639916 0.0435837245293235 0.4463660172534641 - 0.0435837245293235 0.4463660172534641 0.0424735151236536 0.4894031457070530 - 0.0424735151236536 0.4894031457070530 0.0412625632426235 0.5312794640198946 - 0.0412625632426235 0.5312794640198946 0.0399537411327203 0.5718956462026340 - 0.0399537411327203 0.5718956462026340 0.0385501531786156 0.6111553551723933 - 0.0385501531786156 0.6111553551723933 0.0370551285402400 0.6489654712546573 - 0.0370551285402400 0.6489654712546573 0.0354722132568824 0.6852363130542333 - 0.0354722132568824 0.6852363130542333 0.0338051618371416 0.7198818501716109 - 0.0338051618371416 0.7198818501716109 0.0320579283548516 0.7528199072605319 - 0.0320579283548516 0.7528199072605319 0.0302346570724025 0.7839723589433414 - 0.0302346570724025 0.7839723589433414 0.0283396726142595 0.8132653151227975 - 0.0283396726142595 0.8132653151227975 0.0263774697150547 0.8406292962525803 - 0.0263774697150547 0.8406292962525803 0.0243527025687109 0.8659993981540928 - 0.0243527025687109 0.8659993981540928 0.0222701738083833 0.8893154459951141 - 0.0222701738083833 0.8893154459951141 0.0201348231535302 0.9105221370785028 - 0.0201348231535302 0.9105221370785028 0.0179517157756973 0.9295691721319396 - 0.0179517157756973 0.9295691721319396 0.0157260304760247 0.9464113748584028 - 0.0157260304760247 0.9464113748584028 0.0134630478967186 0.9610087996520538 - 0.0134630478967186 0.9610087996520538 0.0111681394601311 0.9733268277899110 - 0.0111681394601311 0.9733268277899110 0.0088467598263639 0.9833362538846260 - 0.0088467598263639 0.9833362538846260 0.0065044579689784 0.9910133714767443 - 0.0065044579689784 0.9910133714767443 0.0041470332605625 0.9963401167719553 - 0.0041470332605625 0.9963401167719553 0.0017832807216964 0.9993050417357722 - 0.0017832807216964 0.9993050417357722 642 mat :

This page was compiled by Mike "Pomax" Kamermans, on June 5th, 2011, as a resource for the primer on Bezier curves, specifically the section that deals with arc length computation. You are free to do what you like with this data. Go make the world a better place than it was when you found it.

select wr63/wr64

x f 4 x 2^1 + / :

Σalgo(lo,mesh,hi,∫) ... poor candidate for this application

x f

w wr63 1 col : (weight, wi ≡ x wr63 2 col : (abscissea,xi ≡ a 0 : b 1 : 12 mat m wr63 rows : 41 line

y i 1 m range y i el b a - 2 / (x i el * b a + 2 / (+ : for y 21 line :

a 1 - : b 0 : 12 mat wr64 ≡

a 0 : b 1 : 12 mat wr63 ≡

t0 1 time :

Area b a - 2 / (w i el y i el f * (i 1 m sum eval * π - : 21015 -^*

1 time t0 - 3.1 s *

f x i N Clear 1

The Gram-Schmidt orthogonal polynomials

style 3 ... composite algo

x r f N S Clear 1

x r Lagrange x r k el - (k 1 r rows product 11 line :

kernel algo, stand alone,collapse Utilities N S i 1 N range f i el x i 1 -^: for S 1 el f 1 el : i 2 f rows range d 0 : a 0 : 12 mat j 1 i 1 - (range a f i el S j el * x 1 - 1 int S j el S j el * x 1 - 1 int / 12 round : d d a S j el * - : 21 line for S i el f i el d + (: 31 line for S 41 line :

command line N 8 :

x Schmidt N S :

evaluate
isolate r x Schmidt N el x 1 - 1 solve :

isolate 'r'

r 0.9491 - 0.7415 - 0.4058 - 0 0.4058 0.7415 0.9491 71 mat :

R i 1 r rows range pts i el . : for i 1 r rows range zero i el 0 : for r zero pts augment 31 line :

R x r Lagrange 2 1 sys

x r Lagrange x 1 - 1 int 11019 -^*

can't be vectorized m n * x 1 - 1 int m m * x 1 - 1 int /

round(▄ , n) 
has vectorizing effect m n * x 1 - 1 int m m * x 1 - 1 int / 12 round

data

data 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 :

. . . successive discrete convolution-derivatives

]]>

data rows 101

Discrete derivative by convolution ... [-1 point at each end]. data scale convol vx data 1 col : vy data 2 col : 12 mat f 1 - 0113 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 :

scale 1 - :

sanity vs Symbolic Derivative_1 data 3 - convol : Derivative_2 Derivative_1 scale convol : Derivative_3 Derivative_2 scale convol : 31 sys

Derivative_1 data 2 1 sys Derivative_2 Derivative_3

t Clear 1

Tools ► Options ► acc[100]

Fresnel integrals

3 appVersion 0.98.6179 ≡

π 2 / t 2^* cos t int maple t FresnelC

They arise in the description of far field Fresnel diffraction phenomena, and are defined through the following integral representations: The simultaneous parametric plot of S(x) and C(x) is the Euler spiral (also known as the Cornu spiral or clothoid). Clothoid and arcs of clothoid have applications in the construction of railway, freeway and roller coaster. Im this Smath document, we propose an exact view and export of these two integrals as well as the Cornu spiral as you drive your car in advanced maths style.

π 2 / t 2^* sin t int maple t FresnelS

x FC π 2 / t 2^* cos t 0 x int :

x FS π 2 / t 2^* sin t 0 x int :

C x 1.5 π * - 1.5 π * (1.5 π * - 0.1 + (range : i 1 x rows range u i el x i el : F i el x i el FC eval : 21 line for u F augment 31 line :

S x 1.5 π * - 1.5 π * (1.5 π * - 0.1 + (range : i 1 x rows range u i el x i el : F i el x i el FS eval : 21 line for u F augment 31 line :

FresnelC

FresnelS

Cornu s π : x s - s 0.99 s * - range : i 1 x rows range Uf i el x i el FS eval : Vf i el x i el FC eval : 21 line for Uf Vf augment 41 line :

box 0.714 - 0.78 0.714 0.78 0.714 0.78 - 0.714 - 0.78 - 0.714 - 0.78 52 mat :

S 0 :

Cornu spiral [clothoid]

Cornu box 0.5 0.5 + 50 black 1 5 mat 3 1 sys Approximation UT

Approximation

http://www.dgp.toronto.edu/~mccrae/projects/clothoid/sbim2008mccrae.pdf

t R 0.506 t * 1 + 1.79 t 2^* 2.054 t * + 2 sqrt + / :

t A 1 0.803 t 3^* 1.886 t 2^* + 2.524 t * + 2 + / :

t C 12 / t R 12 / π * t A t 2^- (* sin * - :

t S 12 / t R 12 / π * t A t 2^- (* cos * - :

x 0 ≥ x C Ignored if x 0 ≥ x S Ignored if lookup(value,table,col)

value table col lookup k 1 : j k table rows range table j col el value ≡ index k el j : k k 1 + : 21 line continue if for k 1 ≡ index NO match : index if 31 line :

clothoid t 0 π 0.01 range : i 1 t rows range Ufr i el t i el C eval : Vfr i el t i el S eval : 21 line for Vfr Ufr augment 31 line :

clothoid 0.4381 0.7796 + 40 red 1 5 mat 2 1 sys

ROW clothoid 2 col Max clothoid 2 lookup : 10111 mat

clothoid ROW row 0.4381 0.7796 12 mat

Project notes:

For high accuracy of point compute FC(x) FS(x) set the numerical integrator at higher accuracy than the default. This work sheet is in default accuracy 100.

Formule approximative de la clothoïde

b 2 sqrt :

convenient 'N' N 10 :

x f 1 b / 1 - (n^x 4 n * 3 +^* 2 n * 1 + (! (4 n * 3 + (* b 4 n * 2 +^* / (n 0 N sum * 11 line :

x g 1 b / 1 - (n^x 4 n * 1 +^* 2 n * (! (4 n * 1 + (* b 4 n *^* / (n 0 N sum * 11 line :

Clothoid t 0 π sqrt 0.01 range : i 1 t rows range Ucl i el t i el f eval : Vcl i el t i el g eval : 21 line for Ucl Vcl augment 31 line :

Clothoid Clothoid :

user u 1.25 : u f u g 13 mat : 1.25 0.2203 0.8314 13 mat

Clothoïde: highway access ramp

Clothoid 1 user 1 2 el user 1 3 el + 20 black 1 5 mat 3 1 sys

n Clothoid rows : 178

Clothoid n row 0.5475 0.9774 12 mat

data Clothoid :

start 2 ... escape divide by 0 data data 2 data rows range 12 range el :

Finite differences derivative X data 1 col : Y data 2 col : 12 mat 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 1 - el : D X deriv augment : Arc integrand .................... xx D 1 col : δx D 2 col : 12 mat i 1 δx rows range yy i el 1 δx i el (2^+ sqrt eval : 11 line for Arc xx yy augment : Arc length ....................... vx Arc 1 col : vy Arc 2 col : 12 mat S 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : S i el S i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for Σ vx S augment : 181 line

Derivative & Arc integrand

D Arc 2 1 sys

Cumulative arc length

Σ data 2 1 sys

km 1 :

τ 1178 range :

Clothoid rows 178

t X Clothoid 1 col t el : t Y Clothoid 2 col t el : 12 mat

t 1 ≡ path t X t Y 1 2 mat : path path t X t Y 1 2 mat stack : if 0.9774 0.5475 0.9774 . 14 black 1 5 mat 3 1 sys

Σ Σ rows row 0.5475 1.2822 12 mat

ClothoidLength 1.2822 :

ClothoidLength km 1.2822

REMEMBER ... Riemann FD integrator is almost insensible to noisy data