plotG(vx,vy,a,b,c)

08783b1b-97e6-4d97-ace5-a0c8eb1afaa1 50 4 5 decimal

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

plotG(vx,vy,a,b,c)

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 :

data n Decimate collects every 'n' points range 1 data rows n range : k 1 : i range u k el data i 1 el : v k el data i 2 el : k k 1 + : 31 line for u v augment 31 line 31 line :

Spline Utilities

Part 2. The [lpc] spline interpolation

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 :

Part 3. The complimentary scalar dspline

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 :

Part 4. The complimentary cumulative integrate

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 eval :

5. The complimentary Simpson-Romberg

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from this l,p,c spline demo f a b n ∫ 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 / : 14 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 :

Golden ratio ..... RootDichotomy

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : 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 71 line :

"RootDichotomy" ...robust companion. Φ a b ε D RootDichotomy xn b a + (2 / : n 0 : xn Φ abs ε ≥ xn Φ a Φ * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 51 line :

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"

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

L 0 : H 10 : Num 51 : 13 mat U L H Num xd : 21 sys

t0 1 time :

4 appVersion 0.98.6179.21440 ≡

x φ 11 x 5 - (2^+ / :

exact 2.746801142328240 x φ x L H int 2.74680153389003

data i 1 Num (range vy i el U i el φ eval : for U vy augment 21 line :

data

===== generalise from here =====

X data 1 col : Y data 2 col : 12 mat L X Min : H X Max : 12 mat x c 1 L x ≤ (x H ≤ (& cases : pts X Y . 9 red plotG : 41 sys

Prepare the cubic splines

This document implements the classical CAS lpc_cubic spline, c/w 1rst/2nd derivatives and the fintegrate. Smath native Simpson does not recognize the interpolation. The best of this devil is replaced by Romberg kernel ∫(f,a,b,n) lot more convenient and accurate.

n data rows : 51

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

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

Initialise the blank
diagonal matrix M 11 el 1 : M n n el 1 : 21 sys

Part 1. Extract the respective l_p_c spline 's'

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

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

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 - (/ :

Part 2, 3, 4, 5. in collapsed utilities

the respective spline coefficients s M 1 -^br * eval :

CWD C:\SmathFile CurrentDirectory :

Disable the 'cspline' tridiagonal to collect 'pspline' ... disable 'pspline & cspline' tridiagonal to collect 'lspline.

Spool to file the spline coefficcients s CWD spline wfile 1

s CWD spline rfile :

isolate Σ s X Y fintegrate eval :

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 :

l_p_c spline cumulative integrate

Σcumul 0 2 1 sys

vx Σcumul 1 col : vy Σcumul 2 col : x ∫cumul vx vy x cinterp : 31 sys

closely exact ... 5 ∫cumul 1.37340057116413

1.37340057116412

Exact value ! Σcumul Σcumul rows row 10.000000000000000 2.746801142328240 12 mat

x f s X Y x interpspline :

[10000] ► 2.74680114232826 ... Simpson ◄ 2.74680153389003 f L H 100 ∫ 2.746801

x D s X Y x dspline :

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

The scalar dspline

x plotD 0 2 1 sys

2nd derivative of [lpc] spline

X s augment 0 2 1 sys

x Spline s X Y x interpspline : x D1 s X Y x dspline : vx X : vy Σcumul 2 col : 12 mat x Σspline s vx vy x interpspline : x D2 s X s x interpspline : x u 1 L x ≤ (x H ≤ (& cases : 61 sys

◄ collect the scalar project

from dichotomy V 5.9891 : U 5.4121 : 21 sys

x Spline x u * x D1 x u * x D2 x u * 0.5 x Σspline * x u * V V Spline + 20 black 1 5 mat U U D1 + 20 black 1 5 mat 6 1 sys $V Spline V D1 V D2 U D2 4 D2 6 D2 61 mat 0.5055 0.5057 - 0.5055 0.6016 - 0.5098 0.5098 61 mat$

2nd derivative "scalar'

x D2 x u * 0 2 1 sys

x Φ x Spline x D2 - : a 5.75 : b 6.25 : ε 106 -^: 13 mat 21 sys

Sources and credits:

1. Mathsoft for the tridiagonal systems

and original dspline [Advisor].

2. Robert Adair for the algos:

...InterpSpline(s,X,Y,t)

...dspline(s,X,Y,t)

...fintegrate(s,vx,vy)

3. Smath adapted: Jean Giraud

]]>

Dichotomy algo Φ a b ε D 5.9891

x Φ x D1 x D2 - : a 5 : b 6 : ε 106 -^: 13 mat 21 sys

Dichotomy algo Φ a b ε D 5.4121

Golden ratio y1 y2 ≤ (min in range ≡ y1 y2 ≥ (max in range ≡ 21 sys

refreshed: 20210711

TOL 109 -^:

x f x D2 :

X f 3.75 4.25 y1 y2 ≥ TOL Extrema 1 el f 5.75 6.25 y1 y2 ≥ TOL Extrema 1 el 21 mat : 4621 mat

iter f 3.75 4.25 y1 y2 ≥ TOL Extrema 2 el f 5.75 6.25 y1 y2 ≥ TOL Extrema 2 el 21 mat : 424221 mat

1 time t0 - 32 s *