<p> plotG(vx,vy,a,b,c) </p>

edd16e6e-ea2e-4c9e-a5e5-730784b79ca6 51 4 5 decimal

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

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 :

x c 12 - x ≤ (x 2 ≤ (& cases :

Gaussian basis 'b-spline'

x b x c *

0 b 0.6667

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 : XY X transpose Y transpose augment : Points X transpose Y transpose . 14 black plotG : 41 line

u v Bsp v m el u m - b * (m 1 v rows sum : u ox u X transpose Bsp : u oy u Y transpose Bsp : 31 mat

h 0.001 : x f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : 21 mat

This document reconciliates => 1. Derivative(s) ... 2. ∫(x)

Smath derivative d/dx (f(x)) is a silent symbolic operator, i.e: it will derive

algebraic functions, exclusively of fake functions [conditional, spline].

Derivative(s) of order 1, 2 are rescued from infinitesimal analysis. It then

behaves as a scalar analytical algorithm . . . 'h' is user.

Unlike Smath symbolic d/dx (f(x)), Mathcad/Mathsoft d/dx (f(x)) is a

numerical-scalar operator. Thus, will take: conditional, spline(s).

The Smath ∫(z) is numerical-scalar, it behaves same as Mathcad/Mathsoft

except in advanced applications [Fourier Quantum ...].

The order of the error of the infinitesimal derivative(s) is ~ 3*h

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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 :

x Σ u b u 2 - x int :

x f x b :

cumulative integration in range

x Σ x c *

infinitesimal deriv in range

x f deriv x c *

user discrete b_spline demo

XY data Points 3 1 sys

More from Smath chained algos

x f x oy :

x f deriv 1 1 sys

x f x ox :

x f deriv 1 1 sys

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 :

Maths(x) function 1rst deriv 2nd deriv antideriv cumul∫f(x) 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

central difference x h + f x h - f - 2 h * /

data CWD b_spline rfile :

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

h 0.0001 : 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

Remember this:

The native Smath QuickPlot is 235 pixels wide.

Plotting simple functions [sin(x), exp(x) ... ] is very quick.

It evaluates the ordinates 235 times from fast Microsoft

code. Same topo for the scalar plots Σ(x), splineΣ(x) ,

i.e: 235 evaluations but the integrator is rather slow

compared to sin(x) ...

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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

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

t0 1 time :

2 splineΣ 0.0938

x splineΣ 1.875 x ≤ ( x 2.125 ≤ ( & cases 2 2 splineΣ + 20 red 1 5 mat 2 1 sys

1 time t0 - 117 s *

faster on native canvas

Reconciliate a difficult scalar integrator

Γ(0.5,x) Plots on native canvas. If you shift/stretch, MAY
NOT replot "t_undefined", i.e: integrand not compatible
with the non native canvas ... manage otherwise. α x Γ t α 1 -^t - exp * t x 100 int :

sanity 1.772 0.5 1012 -^Γ 3.3333 105^*

X 1 :

sanity 0.2788 0.5 X Γ 0.2852

Plots on native canvas

0.5 x Γ

May NOT replot on manipulated native canvas

0.5 x Γ

TENTATIVELLY: manage otherwise 1. call the blank canvas 2. stretch/shift as you would like 3. plug the plot argument, replot/adjust 4. tentativelly: shift/stretch little bit

0.5 x Γ X 0.5 X 0.5 - 2 2 mat 2 1 sys

Reconciliate accuracy of the integrator:

Export this section, set "acc 10000", Romberg increases the range of integration,

it will significantly improve most of the integration, still by far not competive

with Mathcad/Mathsoft adaptive.

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f a b Int x f x a b int 32 / b a - (* w G w 01 int * ≡ (memo ≡ 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 1015 -^1 int * 41 line :

recast Γ(α,x) α 0.5 : t f t α 1 -^t - exp * : 21 sys

sanity 1.7524 t f t 0.0001 10 int 4.7095

sanity 1.772 f 0.00001 10 Int 1.7356