<p> Adaptive Carlos Algorithm </p>

554c3d87-4a29-4767-b2c3-7c1bda175b98 13 4 5 decimal

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

x 0.5 1 -^x - exp * x int maple #

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

indefinite integral x Curve 11 x 2^+ / x int maple : x atan

MCD 11 default adaptive ... 0.611991 x 0.5 1 -^x - exp * x 1015 -^0.1 int 53.3138

<= Wolfram Alpha

3 appVersion 0.98.6179 ≡

ε 1015 -^:

"Romberg singular" ... option integral accuracy: 10000 f a b 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^- (* : 32 / b a - (* w G w ε 1 int * 31 line :

x f x 0.5 1 -^x - exp * :

t0 1 time :

Smath Romberg singular end point f ε 0.1 Int 0.61197

1 time t0 - min 1

==================== Adaptive =====================

Test function x f x 0.5 1 -^x - exp * :

t0 1 time :

'N' iterations
precision N 50 : ε 109 -^: 21 sys

User inputs

lower limit
upper limit a 1015 -^: b 0.1 : 21 sys

Initialise the while loop

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

<p> Adaptive Carlos Algorithm </p>

The adaptive while loop i 0 > fd aa i el h i el 2 / + f : fe aa i el 3 h i el * 2 / + f : 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 / (* : 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 : 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 : h i el v5 2 / : tol i el v6 2 / : s i el s2 : L i el v8 1 + : 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 : L i el L i 1 - el : 171 line if if Solution sum : 151 line while

solution[Simpson Adaptive Quadrature] Solution 0.6119913

1 time t0 - 6 s *

MCD 0.611991366111768 ≡

============== More interesting stuff ================

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'

x b

0 b 0.6667

Can Simpson be an exact integrator ?

Answer is YES from some examples in literature.

Answer is YES also from a random example.

The b_spline is a collection of piecewise cubics.

Conclusion: ========================

Simpson Integrator ranges from poor to exact.

The Simpson efficient companion is Romberg .

Observe ln(3), exact from integration !

]]>

a specific property of b_spline. u b u 2 - 2 int 1.000000000000000

x f 1 x / :

ln(3) => 1.09861228866811 x f x 13 int 1.09861228866811

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 :

x g 0 :

b 3 :

plot x f x g x f x g 1 b 10 hatch 31 sys :

Classroom hatch demo ln(x)

plot

x f x 1 b int b ln 21 mat 1.09861228866811 1.09861228866811 21 mat