f22bf242-15b2-468b-9dd6-1955b1df7b5c 29 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 :

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

a 109 -^: b 0.1 : c 0.25 : 13 mat

a f 31622.7766

b a - (a b + 2 / f * b a - 6 / 2 a * b + 3 / f a 2 b * + 3 / f + (* 21 mat 0.425403 0.148681 21 mat

x f

Observe: @ 1 000 000 points Simpson(f,a,b,n)

1. Example 1 consumes 50 min

2. Example 2 consumes 14 min

. . . . . . . . Explain . . . . . .

exp(x) & ln(x) are both approximations that consumes same.

Example 1 consumes the additional sqrt(x), greedy in the

low range. All numerical functions are approximations.

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f a b n Simpson h b a - n / : 13 / h 2 / a f b f + (* h a k h * + f k 1 n 1 - sum * + (* 23 / h * a h 2 / - k h * + f k 1 n sum * + eval 21 line :

t0 1 time :

50 min S f a b 1000000 Simpson : #

1 time t0 - min 0.5

XY 2.5 0.6313 5 0.6213 10 0.6164 20 0.614 30 0.6132 40 0.6129 50 0.6126 100 0.6122 82 mat :

X XY 1 col : Y XY 2 col : x predict X Y x ainterp : xy X Y . 14 red plotG : 41 line

Simpson(f,a,b,n) converges wrt 'n'

x predict xy 2 1 sys

relative error ~ 0.0003 1 0.6122 0.611991 / - 0.0003 -

Question/Answer:

1. The question is whether the iterative Simpson converges truly ?

2. For this example => Answer is YES. An infinitesimal analysis

will reveal that the smaller 'dx' => closer it tends to true value ...

for this particular example, as demonstrated. For other applications

one may check 1rst the convergence via Simpson(f,a,b,n).

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x 0.1 :

sanity vs Mathematica π sqrt x sqrt erf * 0.611991

http://www.wolframalpha.com/input/?i=integrate+x%5E(0.5-1)*exp(-x)+from+0+to+0.1

ε 11015 -^- : 0.999999999999999

x f 1 x - ln - x / :

a 1021 -^: b ε : 12 mat

x f

sanity Wolfram  [0..1] 1.6449340668 x f x a b int 1.645286

t0 1 time :

14 min S f a b 1000000 Simpson : 1.644937

1 time t0 - min 0

http://www.wolframalpha.com/input/?i=integrate+-(ln(1-x))%2Fx+from+0+to+1

x f 50 sqrt 50 π * x 2^* - exp * :

Watch where you thread !
... x_not defined ...
f(2.1) > max allowed x f x 0 2.1 int #

x f

x f x 02 int x f x 0 0.5 int 21 mat 0.500000000000002 0.499999999999994 21 mat

0 f 0.5 f 1 f 2 f 41 mat 7.07106781186547 6.23419782289092 1017 -^* 4.27235478366182 1068 -^* 9.42354337322279 10273 -^* 41 mat