d8410694-85db-4a9d-a03f-cc08b28258df 29 4 5 decimal

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

set integral accuracy 10000

This document will explore many examples. Some to the merit of "Romberg" and some w/o merit.

===================================================== Example_1

x m x ln x int maple : x 1 - x ln + (*

ε 1015 -^:

a 0 : b 1 : 12 mat

b m 1 -

x f x 1 - x ln + (* :

x m e 0 + 15 red b b m + 15 red 2 5 mat 2 1 sys

Exact => -0.75 x f x ε 1 int 0.749998074594638 -

8 decimals x f x ε e int 1.84724979780443 -

x f x 0 e int e 2^4 / - ≡ (1.84726402473266 - ≡

"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 :

t0 1 time :

Smath Romberg singular end point f a e Int 1.84726399729247 -

1 time t0 - min 0

a 0 : b e : 12 mat w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 31 line

Mathcad Adaptive 1.84726402473266 - ≡

x s f Transform ≡ 12 x * 1 - (2^- (Parabola ≡ x G Integrand ≡ 31 sys

x c 10 x ≤ (x 1 ≤ (& cases :

Romberg cubic kernel

x s x c * x s f 1 2 x * 1 - ( 2 ^ - ( x G 3 1 sys

===================================================== Example_2

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 01 int * 41 line :

Test function x f x sqrt exp :

7 true digits ... insufficient
integration subspaces. x f x 02 int 5.40744702348009

x f 0 2 1 sys

sanity check the quadruplet Mathcad 11 Romberg 5.40752818465632 ≡ Technical accuracy 5.40752818465627 ≡ Smath Romberg 5.40752818465638 ≡ Smath Integrator 5.4075281 ≡ 41 line

f 02 Int 5.40752818690417

a 0 : b 2 : 12 mat w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 31 line

x s f Transform ≡ 12 x * 1 - (2^- (Parabola ≡ x G Integrand ≡ 31 sys

x s f 1 2 x * 1 - ( 2 ^ - ( x G 3 1 sys

Explain Romberg:

1. The Romberg integrand is the product of the transform

by the parabola. It reduces the original [a, b] space to

[0, 1]. The final integration is dimensionned 3/2(b-a).

2. wrt to Smath integrator, it has the virtue of improving

the integration ... 10000 accuracy is now on smaller span.

3. The other virtue results from transforming the original

integrand toward a form more appropriate for the Smath

native Simpson integrator.

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===================================================== Example_3

x f 41 x 2^+ / :

x f

NOT a case for Romberg. The integrand
is perfectly suited for Simpson. π x f x 01 int f 01 Int 31 mat 3.14159265358979 3.14159265358979 3.14159265283962 31 mat

===================================================== Example_4

x f 2 x * cos x exp / :

x c 10 x ≤ (x 2 π * ≤ (& cases :

x f x c * 2 π * 0.5 2 π * 0.5 - 2 2 mat 2 1 sys

x f x 02 π * int f 02 π * Int 21 mat 0.199626452011026 0.199626526204812 21 mat

===================================================== Example_5

x f x - exp x / :

x c 11 x ≤ (x 100 ≤ (& cases :

x f x c *

Integrand suitable for Romberg transform. x f x 1100 int f 1100 Int 0.219383934395520 31 mat 0.22071060184138 0.219386700436576 0.21938393439552 31 mat

===================================================== Example_6

x f 11 x 2^+ / :

x c 10 x ≤ (x 3 ≤ (& cases :

x f x c *

x f x 03 int f 03 Int 21 mat 1.24904577238205 1.249045772194 21 mat