04-18-2012, 12:48 AM

Hi All,

I just posted, on my web site, the following article for a new variant for the Romberg method. The article actually looks at several variants and selects the best one.

Enjoy!!

Namir

*Edited: 18 Apr 2012, 12:49 a.m. *

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04-18-2012, 12:48 AM

Hi All,

I just posted, on my web site, the following article for a new variant for the Romberg method. The article actually looks at several variants and selects the best one.

Enjoy!!

Namir

*Edited: 18 Apr 2012, 12:49 a.m. *

04-18-2012, 12:50 AM

Thanks. This should be intriguing

04-18-2012, 05:17 AM

Using both the HP-15c and Wolfram Alpha I get values that differ from yours for these examples:

ln(x)/x integrate 1 to 100 = 10.60378

x in radians

sin(x) integrate 1e-10 to pi/4 = 0.2928932

Nick

*Edited: 18 Apr 2012, 5:27 a.m. *

04-18-2012, 07:16 AM

Nick,

Thanks for the corrections. I n the case of the sin(x), I meant sin(x)/x. I posted the article with the corrected results.

Namir

*Edited: 18 Apr 2012, 7:29 a.m. after one or more responses were posted*

04-18-2012, 07:24 AM

Quote:

Nick,Thanks for the corrections. I n the case of the sin(x), I meant sin(x)/x. I should posted a corrected article very shortly.

Namir

The standard name for the sin(x)/x function is **sinc(x)**, an abbreviation of *sinus cardinalis* (i.e.: cardinal sine).

Regards from V.

04-18-2012, 07:31 AM

Right you are! And I learned a new function name. Alpha Worlfram recognized the sinc(x) funcion!!

:=)

04-18-2012, 07:39 AM

So does the 34S :-)

- Pauli

04-18-2012, 07:47 AM

This brings back memories as the sinc function was one of the first things I plotted as a teenager on my newly acquired Sinclair ZX81 computer.

Nick

04-18-2012, 09:45 AM

Quote:

Right you are! And I learned a new function name. Alpha Worlfram recognized the sinc(x) funcion!!:=)

I'm glad you did, I also learn new things each and every day.

About the **sinc(x)** function, it has many interesting properties and quirks but the one

I find most uncanny is this: a little computation or theoretical work will quickly stablish the following results:

- I
_{1}= Integral( 0, Infinity, sinc(x) dx) =**Pi/2** - I
_{2}= Integral( 0, Infinity, sinc(x)*sinc(x/3) dx) =**Pi/2** - I
_{3}= Integral( 0, Infinity, sinc(x)*sinc(x/3)*sinc(x/5) dx) =**Pi/2** - I
_{4}= Integral( 0, Infinity, sinc(x)*sinc(x/3)*sinc(x/5)*sinc(x/7) dx) =**Pi/2** - I
_{5}= Integral( 0, Infinity, sinc(x)*sinc(x/3)*sinc(x/5)*sinc(x/7)*sinc(x/9) =**Pi/2** - I
_{6}= Integral( 0, Infinity, sinc(x)*sinc(x/3)*sinc(x/5)*sinc(x/7)*sinc(x/9)*sinc(x/11) =**Pi/2** - I
_{7}= Integral( 0, Infinity, sinc(x)*sinc(x/3)*sinc(x/5)*sinc(x/7)*sinc(x/9)*sinc(x/11)*sinc(x/13) =**Pi/2**

but lo and behold, we unexpectedly find that

- I
_{8}= Integral( 0, Infinity, sinc(x)*sinc(x/3)*sinc(x/5)*sinc(x/7)*sinc(x/9)*sinc(x/11)*sinc(x/13)*sinc(x/15) =**Pi/**!!__2.0000000000294+__

You might want to check this amazing fact by trying and computing said integrals I1, I2, ..., I8 using the 34S' extreme precision capabilities, it would be a fine test for any numerical integration procedure such as yours ! ... XD

Best regards from V.

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