HP Forum Archive 21

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New variant for the Romberg Integration Method Message #1 Posted by Namir on 18 Apr 2012, 12:48 a.m. 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.

Re: New variant for the Romberg Integration Method Message #2 Posted by Matt Agajanian on 18 Apr 2012, 12:50 a.m., in response to message #1 by Namir Thanks. This should be intriguing

Re: New variant for the Romberg Integration Method Message #3 Posted by Nick_S on 18 Apr 2012, 5:17 a.m., in response to message #1 by Namir 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.

Re: New variant for the Romberg Integration Method Message #4 Posted by Namir on 18 Apr 2012, 7:16 a.m., in response to message #3 by Nick_S 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

Re: New variant for the Romberg Integration Method Message #5 Posted by Valentin Albillo on 18 Apr 2012, 7:24 a.m., in response to message #4 by Namir 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.

Re: New variant for the Romberg Integration Method Message #6 Posted by Namir on 18 Apr 2012, 7:31 a.m., in response to message #5 by Valentin Albillo Right you are! And I learned a new function name. Alpha Worlfram recognized the sinc(x) funcion!! :=)

Re: New variant for the Romberg Integration Method Message #7 Posted by Paul Dale on 18 Apr 2012, 7:39 a.m., in response to message #6 by Namir So does the 34S :-) - Pauli

Re: New variant for the Romberg Integration Method Message #8 Posted by Valentin Albillo on 18 Apr 2012, 9:45 a.m., in response to message #6 by Namir 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: I1 = Integral( 0, Infinity, sinc(x) dx) = Pi/2 I2 = Integral( 0, Infinity, sinc(x)*sinc(x/3) dx) = Pi/2 I3 = Integral( 0, Infinity, sinc(x)*sinc(x/3)*sinc(x/5) dx) = Pi/2 I4 = Integral( 0, Infinity, sinc(x)*sinc(x/3)*sinc(x/5)*sinc(x/7) dx) = Pi/2 I5 = Integral( 0, Infinity, sinc(x)*sinc(x/3)*sinc(x/5)*sinc(x/7)*sinc(x/9) = Pi/2 I6 = Integral( 0, Infinity, sinc(x)*sinc(x/3)*sinc(x/5)*sinc(x/7)*sinc(x/9)*sinc(x/11) = Pi/2 I7 = 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 I8 = 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.

Re: New variant for the Romberg Integration Method Message #9 Posted by Nick_S on 18 Apr 2012, 7:47 a.m., in response to message #5 by Valentin Albillo 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

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