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 November 1st, 2013, 08:44 AM #1 Member   Joined: Apr 2013 Posts: 52 Thanks: 0 SOS NUMERICAL ANALYSIS I noticed at a program that is calculating the sum $?_{k=1}^{n}(\frac{1}{k^2})$,that for n<10000 the forward method is better,and for n>=10000,the backward method is better..Why does this happen????
 November 1st, 2013, 11:19 AM #2 Senior Member   Joined: Feb 2012 Posts: 628 Thanks: 1 Re: SOS NUMERICAL ANALYSIS To answer that question, we would have to know how the program is calculating the sum.
 November 1st, 2013, 03:00 PM #3 Member   Joined: Apr 2013 Posts: 52 Thanks: 0 Re: SOS NUMERICAL ANALYSIS forwards: 1+....+1/(n-1)^2+1/n^2 backwars: 1/n^2+1/(n-1)^2+...+1
 November 1st, 2013, 03:28 PM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: SOS NUMERICAL ANALYSIS What program? What (hardware) architecture? What precision (or is this done in rational arithmetic)? How do you decide "better": speed, higher accuracy, etc.?
 November 1st, 2013, 03:44 PM #5 Member   Joined: Apr 2013 Posts: 52 Thanks: 0 Re: SOS NUMERICAL ANALYSIS The program should approximate pi..Also,it uses floats.I look how near the result approximates pi,to decide which way(backward/forward) is better..
 November 1st, 2013, 06:13 PM #6 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: SOS NUMERICAL ANALYSIS Backward should give better answers then, regardless of the size. Speed shouldn't vary much.
 November 2nd, 2013, 03:48 AM #7 Member   Joined: Apr 2013 Posts: 52 Thanks: 0 Re: SOS NUMERICAL ANALYSIS But it does not give better results for n<=1000..why??
November 2nd, 2013, 03:37 PM   #8
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Re: SOS NUMERICAL ANALYSIS

Quote:
 Originally Posted by evinda But it does not give better results for n<=1000..why??
It does for me. With n = 5 and 19 digits of precision, backward gives
1.6429360655148941[color=#FF0000]70[/color]
while forward gives
1.6429360655148941[color=#FF0000]42[/color]
and the correct result is
1.6429360655148941[color=#FF0000]69[/color]802

 November 3rd, 2013, 07:54 AM #9 Member   Joined: Apr 2013 Posts: 52 Thanks: 0 Re: SOS NUMERICAL ANALYSIS Did you find the approximation of pi or of (pi^2)/6 ???
 November 3rd, 2013, 08:22 AM #10 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: SOS NUMERICAL ANALYSIS I found the sum 1 + 1/4 + 1/9 + ... + 1/n^2.

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