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[Maxima-bugs] [ maxima-Bugs-2184396 ] Wrong factorization of sqrt()
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From: SourceForge.net <no...@so...> - 2009-10-07 22:10:55
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Bugs item #2184396, was opened at 2008-10-21 15:05 Message generated for change (Comment added) made by crategus You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2184396&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. Category: None Group: None Status: Open Resolution: None Priority: 5 Private: No Submitted By: Satoshi Adachi (satoshi_adachi) Assigned to: Nobody/Anonymous (nobody) Summary: Wrong factorization of sqrt() Initial Comment: Dear Developers of Maxima, I found a wrong behavihor of sqrt(). It is a principle that sqrt(X*Y) should not be factorized to sqrt(X)*sqrt(Y) unless X and Y can be judged to be non-negative. I found a case in which this principle is broken. Here is a program for demonstration: ------------------------------------------------------------------------------ /* * wrong_factorization_of_sqrt.maxima: * * S.Adachi 2008/10/01 */ display2d:false; /* real for 1 <= x <= 2 */ correct:sqrt((1-1/x)*(2/x-1)); /* real for 2-sqrt(2) <= x <= 2+sqrt(2) */ INCORRECT:sqrt((1-(2-sqrt(2))/x)*((2+sqrt(2))/x-1)); INCORRECT_AT_2:at(INCORRECT,[x=2]); SHOULD_BE_POSITIVE:float(INCORRECT_AT_2); /* END */ ------------------------------------------------------------------------------ The result of execution is as follows: ------------------------------------------------------------------------------ Maxima 5.14.0cvs http://maxima.sourceforge.net Using Lisp GNU Common Lisp (GCL) GCL 2.6.7 (aka GCL) Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. The function bug_report() provides bug reporting information. (%i1) batch(wrong_factorization_of_sqrt.maxima) batching #p/Volumes/HFS+2/home/adachi/work/307/wrong_factorization_of_sqrt.maxima (%i2) display2d : false (%o2) false (%i3) correct:sqrt((1-1/x)*(2/x-1)) (%o3) sqrt((1-1/x)*(2/x-1)) (%i4) INCORRECT:sqrt((1-(2-sqrt(2))/x)*((sqrt(2)+2)/x-1)) (%o4) sqrt((2-sqrt(2))/x-1)*sqrt(1-(sqrt(2)+2)/x) (%i5) INCORRECT_AT_2:at(INCORRECT,[x = 2]) (%o5) sqrt((2-sqrt(2))/2-1)*sqrt(1-(sqrt(2)+2)/2) (%i6) SHOULD_BE_POSITIVE:float(INCORRECT_AT_2) (%o6) -0.70710678118655 (%o7) "wrong_factorization_of_sqrt.maxima" ------------------------------------------------------------------------------ At first, as is seen from (%i2) and (%o2), sqrt(1-1/x)*(2/x-1)) is not factorized to sqrt(...)*sqrt(...) in any automatic way. This is the correct behavihor. Next, as is seen from (%i3) and (%o3), sqrt(1-(2-sqrt(2))/x)*((2+sqrt(2))/x-1)) is factorized to sqrt((2-sqrt(2))/x-1)*sqrt(1-(sqrt(2)+2)/x) in an automatic way. THIS IS AN INCORRECT BEHAVIOR. This expression is non-negative for 2-sqrt(2) <= x <= 2+sqrt(2). However, as is demonstrated by (%i4)-(%o5), the calculated expression takes a negative value at x=2. Please stop the incorrect factorization of sqrt(X*Y) to sqrt(X)*sqrt(Y). Sincerely yours, Satoshi Adachi ---------------------------------------------------------------------- >Comment By: Dieter Kaiser (crategus) Date: 2009-10-08 00:10 Message: Maxima can simplify the sqrt function correctly. No simplification, when the sign of a is not known: (%i1) sqrt(a*z); (%o1) sqrt(a*z) Simplification for a a positive or b a negative value: (%i2) assume(a>0)$ (%i3) sqrt(a*z); (%o3) sqrt(a)*sqrt(z) (%i4) assume(b<0)$ (%i5) sqrt(b*z); (%o5) sqrt(-b)*sqrt(-z) The problem of this bug report is one of the factors of the example. The sign is wrongly determined to be negative: (%i8) sign(1-(2-sqrt(2))/x); (%o8) neg Therefore Maxima does the above simplification, which is wrong. Dieter Kaiser ---------------------------------------------------------------------- Comment By: boud (boud1) Date: 2009-02-19 03:31 Message: There's another bug related to this one IMHO: [ 2202764 ] Taylor series of sqrt(1+x-y) http://sourceforge.net/tracker/index.php?func=detail&aid=2202764&group_id=4933&atid=104933 [This is why i came here - i had a Taylor series bug that i think is equivalent.] HACK SOLUTION: The bug is solved for me (maxima 5.10.0, debian-etch) by setting radexpand : false; ANALYSIS: However, i'm not convinced that this is a sufficient response to the bug. i don't really know maxima well enough to know what would be most reasonable. (1) Set radexpand to false by default? And add more warnings in the documentation (or do we expect users to learn mathematics elsewhere than in software documentation?) (2) Tell maxima not to rewrite sqrt(X) as %i sqrt(-X) when domain:real ? (3) Extend the "is X positive, negative or zero?" question to the cases where it still needs to be asked but so far does not get asked, i.e. when deciding whether sqrt(X*Y) = sqrt(-X)*sqrt(-Y) or sqrt(X*Y) = - sqrt(-X)*sqrt(-Y) ? (4) Decide not to invert the order of an expression inside sqrt( ... ) if domain:real, except when it's sure that the expression is positive? ---------------------------------------------------------------------- You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2184396&group_id=4933 |
