clearing denominators and eliminating square roots in *huge*
in [Mathematica]
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From: dantimatter on 4 Mar 2010 05:28 Hello Community of Mathematica Users, My collaborators and I are having a nasty problem that I'm hoping you might be able to help us move beyond: After spending quite a few hours Solve[]-ing an equation, Mathematica returns an extremely large solution where all the terms contain complicated denominators and square roots. We've set the solution equal to zero and would like to now simplify it by getting rid of all the denominators (e.g., by multiplying through by all of them) and the square roots (e.g., by multiplying by their conjugates). Is there a good way to do this? Mathematica seems to get hung up when we try to use Together[], Collect[], Simplify[], etc. Is there another function that would be better to use? Perhaps there's some other way around this problem? Any suggestions you have would be much appreciated. Many thanks, Dan
From: David Park on 5 Mar 2010 04:32 Dan, "Huge" seems daunting and without knowing more about your expression I'm not certain how much progress can be made. So I tentatively offer the following. In the Presentations package I have developed a number of routines that aid in manipulating and simplifying expressions. The various Mathematica commands are useful, but often they do too much, or too little, or are difficult to precisely control. Some of the extra "Manipulations" commands in Presentations are: CompleteTheSquare FactoOut - any factor, even if it's not in the expression. AddZero - creating a new expression by adding and subtracting the same subexpression. MultiplyByOne - Multiply and operate on Numerator and Denominator by the same factor. LinearBreakout - breakout function patterns on operand patterns PushOnto - push arguments onto selected function patterns. (Much better than Through.) CreateSubexpression - protect subexpressions with a tag for selective manipulation. ReleaseSubexpressions OperateSubexpression MapLevelParts - Map a function to a subset of level parts, for example selected terms in a larger Plus expression. The function operates on the complete group of selected terms and not on individual terms. MapLevelPatterns - same thing except that the parts are selected by a pattern and not by position. EvaluateAt - Evaluate a part of a held or ordinary expression. EvaluateAtPattern - same thing but by pattern. Two useful techniques for manipulating expressions are to: 1) Shield portions of the expression so Mathematica will treat them as a unit and not break them apart. This will often make commands like Simplify work better. You could shield with Hold or HoldForm, and in a more controlled manner with CreateSubexpression. Another method to shield might be to replace the expression by a single symbol, and then later do the reverse replacement. 2) Operate on selected portions of the expression using MapAt, ReplacePart, MapLevelParts, MapLevelPatterns, EvaluateAt, EvaluateAtPattern. Using these methods either alone or combined with the higher level Mathematica operations such as Simplify, Together and Apart can be very effective. You should also consider the use of Rules. The MultiplyByOne command might simplify your denominators by using the complex conjugate. (You could probably implement this yourself) If different terms had different denominators you could probably use MapLevelPatterns to pick out the subgroups of common terms. You could use FactorOut to get a common denominator out of a group of terms. Here is a sample case: Needs["Presentations`Master`"] expr = k1/(a1 + I b1) + k2/(a2 + I b2); (# /. term : k_/(a_ + b_ I) :> MultiplyByOne[a - b I][term]) & /@ expr % // FactorOut[1/((a1^2 + b1^2) (a2^2 + b2^2)), Factor] giving ((a1 - I b1) k1)/(a1^2 + b1^2) + ((a2 - I b2) k2)/(a2^2 + b2^2) ((a1 - I b1) (a2 - I b2) (a2 k1 + I b2 k1 + a1 k2 + I b1 k2))/((a1^2 + b1^2) (a2^2 + b2^2)) David Park djmpark(a)comcast.net http://home.comcast.net/~djmpark/ From: dantimatter [mailto:google(a)dantimatter.com] Hello Community of Mathematica Users, My collaborators and I are having a nasty problem that I'm hoping you might be able to help us move beyond: After spending quite a few hours Solve[]-ing an equation, Mathematica returns an extremely large solution where all the terms contain complicated denominators and square roots. We've set the solution equal to zero and would like to now simplify it by getting rid of all the denominators (e.g., by multiplying through by all of them) and the square roots (e.g., by multiplying by their conjugates). Is there a good way to do this? Mathematica seems to get hung up when we try to use Together[], Collect[], Simplify[], etc. Is there another function that would be better to use? Perhaps there's some other way around this problem? Any suggestions you have would be much appreciated. Many thanks, Dan
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