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Dropping higher order terms
*To*: mathgroup at smc.vnet.net
*Subject*: [mg4121] Dropping higher order terms
*From*: Paul Abbott <paul at earwax.pd.uwa.edu.au>
*Date*: Wed, 5 Jun 1996 01:38:53 -0400
*Organization*: University of Western Australia
*Sender*: owner-wri-mathgroup at wolfram.com
Regarding the question:
I want to be able to drop higher order terms in an algebraic
expression. As a simple example consider two matrices
(mat1 = {{-1,a,b},{a,1,0},{b,0,1}}) // MatrixForm
-1 a b
a 1 0
b 0 1
(mat2 = {{1,0,c},{0,1,d},{c,d,1}}) // MatrixForm
1 0 c
0 1 d
c d 1
where a, b, c, and d are small. The matrix product
(prod = mat1 . mat2) // MatrixForm
-1 + b c a + b d b - c + a d
a 1 a c + d
b + c d 1 + b c
contains terms of first and second order in a, b, c, and d. How can I
retain only the terms first order in in a, b, c, and d?
A: Looking at
FullForm[prod]
List[List[Plus[-1, Times[b, c]], Plus[a, Times[b, d]],
Plus[b, Times[-1, c], Times[a, d]]],
List[a, 1, Plus[Times[a, c], d]],
List[Plus[b, c], d, Plus[1, Times[b, c]]]]
reveals that the terms you want to delete, say Times[b,c], are of the
general form Times[__]. However, you do not want to delete terms like
Times[-1,c] so you can use pattern-matching with
Times[_Symbol,_Symbol..], (i.e., a product of two or more Mathematica
symbols. The repetition operator .. indicates a pattern repeated one or
more times) to retain only the terms first order in in a, b, c, and d:
(prod /. Times[_Symbol,_Symbol..] :> 0) // MatrixForm
-1 a b - c
a 1 d
b + c d 1
Here is a more complicated example:
(mat1 . mat2 . mat1 // Expand) /.
Times[_Symbol,_Symbol..] :> 0
2 2 2 2
{{1 + a + b , 0, -c + b c}, {0, 1 + a , d},
2 2
{-c + b c, d, 1 + b }}
You now want to delete all terms of the general form _^_ (i.e.,
powers). You can again use pattern-matching:
(% /. _^_ :> 0) // MatrixForm
1 0 -c
0 1 d
-c d 1
Combining these rules we have:
firstorder = {Times[_Symbol,_Symbol..] :> 0, _^_ :> 0};
(mat1 . mat2 . mat1 . mat1 . mat2 // Expand) /.
firstorder // MatrixForm
-1 a b - 2 c
a 1 2 d
b + 2 c 2 d 1
_________________________________________________________________
Paul Abbott
Department of Physics Phone: +61-9-380-2734
The University of Western Australia Fax: +61-9-380-1014
Nedlands WA 6907 paul at physics.uwa.edu.au
AUSTRALIA http://www.pd.uwa.edu.au/Paul
_________________________________________________________________
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