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MathGroup Archive 1998

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Re: Linearization

  • To: mathgroup at smc.vnet.net
  • Subject: [mg15185] Re: Linearization
  • From: "Allan Hayes" <hay at haystack.demon.co.uk>
  • Date: Fri, 18 Dec 1998 02:11:01 -0500
  • References: <754vm1$jgc@smc.vnet.net> <75a0ll$rtv@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Daniel Lichtblau emailed me

>>>

Normal[Series[...]] may be more efficient than expansion, especially in
cases where the latter might cause memory problems.

Below is the code, it is a minor modification of your version.

In[1]:= Linearize[poly_, vars_] := Module[{u},
        Normal[Series[poly /. Thread[vars -> (vars u)], {u,0,1}]] /.
u->1]

In[2]:= Linearize[1 + u + (a + b)x + x u + x^2 + u^2, {x, u}]

Out[2]= 1 + u + (a + b) x

<<<

And several other postings used this Series approach So I took another
look:

If we have to start from a general function (which was the original
question) and generate its linearization then the direct use of series
is faster and uses less memory then getting the series and then
truncating it.
However, if we are already working with polynomials in the variables it
seems that
the Expand form is quicker and uses less memory (at least on the example
that I have used):
I include two variants using Collect and CoefficientList that appear to
be faster again and also to use less memory.

Method            Using               Memory Used            
Timing/seconds LinearizeE        Expand              2744              
0.27 LinearizeS        Series                3496                      
0.55 LinearizeC        Collect                2648                     
0.11 LinearizeCL      CoefficientList      2648                        
0.11


LinearizeE[poly_, vars_] :=
  Module[{u},
    Expand[poly, Alternatives@@vars] /. Thread[vars -> (vars u)] /.
      {u^n_ -> 0, u -> 1}]

LinearizeS[poly_, vars_] :=
  Module[{u},
    Normal[Series[poly /. Thread[vars -> (vars u)], {u, 0, 1}]] /. u ->
1]

LinearizeC[poly_, vars_] :=
  Module[{u},
    Collect[poly /. Thread[vars -> (vars u)], u] /.
      {u^n_ -> 0, u -> 1}]

LinearizeCL[poly_, vars_] :=
  Module[{u},
    Plus@@Take[CoefficientList[poly /. Thread[vars -> (vars u)], u], 2]
    ]

pol =
    Collect[Normal[Series[f[x, y, z], {x, 0, 6}, {y, 0, 6}, {z, 0, 6}]],
      {x, y, z}]
  ;

m1 = MemoryInUse[];

(le = LinearizeE[pol, {x, y}]); // Timing // First

0.27 Second

m2 = MemoryInUse[]; m2 - m1

2744

(ls = LinearizeS[pol, {x, y}]); // Timing // First

0.55 Second

m3 = MemoryInUse[]; m3 - m2

3496

(lc = LinearizeC[pol, {x, y}];) // Timing // First

0.11 Second

m4 = MemoryInUse[]; m4 - m3

2648

(lcl = LinearizeCL[pol, {x, y}];) // Timing // First

0.11 Second

m5 = MemoryInUse[]; m5 - m4

2648

Check

Expand[lc] == Expand[le ] == Expand[ls] == Expand[lcl]

True

Allan,

---------------------
Allan Hayes
Mathematica Training and Consulting
www.haystack.demon.co.uk
hay at haystack.demon.co.uk
Voice: +44 (0)116 271 4198
Fax: +44 (0)870 164 0565




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