Re: Alternative to defining 'operator' function?
- To: mathgroup at smc.vnet.net
- Subject: [mg47587] Re: Alternative to defining 'operator' function?
- From: "Peter Pein" <petsie at arcor.de>
- Date: Fri, 16 Apr 2004 05:21:58 -0400 (EDT)
- References: <c5leel$bit$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
"Owen, HL (Hywel)" <H.L.Owen at dl.ac.uk> schrieb im Newsbeitrag news:c5leel$bit$1 at smc.vnet.net... > Hi folks, > > I often have programming problem where I'd like to calculate a set of dot > products, e.g. applying a list of square matrices {R1,R2,R3...} to a vector > v to obtain: > > {R1.v,R2.R1.v,R3.R2.R1.v,...} > > or other functions like that. > > The method I've been using is to define an 'operator' function, e.g. > > DotOperator[M_] := Dot[M, #] & > > Then we have: > > In: DotOperator[R][v] > Out: R.v > > as wanted, so that we can define a ComposeList as > > In: Rest[ComposeList[DotOperator[#] & /@ {R1, R2, R3}, v]] > Out: {R1.v, R2.R1.v, R3.R2.R1.v} > > to obtain the result we want. > > Is there a simpler way than this that doesn't involve defining functions > like DotOperator? > > Thanks, > > Hywel > Hi Hywel, your method gives the same result as Rest[FoldList[#2.#1&,v,{R1,R2,R3...}]] but you should consider the deinition of ComposeDot[mlist_, v_] := Nest[ReleaseHold, Rest[FoldList[Hold[#2.#1] &, v, mlist]], Length[mlist]]; In[2]:= ComposeDot[{M1, M2, M3}, v] Out[2]= {M1.v, M2.M1.v, M3.M2.M1.v} this is orders of magnitudes faster and uses memory more economical (see the attached notebook for details). Peter (*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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M1 . v}\)], "Output"] }, Open ]], Cell["This looks complicated compared to", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(FoldDot[mlist_, v_] := Rest[FoldList[#2 . #1 &, v, mlist]];\)\), "\[IndentingNewLine]", \(FoldDot[{M1, M2, M3}, v]\)}], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({M1 . v, M2 . M1 . v, M3 . M2 . 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And the second advantage could be even more important:\ \>", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[{ \(LeafCount /@ {r2, r1}\), "\[IndentingNewLine]", \(Quotient @@ %\)}], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({2108798, 1121}\)], "Output"], Cell[BoxData[ \(1881\)], "Output"] }, Open ]], Cell["or (same order of magnitude):", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[{ \(ByteCount /@ {r2, r1}\), "\[IndentingNewLine]", \(Quotient @@ %\)}], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({35916904, 18752}\)], "Output"], Cell[BoxData[ \(1915\)], "Output"] }, Open ]], Cell["Your method (very similar to FoldDot):", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[{ \(DotOperator[M_] := M . #1 &; \), "\n", \(mem = MaxMemoryUsed[]; \), "\n", \(First /@ {Timing[\(r3 = Rest[ComposeList[\((DotOperator[#1] &)\) /@ \((\(R\_#1 &\) /@ Range[2, 8])\), {x, y}]];\)], Timing[z3 = Expand[Last[r3]]; ]}\), "\n", \(\(Print[MaxMemoryUsed[] - mem];\)\)}], "Input", CellAutoOverwrite->False], Cell[BoxData[ \({0.6709999999999994`\ Second, 38.114000000000004`\ Second}\)], "Output"], Cell[BoxData[ \(1464\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(z3 \[Equal] z2\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ByteCount[r3]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(35916904\)], "Output"] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowSize->{897, 612}, WindowMargins->{{0, Automatic}, {Automatic, 0}} ] (*********************************************************************** Cached data follows. 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