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Verbeia.com Mathematica Tips

The following are some of the tips Luci has picked up in the years she has been using Mathematica. See Ted’s Tricks, also on this site, for more comprehensive Mathematica tips.

• If you’re getting bracket matching errors in version 3, even when you are sure you had them right, try deleting all the linefeeds in the cell. Sometimes they confuse the kernel.

• Functional Programming works faster, although the code is harder to understand. Instead of Tables, consider Mapping a function to a Range[] statement. Use FoldList[] and NestList[] instead of loops. This Mathematica notebook shows concrete evidence of the difference. Functional programming can be powerful, but sometimes it gets difficult to handle when you have multiple nested pure functions. For example, to get list of lists of labels, with a different number of labels in each sublist.

  LagNums =ToString/@Range[16];
  Var = {"Y ", "DGDP ", "CommclPropInf ", 
    "CredittoGDP "};
  LagList = {6, 7, 8, 6};
  
  VaRLagLabelFn[VarString_, Lag_] := 
    Map[StringJoin[VarString,  #] &,
     Take[LagNums, {1, Lag}]]
  
  ModelLabels = 
    MapThread[VaRLagLabelFn[#1, #2] &, {Var , LagList}];
  

  You can’t make this into a single step because the pure function labels don’t nest properly. Instead of using MapThread[Map, use MapThread[Inner:

  ModelLabels[Var : {__String}, 
    LagList : {__Integer?Positive}] := 
    MapThread[
      Inner[StringJoin, Table[#1, {#2}], 
      (ToString /@ Range[#2]), List] &, {Var, LagList}]

• Don’t forget the & in functional programming!!

• When creating matrices and arrays, don’t use formatting commands like //TableForm and //MatrixForm in the initial assignment statements. This just won’t work if you then want to manipulate your matrix like a normal list. Instead, try defining the matrix, suppressing the output of the definition by putting a semicolon at the end of the line. Then have a command that just reads nameOfMatrix//MatrixForm -- you can even put it on the same line after the semicolon.

  The reason for this is that if you define the object with a //MatrixForm at the end, it has the form MatrixForm[List[...]], instead of just List[..], and so can’t be manipulated like a list.

• Which statements are much more effective than nested If statements.

• If you are writing a package that depends on other (built-in) packages (standard add-on), you must ensure that your BeginPackage statement includes imports not only for the package that you think you need, but all the packages that it imports when it is read it. You can find out what those other packages are by reading in the package you want in a fresh Mathematica session, and then evaluating the $Packages command. Put all of the package contexts that come up into your BeginPackage statement.

• Sometimes, if you are multiplying a matrix containing some zeros and some approximate real numbers, with another matrix of algebraic expressions, you will end up with approximate zeros multiplied by the algebraic elements in the result. But of course you want them to cancel out, right?
  The fix is to map Chop[] onto the matrix. Chop[] gets rid of elements smaller than a specified level (the default is 10^-10). Don’t forget that to Map[] a function to the elements of a matrix, you have to use Map[function,matrix,{2}], not function/@matrix.

• Be careful with Applying Plus functions to matrices. Map and Apply have higher precedence than most other operators, including powers. So if, for example, you want to sum the squares of a vector, use Plus @@ (vector^2) not Plus @@ vector^2.

• If you need to use single-letter local or place-holder variables in a function or package, consider using the non-standard character sets, such as the script, gothic or double-struck letters. That way, they are less likely to conflict with other global variables.

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