BEGINNING MATHEMATICA PROGRAMMING FOR CSE/MATH 467

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1 BEGINNING MATHEMATICA PROGRAMMING FOR CSE/MATH 467 W. DALE BROWNAWELL 1. BEGINNING A MATHEMATICA SESSION 1.1. Using Mathematica s Graphical Interface. On a workstation with Mathematica installed, you can start a Mathematica Xsession after logging on simply by entering the following command at an xterm prompt: mathematica & There IS a lot of help on-line when you use this interface. Many people like the many features of Mathematica s notebooks. I had some flaky response years ago and do not use them, preferring the solid (clunky?) Emacs interface Using the Emacs Interface. In Emacs (my favorite editor), you can open Mathematica by simply typing Esc -x math. This requires making emacs aware of Mathematica, which you can do yourself by via the files found at or, as I did, mccann/mathematica.el. This is especially helpful if you are already an experienced command-line Emacs user and want to run Mathematica remotely without the overhead penalty of a GUI over the internet. Then you enter emacs -nw to open a non-graphics version of Emacs Falling into a Terminal State. If you start Mathematica from a command line with the command math, you will have only a command-line, non-graphics, version. My advice: Don t do it this way, as you lose the ability to cut and paste; it is very cumbersome to not be able to edit input easily or to enter output.. 2. WRITING AND ENTERING PROGRAMS 2.1. Writing Mathematica Programs. Mathematica programs are best stored in files with names like: gcd.m. These files are plain ascii files created by your texteditor and saved by you. You may store several, indeed all, your functions in one file. Date: 21 August,

2 2 W. DALE BROWNAWELL 2.2. Loading Mathematica Programs. The programs we will write are actually all definitions of functions which take the input, usually numbers or vectors (lists) of numbers, and produce some output, again usually numbers or vectors of numbers, using various other functions, some of which we have to define as well. The functions in a file are made available to you in a Mathematica session by loading the file at the Mathematica prompt as in In[1]:= << gcd.m These functions then remain available to you during the whole session. So it is important to have distinct names for distinct functions. When working with Mathematica under Emacs, you need to enter commands or load files using Esc - Return, not simply Return. If you load a file under Emacs and a syntax error is reported, you can go to the error by typing Ctrl -c Ctrl -e, provided you have started Emacs in the directory which contains your Mathematica files. Emacs also offers rudimentary syntax checking as you type, the ability to copy previous cells, complete Mathematica symbols, help with Mathematica symbols, and a few more things Local Variables in Mathematica. Because Mathematica remembers anything already used or defined in a session, you need to keep dummy variables involved in defining functions free from the influence of any previous use of the variables. This is done by using local variables in Modules. In the following example, x,y,t are the local variables: gcd.m gcd[m,n ] := Module[{x=Abs[m],y=Abs[n],t}, While[y!= 0, t = y; y = Mod[x,y]; x = t ]; Return[x] ] Notice the following features: Parentheses ( ) are used for grouping mathematical expressions, braces { } are used for lists, and brackets [ ] are used for function evaluation. The values to be fed into the function are indicated in the definition by trailing underlines. These values are then used in the program via local variables.

3 BEGINNING MATHEMATICA PROGRAMMING FOR CSE/MATH The local variables are named in Module[{... }, which is also the place to initialize them (no underlines here). The punctuation of a Mathematica command is usually a comma or a semi-colon and often the opposite of C usage. 3. HELPFUL HINTS 3.1. Putting Several Functions into a Single File. It s often a good idea so you don t have to remember later what functions are used in the definition of a function you haven t seen in a couple of weeks. You can also organize your functions this way by grouping related functions together into a single.m file. The best reason is perhaps that then you don t have to enter all the.m files singly by hand at the Mathematica prompt! There is another way to keep all your.m files in one directory and still to allow functions to rely on other functions. If the definition being relied upon occurs in say, prelim.m, you simply include a line << prelim.m in your new function before the earlier function is evaluated. (Of course, punctuation such as a comma or semicolon may be necessary at the end of this line.) 3.2. Many Happy Returns. When Bressoud uses WRITE or PRINT, it is better for us to Return so the output can be used as input for another function if desired. However, for debugging purposes, you may want to use the Print function. See Section 2.8 of the Mathematica book for syntax: Print[ Here the value of x =,x] The use of Return can be subtle. It seems Return that will break out of the innermost construct only. Here is more information (than I want to digest): Getting Hardcopy. I prefer to run everything within Emacs, whence you can save copies of files and of sessions, as well as printing them without leaving the buffer. When you are in one of the Linux labs, you can print a file, say named myfile by entering lpr myfile. You can check on how much paper etc you have available on your account at: Bressoud 4. A SMALL PASCAL/REXX TO MATHEMATICA DICTIONARY Mathematica READ a,b := Module[{x=a,y=b},... WHILE b 0 DO While[y!= 0,..., ] temp b temp = b WRITE a Return[a]

4 4 W. DALE BROWNAWELL u 1 1, u 2 0, u 3 a u = {1,0,a} q u 3 /v 3 q = Floor[u[[3]]/v[[3]]] IF x = 0 THEN DO If[x==0,... IF x is even DO If[EvenQ[x],... IF b is odd THEN If[OddQ[b],... c a-b c = Abs[a-b] FOR i = 2 to n DO For[i=2, i<= n, i++;... ] FOR i = 2 to n DO a = Range[n] a i i f MOD d Mod[f,d] i i+1 i = i+1 or ++i AND && OR 5. ARRAYS TO THE FINISH Multiple outputs are often best collected and output as a single list, as for Algorithm 2.3 and Algorithm 2.4 (a list of pairs in the latter case), whose entries can then be utilized in further programs. Array[a,{m,n}] declares an m n matrix with entries which can be accessed as either a[[i,j]] or a[[i]][[j]]. Often you won t need to declare things very precisely in Mathematica; just declare (i.e. list) them as dummy variables and define them as you go. Since matrices in Mathematica are just lists of rows, the ith row of a can be accessed as a[[i]]. You can display the matrix a in matrix form by using MatrixForm[a]. An n n identity matrix can be created using IdentityMatrix[n]. The ith row of the matrix M can be used via M[[i]] or Part[M,i]. Lists can also avoid the need for function calls as in Algorithm Defining a list. Defining the entries of an array can be a bit confusing. See sections of the Mathematica manual on such things as Making List out of Functions, Transformation Rules, Array, List for details. Basically there are three families of ways of defining arrays (lists): Table and built-in functions. The commands Table[1, {k, 1, n}] and Table[h[i], {i, 1, n}] create vectors of length n. In the first case, all entries equal 1; in the second, entries are obtained by evaluating the function value h at the values i = 1, 2,..., n. So Range[n] is the same as Table[i,{i, 1, n}]. If you have set T = Table[0,{k,1,5}] and then T = Table[T,{k,1,7}], you will have defined a 5 7 matrix T with all entries equal to 0, because Mathematica thinks of a matrix as a list of lists.

5 BEGINNING MATHEMATICA PROGRAMMING FOR CSE/MATH Append. You can build up a list l recursively by naming it in Module[{ l = { },... },... and then, whenever you want to tack on a new element e at the end of l, set l = Append[l,e], and the entries can be accessed. Prepend and a host of commands such as Join, Union are also available. For example if l = {x,y}, then setting l = Append[l,e] gives l = {x,y,e}, and we can access entries, since l[[1]] = x, l[[2]] = y, l[[3]] = e Table and user-defined functions. Another way to build up a list is to define function values for a function, say h, by setting, e.g. h[i] = p in a loop and later making up an array with these values as entries, as in the command h = Table[ h[i], {i, j-1}]. Then the various terms, such as, h[[2]] have the same meaning (value) as their counterparts h[2] Common pitfalls. Once the array v has been defined, its jth entry can be re-defined by, e.g., v[[j]] = 2 + i. But the initial definition of a list apparently CAN- NOT be made this way! The Do command can be tricky. In order to return a value from it, you probably need to first exit the Do-loop and only then Return the value. The Mathematica manual puts it this way: The result of Do is Null. You may often find it easier to instead use the commands While or For A useful application. Length[list] returns the number of elements in the list named list. This might come in handy. say, when you have stored a list l consisting of pairs of primes and their exponents and want to reconstitute the corresponding product as p = 1; Do[p = p*(l[[i,1]]ˆl[[i,2]]), {i, 1, Length[l]}]. 6. INPUTTING OUTPUT FROM PREVIOUS SESSIONS If your program is too large to run in one session or you want to debug your package in sections, you can save intermediate output to a file, say myoutput.in in a form suitable (after editing away any extraneous lines the Mathematica session produced) for reading back into another session as input. One way to do this is to save your output in InputFormat by entering the following before starting the program that will generate the output: SetOptions["stdout", FormatType->InputForm] This makes sure that the output looks like aˆ2-2*a*b instead of the usual style for Mathematics output with exponents on a higher line and multiplication indicated by a space. That way you can easily edit the output to define, say, a matrix M. Then by entering << myoutput.in in a session, the matrix will be

6 6 W. DALE BROWNAWELL defined and available for your programs. On long jobs, you could even put the commands to load functions, read input, e.g myoutput.in, and evaluate the functions into a script and run in the background. See [RM], pp , for example. For short jobs, it will not be worth the trouble. But sieving is an example of an important process which takes a long time, and you will not want to repeat it more often than is absolutely necessary. 7. REDUCED PRECISION FOR FASTER SIEVING In order to speed Mathematica up, you can suppress the infinite precision by, say, for each prime p setting lp = N[Log[p],5]. This truncates the precision to 5 decimal places. Then you can add l p to all the locations corresponding to the r for which r ±t p mod p, where t 2 p n mod p. 8. MATRIX REDUCTION For the quadratic sieve, you need to reduce a huge matrix, say m, modulo 2. Mathematica s built-in function for that is: RowReduce[m, Modulus -> 2] 9. DOCUMENTATION You are required to document, i.e. explain, your programs. You can do this by putting comments into your program, by beginning with (* and ending with *). Mathematica will ignore what lies in between. Or you can simply write (in English sentences, please) explanations neatly in the margin. 10. CORRECTIONS I would be thankful for any corrections or comments. Please send them to me at wdb@math.psu.edu. Note: Wolfram s book below is the standard reference, and it seems to be on permanent reserve in the Mathematics/Physics Library. There is now an on-line version at the Mathematica website: REFERENCES [1] R. Maeder, Programming in Mathematica, 3rd edition, Addison Wesley, Redwood City, Ca, [2] P. Wellin, R. Gaylord, and S. Kamin, An Introduction to Programming in Mathematica, Cambridge University Press, Cambridge, [3] S. Wolfram, Mathematica, Wolfram Media, 2003, Fifth Edition. [4] address: wdb@@math.psu.edu