CSE 413 Au12 Assignment 3 - Symbolic Math

Due: Online via the Catalyst Dropbox by 11 pm, Thursday, Oct. 18, 2012.

For this project you will write a racket program to differentiate simple expressions.  In case calculus is a distant memory, the rules for differentiation go something like:

Basic rules:

(d/dx) (constant) = 0

(d/dx) (x) = 1

(d/dx) (y) = 0        (if y does not depend on x)

(d/dx) (E1 + E2 + E3) = (d/dx)(E1) + (d/dx)(E2) + (d/dx)(E3)   	Differentiation 
								  of sums

(d/dx) (E1 * E2) = (E1 * (d/dx)(E2)) + (E2 * (d/dx)(E1)) 	Product rule

(d/dx) (xr) = r * (x)r-1                               		 Power rule

More complex rules:

(d/dx) (ex) = ex     

(d/dx) (ln x) = 1/x

(da/dx) = (da/db) * (db/dx)                       	Chain rule

(d/dx) ((f(x))r) = r * (f(x))r-1 * (d/dx)(f(x))		 Applying the chain rule. 

(d/dx) (ef(x)) = ef(x) *  (d/dx)(f(x))               	  Applying the chain rule.

Your program should be able to differentiate expressions containing constants, variables, sums with an arbitrary number of terms (+ E1 E2 E3 ... En), products with two terms (* E1 E2), and exponents of the form x^y where y can be an integer or a variable other than x (expt x y). You can add additional operators for extra credit once you've implemented these basic requirements; see below for details.

You should implement the function (diff x E) to differentiate the expression E with respect to the variable x.  Expressions should be represented in list format.  That is,

Examples: (Your exact output may differ, but should be algebraically equivalent.)

> (diff 'x '4)			=>  0

> (diff 'x '(* 2 x))		=> (+ (* 0 (* x)) (* 2 1)) 	   (i.e. 2)

> (diff 'y '(* 2 y))		=> (+ (* 0 (* y)) (* 2 1)) 	   (i.e. 2)

> (diff 'x '(+ x (* x x)))	=> (+ 1 (+ (* 1 (* x)) (* x 1)))   (i.e. 1 + x + x)

> (diff 'x '(expt x 4))		=> (* 4 (expt x 3))		   (i.e. 4 * x3)

Implementation:

You should implement function (diff v E) to differentiate the expression E with respect to the variable v. Note that v is an argument to diff that can be given any variable, not just 'v.

Your program must include functions to differentiate individual kinds of expressions (i.e., one function per top-level operator), and a dispatch table that function diff will use to determine the appropriate sub-function to handle an expression, based on the expression's operator.  Use these fragments as starting points for your code. 

(define (diff-sum x E) ...)        ; differentiate (+ x1 x2 ...)
(define (diff-product x E) ...)    ;               (* x y)
(define (diff-expt x E) ...)       ;               (expt x y)

;; Dispatch Table of supported operators.
 (define diff-dispatch
   (list (list '+ diff-sum)
         (list '* diff-product)
         (list 'expt diff-expt)
         ))

Be sure that the functions diff-sum, diff-product and diff-expt are defined before your dispatch table in your source file.   To expand the program to differentiate other functions, you should only have to write an appropriate function to do the transformation, and then add an entry to the dispatch table with the operator symbol and the function name in a list. Once the code for diff is working on the basic cases, it should not need further changes to add additional kinds of expressions.

Your main diff function will look something like this:

;; Differentiate expression E w.r.t. x.
(define (diff x E)
  (cond ((number? E) (diff-constant x E))
        ;; insert code here to handle the other base case - variables
	...
        (else 	; insert code here to lookup the appropriate function in the 
		; dispatch table based on the operator in the expression,
                ; then call that function to differentiate the expression
                    (diff-func x E)))) ))

You need to implement the functions: diff-sum, diff-product and diff-expt.  Your diff function should handle differentiation of numbers and symbols (single variables such as x or y) directly, as well as looking up and applying the appropriate differentiating function for sums, products, expt and any other functions you add.

Racket Style:

Be sure to include the code above in your program exactly as written (with, of course, additions needed to implement the various functions).   That is, you should use the function names diff-sum, diff-product etc with dashes (NOT underscores). 

Good racket style is to define functions to abstract away from representation details. So instead of (car E) to access the operator of an expression, a better way of doing this is to define a function get-op to extract the operator from an expression, as follows:.

	(define (get-op E) (car E))

You should define similar functions to access other parts of expressions, create various kinds of expressions, and so forth.  Example: if you need to create a sum given a list of arguments, you could use a function like this:

	(define (make-sum alist) (cons '+ alist))

If you include appropriate constructor and access functions, the code that differentiates expressions can be written in terms of arguments and operators, not car, cdr, cadar, and similar things, which should make it much more readable.

Use higher-order functions and library functions when appropriate; don't implement special-purpose map functions when you can use library functions and functional parameters to do the job.  For example, do not write your own specialized version of map that differentiates sums (instead use map if appropriate to implement diff-sum).

Testing:

Part of this assignment is to demonstrate that your program works using well-chosen test cases.   A good set of test cases verifies basic and edge cases with a reasonably small set of carefully thought out test data, not just a large scattershot test with a bunch of random expressions, but still comprehensively tests all of the functions and code.  The examples listed above are only meant to illustrate how your program should work, they do not comprise a reasonable test suite!  For full credit, you must 1) create a good suite of test cases and 2) write appropriate function(s) to simplify testing.

Here is one way you could do this.  Define individual tests and a list that contains all of the tests.

(define test1 '(+ x 4))
(define test2 '(+ 9 8))
(define test-suite (list test1 test2))

Then define a function (runtest atest) that has a test as an argument, and prints its argument and the result of applying (diff 'x arg) to that argument all neatly formatted.  Then you can use that function to run an individual test, or map it over the list of tests to run all of them.  Example:

> (runtest test1)
diff x (+ x 4) => (+ 1 0)

> (map runtest test-suite)
diff x (+ x 4) => (+ 1 0)
diff x (+ 9 8) => (+ 0 0)

This makes it easy to re-run your test cases after modifying your program.   Include the test cases and functions to process them in your racket file.  Be sure that it is obvious how to run the tests, and also how to turn them off so we can run the code with additional tests.

Alternative:  Instead of differentiating everything with respect to x, you might want a test cases to include both the expression and the differentiation variable.  Example:

(define test10 '(x (+ x 4)))
(define test11 '(y (+ x 4)))

Optional Extra Credit Extensions:

Be sure to have the original version of the program in perfect working order before attempting any extra credit options.  Turn in your code in a file named hw3.rkt once you have the basic part of the assignment working. After adding any extra credit options you so desire clearly label those extra credit options added and resubmit your files via electronic turnin using a different file name, hw3-extra.rkt.

• Simplify the output:  If you simply run diff on a large expression, the result can be rather hard to read.  Write an algebraic simplifier that will make your output easier to read.  The simplifier should NOT be folded in with the basic diff code.  That is (diff x exp) should give an unsimplified answer.   Execution of (simplify (diff x exp)) should produce the simplified expression.  An example of simplification include "flattening" sums and products such as converting: (+ x (+ 1 x) 1) to (+ x 1 x 1).  Another simplification is to remove the identity operand from expressions, for instance converting: (* 1 (+ x 0) x) to (* (+ x) x).   You could also fold constants together or do fancier things, for example converting (+ x (+ 1 x) 1 x) to (+ (* 3 x) 2).  Suggestion: look at some of the output produced by diff and try to identify simple, high-impact transformations that make a big difference in the readability of the output.

• Add more functions: Add in trig functions: cos, sin, tan, or other functions: exp, log, sqrt, quotients, or whatever you want that expands the capabilities of your diff function.  See the racket reference pages for a list of functions found in racket, or invent notation for functions of your own (but be sure to clearly document your extensions if you invent new notation).
  
  
• Automate testing: Augment the test suite so it not only runs tests but also verifies that the results are correct. You should figure out how best to do this, but it should still be possible to run tests to see the output they produce.
  

What to Hand In

Electronic Submission

Turn in a copy of your racket source file using the regular online collection dropbox. Please put all of your functions in one file named hw3.rkt. If you attempted extra credit please turn in two versions of your program, the first one named hw3.rkt without extra credit, and a second one named hw3-extra.rkt with extra credit.

Be sure that your name is included at the beginning of the file in a comment.