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Goal

This lab should teach you how to create and use new arithmetic functions. We will tackle LISP data structures in the next lab.

Deliverables

You should record your insights and confusions in a new cs320 lab page and link it to you home/cs320/index page.

To earn full credit the work must be done before the end of the lab and should contain a list of at least 3 notes. Each note is a short paragraph with one or two sentences and a new LISP function (or link to a file containing the function). The sentences should say what the function does and what you learned by writing it.

Let me know by calling me across to your workstation when done.

Hints


Net
1. To run LISP open a terminal window. No source to edit..... in this lab!
2. To leave XLISP send EOT by holding down CTRL and tapping D
3. Parentheses (...) are the must important things in LISP.
4. A comment in XLISP starts with a ";" and runs to the end of the line.
5. Keep your handout on LISP open beside you and look in it for the rules of LISP.
6. Open many windows and use your hilight-and-paste feature to save time.
7. Use an editor to write your functions and their test cases then either
   Any test editor is good for LISP, but vi can help you track the parentheses.
8. Put XLISP code in files that end ".lsp".

   ---

   1. In 'vi' you can track parentheses if you do the following

       		:set showmatch

      • Copy and paste into a window running XLisp to run them...
      • or
      • Save the version ( :w in vi) and run xlisp on it. (:!xlisp % in vi)

   ---

9. Some browser may panic if you link to a ".lsp" file. If you put source code in a ".txt" file every browser in the world will read it and your user's will love you.

(End of Net)



Review

Start by rereading your last published page of notes!

Here are some simple LISP expressions/commands. Load the LISP interpreter and input each in turn. Try to predict what each will return as a value before inputting it:

 		()

 		(+ 1 2)

 		(1 + 2)

 		(* (+ 1 2) (+ 3 4))

 		(+ 1 2 3 4)

 		(A B C)

 		'(A B C)

 		A

 		'A

 		(A)

If you get one wrong... you may need to go back to [ 11.html ] again.

Constructing Lists with back-quotations

Here is a wierd but elegant feature of LISP.

You know that LISP has double quoted strings:

 		"I am a string!"

You already know that a single quotation mark stops the interpreter from evaluating a list.

 		'(+ 1 2)

 		(setq day 'wednsday)

 		'(Today is day)

A backquoted list starts with a backquote (`) and is treated as a constant list that contains some expressions. All you do is put a comma(,) in front of the expressions.

 		`(Today is ,day)

(day is replaced by the value of the variable day)

Or

 		`((+ 1 2) = ,(+ 1 2))

Or

 		`( example1 ( (sqrt 4.0) = ,(sqrt 4.0) ) ( day = ,day))

Note. This is a feature that I only learned about in 2005... it looks like it may be useful in the future.

Creating new functions

Here is a function with no arguments:

 		( define  (a) 4321)

 		(a)

 		a

 		'a

 		(a 1)

Copy and paste the above XLISP commands into XLISP in a terminal window.

Then define and test a new function called answer that returns the value 42.

Functions with a single parameter

Here is how we would define a LISP function with a single argument that returns a list:

 		(DEFINE (square x)           (* x x))

Here is how you test it...

 		(square 3)

 		(square 4)

 		(square 5)

Here is how you can use it:

 		(square (+ 1 2 3))

 		(+ (square  3) (square 4) )

 		(+ 3 (square 3) )

Test the above!

Here is how XLISP can list it:

 		square

XLISP does not let you edit a function however!

Do not leave LISP until you complete the next two steps.

Here is a function for the cube:

 		(DEFINE (cube x)           (* x (square x))

Test it.

Define a function called fourth that returns the fourth power of a number. Use the fact that the fourth power on n is the square of the square of n:

1. n^4 = (n^2)^2.

   Test it. And save it...

   Functions with two parameters

   Functions of more arguments/parameters are done the same way using the syntax
2. ( DEFINE (name w x y z ...) expression)

   Here is a definition of a function with two arguments in LISP:

    		(DEFINE (pythagoras x y) (+ (square x) (square y)))

    		(pythagoras 3 4)

   Some common errors

    		(pythagoras 4)

    		(pythagoras 1 3 4)

    		(pythagoras )

   In our XLISP the value of the function name is an expression defining a function:

    		pythagoras

   Here is a function that return the larger of two expressions:

    		(define (max2 a b) (if (> a b) a b))

    		(max2 1 17)

    		(max2 17 1)

    		(max2 (max2 3 5) (max2 4 1))

    		(max2 (square 3) (cube 2))

   Define a function called min2 that returns the smallest of two arguments. Test it and save it....

   A Binary Search

   Here is a function that searches for the positive square root of a positive number. It uses the square function. It has four arguments:
   
   1. target: the number for which you need a root. Must be >=0.
   2. lo: a number that is less than the root.
   3. hi: a number that is greater than the root.
   4. error: a number indicating the desired accuracy of the approximation.
   
   

    (define (binroot target lo hi error )

    	(let (( mid (/ (+ lo hi) 2.0))) ; this saves the time to recalculate mid

    	 (if (<= (- hi lo) error)

    	     mid

    	     (if (< (square mid) target)

    	         (binroot target mid hi  error)

    	         (binroot target lo  mid error)

                 )

    	 )

    	)

    )

   Here is how I tested it:

    		(binroot 50 0 100 0.05)

   Test it further and trace it.

   The above algorithm will find roots of any monotonic increasing function. Modify it to find cube roots and fourth roots of positive numbers.

   Optional experiments if you have time

   Higher powers

   The following function works out the value of x to the power n when n is a whole number greater than or equal to 0.

    		(define (power x n)

    		   (cond

    			((= n 0) 1)

    			((= n 1) n)

    			(T (* x (power x (- n 1))))

    		   )

    		)

   Here are two test cases

    		(power 2 3)

    		(power 3 2)

   Test with the trace function....

   However this is not a very fast way to calculate powers. There is another one based on these facts:
   Net

   1. If n is even then n=2*k for some k and so power(x,n)=square(power(x,k)).
   2. If n is odd then n=2*k+1 for some k and so power(x,n)=x*square(power(x,k)).
   
   (End of Net)
   
   We have LISP function EVENP and ODDP to detect parity and (/ n 2) works out a k for us by rounding down.

   Can you figure out how to speed up the original power function?

   Classic factorial

   We now define a new function, command and kind of expression that calculates the factorial of a positive integer. The factorial of the number 0 is 1, and for all other positive integers n you work out n times the factorial of n-1. In LISP we say:

    		(define (factorial n)

    		 (if (= n 0) 1 (* n ( factorial ( - n 1 ) ))

    		 )

    		)

   Here you print out the definition of factorial:

    		factorial

   Here you input expressions that apply factorial to simple numbers:

    		(factorial 0)

    		(factorial 3)

    		(factorial 5)

   Trace how it works:

    		(trace factorial)

    		(factorial 0)

    		(factorial 3)

    		(factorial 5)

    		(untrace factorial)

   Below you can use factorial is more complicated ways:

    		(factorial (factorial 3))

    		(setq n 3)

    		(factorial n)

    		(setq f (factorial n))

    		f

    		(setq f (factorial f))

    		f

   The problem with this factorial is that n! quite a small n is too large to be represented as an integer. Worse n! when n<0 is infinitely small! As a result we should define a safe factorial that returns a string "error" when n<0 or n is too big.

   Program and test this one.

   An example of top-down design in LISP

   [ primes.html ]

   . . . . . . . . . ( end of section Optional experiments if you have time) <<Contents | End>>

   Leave LISP

   To exit lisp, input the EOT character CTRL/D

. . . . . . . . . ( end of section CS320 Lab 12 LISP Laboratory Number 2) <<Contents | End>>

Check the Preparation for next class