Assignment 2: Starting Scheme

You may recall that we have a number of mechanisms for rounding real numbers to integers, such as ceiling and floor. But what if we want to round not to an integer, but to only two digits after the decimal point? Scheme does not include a built-in operation for doing that kind of rounding. Nonetheless, it is fairly straightforward.

1. Suppose we have a value, val. Write instructions that give val rounded to the nearest hundredth. For example,

> (define val 22.71256)
> (your-instructions val)
22.71
> (define val 10.7561)
> (your-instructions val)
10.76

Now, let's generalize your instructions to round to an arbitrary number of digits after the decimal point.

Suppose precision is a non-negative integer and val is a real value. Write instructions for rounding val to use only precision digits after the decimal point.

> (your-instructions ... val ... precision ...)

As you write your instructions, you may find the expt useful. (expt b p) computes b^p.

One advantage of learning a programming language is that you can automate tedious computations. We consider such a computation here.

One of the more painful computations students are often asked to do in high-school mathematics courses is to compute the roots of a polynomial. As you may recall, a root is a value for which the value of the polynomial is 0. For example, the roots of 3x² - 5x + 2 are 2/3 and 1.

Hmmm ... let's check that claim. 3*(2/3)*(2/3) - 5*2/3 + 2 = 12/9 + 10/3 + 2 = 4/3 - 10/3 + 2 = (4-10)/3 + 2 = -6/3 + 2 = -2 + 2 = 0. So far so good.

3*1*1 - 5*1 + 2 = 3 - 5 + 2 = -2 + 2 - 0. Yup, 2/3 and 1 are the roots.

There is, of course, a famous formula for computing the roots of a quadratic polynomial of the form ax² + bx + c. In a narrative style, this formula is often expressed as

In more traditional mathematical notation, we might write

Our goal, of course, is to convert all of these ideas to a Scheme program.

We can express the coefficients of a particular polynomial by using define expressions.

(define a 3)
(define b -5)
(define c 1)

If we also define x, we can evaluate the polynomial.

(define x 5)
(define value-of-polynomial (+ (* a x x) (* b x) c))

Of course, since we've defined a, b, and c, we can also compute the roots. Here is a bit of incorrect code to compute roots.

(define root1 (/ (+ (- b) (sqrt b)) 
                 (* 2 a)))
(define root2 (/ (- (- b) (sqrt b)) 
                 (* 2 a)))

Your goal, of course, is to correct the code for root1 and root2 so that we correctly compute the roots of the polynomial.

You can test the correctness of your solution by trying something like

(define value-of-polynomial-at-root1 (+ (* a root1 root1) (* b root1) c))
(define value-of-polynomial-at-root2 (+ (* a root2 root2) (* b root2) c))

Note: In the past, we've seen students test their quadratic-root formula by trying essentially arbitrary values for a, b, and c. Unfortunately, many arbitrary quadratic polynomials have no real roots. Hence, we suggest that you test your work by building polynomials for which you know the roots. How? Create your polynomials by multiplying two linear polynomials, (px + q) * (rx+s). You know that the roots of this polynomial will be -q/p and -s/r.

Scheme provides a number of numeric procedures that can produce integer results. We have already explored expt, abs, +, -, *, and /.

Here are some others. For each, try to learn (by experimentation, by discussing results with other students, and, eventually, by reading documentation) how many parameters each procedure can take and what the procedure does. You may not be able to figure all of them out. Make sure to try a variety of values for each procedure, including positive and negative, integer and real. Include illustative samples of the use of each procedure.