The manipulation of symbolic algebraic expressions is a complex process that illustrates many of the hardest problems that occur in the design of large-scale systems. An algebraic expression, in general, can be viewed as a hierarchical structure, a tree of operators applied to operands. We can construct algebraic expressions by starting with a set of primitive objects, such as constants and variables, and combining these by means of algebraic operators, such as addition and multiplication. As in other languages, we form abstractions that enable us to refer to compound objects in simple terms. Typical abstractions in symbolic algebra are ideas such as linear combination, polynomial, rational function, or trigonometric function. We can regard these as compound “types,” which are often useful for directing the processing of expressions. For example, we could describe the expression
as a polynomial in x with coefficients that are trigonometric functions of polynomials in y whose coefficients are integers.
We will not attempt to develop a complete algebraic-manipulation system here. Such systems are exceedingly complex programs, embodying deep algebraic knowledge and elegant algorithms. What we will do is look at a simple but important part of algebraic manipulation: the arithmetic of polynomials. We will illustrate the kinds of decisions the designer of such a system faces, and how to apply the ideas of abstract data and generic operations to help organize this effort.
Arithmetic on polynomials
Our first task in designing a system for performing arithmetic on polynomials is to decide just what a polynomial is. Polynomials are normally defined relative to certain variables (the indeterminates of the polynomial). For simplicity, we will restrict ourselves to polynomials having just one indeterminate (univariate polynomials).[54] We will define a polynomial to be a sum of terms, each of which is either a coefficient, a power of the indeterminate, or a product of a coefficient and a power of the indeterminate. A coefficient is defined as an algebraic expression that is not dependent upon the indeterminate of the polynomial. For example,
is a simple polynomial in x, and
is a polynomial in x whose coefficients are polynomials in y.
Already we are skirting some thorny issues. Is the first of these polynomials the same as the polynomial 5y^{2} + 3y + 7, or not? A reasonable answer might be “yes, if we are considering a polynomial purely as a mathematical function, but no, if we are considering a polynomial to be a syntactic form.” The second polynomial is algebraically equivalent to a polynomial in y whose coefficients are polynomials in x. Should our system recognize this, or not? Furthermore, there are other ways to represent a polynomial — for example, as a product of factors, or (for a univariate polynomial) as the set of roots, or as a listing of the values of the polynomial at a specified set of points.[55] We can finesse these questions by deciding that in our algebraic-manipulation system a “polynomial” will be a particular syntactic form, not its underlying mathematical meaning.
Now we must consider how to go about doing arithmetic on polynomials. In this simple system, we will consider only addition and multiplication. Moreover, we will insist that two polynomials to be combined must have the same indeterminate.
We will approach the design of our system by following the familiar discipline of data abstraction. We will represent polynomials using a data structure called a poly, which consists of a variable and a collection of terms. We assume that we have selectors variable
and term-list
that extract those parts from a poly and a constructor make-poly
that assembles a poly from a given variable and a term list. A variable will be just a symbol, so we can use the same-variable?
procedure of section 2.3.2 to compare variables. The following procedures define addition and multiplication of polys:
(define (add-poly p1 p2)
(if (same-variable? (variable p1) (variable p2))
(make-poly (variable p1)
(add-terms (term-list p1)
(term-list p2)))
(error "Polys not in same var -- ADD-POLY"
(list p1 p2))))
(define (mul-poly p1 p2)
(if (same-variable? (variable p1) (variable p2))
(make-poly (variable p1)
(mul-terms (term-list p1)
(term-list p2)))
(error "Polys not in same var -- MUL-POLY"
(list p1 p2))))
To incorporate polynomials into our generic arithmetic system, we need to supply them with type tags. We’ll use the tag polynomial
, and install appropriate operations on tagged polynomials in the operation table. We’ll embed all our code in an installation procedure for the polynomial package, similar to the ones in
section 2.5.1:
(define (install-polynomial-package)
;; internal procedures
;; representation of poly
(define (make-poly variable term-list)
(cons variable term-list))
(define (variable p) (car p))
(define (term-list p) (cdr p))
<procedures same-variable? and variable? from section 2.3.2>
;; representation of terms and term lists
<procedures adjoin-term …coeff from text below>
(define (add-poly p1 p2) …)
<procedures used by add-poly>
(define (mul-poly p1 p2) …)
<procedures used by mul-poly>
;; interface to rest of the system
(define (tag p) (attach-tag 'polynomial p))
(put 'add '(polynomial polynomial)
(lambda (p1 p2) (tag (add-poly p1 p2))))
(put 'mul '(polynomial polynomial)
(lambda (p1 p2) (tag (mul-poly p1 p2))))
(put 'make 'polynomial
(lambda (var terms) (tag (make-poly var terms))))
'done)
Polynomial addition is performed termwise. Terms of the same order (i.e., with the same power of the indeterminate) must be combined. This is done by forming a new term of the same order whose coefficient is the sum of the coefficients of the addends. Terms in one addend for which there are no terms of the same order in the other addend are simply accumulated into the sum polynomial being constructed.
In order to manipulate term lists, we will assume that we have a constructor the-empty-termlist
that returns an empty term list and a constructor adjoin-term
that adjoins a new term to a term list. We will also assume that we have a predicate empty-termlist?
that tells if a given term list is empty, a selector first-term
that extracts the highest-order term from a term list, and a selector rest-terms
that returns all but the highest-order term. To manipulate terms, we will suppose that we have a constructor make-term
that constructs a term with given order and coefficient, and selectors order
and coeff
that return, respectively, the order and the coefficient of the term. These operations allow us to consider both terms and term lists as data abstractions, whose concrete
representations we can worry about separately.
Here is the procedure that constructs the term list for the sum of two polynomials:[56]
(define (add-terms L1 L2)
(cond ((empty-termlist? L1) L2)
((empty-termlist? L2) L1)
(else
(let ((t1 (first-term L1)) (t2 (first-term L2)))
(cond ((> (order t1) (order t2))
(adjoin-term
t1 (add-terms (rest-terms L1) L2)))
((< (order t1) (order t2))
(adjoin-term
t2 (add-terms L1 (rest-terms L2))))
(else
(adjoin-term
(make-term (order t1)
(add (coeff t1) (coeff t2)))
(add-terms (rest-terms L1)
(rest-terms L2)))))))))
The most important point to note here is that we used the generic addition procedure add
to add together the coefficients of the terms being combined. This has powerful consequences, as we will see below.
In order to multiply two term lists, we multiply each term of the first list by all the terms of the other list, repeatedly using mul-term-by-all-terms
, which multiplies a given term by all terms in a given term list. The resulting term lists (one for each term of the first list) are accumulated into a sum. Multiplying two terms forms a term whose order is the sum of the orders of the factors and whose coefficient is the product of the coefficients of the factors:
(define (mul-terms L1 L2)
(if (empty-termlist? L1)
(the-empty-termlist)
(add-terms (mul-term-by-all-terms (first-term L1) L2)
(mul-terms (rest-terms L1) L2))))
(define (mul-term-by-all-terms t1 L)
(if (empty-termlist? L)
(the-empty-termlist)
(let ((t2 (first-term L)))
(adjoin-term
(make-term (+ (order t1) (order t2))
(mul (coeff t1) (coeff t2)))
(mul-term-by-all-terms t1 (rest-terms L))))))
This is really all there is to polynomial addition and multiplication. Notice that, since we operate on terms using the generic procedures add
and mul
, our polynomial package is automatically able to handle any type of coefficient that is known about by the generic arithmetic package. If we include a coercion mechanism such as one of those discussed in section 2.5.2, then we also are automatically able to handle operations on polynomials of different coefficient types, such as
Because we installed the polynomial addition and multiplication procedures add-poly
and mul-poly
in the generic arithmetic system as the add
and mul
operations for type polynomial
, our system is also automatically able to handle polynomial operations such as
add
and mul
. Since the coefficients are themselves polynomials (in y), these will be combined using add-poly
and mul-poly
. The result is a kind of “data-directed recursion” in which, for example, a call to mul-poly
will result in recursive calls to mul-poly
in order to multiply the coefficients. If the coefficients of the coefficients were themselves polynomials (as might be used to represent polynomials in three variables), the data direction would ensure that the system would follow through another level of recursive calls, and so on through as many levels as the structure of the data dictates.[57]
Representing term lists
Finally, we must confront the job of implementing a good representation for term lists. A term list is, in effect, a set of coefficients keyed by the order of the term. Hence, any of the methods for representing sets, as discussed in section 2.3.3, can be applied to this task. On the other hand, our procedures add-terms
and mul-terms
always access term lists sequentially from highest to lowest order. Thus, we will use some kind of ordered list representation.
How should we structure the list that represents a term list? One consideration is the “density” of the polynomials we intend to manipulate. A polynomial is said to be dense if it has nonzero coefficients in terms of most orders. If it has many zero terms it is said to be sparse. For example,
is a dense polynomial, whereas
is sparse.
The term lists of dense polynomials are most efficiently represented as lists of the coefficients. For example, A above would be nicely represented as (1 2 0 3 -2 -5)
. The order of a term in this representation is the length of the sublist beginning with that term’s coefficient, decremented by 1.[58] This would be a terrible representation for a sparse polynomial such as B: There would be a giant list of zeros punctuated by a few lonely nonzero terms. A more reasonable representation of the term list of a sparse polynomial is as a list of the nonzero terms, where each term is a list containing the order of the term and the coefficient for that order. In such a scheme, polynomial B is efficiently represented as ((100 1) (2 2) (0 1))
. As most polynomial manipulations are performed on sparse polynomials, we
will use this method. We will assume that term lists are represented as lists of terms, arranged from highest-order to lowest-order term. Once we have made this decision, implementing the selectors and constructors for terms and term lists is straightforward:[59]
(define (adjoin-term term term-list)
(if (=zero? (coeff term))
term-list
(cons term term-list)))
(define (the-empty-termlist) '())
(define (first-term term-list) (car term-list))
(define (rest-terms term-list) (cdr term-list))
(define (empty-termlist? term-list) (null? term-list))
(define (make-term order coeff) (list order coeff))
(define (order term) (car term))
(define (coeff term) (cadr term))
where =zero?
is as defined in exercise 2.80. (See also exercise 2.87 below.)
Users of the polynomial package will create (tagged) polynomials by means of the procedure:
(define (make-polynomial var terms)
((get 'make 'polynomial) var terms))
union-set
operation we developed in exercise 2.62. In fact, if we think of the terms of the polynomial as a set ordered according to the power of the indeterminate, then the program that produces the term list for a sum is almost identical to union-set
. [back]To make this work completely smoothly, we should also add to our generic arithmetic system the ability to coerce a “number” to a polynomial by regarding it as a polynomial of degree zero whose coefficient is the number. This is necessary if we are going to perform operations such as
which requires adding the coefficient y + 1 to the coefficient 2.
[back]car
is the symbol scheme-number
. [back]adjoin-term
to simply cons
the new term onto the existing term list. We can get away with this so long as we guarantee that the procedures (such as add-terms
) that use adjoin-term
always call it with a higher-order term than appears in the list. If we did not want to make such a guarantee, we could have implemented adjoin-term
to be similar to the adjoin-set
constructor for the ordered-list representation of sets (exercise 2.61). [back]Exercises
Hierarchies of types in symbolic algebra
Our polynomial system illustrates how objects of one type (polynomials) may in fact be complex objects that have objects of many different types as parts. This poses no real difficulty in defining generic operations. We need only install appropriate generic operations for performing the necessary manipulations of the parts of the compound types. In fact, we saw that polynomials form a kind of “recursive data abstraction,” in that parts of a polynomial may themselves be polynomials. Our generic operations and our data-directed programming style can handle this complication without much trouble.
On the other hand, polynomial algebra is a system for which the data types cannot be naturally arranged in a tower. For instance, it is possible to have polynomials in x whose coefficients are polynomials in y. It is also possible to have polynomials in y whose coefficients are polynomials in x. Neither of these types is “above” the other in any natural way, yet it is often necessary to add together elements from each set. There are several ways to do this. One possibility is to convert one polynomial to the type of the other by expanding and rearranging terms so that both polynomials have the same principal variable. One can impose a towerlike structure on this by ordering the variables and thus always converting any polynomial to a “canonical form” with the highest-priority variable dominant and the lower-priority variables buried in the coefficients. This strategy works fairly well, except that the conversion may expand a polynomial unnecessarily, making it hard to read and perhaps less efficient to work with. The tower strategy is certainly not natural for this domain or for any domain where the user can invent new types dynamically using old types in various combining forms, such as trigonometric functions, power series, and integrals.
It should not be surprising that controlling coercion is a serious problem in the design of large-scale algebraic-manipulation systems. Much of the complexity of such systems is concerned with relationships among diverse types. Indeed, it is fair to say that we do not yet completely understand coercion. In fact, we do not yet completely understand the concept of a data type. Nevertheless, what we know provides us with powerful structuring and modularity principles to support the design of large systems.
Exercises
Extended exercise: Rational functions
We can extend our generic arithmetic system to include rational functions. These are “fractions” whose numerator and denominator are polynomials, such as
The system should be able to add, subtract, multiply, and divide rational functions, and to perform such computations as
(Here the sum has been simplified by removing common factors. Ordinary “cross multiplication” would have produced a fourth-degree polynomial over a fifth-degree polynomial.)
If we modify our rational-arithmetic package so that it uses generic operations, then it will do what we want, except for the problem of reducing fractions to lowest terms.
Exercises
We can reduce polynomial fractions to lowest terms using the same idea we used with integers: modifying make-rat
to divide both the numerator and the denominator by their greatest common divisor. The notion of “greatest common divisor” makes sense for polynomials. In fact, we can compute the GCD of two polynomials using essentially the same Euclid’s Algorithm that works for integers.[60] The integer version is
(define (gcd a b)
(if (= b 0)
a
(gcd b (remainder a b))))
Using this, we could make the obvious modification to define a GCD operation that works on term lists:
(define (gcd-terms a b)
(if (empty-termlist? b)
a
(gcd-terms b (remainder-terms a b))))
where remainder-terms
picks out the remainder component of the list returned by the term-list division operation div-terms
that was implemented in exercise 2.91.
Exercises
We can solve the problem exhibited in exercise 2.95 if we use the following modification of the GCD algorithm (which really works only in the case of polynomials with integer coefficients). Before performing any polynomial division in the GCD computation, we multiply the dividend by an integer constant factor, chosen to guarantee that no fractions will arise during the division process. Our answer will thus differ from the actual GCD by an integer constant factor, but this does not matter in the case of reducing rational functions to lowest terms; the GCD will be used to divide both the numerator and denominator, so the integer constant factor will cancel out.
More precisely, if P and Q are polynomials, let O_{1} be the order of P (i.e., the order of the largest term of P) and let O_{2} be the order of Q. Let c be the leading coefficient of Q. Then it can be shown that, if we multiply P by the integerizing factor c^{1+O1 -O2}, the resulting polynomial can be divided by Q by using the div-terms
algorithm without introducing any fractions. The operation of multiplying the dividend
by this constant and then dividing is sometimes called the pseudodivision of P by Q. The remainder of the division is called the pseudoremainder.
Exercises
Thus, here is how to reduce a rational function to lowest terms:
Compute the GCD of the numerator and denominator, using the version of
gcd-terms
from exercise 2.96.- When you obtain the GCD, multiply both numerator and denominator by the same integerizing factor before dividing through by the GCD, so that division by the GCD will not introduce any noninteger coefficients. As the factor you can use the leading coefficient of the GCD raised to the power 1 + O_{1} - O_{2}, where O_{2} is the order of the GCD and O_{1} is the maximum of the orders of the numerator and denominator. This will ensure that dividing the numerator and denominator by the GCD will not introduce any fractions.
The result of this operation will be a numerator and denominator with integer coefficients. The coefficients will normally be very large because of all of the integerizing factors, so the last step is to remove the redundant factors by computing the (integer) greatest common divisor of all the coefficients of the numerator and the denominator and dividing through by this factor.
Exercises
The GCD computation is at the heart of any system that does operations on rational functions. The algorithm used above, although mathematically straightforward, is extremely slow. The slowness is due partly to the large number of division operations and partly to the enormous size of the intermediate coefficients generated by the pseudodivisions. One of the active areas in the development of algebraic-manipulation systems is the design of better algorithms for computing polynomial GCDs.[62]
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