Iteration and Sequences: Dot Product#

The next function we’ll look at calculates the inner product, or dot product, of two one-dimensional arrays of numbers. This is defined as the sum of the pairwise products of numbers from the arrays. As an added complication, we’ll not fix the length of the arrays, so our Pascal version will use conformant arrays, a late addition to the language.

function dotProduct(x: array[xlo..xhi: integer] of real,
                    y: array[ylo..yhi: integer] of real): real;
  i: integer;
  sum: real;
  if (xhi - xlo <> yhi - ylo)
    error("dotProduct: arrays of different lengths");
  sum := 0.0;
  for i := xlo to xhi do
    sum := sum + x[i] * y[i - xlo + ylo];
  dotProduct := sum

Because a dot product only makes sense for arrays of the same length, we need to check that the arrays have the same length before proceeding. Since Pascal has no built-in error-handling facilities, we’ve invented a minimal interface – a procedure named error which takes a string description of the error – that has to be implemented on every different system using non-standard facilities.

There is one other complication worth mentioning here. Since the two arrays can have different lower bounds and the loop index i starts at the value for the array x, it has to be adjusted before being used as a subscript for y.

Our first Dylan version of this function uses a while loop:

define method dot-product (x, y)
  if (size(x) ~= size(y))
    error("dot-product: arrays of different lengths")
  end if;
  let sum = 0;
  let i = 0;
  while (i < size(x))
    sum := sum + x[i] * y[i]
  end while;
end method dot-product;

Let’s first look at the syntactic differences. Other than using ~= for the “is not equal to” relationship, Dylan is very similar to Pascal for the operations needed in this function. The operator := is used for assignment to an existing variable (as opposed to =, which is used for setting the initial value for a variable created with let) and the notation *array*[*index*] is used for subscripting. As with if, the while statement is always ended with end, and no explicit begin is needed.

The main difference between these two functions is in how they manipulate the arrays. When declaring an array in Pascal, a programmer specifies both the lower and upper bound; these bounds can be integers, characters, or enumerated types. In this regard, Dylan is less flexible than Pascal: arrays are always indexed with increasing integers starting at zero. (Many other languages, such as C, share this restriction. Fortran is similar, but it starts all arrays at one.)

In standard Pascal, the length of all arrays must be known ahead of time. The one exception to this is the conformant array feature of ISO Pascal, where the bounds of the array are treated as an extra set of parameters to a function which accepts an array as an argument. In Dylan, all arrays carry with them their length, so we only have to specify the size of an array when it is not already known, such as when the array is first created. The function size is used to obtain the length of the array. For any array a, the valid indexes of the array are from 0 to size(a) - 1; that is, size(a) elements, starting at zero.

One more thing to note is that, in Pascal, we had to invent the error procedure and say that it is implemented in some platform-dependent way. Our Dylan version also calls a function named error, but this function is a part of the Dylan language. The full behavior of error is rather complicated and beyond the scope of this tutorial – Dylan has a powerful exception handling and recovery mechanism for dealing with and possibly correcting errors discovered while a program is running – but if you’re running the program in a debugger, a call to error will generally result in the debugger stopping the program and reporting the message.

The original Pascal version of this function used a for loop, which has a direct equivalent in Dylan, as we see in this version of the function:

define method dot-product (x, y)
  unless (size(x) = size(y))
    error("dot-product: arrays of different lengths")
  end unless;
  let sum = 0;
  for (i :: <integer> from 0 to size(x) - 1)
    sum := sum + x[i] * y[i]
  end for;
end method dot-product;

We’ve changed the error check at the beginning of the function from an if statement to an unless, which executes its body if the condition is false. Sometimes, often with error checks, it is clearer to write a conditional with unless than with an if that tests the opposite condition. Note that there is no else clause for unless; if you need to do one thing if something is true and another if it is false, use if or case.

The for loop in Dylan takes several forms, one of which we see above, which is called a numeric for clause. The syntax of for in Dylan is similar to the while statement, except instead of a condition inside parentheses, we have a clause that describes the loop. A numeric for clause counts from a number to some other. In this case, the for-clause counts from zero to one less than the size of the array. The syntax of the clause is straightforward:

variable from initial-value to bounding-value

As in Pascal’s for statement and unlike the while loop, the initial value and the bounding value are evaluated exactly once, before the first iteration of the loop.

Unlike the Pascal for loop, Dylan’s for defines the loop variable – i in this example – as part of its job. That is, no declaration of i outside the loop is needed or used. This also means that the loop variable can’t be used outside the body of the loop. (I included a type declaration for i above, just to show that one could be used; typically, it would be omitted.)

The for loop above is not very typical Dylan code. If written first in Dylan, it might look more like this, taking advantage of a few constructs that don’t correspond directly to ones in Pascal:

define method dot-product (x, y)
  let sum = 0;
  for (i from 0 below x.size)
    sum := sum + x[i] * y[i]
  end for
end method dot-product;

(We’ve eliminated the error check at the beginning of the function from this and future versions, not because it is unnecessary, but because it would stay the same – there is no need to repeat it.)

We see here a slight variation on numeric iteration. The word to in the iteration clause has been changed to below. While to means all values between the initial value through, and including, the final value, below means all those values from the initial value that are less than, but not equal to, the bounding value.

In addition to what we’ve seen here, numeric iteration clauses can use the by keyword, which introduces an amount to count by, as in Pascal. The by phrase can be omitted, as we’ve seen, in which case the value is incremented by one each time through the loop. Also, we can identify the ending condition with above, the opposite of below, which means the loop runs while the values are greater than the bound. The ending condition can even be omitted, which leaves a (potentially) infinite loop – something other than finishing the numeric iteration would have to happen to terminate the loop.

The value specified by below in the loop is also a new construct, x.size. This is a Dylan shorthand for calling a function of one argument. That is, expression.function-name is the same thing as function-name(expression). So the expression here is exactly the same as size(x), just written somewhat differently. (The hecklers in the audience might observe that it isn’t much of a shorthand: just one fewer character.)

Pascal and C programmers, among others, are used to using the syntax for referring to fields of a compound object like a record. It makes sense to wonder why this notation was chosen for something else in Dylan, especially when that something else – calling functions – already had a perfectly reasonable syntax. The reason is that calling a function in Dylan is the only way to access fields in compound objects; we’ll see more of this later.

The last thing to note about this version of the function is that the for statement is the last one in the function, so the value returned by dot-product is the value of the for. What value does the loop return? Normally, there is no meaningful value to return from a loop, so an arbitrary one – the false value – was picked in Dylan. But sometimes it’s useful to return a value from a loop, and that’s what the finally clause is for. After the for loop is done, all the statements in the finally clause – here it’s just one expression – are evaluated, and the value of the last is returned. One detail worth knowing is that the statements after finally are still part of the for statement, so the variable from the numeric iteration clause can still be used.

All our versions of dot-product so far have worked in basically the same way: step through the arrays by counting from 0 to one less than the size of the array, and doing something with each element in the array. This is all we use the variable i for, and it’s pretty common that loop variables are used in this way. Dylan provides a very convenient way of doing this kind of iteration, using what’s known as a collection iteration clause:

define method dot-product (x, y)
  let sum = 0;
  for (xi in x, yi in y)
    sum := sum + xi * yi
  end for
end method dot-product;

Here we’ve done away with the numeric iteration clause, and replaced it with two collection iteration clauses. Note that a for loop may have an arbitrary number of iteration clauses, and when any of them completes, the loop is done. The iteration clause variable in expression, where the expression is an array, means execute the loop for each element in the array, from first to last, setting the variable to each value in succession. Thus the first time the loop executes, xi holds the value x[0] and yi holds y[0]; the second time through, xi holds x[1] and yi holds y[1]; and so on.

The use of multiple iteration clauses in a for is relatively common. Note that a single loop can mix numeric and collection iteration clause, along with a third form, general iteration, which we’ll see later. The important thing to realize is that the loop ends when any of the clauses causes it to end.

Again, we haven’t declared types for the parameters x and y and have just referred to them as arrays. This is inaccurate in two ways. The first, and more minor, is that one-dimensional arrays are referred to as vectors in Dylan. While the code will work on multi-dimensional arrays, it doesn’t really make mathematical sense except for vectors – there is no dot product of an array of two or more dimensions.

The more important reason that describing the arguments to dot-product as arrays is wrong is that this function will work for other types of arguments as well. Dylan has several different types known collectively as sequences, all of which share several properties: they hold a collection of objects which are numbered from zero. Vectors are one kind of sequence; others are strings, linked lists, ranges of numbers, and queues. The subscripting notation using [], the size function, and collection iteration clauses – among many other operations – may be used with all sequences.

Sequences are just one kind of collection in Dylan. A collection is a kind of object which can hold other objects. The most commonly used collection which isn’t a sequence is a table, which is Dylan’s built-in version of a hash table. Where sequences use integers starting at zero as indexes, tables can use any object.

Collections are used extensively by most Dylan programs. This should not be a surprise to people who know Pascal: many complex data structures in Pascal programs are collections of other objects, usually built up from arrays, linked lists, etc. One major difference is that where Pascal only defines the array type, in Dylan there are a variety of existing collection types to use. Moreover, there is powerful set of operations predefined on collections and sequences; we’ll see two of these in our next version of dot-product.

Another aspect of collections is that users can define their own collection types. Again, this is no surprise. What is different from Pascal is that, when programmers follow a few rules when creating a new kind of collection, it can be used with any of the predefined functions that operate on collections. We’ll see more about this in a later section.

Turning back to the example at hand, there is one more thing to understand about using collection iteration clauses instead of numeric iteration and indexing: the performance impact of the choice. When we used numeric iteration, to calculate the dot-product we had to evaluate x[i] and y[i] each time through the loop. For vectors this makes perfect sense, as indexing into a vector is an efficient operation. But if x is a linked list, for example, finding x[i] requires stepping through the first i - 1 elements of the list, which makes evaluating x[i] inside the loop an expensive operation. In more precise terms, it changes the computational complexity of the function from O(n) for vectors to O(n^2) for lists, where n is the length of the arguments.

On the other hand, collection iteration is designed to be efficient when possible. For both linked lists and arrays, iterating through the collection takes time proportional to the length of the list, so the version of dot-product which uses collection iteration has O(n) complexity for both vectors and lists. In general, one should use collection iteration when iterating over the elements of a collection without concern for what their index values are.

Now let’s take a look at a very different version of dot-product, this time using two built-in collection operations, reduce and map:

define method dot-product (x, y)
  local method add(a, b)
          a + b
        end method add;
  local method multiply(m, n)
          m * n
        end method mul;
  let x*y-sequence = map(multiply, x, y);
  reduce(add, 0, x*y-sequence)
end method dot-product;

The thing to note about this function is that there is no explicit iteration clause. Instead, the iteration is done by the built-in functions.

Let’s carefully step through what this function does. First, it defines two local functions – add and mul – which, respectively, add and multiply their arguments. It then calls the function map. Map is what is known as a higher-order function, which means it is a function that takes a function as an argument. Map creates a new collection from the results of calling a function on each element of one or more existing collections. The first argument to map is the function to use; all the other arguments are collections to draw elements from. In this case, we’ve applied map to the function multiply and the two sequences which were arguments, so it produces a sequence that has x[0] * y[0] as its first element, x[1] * y[1] as its second, and so on. (The resulting sequence is only as long as the shorter of the original sequences. If one is longer, only the elements which have corresponding members in the shorter sequence are used. As above, we’ll assume that the sequences are of the same length.)

The result of map, a sequence of the products of the elements of x and y is then attached to the local variable named x*y-sequence. There’s nothing special about the use of the asterisk in this name – it’s just one of the characters which is legal in Dylan as part of a name – but sometimes it is convenient to name a variable with a mathematical expression.

Next, we call the function reduce, which is also a higher-order function. Where map applies a function to the elements of one or more collections to produce a new collection, reduce combines the elements of a collection with a function to produce a single value. Reduce takes three arguments: a function, an initial value, and a collection. It keeps track of the result value, which starts off as the initial value argument. For each element of the collection, it computes the result of the function called with the value of the result so far and the current element, and uses that as the next value of the result. When every element of the collection has been seen, the value that has been computed so far is retuned.

In our call to reduce, the function used for combining values is add, which sums two numbers, and the initial value is 0, which is the identity for addition. Therefore, this call to reduce produces the sum of the members of a collection. In this case, the collection being summed contains the products of the elements of the arguments, so the result is the dot product.

However, there is a potential drawback to using this approach for implementing dot-product: it might be less efficient than a version that directly used a loop. The reason is that memory might have to be allocated for the intermediary result x*y-sequence, and allocating that memory takes some time, as does freeing it when it is no longer in use. (Dylan uses garbage collection, also known as automatic storage reclamation, to free memory that isn’t being used anymore, but whether the programmer or the system does it, freeing the memory does take some time. A full discussion of garbage collection is beyond the scope of this essay.) So we potentially have a bigger (more memory) and slower (more time) program. A very clever compiler could eliminate this use of extra memory, perhaps by inlining the functions and rewriting them internally as loops, but programmers concerned with efficiency probably shouldn’t count on such an optimization unless they know their compiler very well.

By the way, there is nothing special about higher-order functions: anyone can write them. Here, for example, is a portable version of reduce, which could be used if it wasn’t already part of the language:

define method reduce
    (function :: <function>, initial, collection :: <collection>)
 => value;
  let result = initial;
  for (element in collection)
    result := function(result, element)
  end for
end method reduce;

This implementation is a direct translation into Dylan of the description given above. For once, I’ve given types for the arguments, because that’s how the types are specified by the language definition. The major difference from other functions we’ve seen so far is an argument, function, is used as a function, inside the for loop; that works just as one might expect it to.

The previous version of dot-product used three locally-bound identifiers (add, multiply, and x*y-sequence) that were only used once. As in Pascal, if an expression is used only once, there is no need to associate it with a name. Unlike Pascal, though, functions in Dylan are expressions, and can be created anonymously, that is, without names. So we can rewrite dot-product as:

define method dot-product (x, y)
  reduce(method (a, b) a + b end,
         0, map(method (m, n) m * n end, x, y))
end method dot-product;

In this version of dot-product, the function we use as the first argument to reduce is the expression:

method (a, b)
  a + b

This expression creates a method – with exactly the same meaning as the add method from our previous version – but doesn’t give it a name. Nonetheless, this method can be used anywhere add could have been. Similarly, multiply has been changed to another anonymous method.

We also eliminated the variable x*y-sequence, substituting the map expression for it in the call to reduce. The meaning of the function hasn’t changed, however, so the potential inefficiency of allocating extra memory for the result of map remains.

The full syntax for an anonymous method is

method (arguments ...) => results ...;
end method

and, as with all other methods, the description of the results and the word method after end can be omitted. Since it doesn’t have a name, there is no name to put after the end method.

These examples of map and reduce have been somewhat artificial in order to introduce the use of higher-order functions. A version of dot-product in Dylan would more likely be written as:

define method dot-product (x, y)
  reduce(\+, 0, map(\*, x, y))
end method dot-product;

In the previous two examples, we created a function which took two arguments and added them. The first time it was named add; the second, it was anonymous and passed as the first argument to reduce. In fact, there is no real need to create such a function, because one already exists: it’s called +.

Because + is normally used as an operator in traditional arithmetic notation, it is easy to forget that it’s a function. In Pascal and C, in fact, operators like + have different rules from functions and can’t just be used like any other function, but in Dylan they can. To avoid syntactic confusion with + used as an operator, when, for example, we refer to the function name in order to pass it as an argument to another function we have to precede it with a backslash, as above.

This function has the same result as the previous two, but we have avoided introducing the intermediate functions and let map and reduce call * and + directly.

This version of dot-product is very concise and, for a programmer used to map and reduce, easy to read. There are some very common idioms in Dylan built around these and other operators. For example, there is no built-in function to sum a sequence: reduce obviates the need for it.

The use of higher-order functions is usually associated with the functional style of programming, which is identified with languages such as Standard ML, Haskell, and some dialects of Lisp, notably Scheme. In Dylan it is often convenient to use this style for parts of programs – often those dealing with operations on collections, as we’ve seen – and use other styles, such as object-oriented or procedural, in other parts. One of the goals for Dylan is to support multiple styles, because there is no one single, “right” way to structure a program. The functional style is a useful one, and is well supported in Dylan; we’ll see more of it later.

On the other hand, it is possible to take this approach to extremes and write large, complicated expressions with no named intermediate values. While some people enjoy writing large blocks of code that way, the result is often an unreadable program. APL, another language which makes pervasive use of higher-order functions, is often criticized for encouraging a “write-only” programming style, with very complex expressions. Introducing a few named values with let or splitting one method into several that calculate different parts of a result can often turn a large, hard-to-follow expression into something more readable and maintainable.

Back – Conditions and Multiple Values: The Quadratic Formula

Copyright © 1995 Paul Haahr. All rights reserved.