Some important quotes from the book:
We began section 1.3 with the observation that compound procedures are a crucial abstraction mechanism, because they permit us to express general methods of computing as explicit elements in our programming language. Now we’ve seen how higher-order procedures permit us to manipulate these general methods to create further abstractions.
[...] In general, programming languages impose restrictions on the ways in which computational elements can be manipulated. Elements with the fewest restrictions are said to have first-class status. Some of the ``rights and privileges’’ of first-class elements are: (1) They may be named by variables; (2) They may be passed as arguments to procedures; (3) They may be returned as the results of procedures; (4) They may be included in data structures; Lisp, unlike other common programming languages, awards procedures full first-class status. This poses challenges for efficient implementation, but the resulting gain in expressive power is enormous.
Here is the code for
newtons-method in CL. Note that this time I’ll follow Lisp’s convention of signifying the names of global variables by asterisks:
(defvar *tolerance* 0.00001) (defun fixed-point (f first-guess) (labels ( (close-enough? (v1 v2) (< (abs (- v1 v2)) *tolerance*)) (try (guess) (let ((next (funcall f guess))) (if (close-enough? guess next) next (try next))))) (try first-guess))) (defvar *dx* 0.00001) (defun deriv (g) (lambda (x) (/ (- (funcall g (+ x *dx*)) (funcall g x)) *dx*))) (defun newton-transform (g) (lambda (x) (- x (/ (funcall g x) (funcall (deriv g) x))))) (defun newtons-method (g guess) (fixed-point (newton-transform g) guess))
Note again that CL’s semantics of handling functions makes the code a little more cumbersome than the Scheme code of the authors. While anonymous functions defined by
lambda can be passed as arguments where functions are expected (for example the second argument in the call to
newtons-method), when applied to arguments explicitly, a
funcall must be used (for example the application of
(deriv g) in the last line of
This is just an exercise in building functions from arguments and returning them.
(defun cubic (a b c) (lambda (x) (+ (cube x) (* a (square x)) (* b x) c)))
cubic is a function builder – each time it’s called with some arguments, it creates a new function and returns is.
Again, this is a function builder. It is even more interesting, because its argument is a function and not numbers. So
double takes a function as an argument and returns a function – it’s almost a full case study of higher-order functions by itself!
(defun double (f) (lambda (x) (funcall f (funcall f x)))) (print (funcall (double #'1+) 1)) => 3
(double double) do ? Applies f twice to its argument, when f is a function that applies some function twice to its argument. Therefore, it applies the function it receives 4 times.
((double (double double)) applies the function it receives 4*4 = 16 times. Therefore:
(print (funcall (funcall (double (double #'double)) #'1+) 5)) => 21
This is very similar to the previous exercise:
(defun compose (f g) (lambda (x) (funcall f (funcall g x))))
(defun repeated (f n) (if (= n 0) #'identity (compose f (repeated f (1- n)))))
This is a process that “accumulates” applications of
f on itself. Note the stop condition –
identity is returned for
n = 0. This means that for
n = 1,
f is composed with
identity, and the result is just applying
f once, which is what we need.
(defvar *dx* 0.00001) (defun smooth (f) (lambda (x) (/ (+ (funcall f (- x *dx*)) (funcall f x) (funcall f (+ x *dx*))) 3))) (defun n-fold-smooth (f n) (funcall (repeated #'smooth n) f))
First, recall the code of section 1.3.3 and the solutions to its exercises. Here is the
dampen-sqrt function again:
(defvar *tolerance* 0.00001) (defun fixed-point (f first-guess) (labels ( (close-enough? (v1 v2) (< (abs (- v1 v2)) *tolerance*)) (try (guess) (let ((next (funcall f guess))) (if (close-enough? guess next) next (try next))))) (try first-guess))) (defun average (a b) (/ (+ a b) 2)) (defun dampen-sqrt (x) (fixed-point (lambda (y) (average y (/ x y))) 1.0))
Now, let’s implement a more general dampening function for any root of x:
(defun dampen-root (x n) (fixed-point (lambda (y) (average y (/ x (expt y (1- n))))) 1.0))
n will be 2 for the square root, 3 for the cube root, 4 for the 4th root and so on. Using this function we can try computing various roots:
(print (dampen-root 2 2)) => 1.4142 (print (dampen-root 2 3)) => 1.2599 (print (dampen-root 2 4)) => *** - Program stack overflow. RESET
Indeed, the simple 1-step dampening doesn’t work for the 4th root – the computation doesn’t converge. So let’s follow the authors’ advice and implement a repeated dampening1 version:
(defun repeated-dampen-root (x nroot nrepeat) (fixed-point-of-transform (lambda (y) (average y (/ x (expt y (1- nroot))))) (repeated #'average-damp nrepeat) 1.0)) (print (repeated-dampen-root 2 4 2)) => 1.89
This is better.
(defun iterative-improve (good-enough? improve) (lambda (first-guess) (labels ( (improve-iter (guess) (let ((improved-guess (funcall improve guess))) (if (funcall good-enough? guess improved-guess) improved-guess (improve-iter improved-guess))))) (improve-iter first-guess)))) (defun improved-sqrt (num) (funcall (iterative-improve (lambda (x y) (let ((ratio (/ x y))) (and (< ratio 1.001) (> ratio 0.999)))) (lambda (guess) (average guess (/ num guess)))) 1.0)) (defvar *tolerance* 0.00001) (defun improved-fixed-point (f first-guess) (funcall (iterative-improve (lambda (x y) (< (abs (- x y)) *tolerance*)) (lambda (guess) (funcall f guess))) first-guess))
1 Thanks to Sean who suggested a bug fix in this function (see Comments to this post).