Comments on: SICP section 1.3.4 http://eli.thegreenplace.net/2007/07/19/sicp-section-134/ Eli Bendersky's personal website Thu, 04 Dec 2008 00:48:31 +0000 http://wordpress.org/?v=2.6.2 By: gorilych http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-133138 gorilych Wed, 01 Oct 2008 10:40:16 +0000 http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-133138 btw, we can use procedure accumulate from ex. 1.32 for ex. 1.43: <code> ; 1.43 ; we need compose from ex. 1.42: (load "1.42.lisp") ; and accumulate from ex. 1.32: (load "1.32.lisp") (defun repeated (f n) (accumulate #'compose (lambda (x) x) ; identity function as a null-value (lambda (i) f) 1 #'1+ n)) </code> Note that we already have 'accumulate' to produce iterative process. btw, we can use procedure accumulate from ex. 1.32 for ex. 1.43:


; 1.43

; we need compose from ex. 1.42:
(load "1.42.lisp")
; and accumulate from ex. 1.32:
(load "1.32.lisp")

(defun repeated (f n)
(accumulate #'compose
(lambda (x) x) ; identity function as a null-value
(lambda (i) f)
1
#'1+
n))

Note that we already have ‘accumulate’ to produce iterative process.

]]> By: eliben http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-132920 eliben Mon, 29 Sep 2008 12:03:07 +0000 http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-132920 @gorilych You're right - the word <i>iterative</i> is confusing in this context, so I removed it. @gorilych
You’re right - the word iterative is confusing in this context, so I removed it.

]]> By: gorilych http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-132917 gorilych Mon, 29 Sep 2008 09:58:32 +0000 http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-132917 Your solution for ex. 1.43 produces recursive process, not iterative as you states. Your solution for ex. 1.43 produces recursive process, not iterative as you states.

]]> By: eliben http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-98568 eliben Sun, 13 Jan 2008 19:19:43 +0000 http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-98568 Sean, you're absolutely right - thanks for the tip! I fixed the code in exercise 1.45 Sean, you’re absolutely right - thanks for the tip!
I fixed the code in exercise 1.45

]]> By: Sean http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-98508 Sean Sun, 13 Jan 2008 12:21:11 +0000 http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-98508 I think your algorithm is bad for 1.45: repeated-dampen-root, repeatedly applies f as well as the average-damp. You should only be repeating average-damp. I investigated with <code> (define (try-ave-damp-root num-ave-damps n x) (fixed-point-of-transform (lambda (y) (/ x (expt y (-1+ n)))) (repeated average-damp num-ave-damps) 1.0)) </code> and found that (2^n)th root takes n average-damps. I took tolerance down to 0.0000001 in my fixed-point test to stop some spurious convergences. This is on MIT/GNU Scheme 7.7.90. So a solution is: <code> (define (log2 n) (if (= 1 n) 0 (1+ (log2 (floor (/ n 2)))))) (define (nth-root n x) (fixed-point-of-transform (lambda (y) (/ x (expt y (-1+ n)))) (repeated average-damp (log2 n)) 1.0)) </code> I think your algorithm is bad for 1.45: repeated-dampen-root, repeatedly applies f as well as the average-damp. You should only be repeating average-damp.

I investigated with


(define (try-ave-damp-root num-ave-damps n x)
(fixed-point-of-transform (lambda (y) (/ x (expt y (-1+ n))))
(repeated average-damp num-ave-damps)
1.0))

and found that (2^n)th root takes n average-damps. I took tolerance down to 0.0000001 in my fixed-point test to stop some spurious convergences. This is on MIT/GNU Scheme 7.7.90.

So a solution is:


(define (log2 n)
(if (= 1 n)
0
(1+ (log2 (floor (/ n 2))))))

(define (nth-root n x)
(fixed-point-of-transform (lambda (y) (/ x (expt y (-1+ n))))
(repeated average-damp (log2 n))
1.0))

]]> By: Jeff http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-93363 Jeff Wed, 19 Dec 2007 17:03:57 +0000 http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-93363 For 1.45, I couldn't find a relationship between n (the root magnitude) and the number of times that damping should be applied either. For 1.45, I couldn’t find a relationship between n (the root magnitude) and the number of times that damping should be applied either.

]]> By: eliben http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-60542 eliben Thu, 19 Jul 2007 15:10:29 +0000 http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-60542 I see you also had trouble with ex. 1.45, right? I see you also had trouble with ex. 1.45, right?

]]> By: Ken Dyck http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-60526 Ken Dyck Thu, 19 Jul 2007 11:30:13 +0000 http://eli.thegreenplace.net/2007/07/19/sicp-section-134/#comment-60526 Yikes! He's catching up. I'd better get a move on. ;) Yikes! He’s catching up. I’d better get a move on. ;)

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