Implement till 3.66

ex-3_50-62.scm

@@ -34,15 +34,6 @@
 
 (display "\nex-3.51\n")
 
-(define (show x)
-  (display-line x)
-  x)
-
-(define (stream-ref s n)
-  (if (= n 0)
-      (stream-car s)
-      (stream-ref (stream-cdr s) (- n 1))))
-
 (define x (stream-map show (stream-enumerate-interval 0 10)))
 
 (stream-ref x 5)
@@ -59,9 +50,6 @@
 
 (display "\nex-3.52\n")
 
-(define (display-stream s)
-  (stream-for-each display-line s))
-
 (define sum 0)
 
 (define (accum x)
@@ -106,19 +94,10 @@
 
 (display "\nex-3.53\n")
 
-(define (add-streams s1 s2)
-  (stream-map + s1 s2))
-
 (define ones (cons-stream 1 ones))
 
 (define integers (cons-stream 1 (add-streams ones integers)))
 
-(define (take n xs)
-  (if (= n 0)
-    '()
-    (cons (stream-car xs)
-          (take (- n 1) (stream-cdr xs)))))
-
 (assert (take 3 integers)
         '(1 2 3))
 
@@ -131,9 +110,6 @@
 (assert (take 7 fibs)
         '(0 1 1 2 3 5 8))
 
-(define (scale-stream stream factor)
-  (stream-map (lambda (x) (* x factor)) stream))
-
 (define double (cons-stream 1 (scale-stream double 2)))
 
 (assert (take 3 double)

ex-3_63-xx.scm

@@ -1,4 +1,104 @@
 (load "util.scm")
 
-(display "\nex-3.63\n")
+(define (pi-summands n)
+  (cons-stream (/ 1. n)
+               (stream-map - (pi-summands (+ n 2)))))
 
+(define pi-stream
+  (scale-stream (partial-sums (pi-summands 1)) 4))
+
+(define (euler-transform s)
+  (let ((s0 (stream-ref s 0))           ; Sn-1
+        (s1 (stream-ref s 1))           ; Sn
+        (s2 (stream-ref s 2)))          ; Sn+1
+    (cons-stream (- s2 (/ (square (- s2 s1))
+                          (+ s0 (* -2 s1) s2)))
+                 (euler-transform (stream-cdr s)))))
+
+(define (make-tableau transform s)
+  (cons-stream s
+               (make-tableau transform
+                             (transform s))))
+
+(define (accelerated-sequence transform s)
+  (stream-map stream-car
+              (make-tableau transform s)))
+
+; (display (take 5 pi-stream))
+; (newline)
+; (display (take 5 (euler-transform pi-stream)))
+; (newline)
+; (display (take 5 (accelerated-sequence euler-transform pi-stream)))
+; (newline)
+
+(display "\nex-3.63 - sqrt-stream\n")
+
+(define (sqrt-improve guess x)
+  (average guess (/ x guess)))
+
+(define (sqrt-stream x)
+  (cons-stream 1.0
+               (stream-map (lambda (guess)
+                             (sqrt-improve guess x))
+                           (sqrt-stream x))))
+
+(define (sqrt-stream x)
+  (define guesses
+    (cons-stream 1.0
+                 (stream-map (lambda (guess)
+                               (sqrt-improve guess x))
+                             guesses)))
+  guesses)
+
+(display (stream-ref (sqrt-stream 2) 1000))
+(newline)
+
+; The first implementation of sqrt-stream computes each value of the stream
+; only once. Louis' suggestion computes all previous values because of the
+; recursive calls to sqrt-stream. If memoization was not used the two solutions
+; would behave in the same way.
+
+(display "\nex-3.64 - stream-limit\n")
+
+(define (stream-limit stream tolerance)
+  (if (< (abs (- (stream-car stream)
+                 (stream-car (stream-cdr stream))))
+         tolerance)
+    (stream-car (stream-cdr stream))
+    (stream-limit (stream-cdr stream) tolerance)))
+
+(define (sqrt-tol x tolerance)
+  (stream-limit (sqrt-stream x) tolerance))
+
+(assert (< (abs (- 1.4142135623730951 (sqrt-tol 2 0.01)))
+           0.01) #t)
+
+(assert (< (abs (- 4.795831523312719 (sqrt-tol 23 0.001)))
+           0.001) #t)
+
+(display "\nex-3.65 - ln2\n")
+
+(define (ln2-summands n)
+  (cons-stream (/ 1. n)
+               (stream-map - (ln2-summands (+ n 1)))))
+
+(define ln2-stream
+  (partial-sums (ln2-summands 1)))
+
+; slow
+(define (ln2-tol tolerance)
+  (stream-limit ln2-stream tolerance))
+
+; fast
+(define (ln2-tol tolerance)
+  (stream-limit (accelerated-sequence euler-transform ln2-stream) tolerance))
+
+(assert (ln2-tol 0.00000000001)
+        0.6931471805599445)
+
+; The series converges slowly. Only with acceleration we get a good result in
+; reasonable time.
+
+(display "\nex-3.66\n")
+
+; (display "\nex-3.67\n")

util.scm

@@ -68,4 +68,33 @@
   (display x)
   (newline))
 
+(define (take n xs)
+  (if (= n 0)
+    '()
+    (cons (stream-car xs)
+          (take (- n 1) (stream-cdr xs)))))
+
+(define (display-stream s)
+  (stream-for-each display-line s))
+
+(define (show x)
+  (display-line x)
+  x)
+
+(define (stream-ref s n)
+  (if (= n 0)
+      (stream-car s)
+      (stream-ref (stream-cdr s) (- n 1))))
+
+(define (partial-sums xs)
+  (cons-stream (stream-car xs)
+               (add-streams (partial-sums xs)
+                            (stream-cdr xs))))
+
+(define (scale-stream stream factor)
+  (stream-map (lambda (x) (* x factor)) stream))
+
+(define (add-streams s1 s2)
+  (stream-map + s1 s2))
+
 'util-loaded