Initial commit

.gitignore

@@ -0,0 +1 @@
+_instructions.txt

1_11.scm

@@ -0,0 +1,9 @@
+(define (fn-rec n)
+  (cond ((< n 3) n)
+        (else (+ (fn-rec (- n 1))
+                 (* 2 (fn-rec (- n 2)))
+                 (* 3 (fn-rec (- n 3)))))))
+
+(fn-rec 3)
+
+; TODO: iterative

1_21.scm

@@ -0,0 +1,20 @@
+(define (square n)
+  (* n n))  
+
+(define (divides? a b)
+  (= (remainder b a) 0))
+
+(define (find-divisor n test-divisor)
+  (cond ((> (square test-divisor) n) n)
+        ((divides? test-divisor n) test-divisor)
+        (else (find-divisor n (+ test-divisor 1)))))
+
+(define (smallest-divisor n)
+  (find-divisor n 2))
+
+(define (prime? n)
+  (= n (smallest-divisor n)))
+
+(smallest-divisor 199)
+(smallest-divisor 1999)
+(smallest-divisor 19999)

1_22.scm

@@ -0,0 +1,38 @@
+; Chicken Scheme has been used.
+
+(define (square n)
+  (* n n))
+
+(define (divides? a b)
+  (= (remainder b a) 0))
+
+(define (find-divisor n test-divisor)
+  (cond ((> (square test-divisor) n) n)
+        ((divides? test-divisor n) test-divisor)
+        (else (find-divisor n (+ test-divisor 1)))))
+
+(define (smallest-divisor n)
+  (find-divisor n 2))
+
+(define (prime? n)
+  (= n (smallest-divisor n)))
+
+(define (report-prime elapsed-time)
+  (display " *** ")
+  (display elapsed-time))
+
+(define (start-prime-test n start-time)
+  (if (prime? n)
+    (report-prime (- (current-milliseconds) start-time))))
+
+(define (timed-prime-test n)
+  (newline)
+  (display n)
+  (newline)
+  (start-prime-test n (current-milliseconds)))
+
+(define (search-for-primes n)
+  (cond ((prime? n) n)
+        (else (search-for-primes (+ n 2)))))
+
+(time (search-for-primes 10000000001 ))

1_23.scm

@@ -0,0 +1,37 @@
+; Chicken Scheme has been used.
+
+(define (square n)
+  (* n n))
+
+(define (next n)
+  (if (= n 2) 3
+    (+ n 2)))
+
+(define (divides? a b)
+  (= (remainder b a) 0))
+
+(define (find-divisor n test-divisor)
+  (cond ((> (square test-divisor) n) n)
+        ((divides? test-divisor n) test-divisor)
+        (else (find-divisor n (next test-divisor)))))
+
+(define (smallest-divisor n)
+  (find-divisor n 2))
+
+(define (prime? n)
+  (= n (smallest-divisor n)))
+
+(define (report-prime elapsed-time)
+  (display " *** ")
+  (display elapsed-time))
+
+(define (start-prime-test n start-time)
+  (if (prime? n)
+    (report-prime (- (current-milliseconds) start-time))))
+
+(define (search-for-primes n)
+  (cond ((prime? n) n)
+        (else (search-for-primes (+ n 2)))))
+
+(time (search-for-primes 10000000001 ))
+(time (search-for-primes 100000000001 ))

1_24.scm

@@ -0,0 +1,29 @@
+; Chicken Scheme has been used.
+
+(define (square n)
+  (* n n))
+
+(define (expmod base exp m)
+  (cond ((= exp 0) 1)
+        ((even? exp)
+          (remainder (square (expmod base (/ exp 2) m))
+                  m))
+        (else
+          (remainder (* base (expmod base (- exp 1) m))
+                  m))))
+
+(define (fermat-test n)
+  (define (try-it a)
+    (= (expmod a n n) a))
+  (try-it (+ 1 (random (- n 1)))))
+
+(define (fast-prime? n times)
+  (cond ((= times 0) #t)
+        ((fermat-test n) (fast-prime? n (- times 1)))
+        (else #f)))
+
+(define (search-for-primes n)
+  (cond ((fast-prime? n 1000) n)
+        (else (search-for-primes (+ n 2)))))
+
+(time (search-for-primes 1000000001 ))

1_27.scm

@@ -0,0 +1,32 @@
+(define (square n)
+  (* n n))
+
+(define (expmod base exp m)
+  (cond ((= exp 0) 1)
+        ((even? exp)
+          (remainder (square (expmod base (/ exp 2) m))
+                  m))
+        (else
+          (remainder (* base (expmod base (- exp 1) m))
+                  m))))
+
+(define (fermat-test n)
+  (define (try-it a)
+    (= (expmod a n n) a))
+  (try-it (+ 1 (random (- n 1)))))
+
+(define (fast-prime? n times)
+  (cond ((= times 0) #t)
+        ((fermat-test n) (fast-prime? n (- times 1)))
+        (else #f)))
+
+(define (search-for-primes n)
+  (cond ((fast-prime? n 100) n)
+        (else (search-for-primes (+ n 2)))))
+
+(fast-prime? 561 100)
+(fast-prime? 1105 100)
+(fast-prime? 1729 100)
+(fast-prime? 2465 100)
+(fast-prime? 2821 100)
+(fast-prime? 6601 100)

1_29.scm

@@ -0,0 +1,32 @@
+(define (cube x) (* x x x))
+
+(define (sum term a next b)
+  (if (> a b)
+    0
+    (+ (term a)
+      (sum term (next a) next b))))
+
+(define (simpson-coef i n)
+  (cond ((= i 0) 1)
+        ((= i n) 1)
+        ((= (modulo i 2) 0) 2)
+        ((= (modulo i 2) 1) 4)
+      ))
+
+(define (integral f a b dx)
+  (define (add-dx x) (+ x dx))
+  (* (sum f (+ a (/ dx 2.0)) add-dx b)
+      dx))
+
+(define (integral-simpson f a b n)
+  (define h (/ (- b a) n))
+  (define (step x) (+ x 1))
+  (define (func k)
+    (define (y_k k)
+      (f (+ a (* k h))))
+    (cond ((or (= k 0) (= k n)) (y_k k))
+          ((even? k) (* 2 (y_k k)))
+          (else (* 4 (y_k k)))))
+  (* (/ h 3) (sum func a step n)))
+
+(integral-simpson cube 0 1 101)

1_30.scm

@@ -0,0 +1,33 @@
+(define (cube x) (* x x x))
+
+(define (sum term a next b)
+  (define (iter a result)
+    (if (> a b)
+      result
+      (iter (next a) (+ result (term a)))))
+  (iter a 0))
+
+(define (simpson-coef i n)
+  (cond ((= i 0) 1)
+        ((= i n) 1)
+        ((= (modulo i 2) 0) 2)
+        ((= (modulo i 2) 1) 4)
+      ))
+
+(define (integral f a b dx)
+  (define (add-dx x) (+ x dx))
+  (* (sum f (+ a (/ dx 2.0)) add-dx b)
+      dx))
+
+(define (integral-simpson f a b n)
+  (define h (/ (- b a) n))
+  (define (step x) (+ x 1))
+  (define (func k)
+    (define (y_k k)
+      (f (+ a (* k h))))
+    (cond ((or (= k 0) (= k n)) (y_k k))
+          ((even? k) (* 2 (y_k k)))
+          (else (* 4 (y_k k)))))
+  (* (/ h 3) (sum func a step n)))
+
+(integral-simpson cube 0 1 101)

1_31.scm

@@ -0,0 +1,27 @@
+(define (product term a next b)
+  (define (iter a result)
+    (if (> a b)
+      result
+      (iter (next a) (* result (term a)))))
+  (iter a 1))
+
+(define (factorial n)
+  (define (inc x) (+ x 1))
+  (define (func x) x)
+  (product func 1 inc n))
+
+(define (ceiled-even n)
+  (if (even? n) n
+    (+ n 1)))
+
+(define (ceiled-odd n)
+  (if (even? n) (- n 1)
+    n))
+
+(define (calculate-pi n)
+  (define (inc x) (+ x 1))
+  (define numerator (product ceiled-even 2 inc (+ 2 n)))
+  (define denominator (product ceiled-odd 3 inc (+ 3 n)))
+  (* 4 (/ numerator denominator)))
+
+(calculate-pi 160)

1_32.scm

@@ -0,0 +1,48 @@
+(define (accumulate combiner null-value term a next b)
+  (define (iter a result)
+    (if (> a b)
+      result
+      (iter (next a) (combiner result (term a)))))
+  (iter a null-value))
+
+(define (sum term a next b)
+  (if (> a b)
+    0
+    (+ (term a)
+      (sum term (next a) next b))))
+
+(define (accumulate-rec combiner null-value term a next b)
+  (if (> a b)
+    null-value
+    (combiner (term a) (accumulate-rec combiner null-value term (next a) next b))))
+
+(define (product term a next b)
+  (define (func x y) (* x y))
+  ;(display (func 2 3))
+  (accumulate func 1 term a next b))
+
+(define (sum term a next b)
+  (define (func x y) (+ x y))
+  (accumulate func 0 term a next b))
+
+
+(define (factorial n)
+  (define (inc x) (+ x 1))
+  (define (func x) x)
+  (product func 1 inc n))
+
+(define (ceiled-even n)
+  (if (even? n) n
+    (+ n 1)))
+
+(define (ceiled-odd n)
+  (if (even? n) (- n 1)
+    n))
+
+(define (calculate-pi n)
+  (define (inc x) (+ x 1))
+  (define numerator (product ceiled-even 2 inc (+ 2 n)))
+  (define denominator (product ceiled-odd 3 inc (+ 3 n)))
+  (* 4 (/ numerator denominator)))
+
+(calculate-pi 1)

1_35.scm

@@ -0,0 +1,19 @@
+(define (average x1 x2) (/ (+ x1 x2) 2))
+
+(define tolerance 0.00001)
+
+(define (fixed-point f first-guess)
+  (define (close-enough? v1 v2)
+    (< (abs (- v1 v2)) tolerance))
+  (define (try guess)
+    (let ((next (f guess)))
+      (if (close-enough? guess next)
+        next
+        (try next))))
+  (try first-guess))
+
+(define (sqrt x)
+(fixed-point (lambda (y) (y (/ x y)))
+            1.0))
+
+(fixed-point (lambda (x) (+ 1 (/ 1 x))) 1.0)

1_36.scm

@@ -0,0 +1,13 @@
+(define tolerance 0.00001)
+
+(define (fixed-point f first-guess)
+  (define (close-enough? v1 v2)
+    (< (abs (- v1 v2)) tolerance))
+  (define (try guess)
+    (let ((next (f guess)))
+      (if (close-enough? guess next)
+        next
+        (try next))))
+  (try first-guess))
+
+(fixed-point (lambda (x) (/ (log 1000) (log x))) 2)

1_37.scm

@@ -0,0 +1,12 @@
+(define (cont-frac n d k)
+  (define (cont-frac-iter i result)
+    (let ((n_k (n i))
+        (d_k (d i)))
+        (cond ((= i 0) result)
+              ((= i k) (/ n_k d_k))
+              (else (cont-frac-iter (- i 1) result)))))
+  (cont-frac-iter k 0))
+
+(/ 1 (cont-frac (lambda (i) 1.0)
+           (lambda (i) 1.0)
+           7))

1_39.scm

@@ -0,0 +1,12 @@
+(define (tan-cf x k)
+  (define (numerator x i)
+    (if (= i 0) x
+      (* x x)))
+  (define (denominator i)
+    (- (* (+ i 1) 2) 1))
+  (define (sub x k i)
+    (cond ((= i k) (/ (numerator x i) (denominator i)))
+      (else (/ (numerator x i) (- (denominator i) (sub x k (+ i 1)))))))
+  (sub x k 0))
+
+(tan-cf 0.5 100)