<TeXmacs|2.1.2>
<style|<tuple|generic|no-page-numbers|chinese>>
<\body>
<subsection*|1.3.1>
<\session|scheme|default>
<\unfolded-io|Scheme] >
(define (cube x) (* x x x))
<|unfolded-io>
cube
</unfolded-io>
<\unfolded-io|Scheme] >
(define (sum-integers a b)
\ \ (if (\<gtr\> a b)
\ \ \ \ \ \ 0
\ \ \ \ \ \ (+ a (sum-integers (+ a 1) b))))
<|unfolded-io>
sum-integers
</unfolded-io>
<\unfolded-io|Scheme] >
(define (sum-cubes a b)
\ \ (if (\<gtr\> a b)
\ \ \ \ \ \ 0
\ \ \ \ \ \ (+ (cube a) (sum-cubes (+ a 1) b))))
<|unfolded-io>
sum-cubes
</unfolded-io>
<\unfolded-io|Scheme] >
(define (pi-sum a b)
\ \ (if (\<gtr\> a b)
\ \ \ \ \ \ 0
\ \ \ \ \ \ (+ (/ 1.0 (* a (+ a 2))) (pi-sum (+ a 4) b))))
<|unfolded-io>
pi-sum
</unfolded-io>
<\unfolded-io|Scheme] >
(sum-integers 1 3)
<|unfolded-io>
6
</unfolded-io>
<\unfolded-io|Scheme] >
(sum-cubes 1 3)
<|unfolded-io>
36
</unfolded-io>
<\unfolded-io|Scheme] >
(pi-sum 1 3)
<|unfolded-io>
0.3333333333333333
</unfolded-io>
<\unfolded-io|Scheme] >
(define (sum term a next b)
\ \ (if (\<gtr\> a b)
\ \ \ \ \ \ 0
\ \ \ \ \ \ (+ (term a)
\ \ \ \ \ \ \ \ \ (sum term (next a) next b))))
<|unfolded-io>
sum
</unfolded-io>
<\unfolded-io|Scheme] >
(define (inc n) (+ n 1))
<|unfolded-io>
inc
</unfolded-io>
<\unfolded-io|Scheme] >
(define (sum-cubes-2 a b)
\ \ (sum cube a inc b))
<|unfolded-io>
sum-cubes-2
</unfolded-io>
<\unfolded-io|Scheme] >
(sum-cubes-2 1 3)
<|unfolded-io>
36
</unfolded-io>
<\input|Scheme] >
\;
</input>
<\unfolded-io|Scheme] >
(define (identity x) x)
<|unfolded-io>
identity
</unfolded-io>
<\unfolded-io|Scheme] >
(define (sum-integers-2 a b)
\ \ (sum identity a inc b))
<|unfolded-io>
sum-integers-2
</unfolded-io>
<\unfolded-io|Scheme] >
(sum-integers-2 1 10)
<|unfolded-io>
55
</unfolded-io>
<\unfolded-io|Scheme] >
(define (pi-sum-2 a b)
\ \ (define (pi-term x)
\ \ \ \ (/ 1.0 (* x (+ x 2))))
\ \ (define (pi-next x)
\ \ \ \ (+ x 4))
\ \ (sum pi-term a pi-next b))
<|unfolded-io>
pi-sum-2
</unfolded-io>
<\unfolded-io|Scheme] >
(* 8 (pi-sum-2 1 1000))
<|unfolded-io>
3.139592655589783
</unfolded-io>
<\input|Scheme] >
\;
</input>
</session>
<subsection*|1.3.2>
<\session|scheme|default>
<\unfolded-io|Scheme] >
(define (square x) (* x x))
<|unfolded-io>
square
</unfolded-io>
<\unfolded-io|Scheme] >
((lambda (x y z) (+ x y (square z))) 1 2 3)
<|unfolded-io>
12
</unfolded-io>
<\input|Scheme] >
\;
</input>
</session>
<subsection*|1.3.3>
<\session|scheme|default>
<\unfolded-io|Scheme] >
(define (average x y) (/ (+ x y) 2))
<|unfolded-io>
average
</unfolded-io>
<\unfolded-io|Scheme] >
(define (search f neg-point pos-point)
\ \ (let ((midpoint (average neg-point pos-point)))
\ \ \ \ (if (close-enough? neg-point pos-point)
\ \ \ \ \ \ \ \ midpoint
\ \ \ \ \ \ \ \ (let ((test-value (f midpoint)))
\ \ \ \ \ \ \ \ \ \ (cond ((positive? test-value)
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (search f neg-point midpoint))
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ((negative? test-value)
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (search f midpoint pos-point))
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (else midpoint))))))
<|unfolded-io>
search
</unfolded-io>
<\unfolded-io|Scheme] >
(define (close-enough? x y)
\ \ (\<less\> (abs (- x y)) 0.001))
<|unfolded-io>
close-enough?
</unfolded-io>
<\unfolded-io|Scheme] >
(define (half-interval-method f a b)
\ \ (let ((a-value (f a))
\ \ \ \ \ \ \ \ (b-value (f b)))
\ \ \ \ (cond ((and (negative? a-value) (positive? b-value))
\ \ \ \ \ \ \ \ \ \ \ (search f a b))
\ \ \ \ \ \ \ \ \ \ ((and (negative? b-value) (positive? a-value))
\ \ \ \ \ \ \ \ \ \ \ (search f b a))
\ \ \ \ \ \ \ \ \ \ (else
\ \ \ \ \ \ \ \ \ \ \ (error "Values are not of opposite sign" a b)))))
<|unfolded-io>
half-interval-method
</unfolded-io>
<\unfolded-io|Scheme] >
(half-interval-method sin 2.0 4.0)
<|unfolded-io>
3.14111328125
</unfolded-io>
<\unfolded-io|Scheme] >
(half-interval-method (lambda (x) (- (* x x x) (* 2 x) 3))
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.0
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.0)
<|unfolded-io>
1.89306640625
</unfolded-io>
<\input|Scheme] >
\;
</input>
<\unfolded-io|Scheme] >
(define tolerance 0.00001)
<|unfolded-io>
0.00001
</unfolded-io>
<\unfolded-io|Scheme] >
(define (fixed-point f first-guess)
\ \ (define (close-enough? v1 v2)
\ \ \ \ (\<less\> (abs (- v1 v2)) tolerance))
\ \ (define (try guess)
\ \ \ \ (let ((next (f guess)))
\ \ \ \ \ \ (if (close-enough? guess next)
\ \ \ \ \ \ \ \ \ \ next
\ \ \ \ \ \ \ \ \ \ (try next))))
\ \ (try first-guess))
<|unfolded-io>
fixed-point
</unfolded-io>
<\unfolded-io|Scheme] >
(fixed-point cos 1.0)
<|unfolded-io>
0.7390822985224024
</unfolded-io>
<\unfolded-io|Scheme] >
(fixed-point (lambda (y) (+ (sin y) (cos y)))
\ \ \ \ \ \ \ \ \ \ \ \ \ 1.0)
<|unfolded-io>
1.2587315962971173
</unfolded-io>
<\input|Scheme] >
\;
</input>
</session>
</body>
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<\associate|toc>
<with|par-left|<quote|1tab>|1.3.1 <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1>>
<with|par-left|<quote|1tab>|1.3.2 <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2>>
<with|par-left|<quote|1tab>|1.3.3 <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-3>>
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