Download presentation
Presentation is loading. Please wait.
1
CS220 Programming Principles
프로그래밍의 이해 2003 가을학기 Class 3 한 태숙
2
Higher Order Procedure
Building Abstraction : assign a name to common pattern and then work in terms of the abstraction directly Accept procedure as arguments or return procedures as values (define (cube x) (* x x x)) (define (hf f x) (f x))
3
A simple common pattern
(* 2 2) (* 3 3) (* 4 4) is captured by (define square (lambda (x) (* x x)) (define (square x) (* x x)) Provides a NAME to the idea of multiplying something by itself
4
Using lambda for Local Name
Want to compute f(x) = x2/1-x2 We don’t want to compute x2 twice (define (f x) ((lambda (xsq) (/ xsq (- 1 xsq))) (square x)))
5
Syntactic sugar (define (inc x) (+ 1 x))
(define inc (lambda (x) (+ 1 x))) (define (f x) (let ((xsq (square x))) (/ xsq (- 1 xsq))))
6
Three Sums 1 + 2 + … + 100 12+ 22+ … + 1002 1/12 + 1/32 + … + 1/992
(define (sum-integers a b) (if (> a b) (+ a (sum-integers (+ a 1) b))))
7
(define (sum-squares a b)
(if (> a b) (+ (square a) (sum-squares (+ a 1) b)))) (define (pi-sum a b) (+ (/ 1 (square a)) (pi-sum (+ a 2) b)))) Sch-Num x Sch-Num -> Sch-Num
8
Type and Pattern of Procedure
(define (sum term a next b) (if (> a b) (+ (term a) (sum term (next a) next b)))) F x Sch-Num x F x Sch-Num -> Sch-Num where F =(Sch-Num -> Sch-Num)
9
Sum as a high-order function
(define (inc i) (+ i 1)) (define (sum-integers a b) (sum (lambda (x) x) a inc b)) (define (sum-squares a b) (sum square a inc b)) (define (pi-sum a b) (sum (lambda (i)(/ 1 (square i))) a (lambda (i) (+ i 2)) b))
10
Numerical Integration
òabf dx =[f(a)+f(a+dx)+…+f(b)]dx (define (integral f a b dx) (define (add-dx x) (+ x dx) (* (sum f a add-dx b) dx)) (integral square ) (integral (lambda (x) (* x x x)) )
11
Exercise 1.30 (define (sum term a next b) (define (iter i result)
(if < > < > (iter < > < >))) (iter < > < >)) ( > i b) result (next i) (+ (term i) result) a
12
Constructing with Lambda
(lambda (<formal-param>) <body>) (define (plus4 x) (+ x 4)) (define plus4 (lambda (x) (+ x 4))) ((lambda (x y z)(+ x y (square z))) 1 2 3) (define (integral f a b dx) (* (sum f (+ a dx) (lambda (x)(+ x dx)) b) dx))
13
Defining Local Variables(I)
(define (f x y) (define a (+ 1 (* x y))) (define b (- 1 y)) (+ (* x (square a)) (* y b) (* a b)))
14
Defining Local Variables(II)
(define (f x y) (define (f-helper a b) (+ (* x (square a)) (* y b) (* a b))) (f-helper (+ 1 (* x y)) (- 1 y)))
15
Defining Local Variables(III)
(define (f x y) ((lambda (a b) (+ (* x (square a)) (* y b) (* a b))) (+ 1 (* x y)) (- 1 y)))
16
Defining Local Variables(IV)
(define (f x y) (let ((a (+ 1 (* x y))) (b (- 1 y))) (+ (* x (square a)) (* y b) (* a b))))
17
Let expression (let ((<var1> <exp1>)
… (<varn> <expn>)) <body>) is syntactic sugar for ((lambda (<var1> … <varn>) <body>) <exp1> … <expn>)
18
Let Expression - Semantics
Scope of a variable is the body of the let (define x 5) (+ (let ((x 3))(+ x (* x 10))) x) ; --> 38 Variables are bound simultaneously, using values computed outside the let (define x 2) (let ((x 3) (y (+ x 2))) (* x y)) ; -> 12
19
Exercise 1.34 (define (f g) (g 2)) > (f square) 4
> (f (lambda (z)(* z (+ z 1)))) 6 > (f f) ???
20
Compute Square Roots - Revisit
Computing Square root of x is a fixed point of the function f(y) = x/y ( = y ) Fixed point of function f mean find y such that f(y) = y start with a guess for y keep applying f over and over until the result doesn’t change very much
21
Fix-point Operator Fixed-point
(define tolerance 0.001) (define (close? u v) (< (abs (- u v)) tolerance)) (define (fixed-point f i-guess) (define (try guess) (let ((next-guess (f guess))) (if (close? next-guess guess) next-guess (try next-guess)))) (try i-guess))
22
Square root of X as a fixed point
Fixed Point of the function f(y) = x/y (define (sqrt x) (fixed-point (lambda (y) (/ x y)) 1)) It does not work, because it doesn’t converge - it oscillates
23
Average Damping (define (average-damp f)
(lambda (x) (average x (f x)))) (define (average-damp (lambda (f) (lambda (x) (average x (f x))))) (Sch-Num -> Sch-Num) -> (Sch-Num -> Sch-Num)
24
Safe Square Root as a fixed point
(define (sqrt x) (fixed-point (average-damp (lambda (y) (/ x y))) 1)) Linear Iterative Process?
25
Conventional Approach
(define (sqrt x) (define (improve guess) (average guess (/ x guess))) (define (sqrt-iter guess) (if (close? (square guess) x) guess (sqrt-iter (improve guess)))) (sqrt-iter 1))
26
Why use Fixed-point? Using fixed-point approach makes us see the idea more clearly Find cube root -> Fixed-point of y -> x/y2 (define (cuberoot x) (fixed-point (average-damp (lambda (y)(/ x (square y)))) 1))
27
Procedures as Returned Values
Derivative of function f : Df Df(x) = [f(x+dx) - f(x)] / dx (Sch-Num -> Sch-Num) -> (Sch-Num -> Sch-Num) (define (deriv f) (lambda (x) (/ (- (f (+ x dx)) (f x)) dx)))
28
Derivative of Fun : Df (define deriv (lambda (f) (let ((dx 0.00001))
(lambda (x) (/ (- (f (+ x dx)) (f x)) dx))))) ((deriv square) 10) ----> …….
29
Newton’s Method To find a zero of a function g such that g(y)=0
Find a fixed-point of the function f(y) f(y) = y - g(y)/Dg(y)
30
Newton’s Method: program
(define (newton-transform g) (lambda (y) (- y (/ (g y)((deriv g) y))))) (define (newtons-method g guess) (fixed-point (newton-transform g) guess)) Example: Square root of x is zero of g(y)=y2-x
31
Example :Square Root (define (sqrt x) (newtons-method
(lambda (y) (- (square y) x)) 1.0)) (fixed-point (average-damp (lambda (y) (/ x y))) 1))
32
Abstraction and First-class Proc
More general idea for square root as fixed points (define (fixed-point-of-transform g transform guess) (fixed-point (transform g) guess))
33
(define (sqrt x) (fixed-point-of-transform (lambda (y) (- (square y) x)) newton-transform 1)) (lambda (y) (/ x y)) average-damp
34
First Class Citizens May be named by variables
May be passed as arguments to procedures May be returned as the results of procedures May be included in data structures
Similar presentations
