or a 3-1a (0 b ) : max: a b a > b result a result b ( ) result Python : def max(a, b): if a > b: result = a else: result = b ret

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1 or a 3-1a (0 b ) : max: a b a > b result a result b result Python : def max(a, b): if a > b: result = a result = b return(result) : max2: a b result a b > result result b result 1

2 Python : def max2(a, b): result = a if b > result: result = b return(result)? 2 (2 ) (a) (b) ( ) (a) (b) ( ) ( ) ( ) b 3-1b a b c : max3: a b c a > b a > c result a result c b > c result b result c result 2

3 Python : def max3(a, b, c): if a > b: if a > c: result = a result = c if b > c: result = b result = c return(result) ( ) if elif else { } begin end Python / / begin/end? max3a: a b c result a b > result result b c > result result c result Python (?): def max3a(a, b, c): result = a if b > result: result = b if c > result: result = c return(result) (if) N max2 def max3b(a, b, c): return(max2(a, max2(b, c)) 3

4 c 3-1c 3 if if Python : def sign1(x): if x > 0: return("positive.") if x < 0: return("negative.") return("zero.") if if elif : : def sign2(x): if x > 0: return("positive.") elif x < 0: return("negative.") return("zero.") : x x > 0 positive. x < 0 negative. zero. 1 2 a > b and a > c a elif 3 ( ): def max3c(a, b, c): if (a > b) and (a > c): return(a) elif b > c: return(b) return(c) 1 elseif elif 2 ( ) / else 4

5 3 N N 4 5 ( N 2 ) N ( ) ( ) ax 2 + bx + c = 0 x = ( b ± D)/2a, D = b 2 4ac 2 ( 2 ) def fa(a, b, c): D = b**2-4.0*a*c x1 = (-b + math.sqrt(d)) / (2.0*a) x2 = (-b - math.sqrt(d)) / (2.0*a) return(x1, x2) ( ) 2 b D / 2 α, β αβ = c a a, b, β ( ) fa x1, x2 x1 x2 -(b + math.sqrt(d)) + b math.sqrt(d) math.sqrt(d) ( )b 2 x1, x2 b fa fb ( ): and 2 or 3 1 and ( 2 or 3) 5

6 import math def fb(a, b, c): D = b**2-4.0*a*c x1 = (-b + math.sqrt(d)) / (2.0*a) x2 = (-b - math.sqrt(d)) / (2.0*a) if b > 0: x1 = c / (a*x2) x2 = c / (a*x1) return(x1, x2) fb 2 x 2 100x + 1 = 0 check >>> x1, x2 = fb(1, -100, 1) >>> x >>> x >>> check(1, -100, 1, x1, x2) (0.0, 0.0) x1 fa fb x2 ( ) fb 0 ( check ) testdiv2 testdiv2 2 2 : import math def testdouble(x): while x!= math.inf: print(x) x = x * 2.0 testdouble 1.0 : >>> testdouble(1.0) e e e e+307 >>>

7 2 ( ) ( ): : 0 3 fizz 4 : fizz1: 3 fizz i 0 i 3 fizz i Python ( ): def fizz1(n): i = 0 while i <= n - 1: if i % 3 == 0: print("fizz") print(i) i = i + 1 : >>> fizz1(20) fizz 1 2 fizz ( ) fizz 19 ( ) 4 fizzbuzz 1, 2,... 3 fizz 5 buzz 3 5 fizzbuzz ( ) 7

8 4-1 fizz1 Python a b. 0 3 fizz 5 buzz 3 5 fizzbuzz ( ) (fizzbuzz ) 5 c. 0 3 hoge : 3 33 hoge 1 3 ( ) 3 True 3 or 6 (numerical integration ) y = f(x) a b 1: : y = f(x) x = a x = b 1 [a, b] 1 [a, b] n ( dx ) f(x) f(x) ( (analytical) (numerical) ) 5 fizzbuzz 6 or ( / ) 8

9 y = x x3 [a, b] [ 1 3 x3 ] b 1000 a [1, 10] = = 333 : integ1: x 2 [a, b] n dx b a n s 0 x a x < b : y x 2 # f(x) s s + y dx x x + dx s x a x x + dx x dx x x (dx ) b s 0 Python # (comment) Python ( ) ( : ) ( ) (comment out) : def integ1(a, b, n): dx = (b - a) / n s = 0.0 x = a # count = 0 while x < b: y = x**2 s = s + y * dx x = x + dx # count = count + 1 # f(x) # print("count=%d x=%.20f" % ((count), (x))) return(s) 333? : >>> integ1(1.0, 10.0, 100) ? >>> integ1(1.0, 10.0, 1000) >>> integ1(1.0, 10.0, 10000)

10 # 7 : >>> integ1(1.0, 10.0, 100)... count=98 x= count=99 x= count=100 x= count=101 x= x >>> 100 1? x x + dx x b dx ( ) 100 b (2 ) x ( (counter) ): i = 0 # i while i < n: # n... # ( ) i = i + 1 # 1 (counting loop) (while ) Python for (for statement) 9 while : for i in range(n):... i 0 n-1 i 1 n 0 n ( ): 7 count x ( ) x print %.20f for for (for loop) 10

11 i 0 n... # 5 ( ) : integ1: x 2 [a, b] n dx b a n s 0 i 0 n x a + i dx y x 2 # f(x) s s + y dx s x i Python : def integ2(a, b, n): dx = (b - a) / n s = 0.0 for i in range(n): x = a + i * dx y = x**2 s = s + y * dx return(s) : >>> integ2(1.0, 10.0, 100) >>> integ2(1.0, 10.0, 1000) >>> integ2(1.0, 10.0, 10000) >>> # f(x) 333? x 2 ( ) ( ) 4-2 a. f(x) f(x) b. f(x) c. a b ( ) 11

12 2: 4-3 a. n 2 n b. n n! = n (n 1) 2 1 c. n r ( n) n C r = n (n 1) (n r + 1) r (r 1) 1 d. x n (Taylor expansion) cos x = x0 0! x2 2! + x4 4! x6 6! + sin x = x1 1! x3 3! + x5 5! x7 7! + x ±10π? n? 12

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P 6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ P