load(os.environ['SAGE_STARTUP_FILE'])
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Sage는 대수적 계산을 할수있는 프로그램(Computer Algebra System (CAS))으로 무료로 sagemath.org 에서 다운로드하여 설치 할 수 있다. 웹서버를 인터페이서로 사용하며 브라우저를 통해 접속하여 사용하기 때문에 인터넷이되는 모든 장소에서 이용할 수 있다. 세이지의 기본언어는 객체지향 언어인 파이썬(Python)이며 이를 사용하여 프로그래밍도 할 수 있다. http://sage.kunsan.ac.kr 에 접속하여 계정을 만들고 사용한다.
대수(Algebra)
덧셈
2+3
5 5 |
2-3
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2*3
6 6 |
1.5/5
0.300000000000000 0.300000000000000 |
2^3
8 8 |
2**3
8 8 |
2^(-3)
1/8 1/8 |
인수분해
factor(10021567679000000000000797089)
3 * 13 * 443 * 121367 * 4779326644317538771 3 * 13 * 443 * 121367 * 4779326644317538771 |
전개하기
var('a,b,c') #변수를 처음 도입할때 선언을 한다. a,b,c 는 이제 변수로 사용가능. 주의: #뒤의 모든것은 무시된다.
(a, b, c) (a, b, c) |
(a+b)^2
(a + b)^2 (a + b)^2 |
var('u,v') #u,v 는 변수가 된다
(u, v) (u, v) |
expand((a+b)^10)
a^10 + 10*a^9*b + 45*a^8*b^2 + 120*a^7*b^3 + 210*a^6*b^4 + 252*a^5*b^5 + 210*a^4*b^6 + 120*a^3*b^7 + 45*a^2*b^8 + 10*a*b^9 + b^10 a^10 + 10*a^9*b + 45*a^8*b^2 + 120*a^7*b^3 + 210*a^6*b^4 + 252*a^5*b^5 + 210*a^4*b^6 + 120*a^3*b^7 + 45*a^2*b^8 + 10*a*b^9 + b^10 |
방정식 해 구하기
solve(x^2-3*x+1==0,x) #방정식을 나타낼 때 == 을 사용함에 주의.
[x == -1/2*sqrt(5) + 3/2, x == 1/2*sqrt(5) + 3/2] [x == -1/2*sqrt(5) + 3/2, x == 1/2*sqrt(5) + 3/2] |
solve(x^5-3*x^2-x+3==0,x)
[x == -1/2*(1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3)*(I*sqrt(3) + 1) + 1/6*(-I*sqrt(3) + 1)/(1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3), x == -1/2*(1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3)*(-I*sqrt(3) + 1) + 1/6*(I*sqrt(3) + 1)/(1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3), x == (1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3) - 1/3/(1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3), x == 1, x == -1] [x == -1/2*(1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3)*(I*sqrt(3) + 1) + 1/6*(-I*sqrt(3) + 1)/(1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3), x == -1/2*(1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3)*(-I*sqrt(3) + 1) + 1/6*(I*sqrt(3) + 1)/(1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3), x == (1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3) - 1/3/(1/18*sqrt(247)*sqrt(3) + 3/2)^(1/3), x == 1, x == -1] |
Numerical solution
find_root(sin(x)-cos(x),0,2)
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연립방정식
var('x,y,z') #변수 y를 도입하자
(x, y, z) (x, y, z) |
solve([x-y==3,x+y==5],x,y)
[[x == 4, y == 1]] [[x == 4, y == 1]] |
solve([2*x-y+z==2,x+y+z==-3,3*x-y+2*z==0],x,y,z)
[[x == 3, y == -1, z == -5]] [[x == 3, y == -1, z == -5]] |
함수 (Function): 함수정의는 아래 두가지 방법을 사용하여 할 수 있다.
f(x)=x*sin(x)
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f(1.)
0.841470984807897 0.841470984807897 |
다음은 파이썬 방법이다.
함수 k(x)=s*sin(x)를 정의한다
def k(x):
return x*sin(x)
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함수 $h(x,y)= (x+y)^{10}+2 x y +x^2$
def h(x,y):
return x**2+2*x*y+(x+y)**10
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def h(x,y):
b=x**2+2*x*y+(x+y)**10
return b
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h(1,2)
59054 59054 |
def h1(x,y):
return x**2+2*x*y+(x+y)**10
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h(1,2)
59054 59054 |
h1(1,2)
59054 59054 |
# e^(x+y)
def g(x,y):
return exp(x+y)
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g(1,1)
e^2 e^2 |
var('y')
y y |
미분(Differentiation)
$\frac{d}{dx} k(x)$
def k(x):
return sin(x**2*cos(x))
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diff(k(x),x)
-(x^2*sin(x) - 2*x*cos(x))*cos(x^2*cos(x)) -(x^2*sin(x) - 2*x*cos(x))*cos(x^2*cos(x)) |
$\frac{d}{dx} f(x)$
diff(f(x),x)
x*cos(x) + sin(x) x*cos(x) + sin(x) |
$\frac{d}{dx} \sin(x)$
diff(sin(x),x)
cos(x) cos(x) |
diff(cos(x),x,97)
-sin(x) -sin(x) |
편미분(Partial Differentiation)
$\frac{\partial}{\partial x} \sin(x y)$
y=var('y')
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diff(sin(x*y),x)
y*cos(x*y) y*cos(x*y) |
diff(sin(x*y),x,20)
y^20*sin(x*y) y^20*sin(x*y) |
diff(sin(x*y),x,2,y,3)
x^3*y^2*cos(x*y) + 6*x^2*y*sin(x*y) - 6*x*cos(x*y) x^3*y^2*cos(x*y) + 6*x^2*y*sin(x*y) - 6*x*cos(x*y) |
diff(sin(x*y),y,3,x,2)
x^3*y^2*cos(x*y) + 6*x^2*y*sin(x*y) - 6*x*cos(x*y) x^3*y^2*cos(x*y) + 6*x^2*y*sin(x*y) - 6*x*cos(x*y) |
$\frac{\partial^{10}}{\partial y^{10}} \sin(xy)$
diff(sin(x*y),y,10)
-x^10*sin(x*y) -x^10*sin(x*y) |
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diff(sin(x*y),x,2,y,3)-diff(sin(x*y),y,3,x,2)
0 0 |
$\frac{\partial^2}{\partial x^2} \sin(x y)$
diff(sin(x*y),x,2)
-y^2*sin(x*y) -y^2*sin(x*y) |
$\frac{\partial^3}{\partial x \partial y^2} \sin(x y)$
diff(sin(x*y),y,2,x)
-x^2*y*cos(x*y) - 2*x*sin(x*y) -x^2*y*cos(x*y) - 2*x*sin(x*y) |
diff(sin(x*y),x,1,y,2)
-x^2*y*cos(x*y) - 2*x*sin(x*y) -x^2*y*cos(x*y) - 2*x*sin(x*y) |
diff(sin(x*y),x,y)
-x*y*sin(x*y) + cos(x*y) -x*y*sin(x*y) + cos(x*y) |
diff(exp(x**2+y**3*sin(x)),x,2,y,5).subs(x=1,y=2).n()
3.80304538819769e10 3.80304538819769e10 |
$x^2\sin(xy^7)$ 의 $x=2,y=5$에서의 값을 소수로 구하시오
(x^2*sin(x*y^7)).subs(x=2,y=5).n()
-0.998213235373344 -0.998213235373344 |
적분(Integration)
$\int k(x) dx$
k(x)
sin(x^2*cos(x)) sin(x^2*cos(x)) |
ff=integral(k(x),x)
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integral(sin(x),x)
-cos(x) -cos(x) |
integral(sin(x),x,0,1).n()
0.459697694131860 0.459697694131860 |
diff(ff,x)
sin(x^2*cos(x)) sin(x^2*cos(x)) |
$\int_1^2 k(x) dx$
integral(k(x),x,1,2) #정적분
integrate(sin(x^2*cos(x)), x, 1, 2) integrate(sin(x^2*cos(x)), x, 1, 2) |
integral(k(x),x,2,1)
-integrate(sin(x^2*cos(x)), x, 1, 2) -integrate(sin(x^2*cos(x)), x, 1, 2) |
위의 결과를 소수로 나타내기 위해서는 함수 $n()$을 점(.)으로 덧붙이면 된다 (객체지향 언어의 특성).
integral(k(x),x,1,2).n()
-0.032996004851711926 -0.032996004851711926 |
수치 적분(Numerical integration)
$\int_0^\pi \sin(x)^2 dx$ 를 수치계산과 정확한 계산으로 구해보고 비교해 보자. 두 계산 결과가 오차범위내에서 일치함을 주목하라.
numerical_integral(sin(x)^2,0,pi) #수치계산. 앞의 수가 결과를 나타내며 두번째 수는 오차를 나타내며 e-14는 10^(-14)를 의미한다.
(1.570796326794897, 1.7439342490043163e-14) (1.570796326794897, 1.7439342490043163e-14) |
integral(sin(x)^2,x,0,pi) #정확한(exact) 계산. 이경우 적분변수 (x)를 표시함에 주의 하시오.
1/2*pi 1/2*pi |
integral(sin(x)^2,x,0,pi).n(digits=20) #결과를 소수로 나타낸다.
1.5707963267948966192 1.5707963267948966192 |
integral(exp(-x**2),x,-Infinity,Infinity)
sqrt(pi) sqrt(pi) |
$\int_0^\infty e^{-\sqrt{x}} dx$
f=exp(-sqrt(x))
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integral(f,x,0,Infinity)
2 2 |
numerical_integral(f,0,Infinity) #Infinity는 무한대를 나타내는 기호. 대문자로 시작함에 주의.
(2.000000014718695, 7.871724825303517e-07) (2.000000014718695, 7.871724825303517e-07) |
integral(f,x,0, Infinity)
2 2 |
numerical_integral(sin(f),0,pi)
(1.0226025498810094, 6.629450086680393e-07) (1.0226025498810094, 6.629450086680393e-07) |
integral(exp(-x**2),x,0,Infinity).n()
0.886226925452758 0.886226925452758 |
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plot(sin(cos(x)),x,1,pi,figsize=4)
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plot(exp(-x**2),x,0,1,figsize=4)
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plot(exp(-x**2),x,-10,10,figsize=4)
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numerical_integral(sin(f),0,pi)
(1.0226025498810094, 6.629450086680393e-07) (1.0226025498810094, 6.629450086680393e-07) |
plot(exp(sin(x)),x,0,100,figsize=4)
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대입(substitution)
k(x).subs(x=2)
sin(4*cos(2)) sin(4*cos(2)) |
k(x)=x*sin(x)
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1/(1+x**2).subs(x=pi).n()
0.0919996683503752 0.0919996683503752 |
var('a,b')
(a, b) (a, b) |
g=(a^2+b^2)/a/b
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g
(a^2 + b^2)/(a*b) (a^2 + b^2)/(a*b) |
g.subs(a=1,b=2)
5/2 5/2 |
f2=(a+b)^5/a^2/b^3
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f2
(a + b)^5/(a^2*b^3) (a + b)^5/(a^2*b^3) |
f2.subs(a=2,b=3).n()
28.9351851851852 28.9351851851852 |
y=var('y')
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a=x^2+y^3+2*x*y
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a
y^3 + x^2 + 2*x*y y^3 + x^2 + 2*x*y |
a.subs(x=2,y=1.5)
13.3750000000000 13.3750000000000 |
k(2)
2*sin(2) 2*sin(2) |
diff(k(x),x).subs(x=2)
2*cos(2) + sin(2) 2*cos(2) + sin(2) |
diff(k(x),x).subs(x=2).n() # x=2에서의 k(x)의 기울기
0.0770037537313969 0.0770037537313969 |
식을 대입하기
f=sin(x)^2+2*sin(x)+exp(-sin(x))
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var('z')
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f.subs(sin(x)==z)
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Simplification
a=x*(1-x)/x**2
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a.simplify_full()
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b=(x-1)**3/(x**2-2*x+1)
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b.simplify_full()
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실수값얻기
pi #원주율
pi pi |
pi.n(digits=100)
3.1415926535897932384626433832795028841971693993751058209749445923078164\ 06286208998628034825342117068 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 |
e.n()
2.71828182845905 2.71828182845905 |
k(2)
2*sin(2) 2*sin(2) |
k(2).n()
1.81859485365136 1.81859485365136 |
k(2).n(digits=40) #정확도를 지정할 때
1.818594853651363390792039731823489685405 1.818594853651363390792039731823489685405 |
List (데이터 저장과 처리에 매우 중요)
var('a,b,c,d,e') #변수 선언
(a, b, c, d, e) (a, b, c, d, e) |
L1=[a,b,3]
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L1
[a, b, 3] [a, b, 3] |
len(L1) #리스트의 원소 갯수, length
3 3 |
L1.append(c) #list에 원소를 추가할때
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L1
[a, b, 3, c] [a, b, 3, c] |
L1.append(b)
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L1.append(b)
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L1
[a, b, 3, c, b, b] [a, b, 3, c, b, b] |
L1.remove(b) #원소 b를 제거
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L1
[a, 3, c, b, b] [a, 3, c, b, b] |
L1[2]
c c |
L1.remove(d); L1
Traceback (click to the left of this block for traceback) ... ValueError: list.remove(x): x not in list Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_112.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("TDEucmVtb3ZlKGQpOyBMMQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmp3JDNKn/___code___.py", line 2, in <module>
exec compile(u'L1.remove(d); L1
File "", line 1, in <module>
ValueError: list.remove(x): x not in list
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L1[0];L1[1];L1[2]# 리스트의 원소를 뽑아보자.원소를 지칭하는 인덱스는 0에서 시작함을 주의한다.
a 3 c a 3 c |
L1[2]
c c |
l2=[9,-1,2,-3,5,2,7]
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l2[0]*l2[6]
63 63 |
zero=[]
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zero
[] [] |
zero.append(3)
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zero
[3] [3] |
l2.sort() #l2를 순서대로 정렬
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l2.reverse()
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l2
[9, 7, 5, 2, 2, -1, -3] [9, 7, 5, 2, 2, -1, -3] |
range(11) #파이썬 내장 정수 리스트
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10] [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10] |
range(20)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] |
range(-20,9,5)
[-20, -15, -10, -5, 0, 5] [-20, -15, -10, -5, 0, 5] |
range(11,30,1)
[11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] |
range(2,20,3)
[2, 5, 8, 11, 14, 17] [2, 5, 8, 11, 14, 17] |
range(1,50,3)
[1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49] [1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49] |
srange(2,3,0.1) #세이지 내장 리스트의 하나
[2.00000000000000, 2.10000000000000, 2.20000000000000, 2.30000000000000, 2.40000000000000, 2.50000000000000, 2.60000000000000, 2.70000000000000, 2.80000000000000, 2.90000000000000] [2.00000000000000, 2.10000000000000, 2.20000000000000, 2.30000000000000, 2.40000000000000, 2.50000000000000, 2.60000000000000, 2.70000000000000, 2.80000000000000, 2.90000000000000] |
import numpy #numpy 수치계산 패키지를 불러들인다
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numpy.arange(2,10,0.1) #0.1 단위로 원소가 증가 함을 주의
array([ 2. , 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3. ,
3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 4. , 4.1,
4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 5. , 5.1, 5.2,
5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 6. , 6.1, 6.2, 6.3,
6.4, 6.5, 6.6, 6.7, 6.8, 6.9, 7. , 7.1, 7.2, 7.3, 7.4,
7.5, 7.6, 7.7, 7.8, 7.9, 8. , 8.1, 8.2, 8.3, 8.4, 8.5,
8.6, 8.7, 8.8, 8.9, 9. , 9.1, 9.2, 9.3, 9.4, 9.5, 9.6,
9.7, 9.8, 9.9])
array([ 2. , 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3. ,
3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 4. , 4.1,
4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 5. , 5.1, 5.2,
5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 6. , 6.1, 6.2, 6.3,
6.4, 6.5, 6.6, 6.7, 6.8, 6.9, 7. , 7.1, 7.2, 7.3, 7.4,
7.5, 7.6, 7.7, 7.8, 7.9, 8. , 8.1, 8.2, 8.3, 8.4, 8.5,
8.6, 8.7, 8.8, 8.9, 9. , 9.1, 9.2, 9.3, 9.4, 9.5, 9.6,
9.7, 9.8, 9.9])
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Tuples: 바꿀 수 없는 리스트
a=(2,3,4)
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a[0]
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a[1]
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a[2]
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len(a)
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-----------------------------------------------------------
그래프
k(x)=x*sin(x)
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plot(k(x),x,0,10,figsize=4)+plot(x*k(x),x,0,10,figsize=4,color='red')
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plot(e^sin(x),x,0,10,figsize=4,color='red')+plot(e^cos(x),x,0,10,figsize=4,color='blue')
Traceback (click to the left of this block for traceback) ... ValueError: Variable 'e' not found Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_135.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cGxvdChlXnNpbih4KSx4LDAsMTAsZmlnc2l6ZT00LGNvbG9yPSdyZWQnKStwbG90KGVeY29zKHgpLHgsMCwxMCxmaWdzaXplPTQsY29sb3I9J2JsdWUnKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmp1a1nhd/___code___.py", line 3, in <module>
exec compile(u"plot(e**sin(x),x,_sage_const_0 ,_sage_const_10 ,figsize=_sage_const_4 ,color='red')+plot(e**cos(x),x,_sage_const_0 ,_sage_const_10 ,figsize=_sage_const_4 ,color='blue')" + '\n', '', 'single')
File "", line 1, in <module>
File "/home/sagerunner/sage-8.4/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 567, in wrapper
return func(*args, **options)
File "/home/sagerunner/sage-8.4/local/lib/python2.7/site-packages/sage/plot/plot.py", line 1941, in plot
G = funcs.plot(*args, **original_opts)
File "sage/symbolic/expression.pyx", line 12064, in sage.symbolic.expression.Expression.plot (build/cythonized/sage/symbolic/expression.cpp:68173)
File "sage/symbolic/expression.pyx", line 12105, in sage.symbolic.expression.Expression._plot_fast_callable (build/cythonized/sage/symbolic/expression.cpp:68463)
File "sage/ext/fast_callable.pyx", line 495, in sage.ext.fast_callable.fast_callable (build/cythonized/sage/ext/fast_callable.c:4871)
File "sage/symbolic/expression.pyx", line 11942, in sage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:67091)
File "/home/sagerunner/sage-8.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1851, in fast_callable
return FastCallableConverter(ex, etb)()
File "/home/sagerunner/sage-8.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 217, in __call__
return self.arithmetic(ex, operator)
File "/home/sagerunner/sage-8.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1779, in arithmetic
return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
File "/home/sagerunner/sage-8.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1779, in <lambda>
return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
File "sage/ext/fast_callable.pyx", line 779, in sage.ext.fast_callable.ExpressionTreeBuilder.call (build/cythonized/sage/ext/fast_callable.c:7484)
File "sage/ext/fast_callable.pyx", line 656, in sage.ext.fast_callable.ExpressionTreeBuilder.__call__ (build/cythonized/sage/ext/fast_callable.c:6570)
File "sage/symbolic/expression.pyx", line 11942, in sage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:67091)
File "/home/sagerunner/sage-8.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1851, in fast_callable
return FastCallableConverter(ex, etb)()
File "/home/sagerunner/sage-8.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 211, in __call__
return self.symbol(ex)
File "/home/sagerunner/sage-8.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1800, in symbol
return self.etb.var(SR(ex))
File "sage/ext/fast_callable.pyx", line 728, in sage.ext.fast_callable.ExpressionTreeBuilder.var (build/cythonized/sage/ext/fast_callable.c:7135)
ValueError: Variable 'e' not found
|
plot(k(x)*sin(10*x),x,0,10,color='green',figsize=5) # x-축 영역 지정
|
한 그림창에 여러개 그리기
plot(k(x)*sin(10*x),x,0,10)+plot(k(x),x,0,10, color='red')+plot(k(x)*sin(20*x),x,0,10,color='green',figsize=5) #동시에 그릴때는 + 를 사용한다.
|
여러개 그림창을 동시에 그리기
fig1=plot(k(x)*sin(10*x),x,0,10)+plot(k(x),x,0,10, color='red')
fig2=plot(k(x)*sin(20*x),x,0,10,color='green',figsize=5)
graphics_array((fig1,fig2))
|
히스토그램
h_data=[sin(i) for i in range(100)] #데이타셑을 만들자
histogram(h_data,bins=50,figsize=5)
|
h_data
[0, sin(1), sin(2), sin(3), sin(4), sin(5), sin(6), sin(7), sin(8), sin(9), sin(10), sin(11), sin(12), sin(13), sin(14), sin(15), sin(16), sin(17), sin(18), sin(19), sin(20), sin(21), sin(22), sin(23), sin(24), sin(25), sin(26), sin(27), sin(28), sin(29), sin(30), sin(31), sin(32), sin(33), sin(34), sin(35), sin(36), sin(37), sin(38), sin(39), sin(40), sin(41), sin(42), sin(43), sin(44), sin(45), sin(46), sin(47), sin(48), sin(49), sin(50), sin(51), sin(52), sin(53), sin(54), sin(55), sin(56), sin(57), sin(58), sin(59), sin(60), sin(61), sin(62), sin(63), sin(64), sin(65), sin(66), sin(67), sin(68), sin(69), sin(70), sin(71), sin(72), sin(73), sin(74), sin(75), sin(76), sin(77), sin(78), sin(79), sin(80), sin(81), sin(82), sin(83), sin(84), sin(85), sin(86), sin(87), sin(88), sin(89), sin(90), sin(91), sin(92), sin(93), sin(94), sin(95), sin(96), sin(97), sin(98), sin(99)] [0, sin(1), sin(2), sin(3), sin(4), sin(5), sin(6), sin(7), sin(8), sin(9), sin(10), sin(11), sin(12), sin(13), sin(14), sin(15), sin(16), sin(17), sin(18), sin(19), sin(20), sin(21), sin(22), sin(23), sin(24), sin(25), sin(26), sin(27), sin(28), sin(29), sin(30), sin(31), sin(32), sin(33), sin(34), sin(35), sin(36), sin(37), sin(38), sin(39), sin(40), sin(41), sin(42), sin(43), sin(44), sin(45), sin(46), sin(47), sin(48), sin(49), sin(50), sin(51), sin(52), sin(53), sin(54), sin(55), sin(56), sin(57), sin(58), sin(59), sin(60), sin(61), sin(62), sin(63), sin(64), sin(65), sin(66), sin(67), sin(68), sin(69), sin(70), sin(71), sin(72), sin(73), sin(74), sin(75), sin(76), sin(77), sin(78), sin(79), sin(80), sin(81), sin(82), sin(83), sin(84), sin(85), sin(86), sin(87), sin(88), sin(89), sin(90), sin(91), sin(92), sin(93), sin(94), sin(95), sin(96), sin(97), sin(98), sin(99)] |
y=var('y')
|
|
g=exp(-x^2-2*y^2)
|
|
plot3d(g,(x,-5,5),(y,-5,5))
|
|
[i^2 for i in range(10)]
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81] [0, 1, 4, 9, 16, 25, 36, 49, 64, 81] |
l2=[sin(0.3*i) for i in srange(0,21,0.1)] #l2의 원소는 range(21) 안의 모든 원소를 i에 대입하여 얻어진다.
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|
l2
WARNING: Output truncated!full_output.txt [0.000000000000000, 0.0299955002024957, 0.0599640064794446, 0.0898785491980111, 0.119712207288919, 0.149438132473599, 0.179029573425824, 0.208459899846100, 0.237702626427135, 0.266731436688831, 0.295520206661339, 0.324043028394868, 0.352274233275090, 0.380188415123161, 0.407760453059570, 0.434965534111230, 0.461779175541483, 0.488177246882908, 0.514135991653113, 0.539632048733969, 0.564642473395035, 0.589144757942270, 0.613116851973434, 0.636537182221968, 0.659384671971473, 0.681638760023334, 0.703279419200410, 0.724287174370143, 0.744643119970860, 0.764328937025505, 0.783326909627484, 0.801619940883777, 0.819191568300998, 0.836025978600521, 0.852108021949363, 0.867423225594017, 0.881957806884948, 0.895698685680048, 0.908633496115884, 0.920750597736136, 0.932039085967227, 0.942488801931698, 0.952090341590516, 0.960835064206073, 0.968715100118265, 0.975723357826659, 0.981853530372360, 0.987100101013850, 0.991458348191686, 0.994924349777581, 0.997494986604054, 0.999167945271476, 0.999941720229966, 0.999815615134291, 0.998789743470524, 0.996865028453919, 0.994043202198076, 0.990326804156158, 0.985719178835554, ... -0.977530117665095, -0.983413187547309, -0.988411251939129, -0.992519812919962, -0.995735173062244, -0.998054438758879, -0.999475522827284, -0.999997146387718, -0.999618840014186, -0.998340944156888, -0.996164608835842, -0.993091792605937, -0.989125260794372, -0.984268583012044, -0.978526129941141, -0.971903069401823, -0.964405361701533, -0.956039754271121, -0.946813775592612, -0.936735728424082, -0.925814682327735, -0.914060465507910, -0.901483655966358, -0.888095571982758, -0.873908261929026, -0.858934493426595, -0.843187741856420, -0.826682178232039, -0.809432656446623, -0.791454699905469, -0.772764487555990, -0.753378839327749, -0.733315200995659, -0.712591628479964, -0.691226771597129, -0.669239857276263, -0.646650672256185, -0.623479545278687, -0.599747328794044, -0.575475380195218, -0.550685542597638, -0.525400125181879, -0.499641883116902, -0.473433997081934, -0.446800052405429, -0.419764017839857, -0.392350223991451, -0.364583341424299, -0.336488358458502, -0.308090558682375, -0.279415498198922, -0.250488982627071, -0.221337043878354, -0.191985916729950, -0.162462015215149, -0.132791908852511, -0.103002298735091, -0.0731199935012560, -0.0431718852087216, -0.0131849251335141] WARNING: Output truncated!full_output.txt [0.000000000000000, 0.0299955002024957, 0.0599640064794446, 0.0898785491980111, 0.119712207288919, 0.149438132473599, 0.179029573425824, 0.208459899846100, 0.237702626427135, 0.266731436688831, 0.295520206661339, 0.324043028394868, 0.352274233275090, 0.380188415123161, 0.407760453059570, 0.434965534111230, 0.461779175541483, 0.488177246882908, 0.514135991653113, 0.539632048733969, 0.564642473395035, 0.589144757942270, 0.613116851973434, 0.636537182221968, 0.659384671971473, 0.681638760023334, 0.703279419200410, 0.724287174370143, 0.744643119970860, 0.764328937025505, 0.783326909627484, 0.801619940883777, 0.819191568300998, 0.836025978600521, 0.852108021949363, 0.867423225594017, 0.881957806884948, 0.895698685680048, 0.908633496115884, 0.920750597736136, 0.932039085967227, 0.942488801931698, 0.952090341590516, 0.960835064206073, 0.968715100118265, 0.975723357826659, 0.981853530372360, 0.987100101013850, 0.991458348191686, 0.994924349777581, 0.997494986604054, 0.999167945271476, 0.999941720229966, 0.999815615134291, 0.998789743470524, 0.996865028453919, 0.994043202198076, 0.990326804156158, 0.985719178835554, ... -0.977530117665095, -0.983413187547309, -0.988411251939129, -0.992519812919962, -0.995735173062244, -0.998054438758879, -0.999475522827284, -0.999997146387718, -0.999618840014186, -0.998340944156888, -0.996164608835842, -0.993091792605937, -0.989125260794372, -0.984268583012044, -0.978526129941141, -0.971903069401823, -0.964405361701533, -0.956039754271121, -0.946813775592612, -0.936735728424082, -0.925814682327735, -0.914060465507910, -0.901483655966358, -0.888095571982758, -0.873908261929026, -0.858934493426595, -0.843187741856420, -0.826682178232039, -0.809432656446623, -0.791454699905469, -0.772764487555990, -0.753378839327749, -0.733315200995659, -0.712591628479964, -0.691226771597129, -0.669239857276263, -0.646650672256185, -0.623479545278687, -0.599747328794044, -0.575475380195218, -0.550685542597638, -0.525400125181879, -0.499641883116902, -0.473433997081934, -0.446800052405429, -0.419764017839857, -0.392350223991451, -0.364583341424299, -0.336488358458502, -0.308090558682375, -0.279415498198922, -0.250488982627071, -0.221337043878354, -0.191985916729950, -0.162462015215149, -0.132791908852511, -0.103002298735091, -0.0731199935012560, -0.0431718852087216, -0.0131849251335141] |
list_plot(l2,figsize=4) #데이터를 그릴때
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list_plot(l2,plotjoined=True,figsize=4,color='green') #점들을 연결해 보자.
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l5=[(cos(0.1*i),sin(0.1*i)) for i in range(70)] # range(20)안의 모든 원소를 i에 대입하여 리스트를 만든다.
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list_plot(l5,figsize=4,aspect_ratio=1)
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list_plot(l5,aspect_ratio=1,plotjoined=True ,figsize=4) #x, y축의 비율을 조절하기 위해서는 aspect_ratio를 사용한다.
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3D List Plot
#Generate some 3D data
a=2.0
w=1.5
data=[]
for t in srange(0,15,0.1):
data.append((a*cos(w*t),a*sin(w*t),t))
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list_plot(data,plotjoined=True)
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Parametric Plot
t=var('t')
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parametric_plot((sin(t^2),exp(0.1*t)),(t,0,3),aspect_ratio=1,figsize=5)
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parametric_plot3d((cos(t),sin(t),t),(t,0,6*pi))
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|
-------------------------------------------------------------------------
선형대수 (Linear Algebra)
u=vector([1,2,3]) #List가 벡터함수의 input임을 주의
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|
u
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v=vector([-1,3,5]);v
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u.dot_product(v) #내적
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u.cross_product(v) #외적
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v.cross_product(u) #외적의 순서를 바꾸면 부호가 바뀐다.
(-1, 8, -5) (-1, 8, -5) |
M=matrix([[1,2,3],[2,5,6],[-1,0,3]]);M #행렬함수도 리스트를 받아들임을 주의
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M*u
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u*M
(2, 12, 24) (2, 12, 24) |
A=vector([1,0,2])
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|
K=matrix([[2,5,6],[1,0,1],[-1,1,2]]);K
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K*A
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A*K
(0, 7, 10) (0, 7, 10) |
J=M.inverse() ;J #역행렬을 구한다.
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|
J*M
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M*J
[1 0 0] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [0 0 1] |
iM=M.inverse()
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|
M*iM
[1 0 0] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [0 0 1] |
iM*M
[1 0 0] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [0 0 1] |
M.det() #determinant 를 구한다.
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|
K.det()
-11 -11 |
K.inverse()
[ 1/11 4/11 -5/11] [ 3/11 -10/11 -4/11] [ -1/11 7/11 5/11] [ 1/11 4/11 -5/11] [ 3/11 -10/11 -4/11] [ -1/11 7/11 5/11] |
MM=matrix([[1,1,1,1],[2,0,3,1],[1,1,-6,0],[4,0,5,6]])
|
|
MM.inverse()
[-13/58 37/58 13/58 -2/29] [ 61/58 -13/58 -3/58 -4/29] [ 4/29 2/29 -4/29 -1/29] [ 1/29 -14/29 -1/29 7/29] [-13/58 37/58 13/58 -2/29] [ 61/58 -13/58 -3/58 -4/29] [ 4/29 2/29 -4/29 -1/29] [ 1/29 -14/29 -1/29 7/29] |
W=vector([5,-5,7,0])
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|
(MM.inverse())*W
(-159/58, 349/58, -18/29, 68/29) (-159/58, 349/58, -18/29, 68/29) |
S=matrix([[1,2,3],[2,1,4],[3,4,5]]);S #S는 대칭행렬
|
|
S
[1 2 3] [2 1 4] [3 4 5] [1 2 3] [2 1 4] [3 4 5] |
S.det()
|
|
ev=S.eigenvalues();ev #S의 고유값. 대칭행렬의 고유값은 실수임을 유의.
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ev[0]
-1.486979759027261? -1.486979759027261? |
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K=S.eigenvectors_right();K #고유벡터.
[(-1.486979759027261?, [(1, -7.323700036494775?, 4.053473437987430?)], 1), (-0.592545608423468?, [(1, -0.11487917829899307?, -0.4542624172751605?)], 1), (9.07952536745073?, [(1, 1.188579214793769?, 1.900788979287731?)], 1)] [(-1.486979759027261?, [(1, -7.323700036494775?, 4.053473437987430?)], 1), (-0.592545608423468?, [(1, -0.11487917829899307?, -0.4542624172751605?)], 1), (9.07952536745073?, [(1, 1.188579214793769?, 1.900788979287731?)], 1)] |
K[0][0]
-1.486979759027261? -1.486979759027261? |
u2=K[1][1];u2 #u1은 K의 첫번째 고유값에 대응하는 고유벡터를 추출한다.
[(1, -0.11487917829899307?, -0.4542624172751605?)] [(1, -0.11487917829899307?, -0.4542624172751605?)] |
u2[0]
(1, -0.11487917829899307?, -0.4542624172751605?) (1, -0.11487917829899307?, -0.4542624172751605?) |
S*u2[0]/K[1][0]
(1, -0.11487917829899307?, -0.4542624172751605?) (1, -0.11487917829899307?, -0.4542624172751605?) |
v1=u1[0];v1 #u1은 원소가 하나인 리스트이고 그 원소가 바로 고유 벡터이다.
Traceback (click to the left of this block for traceback) ... NameError: name 'u1' is not defined Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_194.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("djE9dTFbMF07djEgI3Ux7J2AIOybkOyGjOqwgCDtlZjrgpjsnbgg66as7Iqk7Yq47J206rOgIOq3uCDsm5DshozqsIAg67CU66GcIOqzoOycoCDrsqHthLDsnbTri6Qu"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmp7MznN_/___code___.py", line 3, in <module>
exec compile(u'v1=u1[_sage_const_0 ];v1 #u1\uc740 \uc6d0\uc18c\uac00 \ud558\ub098\uc778 \ub9ac\uc2a4\ud2b8\uc774\uace0 \uadf8 \uc6d0\uc18c\uac00 \ubc14\ub85c \uace0\uc720 \ubca1\ud130\uc774\ub2e4.
File "", line 1, in <module>
NameError: name 'u1' is not defined
|
S*v1/ev[0] #이것이 바로 v1임을 주의해 보시오.
Traceback (click to the left of this block for traceback) ... NameError: name 'v1' is not defined Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_195.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Uyp2MS9ldlswXSAgI+ydtOqyg+ydtCDrsJTroZwgdjHsnoTsnYQg7KO87J2Y7ZW0IOuztOyLnOyYpC4="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmplHhweJ/___code___.py", line 3, in <module>
exec compile(u'S*v1/ev[_sage_const_0 ] #\uc774\uac83\uc774 \ubc14\ub85c v1\uc784\uc744 \uc8fc\uc758\ud574 \ubcf4\uc2dc\uc624.
File "", line 1, in <module>
NameError: name 'v1' is not defined
|
|
|
*********두번째와째 세번째 고유값에 대해서도 같은 연습을 해보시오.*************
-------------------------------------------------------------------------
프로그래밍
For-loop (반복할때)
s=0
for i in range(1,101):
s=s+i
print s
5050 5050 |
p=1
for i in range (1,101):
p=p*i
print(p).n()
9.33262154439442e157 9.33262154439442e157 |
log(p,10).n()
|
|
10^0.9700
|
|
name=['시형','민정','초현']
|
|
for i in name:
print(i)
시형 민정 초현 시형 민정 초현 |
for i in range(3):
print name[i]
시형 민정 초현 시형 민정 초현 |
s=0
for i in range(0,5001,2):
s=s+i
print s
6252500 6252500 |
s=0
for i in range(50):
s=s+2*i+1
print s
2500 2500 |
s=0
for i in range(1,100,2):
s=s+i
print '1에서100사이의 홀수의합=',s
1에서100사이의 홀수의합= 2500 1에서100사이의 홀수의합= 2500 |
s=1
for i in range(2,11,2):
s=s*i
print s
3840 3840 |
p=1
for i in [1,2,3,5]: #list 안의 모든 원소을 차례로 곱한다.
p=p*i
print p
30 30 |
|
|
p=1
for k in range (1,11):
p=p*k
print p
3628800 3628800 |
p=1
for i in range(1,101,2):
p=p*i
print p
272539213975072950298071324540091863329079633054580341373432882344310620\ 1171875 2725392139750729502980713245400918633290796330545803413734328823443106201171875 |
동시에 합과 곱을 계산해 보자
s=0
p=1
q=0
for i in range (1,10): #range 가 1에서 시작. 즉, 1,2,3,...9
s=s+i #인덴트가 있는데까지 loop 이 적용됨.
p=p*i
q=q+i**2
print s,p,q #인덴트가 없으므로 loop 에서 빠져 나옴.
print '합=',s,',', '그리고 product=',p #문자는 따옴표속에.
45 362880 285 합= 45 , 그리고 product= 362880 45 362880 285 합= 45 , 그리고 product= 362880 |
함수를 사용해 보자
def sumprod(x,y):
return x+y, x*y
def main():
a=2
b=3
u,v=sumprod(a,b)
print '합=',u,',','곱=',v
main()
합= 5 , 곱= 6 합= 5 , 곱= 6 |
def sumprod(x,y):
return x+y, x*y
a=2
b=3
u,v=sumprod(a,b)
print '합=',u,',','곱=',v
합= 5 , 곱= 6 합= 5 , 곱= 6 |
mA=1
mB=3
mC=5
A1=1
A2=2
B1=2
B2=3
C1=-1
C2=5
M=mA+mB+mC
CM1=(mA*A1+mB*B1+mC*C1)/M
CM2=(mA*A2+mB*B2+mC*C2)/M
print(CM1,CM2)
(2/9, 4) (2/9, 4) |
# A program for center of mass calculation
mass=[1,3,5]
CO=[[1,2],[2,3],[-1,5]]
def sumlist(XX):
s=0
for i in XX:
s=s+i
return s
M=sumlist(mass)
X=[CO[i][0] for i in range(len(CO))]
Y=[CO[i][1] for i in range(len(CO))]
def listprod(XX,YY):
ZZ=[XX[i]*YY[i] for i in range(len(XX))]
return ZZ
CM1=sumlist(listprod(mass,X))/M
CM2=sumlist(listprod(mass,Y))/M
print CM1,',',CM2
2/9 , 4 2/9 , 4 |
# A program for center of mass calculation
def sumlist(XX):
s=0
for i in XX:
s=s+i
return s
def listprod(XX,YY):
ZZ=[XX[i]*YY[i] for i in range(len(XX))]
return ZZ
mass=[1,3,5,2,7]
CO=[[2,3],[1,1],[4,5],[-1,-2],[-2,7]]
M=sumlist(mass)
X=[CO[i][0] for i in range(len(CO))]
Y=[CO[i][1] for i in range(len(CO))]
CM1=sumlist(listprod(mass,X))/M
CM2=sumlist(listprod(mass,Y))/M
print CM1,',',CM2
1/2 , 38/9 1/2 , 38/9 |
ex) List 의 원소의 갯수를 헤아려 보자
count=0
mm=[1,2,5,6,777,5,2,100]
for i in mm:
count=count+1 #loop 속에서 인덱스 i 를 쓰지 않아도 됨을 주의 하라.
print '원소의 갯는수는',count,'입니다'
원소의 갯는수는 8 입니다 원소의 갯는수는 8 입니다 |
len(mm)
8 8 |
#Counting sum of matrix lements
M=[[1,2,3],[2,3,4]]
s=0
for i in range(2):
for j in range(3):
s=s+M[i][j]
print s
15 15 |
def sumlist(XX):
s=0
for i in XX:
s=s+i
return s
M=[[1,2,3],[2,3,4]]
s=0
for i in range(2):
s=s+sumlist(M[i])
print s
15 15 |
count=0
mk=range(1,1000000,7)
for i in mk:
count=count+1
print count
142857 142857 |
s=0
for j in range(4,101,4):
s=s+j
print s
1300 1300 |
s=0
for i in range(1,101):
s=s+2**i
print s
2535301200456458802993406410750 2535301200456458802993406410750 |
s=0
for i in range(56):
s=s+i
p=2**s
print p
385649988307365214172818656964530258065934919671310232217548006250441182\ 654688512107053603857175367946151802604942080766057986716607193331995138\ 078062523944232834134301060035963325132466829039948295286901982051209215\ 575337264735857513821939535921274399650502614768108420715736845058788545\ 887066234845739259259035057475454710888677121850041352012892734056144158\ 994382765356263460989042410208779740029161680999518854063792955362004134\ 93190419727789712076165429067776 38564998830736521417281865696453025806593491967131023221754800625044118265468851210705360385717536794615180260494208076605798671660719333199513807806252394423283413430106003596332513246682903994829528690198205120921557533726473585751382193953592127439965050261476810842071573684505878854588706623484573925925903505747545471088867712185004135201289273405614415899438276535626346098904241020877974002916168099951885406379295536200413493190419727789712076165429067776 |
#While-loop
s=0
i=0
while (s<1000000 or i <1000):
i=i+1
s=s+i
print i,s-i
1414 998991 1414 998991 |
ex) 1에서 1000사이의 홀수의 합을 계산해 보자
s=0
for i in range(1,1000,2): #range 는 1에서 시작하여 2만큼 999까지 간다. 즉, 1,3,5,7,....,999
s=s+i
print s
250000 250000 |
s=0
for i in range(1,100,2):
s=s+i**3
print s
12497500 12497500 |
mm=[3,5,2,-1,6,9]
s=0
for i in range(5):
s=s+mm[i]*mm[i+1]
print s
71 71 |
s=0
for i in range(1,1001):
s=s+1/i
print s
533629132822947850455910456240429804096524722803842600971013492484562688\ 894971017575060979019850356914090887315504680983784421721178850094643023\ 443265660225021002784256328520814055449412104425101426727702947747127089\ 179639677796104532246924268664688882815820719848971051107968732493191555\ 293970175089315645199760857344730141832840117244122806490743077037366831\ 700558002936592350885893602352858528081607595747378366554131755081315225\ 17/712886527466509305316638415571427292066835886188589304045200199115432\ 408758111149947644415191387158691171781701957525651298026406762100925146\ 587100430513107268626814320019660997486274593718834370501543445252373974\ 529896314567498212823695623282379401106880926231770886197954079124775455\ 804932647573782992335275179673524804246363805113703433121478174685087845\ 348567802188807537324992199567205693202909939089168748767269795093160352\ 0000 53362913282294785045591045624042980409652472280384260097101349248456268889497101757506097901985035691409088731550468098378442172117885009464302344326566022502100278425632852081405544941210442510142672770294774712708917963967779610453224692426866468888281582071984897105110796873249319155529397017508931564519976085734473014183284011724412280649074307703736683170055800293659235088589360235285852808160759574737836655413175508131522517/7128865274665093053166384155714272920668358861885893040452001991154324087581111499476444151913871586911717817019575256512980264067621009251465871004305131072686268143200196609974862745937188343705015434452523739745298963145674982128236956232823794011068809262317708861979540791247754558049326475737829923352751796735248042463638051137034331214781746850878453485678021888075373249921995672056932029099390891687487672697950931603520000 |
s.n()
7.48547086055035 7.48547086055035 |
응용: 적분
pi.n()
3.14159265358979 3.14159265358979 |
뉴턴 적분 : sine 함수
a=0.0
b=3.14159265358979
N=1000 #구간수
h=(b-a)/N
f(x)=sin(x)
s=0
xi=a
for i in range(N):
aa=h*f(xi)
s=s+aa
xi=xi+h
print s
1.99999835506569 1.99999835506569 |
a=-1.0
b=1.0
N=10 #구간수
h=(b-a)/N
f(x)=x**2*exp(-x**2)
s=0
xi=a
for i in range(N):
aa=h*f(xi)
s=s+aa
xi=xi+h
print s
0.378937973263812 0.378937973263812 |
#while ---loop
a=0.0
b=3.14159265358979
N=1000 #구간수
h=(b-a)/N
f(x)=sin(x)
s=0
xi=a
while xi <= b:
aa=h*f(xi)
s=s+aa
xi=xi+h
print s
1.99999835506569 1.99999835506569 |
integral(f(x),x,-1,1).n()
0.000000000000000 0.000000000000000 |
|
|
L=3.141597/2
h=L/10000
area=0
for i in range(10000):
area=area+h*sin(h*i)
print area
0.999923631223926 0.999923631223926 |
L=pi.n()/4
h=L/100000
int=0
for i in range(100000):
int=int+h*tan(h*i)
print int
0.346569663294292 0.346569663294292 |
integral(tan(x),x,0,pi/4).n()
0.346573590279973 0.346573590279973 |
s=0
N=100000
up=3.0
low=0.0
h=(up-low)/N
x=low
while x <=up :
aa=h*cos(x**2)
s=s+aa
x=x+h
print s
0.702892224492688 0.702892224492688 |
y=var('y')
|
|
integral(cos(y**2),y,0,3).n()
0.702863557730268 0.702863557730268 |
If 조건문 (경우를 따질때)
lst=[-1,-2,0,1,-5,3]
for i in lst:
if i<0:
print i, ':a negative number'
elif i==0: #같은지 아닌지를 따질때 == 사용
print i, ':a zero'
else:
print i, ':a positive number'
-1 :a negative number -2 :a negative number 0 :a zero 1 :a positive number -5 :a negative number 3 :a positive number -1 :a negative number -2 :a negative number 0 :a zero 1 :a positive number -5 :a negative number 3 :a positive number |
lst=[-1,-2,0,1,2,3,-1,7,9,5,6]
for i in lst:
if i<0:
print i, ':a negative number' #elif 나 else 는 선택사항이지 필수가 아니다.
-1 :a negative number -2 :a negative number -1 :a negative number -1 :a negative number -2 :a negative number -1 :a negative number |
s=0
for i in lst:
if i>0:
s=s+i
print s
33 33 |
10%3
|
|
lst=[-1,-2,0,1,-5,3]
s=0
for a in lst:
if 1<=a<=5:
s=s+a
print s
4 4 |
s=0
for a in lst:
if 1<=a:
if a<=5:
s=s+a
print s
4 4 |
for k in range(1,21):
if (k%3==0):
print k, '3 의 배수'
else:
print k, '3 의 배수 아님'
1 3 의 배수 아님 2 3 의 배수 아님 3 3 의 배수 4 3 의 배수 아님 5 3 의 배수 아님 6 3 의 배수 7 3 의 배수 아님 8 3 의 배수 아님 9 3 의 배수 10 3 의 배수 아님 11 3 의 배수 아님 12 3 의 배수 13 3 의 배수 아님 14 3 의 배수 아님 15 3 의 배수 16 3 의 배수 아님 17 3 의 배수 아님 18 3 의 배수 19 3 의 배수 아님 20 3 의 배수 아님 1 3 의 배수 아님 2 3 의 배수 아님 3 3 의 배수 4 3 의 배수 아님 5 3 의 배수 아님 6 3 의 배수 7 3 의 배수 아님 8 3 의 배수 아님 9 3 의 배수 10 3 의 배수 아님 11 3 의 배수 아님 12 3 의 배수 13 3 의 배수 아님 14 3 의 배수 아님 15 3 의 배수 16 3 의 배수 아님 17 3 의 배수 아님 18 3 의 배수 19 3 의 배수 아님 20 3 의 배수 아님 |
8%4
0 0 |
s=0
for i in range(1,256):
if i%3==0:
s=s+i
elif i%5==0:
s=s+i
print s
15300 15300 |
s=0
for i in range(1,256):
if (i%3==0 or i%5==0):
s=s+i
print s
15300 15300 |
s=0
for i in range(1,256):
if i%3==0:
if i%5==0:
s=s+i
print s
2295 2295 |
위 리스트의 음수의 합을 구해보라.
Logical Operators
# AND, OR
MM=[-1,2,-3,-5,0,0,2,3,4,7,11,12]
for i in MM:
if i<0 or i>7:
print 1, i
for i in MM:
if i>0 and i<10:
print 2,i
1 -1 1 -3 1 -5 1 11 1 12 2 2 2 2 2 3 2 4 2 7 1 -1 1 -3 1 -5 1 11 1 12 2 2 2 2 2 3 2 4 2 7 |
While-loop (조건이 맞을때까지 무한 반복)
s=0
j=1
while s<=1000000:
s=s+j
j=j+1
print j-1, s-j
1414 998990 1414 998990 |
s=0
i=0
while s<1000: #조건이 만족하는한 반복해서 계산한다.
i=i+1
s=s+i
print i-1,s-i
44 990 44 990 |
s=0
for i in range(100):
s=s+i
if s>=1000:
break
print i-1,s-i
44 990 44 990 |
s=0
i=0
while s<=1000:
i=i+1
s=s+i
print i,s
45 1035 45 1035 |
s=0
for i in range(45):
s=s+i
print s
990 990 |
s1=0
j=0
while s1<10000:
j=j+1
s1=s1+j
print j-1,s1-i
140 9967 140 9967 |
For-loop 에서 빠져나오기 (break)
합이 1000을 넘지 않을때까지 더해보자
s=0
for i in range(1000000):
s=s+i
if s>1000:
print i-1,s-i
break #loop 에서 빠져나온다.
44 990 44 990 |
s=0
for i in range(45):
s=s+i
print s
990 990 |
Defining a function recursively (순환으로 함수 정의하기)
#factorial 함수를 정의해 해보자.
def nfactorial(n):
if n==0:
return 1
else:
return n*nfactorial(n-1)
|
|
for i in range(6):
print i,nfactorial(i)
0 1 1 1 2 2 3 6 4 24 5 120 0 1 1 1 2 2 3 6 4 24 5 120 |
#factorial 함수를 while loop 을 사용하여 정의해 보자.
def nfacto(n):
if n==0:
return 1
else:
p=n
i=n
while i>1:
i=i-1
p=p*i
return p
|
|
nfacto(3); nfacto(5)
6 120 6 120 |
def myabs(n):
if n>0:
print n
else:
print -n
|
|
myabs(3)
3 3 |
myabs(-5)
5 5 |
def facto(n):
if n<0:
print 'Give a positive number'
else:
p=1
for i in range(1,n+1):
p=p*i
return p
|
|
facto(4)
24 24 |
facto(-100)
Give a positive number Give a positive number |
t=0.0
h=0.001
f=2.0
ft=[(t,f)]
while t<=2.0:
f=f-2*f*h
t=t+h
ft.append((t,f))
|
|
list_plot(ft,figsize=4,plotjoined=True)+plot(2*exp(-2*x),x,0,2,color='red')
|
plot(2*exp(-2*x),x,0,2,figsize=4)
|
행렬곱 계산 프로그램
def matrixmulti(A,B):
p=len(A)
q=len(A[0])
r=len(B)
s=len(B[0])
C=[[0 for i in range(s)] for j in range(p)] #p x s 0-행렬
if q!=r:
print('Not multipliable')
return
else:
for i in range(p):
for j in range(s):
for k in range(q):
C[i][j]=C[i][j]+A[i][k]*B[k][j]
return C
#------------- main program -------------------------------------------
A=[[1,2,3,4],[-1,2,3,5],[-1,-2,0,5]]
B=[[2,3],[-1,2],[0,2],[9,1]]
C=matrixmulti(A,B)
print(C)
[[36, 17], [41, 12], [45, -2]] [[36, 17], [41, 12], [45, -2]] |
Python data types
Integer
2/3 #integer
|
|
float---double precision: 소수점이 있는 숫자
2.0/3
|
|
float(2)/3 #float
|
|
2/float(3) #float
|
|
2+3 #integer
|
|
2+3.0 #float
|
|
Strings--문자열
a='How are you?'
print a
How are you? How are you? |
a="How are you?"
print a
How are you? How are you? |
|
|