2533.sagews

a=5
a

5

2==2

True

2>3

False

2**3

8

2^3

8

5/2

5/2

5//2

2

18%5

3

012

10

09

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_11.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("MDk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpIc0DWu/___code___.py", line 2 _sage_const_09 = Integer(09) ^ SyntaxError: invalid token

sin?

File: /usr/local/sage/local/lib/python2.6/site-packages/sage/functions/trig.py

Type: <class ‘sage.functions.trig.Function_sin’>

Definition: sin(*args, coerce=True, hold=False, dont_call_method_on_arg=False)

Docstring:

The sine function.

EXAMPLES:

sage: sin(0)
0
sage: sin(x).subs(x==0)
0
sage: sin(2).n(100)
0.90929742682568169539601986591
sage: loads(dumps(sin))
sin

sin(0)

0

sin(x).subs(x==0)

0

sin(2).n(100)

0.90929742682568169539601986591

loads(dumps(sin))

sin

def is_even(n):
    return n%2==0

is_even(2)

True

is_even(123)

False

def is_divisible_by(number,divisor=2):
    return number%divisor==0

is_divisible_by(123,5)

False

is_divisible_by(123)

False

def myfunction(x):
    if x<0:
        return -1
    elif x==0:
        return 0
    else:return 1

myfunction(-2)

-1

myfunction(0)

0

myfunction(3)

1

money=10000
years=0
while money<20000:
    years=years+1
    money=money*1.069

years

11

money

20833.1411965483

n=10;ans=1;
for each in range(1,n+1):
    ans=ans*each

ans

3628800

for eachitem in ['name:vincent','age=30','job:teacher']:
    print eachitem

name:vincent age=30 job:teacher

x=fibonacci(10)
    len(x)

x
Fibonacci numbers
list(x)
[1,1,2,3,5,8,13,21,34,55]
x.show(11)

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_97.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("eApGaWJvbmFjY2kgbnVtYmVycwpsaXN0KHgpClsxLDEsMiwzLDUsOCwxMywyMSwzNCw1NV0KeC5zaG93KDExKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpy2ANam/___code___.py", line 4 Fibonacci numbers ^ SyntaxError: invalid syntax

fibonacci?

File: /usr/local/sage/local/lib/python2.6/site-packages/sage/combinat/combinat.py

Type: <type ‘function’>

Definition: fibonacci(n, algorithm=’pari’)

Docstring:

Returns the n-th Fibonacci number. The Fibonacci sequence F_n is defined by the initial conditions F_1=F_2=1 and the recurrence relation F_{n+2} = F_{n+1} + F_n. For negative n we define F_n = (-1)^{n+1}F_{-n}, which is consistent with the recurrence relation.

INPUT:

• algorithm - string:
• "pari" - (default) - use the PARI C library’s fibo function.
• "gap" - use GAP’s Fibonacci function

Note

PARI is tens to hundreds of times faster than GAP here; moreover, PARI works for every large input whereas GAP doesn’t.

EXAMPLES:

sage: fibonacci(10)
55
sage: fibonacci(10, algorithm='gap')
55

sage: fibonacci(-100)
-354224848179261915075
sage: fibonacci(100)
354224848179261915075

sage: fibonacci(0)
0
sage: fibonacci(1/2)
...
TypeError: no conversion of this rational to integer

2^3+123456789*81

9999999917

exp?

File: /usr/local/sage/local/lib/python2.6/site-packages/sage/functions/log.py

Type: <class ‘sage.functions.log.Function_exp’>

Definition: exp(x, coerce=True, hold=False, prec=None, dont_call_method_on_arg=False)

Docstring:

The exponential function, \exp(x) = e^x.

EXAMPLES:

sage: exp(-1)
e^(-1)
sage: exp(2)
e^2
sage: exp(2).n(100)
7.3890560989306502272304274606
sage: exp(x^2 + log(x))
e^(x^2 + log(x))
sage: exp(x^2 + log(x)).simplify()
x*e^(x^2)
sage: exp(2.5)
12.1824939607035
sage: exp(float(2.5))
12.182493960703473
sage: exp(RDF('2.5'))
12.1824939607

sage: exp(pi*I/2)
I
sage: exp(pi*I)
-1
sage: exp(8*pi*I)
1
sage: exp(7*pi*I/2)
-I

TEST:

sage: latex(exp(x))
e^{x}
sage: latex(exp(sqrt(x)))
e^{\sqrt{x}}
sage: latex(exp)
\exp
sage: latex(exp(sqrt(x))^x)
\left(e^{\sqrt{x}}\right)^{x}
sage: latex(exp(sqrt(x)^x))
e^{\left(\sqrt{x}^{x}\right)}

Test simplifications when taking powers of exp, #7264:

sage: var('a,b,c,I')
(a, b, c, I)
sage: model_exp = exp(I)**a*(b)
sage: sol1_l={b: 5.0, a: 1.1}
sage: model_exp.subs(sol1_l)
5.00000000000000*(e^I)^1.10000000000000

sage: exp(3)^I*exp(x)
(e^3)^I*e^x
sage: exp(x)*exp(x)
e^(2*x)
sage: exp(x)*exp(a)
e^(a + x)
sage: exp(x)*exp(a)^2
e^(2*a + x)

Another instance of the same problem, #7394:

sage: 2*sqrt(e)
2*sqrt(e)

exp(1/0.1^2)

2.68811714181610e43

def is_odd(n):
    return n%2==1

is_odd(1)

True

is_odd(2)

False

def myfunction(x,y):
    if x^2+y^2<=1:
        return 1
    elif 1<x^2+y^2<=2:
        return 1/2
    else:
        return 0

myfunction(2,2)

0

myfunction(1,1)

1/2

myfunction(0.2,0.3)

1

n=10
m=0
x=range(n)
while x<=100:
    p=random()
    if p<0.2:
        x[m]=1
    else:
        x[m]=0
    m=m+1

Traceback (most recent call last): x[m]=1 File "", line 1, in <module> File "/tmp/tmpCaXFi7/___code___.py", line 6, in <module> exec compile(u'while x<=_sage_const_100 :\n p=random()\n if p<_sage_const_0p2 :\n x[m]=_sage_const_1 \n else:\n x[m]=_sage_const_0 \n m=m+_sage_const_1 ' + '\n', '', 'single') File "", line 6, in <module> IndexError: list assignment index out of range

epsilon=0.01
x_pre=infinity
x=0.0
while abs(x-x_pre)<epsilon:
    e_pre=x
    x=(1-2*x)^(1/3)

x=0;x+=1;x

1

x=2;x**=2;x

4

x=[123,'xyz'];x+=[011];x

[123, 'xyz', 9]

x=y=z=123

x

123

y

123

z

123

x,y,z=1,2,3

x

1

y

2

z

3

x=666;y=888

x,y=y,x

x

888

y

666

x=1
x+=100

x

101

load myadd.sage
x

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_7.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("bG9hZCBteWFkZC5zYWdlCng="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmplYmkAf/___code___.py", line 2, in <module> sage.misc.preparser.load(sage.misc.preparser.base64.b64decode("bXlhZGQuc2FnZQ=="),globals(),False) File "/usr/local/sage2/local/lib/python2.6/site-packages/sage/misc/preparser.py", line 1503, in load exec(preparse_file(open(filename).read()), globals) IOError: [Errno 2] No such file or directory: 'myadd.sage'

attach myadd.sage
x

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_165.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("YXR0YWNoIG15YWRkLnNhZ2UKeA=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmp5WtWxM/___code___.py", line 2, in <module> sage.misc.preparser.load(sage.misc.preparser.base64.b64decode("bXlhZGQuc2FnZQ=="),globals(),True) File "/usr/local/sage/local/lib/python2.6/site-packages/sage/misc/preparser.py", line 1503, in load exec(preparse_file(open(filename).read()), globals) IOError: [Errno 2] No such file or directory: 'myadd.sage'

n=10
x=[int(random()*100 for each in range(n)]
print "random numbers is",x
for each in range(n):
    for k in range(n-1):
        if x[k]<x[k+1]:
            x[k],x[k+1]=x[k+1],x[k]
print "Sorted is",x

Traceback (most recent call last): x[k],x[k+1]=x[k+1],x[k] File "", line 1, in <module> File "/tmp/tmpzYGE3B/___code___.py", line 4 x=[int(random()*_sage_const_100 for each in range(n)] ^ SyntaxError: invalid syntax

cdef extern from "test.c":
    int myfactorial(int n)
def test(n):
    return myfactorial(n)

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_174.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Y2RlZiBleHRlcm4gZnJvbSAidGVzdC5jIjoKICAgIGludCBteWZhY3RvcmlhbChpbnQgbikKZGVmIHRlc3Qobik6CiAgICByZXR1cm4gbXlmYWN0b3JpYWwobik="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpGh86Xl/___code___.py", line 2 cdef extern from "test.c": ^ SyntaxError: invalid syntax

sttach test.apyx
Compiling test.spyx...
test(8)

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_176.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("c3R0YWNoIHRlc3QuYXB5eApDb21waWxpbmcgdGVzdC5zcHl4Li4uCnRlc3QoOCk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpbkV1ro/___code___.py", line 3 sttach test.apyx ^ SyntaxError: invalid syntax

n=var('n')
a_n=(1+1/n)**n
a_n.limit(n=oo)

e

limit(sin(n*pi/3),n=oo)

ind

n=var('n')
limit((n**2-1)/(n+5),n=oo)

+Infinity

x=var('x')
diff(sin(x),x)

cos(x)

diff(sin(x),x,2)

-sin(x)

x,y=var('x,y')
f=x^2*y+x*exp(y)
f.diff(x,y)

2*x + e^y

var('x,a')
f=exp(sin(a-a^2))/x
f.derivative(x)

-e^(sin(-a^2 + a))/x^2

x=var('x')
f=sin(x)
taylor(f,x,0,6)

1/120*x^5 - 1/6*x^3 + x

x,y=var('x,y')
f=sin(x*y)
taylor(f,(x,0),(y,-1),4)

-1/2*(y + 1)*x^3 + 1/6*x^3 + (y + 1)*x - x

x=var('x')
integral(x*sin(x**2),x)

-1/2*cos(x^2)

integral(x^2,x,0,1)

1/3

x=var('x')
integral(x**(-2),1,infinity)

1

var('x,k,w')

(x, k, w)

f=x^3*exp(k*x)*sin(w*x)
f.integrate(x)

-(((k^6*w + 3*k^4*w^3 + 3*k^2*w^5 + w^7)*x^3 - 24*k^3*w + 24*k*w^3 - 6*(k^5*w + 2*k^3*w^3 + k*w^5)*x^2 + 6*(3*k^4*w + 2*k^2*w^3 - w^5)*x)*e^(k*x)*cos(w*x) - ((k^7 + 3*k^5*w^2 + 3*k^3*w^4 + k*w^6)*x^3 - 6*k^4 + 36*k^2*w^2 - 6*w^4 - 3*(k^6 + k^4*w^2 - k^2*w^4 - w^6)*x^2 + 6*(k^5 - 2*k^3*w^2 - 3*k*w^4)*x)*e^(k*x)*sin(w*x))/(k^8 + 4*k^6*w^2 + 6*k^4*w^4 + 4*k^2*w^6 + w^8)

m=var('m')
sum(1/(m*(m+1)),m,1,oo)

1

sum(1/(m*(m+1)),m,1,10)

10/11

sum(1/(m*(m+1)),m,1,100)

100/101

t=var('t')
x=function('x',t)
a,b=var('a,b')
DE=a*diff(x,t)+b*x-1
desolve(DE,[x,t])

(c + e^(b*t/a)/b)*e^(-b*t/a)

plot(sin,(0,2*pi),color='red')

p=plot(sin(x),color=hue(1.0)) 
for a in range(2,8+1): 
    p+=plot(sin(a*x),color=hue(1.0/a)) 
p.show(xmin=-1,xmax=1,ymin=-1,ymax=1)

var('t')
parametric_plot((1-2*sin(t),t^2),(t,-4,4))

var('x,y')
implicit_plot(x^2/4-y^2/9-1,(x,-5,5),(y,-6,6)).show(aspect_ratio=1)

var('a')
polar_plot(a/5,(a,0,6*pi),color='green').show(aspect_ratio=1)

var('x,y')
plot3d(x^2/16-y^2/9,(x,-10,10),(y,-10,10))

p=plot(x/(1+x),color=hue(1.0)) 
for m in range(2,8+1): 
    p+=plot(m*x/(1+m^2*x^2),color=hue(1.0/m)) 
p.show(xmin=-1,xmax=1,ymin=-1,ymax=1)

var('t')
parametric_plot((sin(3*t)*cos(t),sin(3*t)*sin(t)),(t,0,pi))

var('x,y')
implicit_plot(x^2/16+y^2/4-1,(x,-8,8),(y,-4,4)).show(aspect_ratio=1)

var('a')
polar_plot(a^2*sin(a^2),(a,0,6*pi),color='red').show(aspect_ratio=1)

m1=matrix(ZZ,[[1,2,3,],[4,5,6]])
m1.parent()
m1

[1 2 3] [4 5 6]

m1/2

[1/2 1 3/2] [ 2 5/2 3]

m2=matrix(QQ,[[1,2,3,],[4,5,6]])
m2.parent()
m2

[1 2 3] [4 5 6]

m2/2

[1/2 1 3/2] [ 2 5/2 3]

m3=matrix(RR,[[1,2,3,],[4,5,6]])
m3.parent()
m3

[1.00000000000000 2.00000000000000 3.00000000000000] [4.00000000000000 5.00000000000000 6.00000000000000]

m3/2

[0.500000000000000 1.00000000000000 1.50000000000000] [ 2.00000000000000 2.50000000000000 3.00000000000000]

a=matrix(ZZ,[[3,1,1],[2,1,2]])
b=matrix(ZZ,[[1,-2,3],[4,5,-6]])
a+b

[ 4 -1 4] [ 6 6 -4]

a-b

[ 2 3 -2] [-2 -4 8]

a=matrix(ZZ,[[3,1,1],[2,1,2]])
a.transpose()

[3 2] [1 1] [1 2]

transpose(a.transpose())

[3 1 1] [2 1 2]

b=matrix(ZZ,[[1,-2,3],[4,5,-6]])
a*b.transpose()

[ 4 11] [ 6 1]

transpose(b)*a

[11 5 9] [ 4 3 8] [-3 -3 -9]

A=matrix(QQ,3,range(9));A

[0 1 2] [3 4 5] [6 7 8]

b=matrix(QQ,3,1,[1,2,3]);b

[1] [2] [3]

x=A\b;x

[-2/3] [ 1] [ 0]

A*x

[1] [2] [3]

Q=matrix\
([[2,1,0,0,0,],[0,2,1,0,0],[0,0,2,1,0],[0,0,0,2,1],[0,0,0,0,2]])
Q^3

[ 8 12 6 1 0] [ 0 8 12 6 1] [ 0 0 8 12 6] [ 0 0 0 8 12] [ 0 0 0 0 8]

rank(Q)

5

Q.inverse()

[ 1/2 -1/4 1/8 -1/16 1/32] [ 0 1/2 -1/4 1/8 -1/16] [ 0 0 1/2 -1/4 1/8] [ 0 0 0 1/2 -1/4] [ 0 0 0 0 1/2]

Q^(-1)*Q

[1 0 0 0 0] [0 1 0 0 0] [0 0 1 0 0] [0 0 0 1 0] [0 0 0 0 1]

A=matrix(RR,[[1,2,5],[1,3,-2],[3,7,8],[1,4,-9]])
b=matrix(RR,[[1],[1],[3],[1]])
rank(A)

2

AA=matrix(RR,4,4)
AA.set_block(0,0,A)
AA.set_block(0,3,b)
rank(AA)

2

A.rref()

[ 1.00000000000000 0.000000000000000 19.0000000000000] [0.000000000000000 1.00000000000000 -7.00000000000000] [0.000000000000000 0.000000000000000 0.000000000000000] [0.000000000000000 0.000000000000000 0.000000000000000]

s=A.right_kernel()
ss=s.basis_matrix();ss

[ 1.00000000000000 -0.368421052631579 -0.0526315789473684]

x=A.solve_right(b);x

[ 1.00000000000000] [0.000000000000000] [0.000000000000000]

A=matrix(QQ,3,range(9));A[0,0]=2;A

[2 1 2] [3 4 5] [6 7 8]

b=matrix(QQ,1,3,[1,2,3]);b

[1 2 3]

x=b*A^(-1);x

[ 0 5/3 -2/3]

D=matrix(RR,4,[random() for each in range(16)]);D

[ 0.940118813085686 0.965476202692564 0.308800560307966 0.831573569224472] [ 0.830234709624040 0.450246242252960 0.570598270811050 0.454395010787615] [ 0.282073726465788 0.714070529426863 0.328100651923947 0.0721480109333678] [ 0.929933648170211 0.269682898269158 0.520346698401252 0.355973515003413]

D.det()

0.0501581984964803

var('a,b,c,d')

(a, b, c, d)

D=matrix\
([[a^2,(a+1)^2,(a+2)^2,(a+3)^2],[b^2,(b+1)^2,(b+2)^2,(b+3)^2],\
[c^2,(c+1)^2,(c+2)^2,(c+3)^2],[d^2,(d+1)^2,(d+2)^2,(d+3)^2]])
D

[ a^2 (a + 1)^2 (a + 2)^2 (a + 3)^2] [ b^2 (b + 1)^2 (b + 2)^2 (b + 3)^2] [ c^2 (c + 1)^2 (c + 2)^2 (c + 3)^2] [ d^2 (d + 1)^2 (d + 2)^2 (d + 3)^2]

det(D)

0

A=matrix(RR,3,[random() for each in range(9)]);A

[ 0.864356202559331 0.633096841879541 0.924103099577427] [ 0.999474720641427 0.681717900332826 0.0881311159313142] [ 0.456144951079067 0.00281743139502155 0.904914069422016]

B=copy(A)
B.set_row(0,[1,2,3]);B

[ 1.00000000000000 2.00000000000000 3.00000000000000] [ 0.999474720641427 0.681717900332826 0.0881311159313142] [ 0.456144951079067 0.00281743139502155 0.904914069422016]

C=copy(A);C[0,:]=A[0,:]+B[0,:];C

[ 1.86435620255933 2.63309684187954 3.92410309957743] [ 0.999474720641427 0.681717900332826 0.0881311159313142] [ 0.456144951079067 0.00281743139502155 0.904914069422016]

det(A)+det(B)

-2.33516918770469

det(C)

-2.33516918770469

D=matrix(RR,3,[1,10,100,2,40,800,3,90,2700])
b=matrix(RR,3,1,[-0.003,-0.005,-0.008])
D1=copy(D);D1[:,0]=b;a1=det(D1)/det(D);a1

-0.00416666666666667

D2=copy(D);D2[:,1]=b;a2=det(D2)/det(D);a2

0.000150000000000000

D3=copy(D);D3[:,2]=b;a3=det(D3)/det(D);a3

-3.33333333333334e-6

def h(t):
    return 13.6+a1*t+a2*t^2+a3*t^3

h(15)

13.5600000000000

h(40)

13.4600000000000