4-Sage Demo.sagews

Sage Demo

Calculus

2 + 3

5

x = var('x')
show(plot(sin(x^2), 0, pi))

a = integrate(sin(x^2),x); a

1/16*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x) + (I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)*x) - (I - 1)*sqrt(2)*erf(sqrt(-I)*x) + (I + 1)*sqrt(2)*erf((-1)^(1/4)*x))

show(a)

$\displaystyle \frac{1}{16} \, \sqrt{\pi} {\left(\left(i + 1\right) \, \sqrt{2} \operatorname{erf}\left(\left(\frac{1}{2} i + \frac{1}{2}\right) \, \sqrt{2} x\right) + \left(i - 1\right) \, \sqrt{2} \operatorname{erf}\left(\left(\frac{1}{2} i - \frac{1}{2}\right) \, \sqrt{2} x\right) - \left(i - 1\right) \, \sqrt{2} \operatorname{erf}\left(\sqrt{-i} x\right) + \left(i + 1\right) \, \sqrt{2} \operatorname{erf}\left(\left(-1\right)^{\frac{1}{4}} x\right)\right)}$

latex(a)

\frac{1}{16} \, \sqrt{\pi} {\left(\left(i + 1\right) \, \sqrt{2} \operatorname{erf}\left(\left(\frac{1}{2} i + \frac{1}{2}\right) \, \sqrt{2} x\right) + \left(i - 1\right) \, \sqrt{2} \operatorname{erf}\left(\left(\frac{1}{2} i - \frac{1}{2}\right) \, \sqrt{2} x\right) - \left(i - 1\right) \, \sqrt{2} \operatorname{erf}\left(\sqrt{-i} x\right) + \left(i + 1\right) \, \sqrt{2} \operatorname{erf}\left(\left(-1\right)^{\frac{1}{4}} x\right)\right)}

var('a,b,c,X')
s = solve(a*X^2 + b*X + c == 0, X)
show(s[0])

(a, b, c, X)

$\displaystyle X = -\frac{b + \sqrt{b^{2} - 4 \, a c}}{2 \, a}$

Linear Algebra

A = random_matrix(GF(127),3000,3000,density=1.0/3000,sparse=True)
show(A)
A.echelonize()
A.rank()

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WARNING: Output: 71997521 truncated by MAX_HTML_SIZE to 40000. Type 'smc?' to learn how to raise the output limit.

1627

n = 300
A = random_matrix(GF(127),n,n+100,sparse=True,density=2/n)
A.echelonize()
A.visualize_structure()

400x300px 24-bit RGB image

sr = mq.SR(4,2,2,4,gf2=True, allow_zero_inversions=True)
F,s = sr.polynomial_system()
A,v = F.coefficient_matrix()
A.visualize_structure(maxsize=600)

600x414px 24-bit RGB image

Factoring

time factor(next_prime(2^40) * next_prime(2^300),verbose=0)

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "<string>", line 1 time factor(next_prime(Integer(2)**Integer(40)) * next_prime(Integer(2)**Integer(300)),verbose=Integer(0)) ^ SyntaxError: invalid syntax

time ecm.factor(next_prime(2^40) * next_prime(2^300))

[1099511627791, 2037035976334486086268445688409378161051468393665936250636140449354381299763336706183397533] CPU time: 0.25 s, Wall time: 0.82 s

v,t = qsieve(next_prime(2^90)*next_prime(2^91),time=True)
print v, t[:4]

[1237940039285380274899124357, 2475880078570760549798248507] 4.17

Elliptic Curves

e = EllipticCurve("37a") # Cremona Label
show(e)

$\displaystyle y^2 + y = x^{3} - x$

show(plot(e))

E = e.change_ring(GF(997))
show(E.plot())

Graph Theory

D = graphs.DodecahedralGraph()
D.show()

D.show3d()

gamma = SymmetricGroup(20).random_element()
E = D.copy()
E.relabel(gamma)
D.is_isomorphic(E)

True

D.radius()

5

Multivariate Polynomial Rings

P.<x,y,z> = PolynomialRing(QQ,3)
p = (x + y + z + 1)^20 # the Fateman fastmult benchmark
q = p + 1
time r = p*q

Time: CPU 2.46 s, Wall: 2.46 s

P.<x,y,z> = PolynomialRing(GF(32003),3)
p = (x + y + z + 1)^20 # the Fateman fastmult benchmark
q = p + 1
time r = p*q

Time: CPU 0.29 s, Wall: 0.29 s

P = PolynomialRing(GF(32003),10,'x')
I = sage.rings.ideal.Cyclic(P,7)
t = cputime()
gb = I.groebner_basis('libsingular:std')
print 'Sage/Singular', cputime(t)

I = sage.rings.ideal.Cyclic(P,7)
t = magma.cputime()
gb = I.groebner_basis('magma:GroebnerBasis')
print 'MAGMA', magma.cputime(t)

Sage/Singular 4.872305

Traceback (most recent call last): print 'MAGMA', magma.cputime(t) File "/usr/local/sage/local/lib/python2.5/site-packages/sage/interfaces/magma.py", line 349, in cputime return float(self.eval('Cputime()')) File "/usr/local/sage/local/lib/python2.5/site-packages/sage/interfaces/magma.py", line 278, in eval ans = Expect.eval(self, x).replace('\\\n','') File "/usr/local/sage/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 917, in eval return '\n'.join([self._eval_line(L, **kwds) for L in code.split('\n') if L != '']) File "/usr/local/sage/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 613, in _eval_line self._start() File "/usr/local/sage/local/lib/python2.5/site-packages/sage/interfaces/magma.py", line 285, in _start Expect._start(self) File "/usr/local/sage/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 450, in _start self.__name, cmd, self._install_hints()) RuntimeError: Unable to start magma because the command 'magma -n' failed.

Combinatorics

C = Combinations(range(5)); C

Combinations of [0, 1, 2, 3, 4]

C.list()

[[], [0], [1], [2], [3], [4], [0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4], [0, 1, 2], [0, 1, 3], [0, 1, 4], [0, 2, 3], [0, 2, 4], [0, 3, 4], [1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4], [0, 1, 2, 3], [0, 1, 2, 4], [0, 1, 3, 4], [0, 2, 3, 4], [1, 2, 3, 4], [0, 1, 2, 3, 4]]

C.unrank(10)

[1, 2]

Interfaces

%gap
a := 1;
for i in [1..100] do if IsPrime(i) then a:=a+1; else a:=a+2; fi;  od; a;

1 176

%singular
int a = 1; int i = 1;
for(i=1; i<=100; i=i+1) { if(prime(i) == i){  a=a+1; } else {a=a+2;} };
a;

176

%gp
a=1; for(X=1,100,if(isprime(X),a+=1,a+=2)); a

176

a= gap(1) # or gp(1), magma(1), singular(1)
for i in range(100):
    if gap(i+1).IsPrime():
        a+=1
    else:
        a+=2
a

176

a = gap(176)
type(a)

<class 'sage.interfaces.gap.GapElement'>

pari(a).factor()

[2, 4; 11, 1]

Curve Fitting

import scipy.linalg.basic

xdata = [5.357, 5.457, 5.797, 5.936, 6.161, 6.697, 6.731, 6.775, 8.442, 9.769, 9.861]
ydata = [0.376, 0.489, 0.874, 1.049, 1.327, 2.054, 2.077, 2.138, 4.744, 7.068, 7.104]
matrix = []

for x in xdata:
    matrix.append([1.0, x, x*x]) # for y = a + bx + cx^2

coeffs = scipy.linalg.basic.lstsq(matrix, ydata)[0]

print "scipy.linalg.basic.lstsq curve fitting example"
print "fitting data to quadratic equation y = a + bx + cx^2"

print "yields:  x data     y data    calc value   error"
for i in range(len(xdata)):
    ycalc = coeffs[0] + coeffs[1] * xdata[i] + coeffs[2] * xdata[i] * xdata[i]
    error = ycalc - ydata[i]
    print "         % .3f    % .3f      % .3f    % .3f" % (xdata[i], ydata[i],
ycalc, error)
print

x = var('x')
f1 = line(zip(xdata,ydata), rgbcolor=(1,0,0))
f2 = plot(sum([coeffs[i]*x^i for i in range(len(coeffs))]), 5, 10)
(f1 + f2).show(xmin=5)

scipy.linalg.basic.lstsq curve fitting example fitting data to quadratic equation y = a + bx + cx^2 yields: x data y data calc value error 5.357 0.376 0.355 -0.021 5.457 0.489 0.472 -0.017 5.797 0.874 0.882 0.008 5.936 1.049 1.055 0.006 6.161 1.327 1.341 0.014 6.697 2.054 2.054 0.000 6.731 2.077 2.101 0.024 6.775 2.138 2.162 0.024 8.442 4.744 4.688 -0.056 9.769 7.068 7.010 -0.058 9.861 7.104 7.181 0.077

3D Plotting

t = Tachyon(xres=500, yres=500, camera_center=(2,7,4), look_at=(2,0,0), raydepth=4)
t.light((10,3,2), 1, (1,1,1))
t.light((10,-3,2), 1, (1,1,1))
t.texture('black', color=(0,0,0))
t.texture('red', color=(1,0,0))
t.texture('grey', color=(.9,.9,.9))
t.plane((0,0,0),(0,0,1),'grey')
t.cylinder((0,0,0),(1,0,0),.01,'black')
t.cylinder((0,0,0),(0,1,0),.01,'black')
E = EllipticCurve('37a')
P = E([0,0])
Q = P
n = 100
for i in range(n):
   Q = Q + P
   c = i/n + .1
   t.texture('r%s'%i,color=(float(i/n),0,0))
   t.sphere((Q[0], -Q[1], .01), .04, 'r%s'%i)

t.show()

t = Tachyon(xres=800,yres=800, camera_center=(2,5,2), look_at=(2.5,0,0))
t.light((0,0,100), 1, (1,1,1))
t.texture('r', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1,0,0))
for i in srange(0,50,0.1):
   t.sphere((i/10,sin(i),cos(i)), 0.05, 'r')
t.texture('white', color=(1,1,1), opacity=1, specular=1, diffuse=1)
t.plane((0,0,-100), (0,0,-100), 'white')
t.show()

Sequences of Integers

for sq in sloane_find([2,3,5,7], 2):
  print sq[0], sq[1]

Searching Sloane's online database... 40 The prime numbers. 41 a(n) = number of partitions of n (the partition numbers).