x=vector(QQ, [1, 2, -3, 4])
y=vector(QQ, [-2, 4, 1, 0])
z=vector(QQ, [5, -2, 3, -7])
t=vector(QQ, [2, 2, 3, 5])
print "2*x-3*y+z-t=", 2*x-3*y+z-t
vectors = [x, y, z, t]
scalars = [2, -3, 1, -1]
multiples = [scalars[i]*vectors[i] for i in range(4)]
print "a*x+b*y+c*z+d*t=", sum(multiples)
2*x-3*y+z-t= (11, -12, -9, -4) a*x+b*y+c*z+d*t= (11, -12, -9, -4) 2*x-3*y+z-t= (11, -12, -9, -4) a*x+b*y+c*z+d*t= (11, -12, -9, -4) |
x=vector([2, -1, 3, 2])
y=vector([3, 2, 1, -4])
print x.inner_product(y)
print x.norm()
print "cos(theta)= ", x.inner_product(y)/(x.norm()*y.norm())
-1 3*sqrt(2) cos(theta)= -1/180*sqrt(2)*sqrt(30) -1 3*sqrt(2) cos(theta)= -1/180*sqrt(2)*sqrt(30) |
A=matrix(QQ,[[25, 15, -5], [15, 18, 0], [-5, 0, 11]])
L=A.cholesky()
print L
[ 5 0 0] [ 3 3 0] [-1 1 3] [ 5 0 0] [ 3 3 0] [-1 1 3] |
print A.inverse()
L*L.transpose()==A
[ 22/225 -11/135 2/45] [-11/135 10/81 -1/27] [ 2/45 -1/27 1/9] True [ 22/225 -11/135 2/45] [-11/135 10/81 -1/27] [ 2/45 -1/27 1/9] True |
b=vector([1, 2, 3])
A.solve_right(b)
(46/675, 22/405, 41/135) (46/675, 22/405, 41/135) |
f(x) = x^2
a = 0.0
b = 2.0
table = []
exact = integrate(f(x), x, a, b)
for n in [4, 10, 20, 50, 100]:
h = (b-a)/n
midpoint = sum([f(a+(i+1/2)*h)*h for i in range(n)])
trapezoid = h/2*(f(a) + 2*sum([f(a+i*h) for i in range(1,n)])+ f(b))
simpson = h/3*(f(a) + sum([4*f(a+i*h) for i in range(1,n,2)])+ sum([2*f(a+i*h) for i in range (2,n,2)]) + f(b))
table.append([n, h.n(digits=2), (midpoint-exact).n(digits=6),
(trapezoid-exact).n(digits=6), (simpson-exact).n(digits=6)])
html.table(table, header=["n", "h", "Midpoint rule",
"Trapezoidal rule", "Simpson’s rule"])
Traceback (click to the left of this block for traceback) ... IndentationError: expected an indented block Traceback (most recent call last): h = (b-a)/n
File "", line 1, in <module>
File "/tmp/tmp2E47EY/___code___.py", line 9
h = (b-a)/n
^
IndentationError: expected an indented block
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f(x) = x^2-5*x+10
f = Piecewise([[(0,10), f]])
g = f.riemann_sum(6, mode="midpoint")
F = f.plot(color="blue")
R = add([line([[a,0],[a,f(x=a)],[b,f(x=b)],[b,0]], color="red")
for (a,b), f in g.list()])
show(F+R)
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(4^25+5^16).factor (
-1/(u + 1) + 1/u -1/(u + 1) + 1/u |
var('x,y')
plot3d((16-x^2)^(1/2), (y,0,4), (x,0,4))
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Sleeping...
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(4^125+5^64).factor ()
1547881 * 2291213 * 18564532093913667026708643550789 * 27479692856067389975525373924697 1547881 * 2291213 * 18564532093913667026708643550789 * 27479692856067389975525373924697 |
circle((1,1), 1) + plot(x^2, (x,0,5))
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