On 7/27/07, Roger Mason <[EMAIL PROTECTED]> wrote:
> I'm still trying to do some linear algebra with sage.
>
> sage: var('x1,y1,z1, x2,y2,z2, x3,y3,z3')
> (x1, y1, z1, x2, y2, z2, x3, y3, z3)
> sage: var('p1,p2,p3')
> (p1, p2, p3)
> sage: var('x,y,z')
> (x, y, z)
> sage: var('alpha, beta')
> (alpha, beta)
> sage: p1 = transpose(matrix([[x1,y1,z1]]))
> sage: p1
>
> [x1]
> [y1]
> [z1]
> sage: p2 = transpose(matrix([[x2,y2,z2]]))
> sage: p3 = transpose(matrix([[x3,y3,z3]]))
> sage: alpha = p2 - p1
> sage: alpha
>
> [x2 - x1]
> [y2 - y1]
> [z2 - z1]
> sage: beta = p3 - p1
> sage: beta
>
> [x3 - x1]
> [y3 - y1]
> [z3 - z1]
> sage: alpha[0]
> ( x2 - x1)
>
> At this point I wnat to make a matrix that looks like this:
>
> x y z
> alpha[0] alpha[1] alpha[2]
> beta[0] beta[1] beta[2]
>
> My efforts always end in an error.
alpha[0] is the first row of alpha, so you probably want
alpha[0][0].
I should warn you that working with symbolic matrices like
you're doing is potentially very slow, since so far no work
has gone into making it fast. In contrast, if you work with
matrices over a multivariate polynomial ring it could be
quite fast.
sage: # create your variables but in a polynomial ring
sage: # so arithmetic with them is *very* fast
sage: R.<x,y> = QQ[]
sage: p1 = transpose(matrix([[x1,y1,z1]]))
sage: p2 = transpose(matrix([[x2,y2,z2]]))
sage: p3 = transpose(matrix([[x3,y3,z3]]))
sage: alpha = p2 - p1; alpha
[-x1 + x2]
[-y1 + y2]
[-z1 + z2]
sage: beta = p3 - p1
sage: alpha[0]
(-x1 + x2)
sage: m = matrix(3,3,[x,y,z, alpha[0][0], alpha[1][0], alpha[2][0],
beta[0][0], beta[1][0], beta[2][0]])
sage: m
[ x y z]
[-x1 + x2 -y1 + y2 -z1 + z2]
[-x1 + x3 -y1 + y3 -z1 + z3]
Note that arithmetic with m will be billions of times faster than with the
formal symbolic variables. E.g., the following is instant, and results
in a matrix each of whose entries is over 4000 characters long:
sage: time n = m^5
CPU time: 0.00 s, Wall time: 0.00 s
sage: len(str(n[0,0]))
4788
-----
Here is a SAGE notebook version of the above session.
poly matrix
system:sage
{{{id=0|
# create your variables but in a polynomial ring
# so arithmetic with them is *very* fast
R.<x1,y1,z1, x2,y2,z2, x3,y3,z3, p1,p2,p3, x,y,z, alpha, beta> = QQ[]
}}}
{{{id=1|
p1 = transpose(matrix([[x1,y1,z1]]))
}}}
{{{id=2|
p2 = transpose(matrix([[x2,y2,z2]]))
}}}
{{{id=3|
p3 = transpose(matrix([[x3,y3,z3]]))
}}}
{{{id=4|
alpha = p2 - p1; alpha
///
[-x1 + x2]
[-y1 + y2]
[-z1 + z2]
}}}
{{{id=5|
beta = p3 - p1
}}}
{{{id=6|
alpha[0]
///
(-x1 + x2)
}}}
{{{id=7|
m = matrix(3,3,[x,y,z, alpha[0][0], alpha[1][0], alpha[2][0],
beta[0][0], beta[1][0], beta[2][0]])
}}}
{{{id=8|
m
///
[ x y z]
[-x1 + x2 -y1 + y2 -z1 + z2]
[-x1 + x3 -y1 + y3 -z1 + z3]
}}}
{{{id=9|
time n = m^5
///
CPU time: 0.00 s, Wall time: 0.00 s
}}}
{{{id=10|
len(str(n[0,0]))
///
4788
}}}
{{{id=11|
}}}
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