Add a method to find a vector x such that Q(x) = C, where Q is a quadratic form and C is a constant.

[sagetrac-tgaona]

This algorithm can be implemented using PARI's qfsolve(). Will be used as a component for the method implemented in Ticket [ticket:19112]

CC: @annahaensch

Component: quadratic forms

Author: Tyler Gaona

Branch/Commit: c3ef1ed

Reviewer: Jeroen Demeyer

Issue created by migration from https://trac.sagemath.org/ticket/19533

[sagetrac-tgaona]

Branch: u/tgaona/ticket/19533

[sagetrac-tgaona]

New commits:

cfc69b2

Adds the `solve` method for quadratic forms.

[sagetrac-tgaona]

Commit: cfc69b2

[jdemeyer]

comment:3

One quick comment (I have not fully checked the patch): it would be useful to change the function to

def solve(self, c=0)

such that Q.solve() without argument finds a zero of Q. Then adjust the documentation and add a test for this case.

[jdemeyer]

comment:4

Why divide by abs(z) instead of z? I'm not saying it's wrong, there is just no reason for it.

You should limit the length of lines to 72 characters if possible,
so wrap this comment:

    # Case 2: We found a solution self(x) = 0. Let e be any vector such that B(x,e) != 0, where B is the bilinear form corresponding to self.
    # To find e, just try all unit vectors (0,..0,1,0...0). Let a = (c - self(e))/2*B(x,e) and let y = e + a*x.
    # Then self(y) = B(e + a*x, e + a*x) = self(e) + 2B(e, a*x) = self(e) + 2([c - self(e)]/[2B(x,e)]) * B(x,e) = c.

[jdemeyer]

comment:5

Another minor thing: I would change the error message to

raise ArithmeticError("no solution found (local obstruction at {})".format(x))

[jdemeyer]

comment:6

Replace

if z != 0:

by

if z:

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

cb3f891

adds default parameter to solve(). minor formatting changes.

[sagetrac-git]

Changed commit from cfc69b2 to cb3f891

[jdemeyer]

Changed branch from u/tgaona/ticket/19533 to u/jdemeyer/ticket/19533

[jdemeyer]

Changed commit from cb3f891 to c3ef1ed

[jdemeyer]

comment:10

I have made a lot of small changes. I also added a few more examples. If you agree, you can set this to positive_review.

---

New commits:

c3ef1ed

Further fixes to quadratic form solver

[jdemeyer]

Reviewer: Jeroen Demeyer

[vbraun]

Changed branch from u/jdemeyer/ticket/19533 to c3ef1ed