QR decomposition fails for ``qqbar`` matrix

[fredrik-johansson]

Steps To Reproduce

This QR decomposition succeeds for n <= 18 but fails for n = 19 due to an internal error in the qqbar arithmetic.

n = 19
A = matrix(QQbar, [[1/(i+j+1) for i in range(n)] for j in range(n)])
A.QR()

Expected Behavior

The computation should succeed.

Actual Behavior

~/src$ ./sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 10.7, Release Date: 2025-08-09                    │
│ Using Python 3.13.5. Type "help()" for help.                       │
└────────────────────────────────────────────────────────────────────┘
sage: n=19; A = matrix(QQbar, [[1/(i+j+1) for i in range(n)] for j in range(n)]); A.QR();
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Cell In[1], line 1
----> 1 n=Integer(19); A = matrix(QQbar, [[Integer(1)/(i+j+Integer(1)) for i in range(n)] for j in range(n)]); A.QR();

File /var/tmp/sage-10.7/local/lib/python3.13/site-packages/sage/matrix/matrix2.pyx:11311, in sage.matrix.matrix2.Matrix.QR (build/cythonized/sage/matrix/matrix2.c:101881)()
  11309 if hip != 0:
  11310     try:
> 11311         scale = sqrt(hip)
  11312         q = (1/scale)*v
  11313         Q.append(q)

File /var/tmp/sage-10.7/local/lib/python3.13/site-packages/sage/misc/functional.py:1982, in sqrt(x, *args, **kwds)
   1980     return sqrt(x)
   1981 try:
-> 1982     return x.sqrt(*args, **kwds)
   1983 # The following includes TypeError to catch cases where sqrt
   1984 # is called with a "prec" keyword, for example, but the sqrt
   1985 # method for x doesn't accept such a keyword.
   1986 except (AttributeError, TypeError):

File /var/tmp/sage-10.7/local/lib/python3.13/site-packages/sage/rings/qqbar.py:4386, in AlgebraicNumber_base.sqrt(self, all, extend)
   4383     else:
   4384         return []
-> 4386 root = self ** ~ZZ(2)
   4388 if all:
   4389     return [root, -root]

File /var/tmp/sage-10.7/local/lib/python3.13/site-packages/sage/structure/element.pyx:2058, in sage.structure.element.Element.__pow__ (build/cythonized/sage/structure/element.c:23944)()
   2056     return (<Element>left)._pow_(right)
   2057 if BOTH_ARE_ELEMENT(cl):
-> 2058     return coercion_model.bin_op(left, right, pow)
   2059 
   2060 cdef long value

File /var/tmp/sage-10.7/local/lib/python3.13/site-packages/sage/structure/coerce.pyx:1236, in sage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:16548)()
   1234         return (<Action>action)._act_(x, y)
   1235     else:
-> 1236         return (<Action>action)._act_(y, x)
   1237 
   1238 # Now coerce to a common parent and do the operation there

File /var/tmp/sage-10.7/local/lib/python3.13/site-packages/sage/categories/action.pyx:194, in sage.categories.action.Action._act_ (build/cythonized/sage/categories/action.c:6804)()
    192     return self._act_(g, x)
    193 
--> 194 cpdef _act_(self, g, x):
    195     """
    196     Let ``g`` act on ``x`` under this action.

File /var/tmp/sage-10.7/local/lib/python3.13/site-packages/sage/rings/qqbar.py:6580, in AlgebraicNumberPowQQAction._act_(self, e, x)
   6578 val = x._interval_fast(prec)
   6579 if isinstance(val, RealIntervalFieldElement) or not val.crosses_log_branch_cut():
-> 6580     argument = val.argument()
   6581     if val.imag().is_zero() and val.real() < 0:
   6582         argument_is_pi = True

File /var/tmp/sage-10.7/local/lib/python3.13/site-packages/sage/rings/complex_interval.pyx:1681, in sage.rings.complex_interval.ComplexIntervalFieldElement.argument (build/cythonized/sage/rings/complex_interval.c:21825)()
   1679 if which_axes[i % 4] and not which_axes[(i - 1) % 4]:
   1680     if lower is not None:
-> 1681         raise ValueError("Can't take the argument of line-segment interval strictly containing zero")
   1682     lower = i
   1683 

ValueError: Can't take the argument of line-segment interval strictly containing zero

Additional Information

No response

Environment

• OS: Ubuntu
• Sage Version: 10.7

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide