gh-39204: Speed up multimodular algorithm in bad case
    
**Edit**: Now flint has fixed
flintlib/flint#2129 , it should be better to
just switch to flint entirely — according to
flintlib/flint#2129 (comment) ,
one of the possible algorithms by flint is multimodular, which should be
faster than or equal to what we're having now. If it is slower in any
case, bug can be reported upstream.

------

Related to #39197.

Technically the algorithm doesn't deviate from @williamstein  's
original book; however the original book doesn't say *how many*
additional primes to add each time.

The original implementation roughly consider 3 more primes each time.
This can be highly inefficient when there are more columns than rows,
which makes the result's height much higher than the guess.

This increases the length of `M` by roughly a factor of `1.2` each time.
Worst case it makes the algorithm slower by a (hopefully small) constant
factor.

For the added test case, it appears to improve the performance.
(Originally takes 40s, now takes <10s on my machine)

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- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: #39204
Reported by: user202729
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/matrix/misc.pyx

@@ -204,6 +204,44 @@ def matrix_rational_echelon_form_multimodular(Matrix self, height_guess=None, pr
         [                0                 0                 1                 2]
         sage: A.pivots()
         (0, 1, 2)
+
+    A small benchmark, showing that the multimodular algorithm is sometimes faster
+    and sometimes slower than the flint algorithm::
+
+        sage: import copy
+        sage: def benchmark(num_row, num_col, entry_size, timeout=2, integer_coefficient=True):
+        ....:     A = matrix(QQ, [[
+        ....:         randint(1, 2^entry_size) if integer_coefficient else ZZ(randint(1, 2^entry_size))/randint(1, 2^entry_size)
+        ....:         for col in range(num_col)] for row in range(num_row)])
+        ....:     data=[]
+        ....:     for algorithm in ("flint", "padic", "multimodular"):
+        ....:         # classical is too slow
+        ....:         B = copy.copy(A)
+        ....:         t = walltime()
+        ....:         alarm(timeout)
+        ....:         try:
+        ....:             B.echelonize(algorithm=algorithm)
+        ....:         except AlarmInterrupt:
+        ....:             pass
+        ....:         finally:
+        ....:             cancel_alarm()
+        ....:         data.append((round(walltime(t), 4), algorithm))
+        ....:     return sorted(data)
+        sage: benchmark(20, 20, 10000)  # long time (multimodular wins)
+        [...'multimodular'...'flint'...]
+        sage: benchmark(39, 40, 200)  # long time (flint wins)
+        [...'flint'...'multimodular'...]
+
+    In this case, there are more columns than rows, which means the resulting
+    matrix has height much higher than the input matrix. We check that the function
+    does not take too long::
+
+        sage: A = matrix(QQ, [[randint(1, 2^500) for col in range(40)] for row in range(20)])
+        sage: t = walltime()
+        sage: A.echelonize(algorithm="multimodular")  # long time
+        sage: t = walltime(t)  # long time
+        sage: (t < 10, t)  # long time
+        (True, ...)
     """
     if proof is None:
         from sage.structure.proof.proof import get_flag
@@ -221,10 +259,12 @@ def matrix_rational_echelon_form_multimodular(Matrix self, height_guess=None, pr
         height_guess = 10000000*(height+100)
     tm = verbose("height_guess = %s" % height_guess, level=2, caller_name="multimod echelon")
 
+    cdef Integer M
+    from sage.arith.misc import integer_floor as floor
     if proof:
-        M = self._ncols * height_guess * height + 1
+        M = floor(max(1, self._ncols * height_guess * height + 1))
     else:
-        M = height_guess + 1
+        M = floor(max(1, height_guess + 1))
 
     if self.is_sparse():
         from sage.matrix.matrix_modn_sparse import MAX_MODULUS
@@ -309,7 +349,7 @@ def matrix_rational_echelon_form_multimodular(Matrix self, height_guess=None, pr
         except ValueError as msg:
             verbose(msg, level=2)
             verbose("Not enough primes to do CRT lift; redoing with several more primes.", level=2, caller_name="multimod echelon")
-            M = prod * p*p*p
+            M <<= M.bit_length() // 5 + 1
             continue
 
         if not proof:
@@ -321,7 +361,7 @@ def matrix_rational_echelon_form_multimodular(Matrix self, height_guess=None, pr
             verbose("Validity of result checked.", level=2, caller_name="multimod echelon")
             break
         verbose("Validity failed; trying again with more primes.", level=2, caller_name="multimod echelon")
-        M = prod * p*p*p
+        M <<= M.bit_length() // 5 + 1
     #end while
     verbose("total time",tm, level=2, caller_name="multimod echelon")
     return E, tuple(best_pivots)