gh-40711: remove some deprecations in matrix2
    
after

- File: src/sage/matrix/matrix2.pyx ; PR: #36394 ; Closed Date:
2023-10-14
- File: src/sage/matrix/matrix2.pyx ; PR: #29243 ; Closed Date:
2020-08-30
- File: src/sage/matrix/matrix2.pyx ; PR: #31619 ; Closed Date:
2021-06-06

### 📝 Checklist

- [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.
    
URL: #40711
Reported by: Frédéric Chapoton
Reviewer(s): David Coudert

Author: Release Manager

src/doc/en/prep/Quickstarts/Linear-Algebra.rst

@@ -324,16 +324,16 @@ We can easily solve linear equations using the backslash, like in Matlab.
 ::
 
     sage: A = random_matrix(QQ, 3, algorithm='unimodular')
-    sage: v = vector([2,3,1])
-    sage: A,v  # random
+    sage: v = vector([2, 3, 1])
+    sage: A, v  # random
     (
     [ 0 -1  1]
     [-1 -1 -1]
     [ 0  2  2], (2, 3, 1)
     )
-    sage: x=A\v; x  # random
+    sage: x = A.solve_right(v); x  # random
     (-7/2, -3/4, 5/4)
-    sage: A*x  # random
+    sage: A * x  # random
     (2, 3, 1)
 
 For *lots* more (concise) information, see the Sage `Linear Algebra

src/sage/matrix/matrix2.pyx

@@ -105,7 +105,7 @@ from sage.matrix.matrix_misc import permanental_minor_polynomial
 from sage.misc.misc_c import prod
 
 # used to deprecate only adjoint method
-from sage.misc.superseded import deprecation, deprecated_function_alias
+from sage.misc.superseded import deprecated_function_alias
 
 
 # temporary hack to silence the warnings from #34806
@@ -140,35 +140,6 @@ cdef class Matrix(Matrix1):
         [x], [0]
         )
     """
-    def _backslash_(self, B):
-        r"""
-        Used to compute `A \backslash B`, i.e., the backslash solver
-        operator.
-
-        DEPRECATED
-
-        EXAMPLES::
-
-            sage: A = matrix(QQ, 3, [1,2,4,2,3,1,0,1,2])
-            sage: B = matrix(QQ, 3, 2, [1,7,5, 2,1,3])
-            sage: C = A._backslash_(B); C
-            doctest:...: DeprecationWarning: the backslash operator has been deprecated; use A.solve_right(B) instead
-            See https://github.com/sagemath/sage/issues/36394 for details.
-            [  -1    1]
-            [13/5 -3/5]
-            [-4/5  9/5]
-            sage: A*C == B
-            True
-            sage: A._backslash_(B) == A \ B
-            doctest:...: DeprecationWarning: the backslash operator has been deprecated; use A.solve_right(B) instead
-            See https://github.com/sagemath/sage/issues/36394 for details.
-            True
-            sage: A._backslash_(B) == A.solve_right(B)
-            True
-        """
-        deprecation(36394, 'the backslash operator has been deprecated; use A.solve_right(B) instead')
-        return self.solve_right(B)
-
     def subs(self, *args, **kwds):
         """
         Substitute values to the variables in that matrix.
@@ -7347,14 +7318,14 @@ cdef class Matrix(Matrix1):
 
         OUTPUT:
 
-        Returns a list of tuples of the form ``(e,V,n)``,
-        each tuple corresponds to a distinct eigenvalue,
+        A list of tuples of the form ``(e,V,n)``.
+        Each tuple corresponds to a distinct eigenvalue,
         where ``e`` is the eigenvalue, ``V`` is a list of eigenvectors forming a
         basis for the corresponding left eigenspace, and ``n`` is the algebraic
         multiplicity of the eigenvalue.
 
-        If the option ``extend`` is set to ``False``, then only the eigenvalues that
-        live in the base ring are considered.
+        If the option ``extend`` is set to ``False``, then only the
+        eigenvalues that live in the base ring are considered.
 
         EXAMPLES:
 
@@ -7396,13 +7367,6 @@ cdef class Matrix(Matrix1):
             NotImplementedError: generalized eigenvector decomposition is
             implemented for RDF and CDF, but not for Rational Field
 
-        Check the deprecation::
-
-            sage: matrix(QQ, [[1, 2], [3, 4]]).eigenvectors_left(False)                 # needs sage.rings.number_field
-            doctest:...: DeprecationWarning: "extend" should be used as keyword argument
-            See https://github.com/sagemath/sage/issues/29243 for details.
-            []
-
         Check :issue:`30518`::
 
             sage: # needs sage.rings.number_field
@@ -7426,29 +7390,23 @@ cdef class Matrix(Matrix1):
             []
         """
         if other is not None:
-            if isinstance(other, bool):
-                # for backward compatibility
-                from sage.misc.superseded import deprecation
-                deprecation(29243,
-                            '"extend" should be used as keyword argument')
-                extend = other
-                other = None
-            else:
-                raise NotImplementedError('generalized eigenvector '
-                                          'decomposition is implemented '
-                                          'for RDF and CDF, but not for %s'
-                                          % self.base_ring())
+            raise NotImplementedError('generalized eigenvector '
+                                      'decomposition is implemented '
+                                      'for RDF and CDF, but not for %s'
+                                      % self.base_ring())
 
         if algorithm is None:
             R = self.base_ring()
             from sage.rings.abc import RealField, ComplexField
             if isinstance(R, (RealField, ComplexField)):
                 return self._eigenvectors_left(
-                        other, extend=extend, algorithm='flint', suppress_future_warning=True)
+                    other, extend=extend, algorithm='flint',
+                    suppress_future_warning=True)
             else:
                 algorithm = 'sage'
         return self._eigenvectors_left(
-                other, extend=extend, algorithm=algorithm, suppress_future_warning=False)
+            other, extend=extend, algorithm=algorithm,
+            suppress_future_warning=False)
 
     def _eigenvectors_left(self, other=None, *, extend: bool, algorithm: str,
                            suppress_future_warning: bool) -> list:
@@ -15921,7 +15879,7 @@ cdef class Matrix(Matrix1):
         """
         return self._is_positive_definite_or_semidefinite(True)
 
-    def is_positive_definite(self, certificate=False):
+    def is_positive_definite(self):
         r"""
         Determine if a matrix is positive-definite.
 
@@ -15944,9 +15902,6 @@ cdef class Matrix(Matrix1):
         INPUT:
 
         - ``self`` -- a matrix
-        - ``certificate`` -- boolean (default: ``False``); return the
-          lower-triangular and diagonal parts of the :meth:`block_ldlt`
-          factorization when the matrix is positive-definite. Deprecated.
 
         OUTPUT:
 
@@ -15959,14 +15914,6 @@ cdef class Matrix(Matrix1):
         2. Be a subring of the real numbers, complex numbers,
            or symbolic ring.
 
-        If ``certificate`` is ``True``, a triplet ``(b, L, d)`` will
-        be returned instead, with ``b`` containing the result (true or
-        false). If the matrix is positive-definite, then ``L`` and
-        ``d`` will contain the lower-triangular and diagonal parts of
-        the :meth:`block_ldlt` factorization, respectively. Or if the
-        matrix is not positive-definite (that is, if ``b`` is
-        ``False``), then both ``L`` and ``d`` will be ``None``.
-
         .. SEEALSO::
 
             :meth:`block_ldlt`, :meth:`~.Matrix.is_hermitian`,
@@ -16099,22 +16046,7 @@ cdef class Matrix(Matrix1):
             sage: matrix.identity(SR,4).is_positive_definite()                          # needs sage.symbolic
             True
         """
-        result = self._is_positive_definite_or_semidefinite(False)
-        if certificate:
-            from sage.misc.superseded import deprecation
-            msg = "the 'certificate' argument is deprecated; if you "
-            msg += "need the corresponding factorization, you can "
-            msg += "simply compute it yourself (the results are cached)"
-            deprecation(31619, msg)
-            L = None
-            d = None
-            if result:
-                from sage.modules.free_module_element import vector
-                _, L, D = self.block_ldlt()
-                d = vector(D.base_ring(), D.diagonal())
-            return (result, L, d)
-        else:
-            return result
+        return self._is_positive_definite_or_semidefinite(False)
 
     def principal_square_root(self, check_positivity=True):
         r"""

src/sage/misc/all.py

@@ -5,12 +5,10 @@
 from sage.misc.all__sagemath_environment import *
 from sage.misc.all__sagemath_repl import *
 
-from sage.misc.misc import (BackslashOperator,
-                  exists, forall, is_iterator,
-                  random_sublist,
-                  pad_zeros,
-                  newton_method_sizes, compose,
-                  nest)
+from sage.misc.misc import (
+    exists, forall, is_iterator, random_sublist, pad_zeros,
+    newton_method_sizes, compose, nest
+)
 
 from sage.misc.banner import version
 

src/sage/misc/misc.py

@@ -361,95 +361,9 @@ def nest(f, n, x):
 
 
 #################################################################
-# The A \ b operator
+# The A \ b operator has been removed after issue #36394
 #################################################################
 
-
-class BackslashOperator:
-    r"""
-    Implement Matlab-style backslash operator for solving systems::
-
-        A \ b
-
-    The preparser converts this to multiplications using
-    ``BackslashOperator()``.
-
-    EXAMPLES::
-
-        sage: preparse("A \\ matrix(QQ,2,1,[1/3,'2/3'])")
-        "A  * BackslashOperator() * matrix(QQ,Integer(2),Integer(1),[Integer(1)/Integer(3),'2/3'])"
-        sage: preparse("A \\ matrix(QQ,2,1,[1/3,2*3])")
-        'A  * BackslashOperator() * matrix(QQ,Integer(2),Integer(1),[Integer(1)/Integer(3),Integer(2)*Integer(3)])'
-        sage: preparse("A \\ B + C")
-        'A  * BackslashOperator() * B + C'
-        sage: preparse("A \\ eval('C+D')")
-        "A  * BackslashOperator() * eval('C+D')"
-        sage: preparse("A \\ x / 5")
-        'A  * BackslashOperator() * x / Integer(5)'
-        sage: preparse("A^3 \\ b")
-        'A**Integer(3)  * BackslashOperator() * b'
-    """
-
-    def __rmul__(self, left):
-        """
-        EXAMPLES::
-
-            sage: # needs sage.modules
-            sage: A = random_matrix(ZZ, 4)
-            sage: while A.rank() != 4:
-            ....:     A = random_matrix(ZZ, 4)
-            sage: B = random_matrix(ZZ, 4)
-            sage: temp = A * BackslashOperator()
-            doctest:...:
-            DeprecationWarning: the backslash operator has been deprecated
-            See https://github.com/sagemath/sage/issues/36394 for details.
-            sage: temp.left is A
-            True
-            sage: X = temp * B
-            doctest:...:
-            DeprecationWarning: the backslash operator has been deprecated; use A.solve_right(B) instead
-            See https://github.com/sagemath/sage/issues/36394 for details.
-            sage: A * X == B
-            True
-        """
-        from sage.misc.superseded import deprecation
-
-        deprecation(36394, 'the backslash operator has been deprecated')
-        self.left = left
-        return self
-
-    def __mul__(self, right):
-        r"""
-        EXAMPLES::
-
-            sage: # needs scipy sage.modules
-            sage: A = matrix(RDF, 5, 5, 2)
-            sage: b = vector(RDF, 5, range(5))
-            sage: v = A \ b
-            doctest:...:
-            DeprecationWarning: the backslash operator has been deprecated; use A.solve_right(B) instead
-            See https://github.com/sagemath/sage/issues/36394 for details.
-            sage: v.zero_at(1e-19)  # On at least one platform, we get a "negative zero"
-            (0.0, 0.5, 1.0, 1.5, 2.0)
-            sage: v = A._backslash_(b)
-            doctest:...:
-            DeprecationWarning: the backslash operator has been deprecated; use A.solve_right(B) instead
-            See https://github.com/sagemath/sage/issues/36394 for details.
-            sage: v.zero_at(1e-19)
-            (0.0, 0.5, 1.0, 1.5, 2.0)
-            sage: v = A * BackslashOperator() * b
-            doctest:...:
-            DeprecationWarning: the backslash operator has been deprecated; use A.solve_right(B) instead
-            See https://github.com/sagemath/sage/issues/36394 for details.
-            sage: v.zero_at(1e-19)
-            (0.0, 0.5, 1.0, 1.5, 2.0)
-        """
-        from sage.misc.superseded import deprecation
-
-        deprecation(36394, 'the backslash operator has been deprecated')
-        return self.left._backslash_(right)
-
-
 #################################################################
 # is_iterator function
 #################################################################
@@ -495,7 +409,7 @@ def is_iterator(it) -> bool:
         sage: is_iterator(iter(P))                                                      # needs sage.combinat
         True
     """
-    # see trac #7398 for a discussion
+    # see issue #7398 for a discussion
     try:
         return it is iter(it)
     except Exception:
@@ -538,7 +452,7 @@ def random_sublist(X, s):
     return [a for a in X if random.random() <= s]
 
 
-def is_sublist(X, Y):
+def is_sublist(X, Y) -> bool:
     """
     Test whether ``X`` is a sublist of ``Y``.
 

src/sage/repl/preparse.py

@@ -1737,11 +1737,11 @@ def preparse(line, reset=True, do_time=False, ignore_prompts=False,
         8
 
         sage: preparse("A \\ B")
-        'A  * BackslashOperator() * B'
+        'A \\ B'
         sage: preparse("A^2 \\ B + C")
-        'A**Integer(2)  * BackslashOperator() * B + C'
+        'A**Integer(2) \\ B + C'
         sage: preparse("a \\ b \\") # There is really only one backslash here, it is just being escaped.
-        'a  * BackslashOperator() * b \\'
+        'a \\ b \\'
 
         sage: preparse("time R.<x> = ZZ[]", do_time=True)
         '__time__ = cputime(); __wall__ = walltime(); R = ZZ[\'x\']; print("Time: CPU {:.2f} s, Wall: {:.2f} s".format(cputime(__time__), walltime(__wall__))); (x,) = R._first_ngens(1)'
@@ -1853,9 +1853,6 @@ def preparse(line, reset=True, do_time=False, ignore_prompts=False,
     # f(x,y) = x^3 - sin(y)
     L = preparse_calculus(L)
 
-    # Backslash
-    L = re.sub(r'''\\\s*([^\t ;#])''', r' * BackslashOperator() * \1', L)
-
     if do_time:
         # Time keyword
         L = re.sub(r';time;(\s*)(\S[^;\n]*)',

src/sage/tests/books/computational-mathematics-with-sagemath/linalg_doctest.py

@@ -284,13 +284,6 @@
   [  (3*x^2 + 4*x + 3)/(x^3 + 4*x^2 + 4*x)]
   [                                      0]
 
-Sage example in ./linalg.tex, line 1855::
-
-  sage: A.solve_right(b) == A\b
-  doctest:...: DeprecationWarning: the backslash operator has been deprecated; use A.solve_right(B) instead
-  See https://github.com/sagemath/sage/issues/36394 for details.
-  True
-
 Sage example in ./linalg.tex, line 1910::
 
   sage: a = matrix(QQ,3,5,[2,2,-1,-2,-1,2,-1,1,2,-1/2,2,-2,-1,2,-1/2])

src/sage/tests/books/computational-mathematics-with-sagemath/linsolve_doctest.py

@@ -20,7 +20,7 @@
   ....:     A = matrix(QQ, [[1/(i+j-1) for j in [1..n]] for i in [1..n]])
   ....:     y = A*x
   ....:     A[n-1,n-1] = (1/(2*n-1))*(1+1/(10^5))   # modifies the matrix
-  ....:     s = A\y
+  ....:     s = A.solve_right(y)
   ....:     return max(abs(float(s[i]-x[i])) for i in range(0,n))
 
 Sage example in ./linsolve.tex, line 396::