Fix several issues with empty matrices

* make det of null matrix return 1, see e.g.
  https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrix
* fix str/repr

Closes mpmath#745

Co-authored-by: Sergey B Kirpichev <[email protected]>

mpmath/matrices/linalg.py

@@ -146,7 +146,7 @@ def LU_decomp(ctx, A, overwrite=False, use_cache=True):
                 A[i,j] /= A[j,j]
                 for k in range(j + 1, n):
                     A[i,k] -= A[i,j]*A[j,k]
-        if ctx.absmin(A[n - 1,n - 1]) <= tol:
+        if p and ctx.absmin(A[n - 1,n - 1]) <= tol:
             raise ZeroDivisionError('matrix is numerically singular')
         # cache decomposition
         if not overwrite and isinstance(orig, ctx.matrix):
@@ -543,6 +543,12 @@ def det(ctx, A):
         >>> print(det(A))
         1.0
 
+        The determinant of a 0 by 0 matrix is 1 as the product of no factors
+        is by convention the multiplicative identity.
+        >>> A = matrix(0, 0)
+        >>> print(det(A))
+        1
+
         But in general a matrix can have any number as its determinant.
 
         >>> A = matrix([[2, 6, 4],[3, 8, 6],[1, 1, 2]])

mpmath/matrices/matrices.py

@@ -288,7 +288,10 @@ def __init__(self, *args, **kwargs):
                 " If you want to truncate values to integer, use .apply(int) instead.",
                 DeprecationWarning)
         if isinstance(args[0], (list, tuple)):
-            if isinstance(args[0][0], (list, tuple)):
+            if not args[0]:
+                self.__rows = 0
+                self.__cols = 0
+            elif isinstance(args[0][0], (list, tuple)):
                 # interpret nested list as matrix
                 A = args[0]
                 self.__rows = len(A)
@@ -358,12 +361,12 @@ def __nstr__(self, n=None, **kwargs):
                 # Pad each element up to maxlen so the columns line up
                 row[j] = elem.rjust(maxlen[j])
             res[i] = "[" + colsep.join(row) + "]"
-        return rowsep.join(res)
+        return rowsep.join(res) if self.rows or self.cols else ''
 
     def __str__(self):
         return self.__nstr__()
 
-    def _toliststr(self, avoid_type=False):
+    def _toliststr(self):
         """
         Create a list string from a matrix.
 
@@ -375,14 +378,16 @@ def _toliststr(self, avoid_type=False):
         for i in range(self.__rows):
             s += '['
             for j in range(self.__cols):
-                if not avoid_type or not isinstance(self[i,j], typ):
+                if not isinstance(self[i,j], typ):
                     a = repr(self[i,j])
                 else:
                     a = "'" + str(self[i,j]) + "'"
                 s += a + ', '
-            s = s[:-2]
+            if s[-1] != '[':
+                s = s[:-2]
             s += '],\n '
-        s = s[:-3]
+        if s[-1] != '[':
+            s = s[:-3]
         s += ']'
         return s
 
@@ -396,7 +401,7 @@ def __repr__(self):
         if self.ctx.pretty:
             return self.__str__()
         s = 'matrix(\n'
-        s += self._toliststr(avoid_type=True) + ')'
+        s += self._toliststr() + ')'
         return s
 
     def __get_element(self, key):
@@ -989,8 +994,8 @@ def mnorm(ctx, A, p=1):
             p = ctx.convert(p)
         m, n = A.rows, A.cols
         if p == 1:
-            return max(ctx.fsum((A[i,j] for i in range(m)), absolute=1) for j in range(n))
+            return max((ctx.fsum((A[i,j] for i in range(m)), absolute=1) for j in range(n)), default=0)
         elif p == ctx.inf:
-            return max(ctx.fsum((A[i,j] for j in range(n)), absolute=1) for i in range(m))
+            return max((ctx.fsum((A[i,j] for j in range(n)), absolute=1) for i in range(m)), default=0)
         else:
             raise NotImplementedError("matrix p-norm for arbitrary p")

mpmath/tests/test_linalg.py

@@ -82,6 +82,8 @@
               [3, 8, 6],
               [1, 1, 2]])
 
+A14 = matrix(0, 0)
+
 def test_LU_decomp():
     A = A3.copy()
     b = b3
@@ -213,6 +215,7 @@ def test_det():
     assert det(zeros(3)) == 0
     assert det(A11) == 0
     assert absmin(det(A12*1e-30) - 1e-30) < eps
+    assert det(A14) == 1
 
 def test_cond():
     A = matrix([[1.2969, 0.8648], [0.2161, 0.1441]])

mpmath/tests/test_str.py

@@ -1,6 +1,12 @@
 from mpmath import inf, matrix, nstr
 
 
+A1 = matrix([])
+A2 = matrix([[]])
+A3 = matrix(2)
+A4 = matrix([1, 2, 3])
+
+
 def test_nstr():
     m = matrix([[0.75, 0.190940654, -0.0299195971],
                 [0.190940654, 0.65625, 0.205663228],
@@ -13,3 +19,31 @@ def test_nstr():
     '''[    0.75  0.1909   -0.02992]
 [  0.1909  0.6563     0.2057]
 [-0.02992  0.2057  6.445e-21]'''
+
+def test_matrix_repr():
+    assert repr(A1) == \
+    '''matrix(
+[])'''
+    assert repr(A2) == \
+    '''matrix(
+[[]])'''
+    assert repr(A3) == \
+    '''matrix(
+[['0.0', '0.0'],
+ ['0.0', '0.0']])'''
+    assert repr(A4) == \
+    '''matrix(
+[['1.0'],
+ ['2.0'],
+ ['3.0']])'''
+
+def test_matrix_str():
+    assert str(A1) == ''
+    assert str(A2) == '[]'
+    assert str(A3) == \
+    '''[0.0  0.0]
+[0.0  0.0]'''
+    assert str(A4) == \
+'''[1.0]
+[2.0]
+[3.0]'''