Matrix singular values

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

@@ -313,12 +313,8 @@ eigenvectors.
 ::
 
     sage: D,P=H.eigenmatrix_right()
-    sage: P*D-H*P
-    [0 0 0 0 0]
-    [0 0 0 0 0]
-    [0 0 0 0 0]
-    [0 0 0 0 0]
-    [0 0 0 0 0]
+    sage: P*D-H*P == matrix.zero(5)
+    True
 
 Matrix Solving
 ---------------

src/sage/libs/mpmath/ext_main.pyx

@@ -2132,6 +2132,24 @@ cdef class mpf(mpf_base):
         """
         return binop(OP_RICHCMP+op, self, other, global_opts)
 
+    def _mpfr_(self, RR):
+        """
+        Return a Sage ``RealNumber``.
+
+        EXAMPLES::
+
+            sage: from mpmath import mpf
+            sage: mpf(3)._mpfr_(RealField(53))
+            3.00000000000000
+            sage: RR(mpf(3))  # indirect doctest
+            3.00000000000000
+        """
+        signbit, man, exp, bc = self._mpf_
+        result = RR(man) << exp
+        if signbit:
+            result = -result
+        return result
+
 
 cdef class constant(mpf_base):
     """
@@ -2571,6 +2589,21 @@ cdef class mpc(mpnumber):
         """
         return binop(OP_RICHCMP+op, self, other, global_opts)
 
+    def _complex_mpfr_field_(self, CC):
+        """
+        Return a Sage complex number.
+
+        EXAMPLES::
+
+            sage: from mpmath import mpc
+            sage: CC(mpc(1,2))  # indirect doctest
+            1.00000000000000 + 2.00000000000000*I
+            sage: mpc(1,2)._complex_mpfr_field_(CC)
+            1.00000000000000 + 2.00000000000000*I
+        """
+        RR = CC._real_field()
+        return CC((self.real._mpfr_(RR), self.imag._mpfr_(RR)))
+
 
 def hypsum_internal(int p, int q, param_types, str ztype, coeffs, z,
     long prec, long wp, long epsshift, dict magnitude_check, kwargs):

src/sage/matrix/matrix1.pyx

@@ -738,6 +738,33 @@ cdef class Matrix(Matrix0):
         A = numpy.matrix(self.list(), dtype=dtype, copy=copy)
         return numpy.resize(A,(self.nrows(), self.ncols()))
 
+    def _mpmath_(self, prec=None, rounding=None):
+        """
+        Return a ``mpmath`` matrix.
+
+        INPUT: See :meth:`sage.structure.element.Element._mpmath_`.
+
+        EXAMPLES::
+
+            sage: # needs mpmath
+            sage: m = matrix(SR, 2, 2, [1, 2, 3, pi])
+            sage: from mpmath import mp
+            sage: mp.dps = 30
+            sage: mp.matrix(m)  # not tested (doesn't work yet)
+            sage: m._mpmath_(mp.prec)
+            matrix(
+            [['1.0', '2.0'],
+             ['3.0', '3.14159265358979323846264338328']])
+        """
+        if prec is None:
+            R = self.base_ring()
+            try:
+                prec = R.precision()
+            except AttributeError:
+                prec = 53
+        from mpmath import mp
+        return mp.matrix([[item._mpmath_(prec, rounding) for item in row] for row in self])
+
     # Define the magic "__array__" function so that numpy.array(m) can convert
     # a matrix m to a numpy array.
     # See http://docs.scipy.org/doc/numpy/user/c-info.how-to-extend.html#converting-an-arbitrary-sequence-object