Set the precision of imported methods

edit

You can preserve the precision of your matrix while transferring it to numpy as follows:

sage: R = RealField(100)
sage: A = matrix(R ,2,2, [1.746, 0.940, 1.246, 1.898])
sage: import numpy as np
sage: a = np.array(A, dtype=object)
sage: a
array([[1.7460000000000000000000000000, 0.94000000000000000000000000000],
       [1.2460000000000000000000000000, 1.8980000000000000000000000000]], dtype=object)
sage: a[0]
array([1.7460000000000000000000000000, 0.94000000000000000000000000000], dtype=object)
sage: a[0][0]
1.7460000000000000000000000000
sage: a[0][0].parent()
Real Field with 100 bits of precision

Unfortunately, this will not solve your second problem:

sage: from scipy import linalg
sage: b = linalg.svd(a) ; b
(array([[-0.65092234, -0.75914432],
       [-0.75914432,  0.65092234]]),
 array([ 2.92405178,  0.73277362]),
 array([[-0.71216394, -0.70201319],
       [-0.70201319,  0.71216394]]))
sage: b[0][0][0]
-0.65092234401411786
sage: type(b[0][0][0])
<type 'numpy.float64'>

As you can see, the svd operation took your entries back to double precision floating point numbers. If you type:

sage: linalg.svd??

You will see that scipy uses lapack to solve the singular value decomposition, and the lapack library only work with simple or double precision floating-point numbers, not mpfr (arbitrary precision) numbers. For the same reason, Sage offers a SVD method for matrices over RDF and not over RR.

What may be hard to understand is why there is no generic method for doing that in any precision. Actually, it is not so easy since there are a lot of numerical stability issues when doing linear algebra over floating-point numbers, this is why Sage and scipy currently relies on a specialized library.