Use computation dtype in dpnp.linalg.pinv for default tolerance and empty input

[antonwolfy]

This PR updates dpnp.linalg.pinv to take the data type for the default tolerance and for the result of an empty input from the data type the SVD is computed in. This aligns dpnp.linalg.pinv with NumPy 2.6 behavior.

Default tolerance

When rtol=None (the default) and rcond is not given, the cutoff is max(M, N) * eps. Previously eps was taken from dpnp.result_type(a.dtype, dpnp.default_float_type(a.device)), which gives float64:

• for float32/complex64 input on devices with fp64 support;
• for integer input on devices without fp64 support, since result_type called with dtypes only does not know the device.

In both cases the SVD runs in single precision, so the cutoff was about 10⁹ times too small. Singular values at the level of single-precision rounding noise were therefore inverted instead of discarded. The tolerance now uses eps of _common_type(a), which also matches the docstring (max(M, N) * dpnp.finfo(a.dtype).eps).

Note that dpnp.linalg.pinv(a) called without a tolerance now uses the larger cutoff for float32/complex64 input on devices with fp64 support.

Empty input

The empty-input shortcut returned an array of the input data type. An empty integer or boolean array therefore produced an integer/boolean result, while a non-empty one produced a floating-point result. The shortcut now returns the computation data type: float64, or float32 on devices without fp64 support.

import dpnp as np

a = np.diag(np.array([1.0, 1e-9], dtype=np.float32))
np.linalg.pinv(a)  # was [[1, 0], [0, 1e9]], now [[1, 0], [0, 0]]

np.linalg.pinv(np.empty((0, 3), dtype=np.int64)).dtype  # was int64, now float64

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[github-actions]

View rendered docs @ https://intelpython.github.io/dpnp/index.html

[github-actions]

Array API standard conformance tests for dpnp=0.21.0dev13=np2py314h8d9cdd5_7 ran successfully.
Passed: 1376
Failed: 0
Skipped: 6

[coveralls]

coverage: 78.669%. remained the same — align-pinv-empty-dtype-with-numpy into master

[vlad-perevezentsev]

LGTM
Thank you @antonwolfy