diff options
Diffstat (limited to 'numpy/linalg/linalg.py')
| -rw-r--r-- | numpy/linalg/linalg.py | 23 |
1 files changed, 15 insertions, 8 deletions
diff --git a/numpy/linalg/linalg.py b/numpy/linalg/linalg.py index 383c51387..f6547aac3 100644 --- a/numpy/linalg/linalg.py +++ b/numpy/linalg/linalg.py @@ -1352,7 +1352,7 @@ def cond(x, p=None): ---------- x : (M, N) array_like The matrix whose condition number is sought. - p : {None, 1, -1, 2, -2, inf, -inf, 'fro'}, optional + p : {None, 1, -1, 2, -2, inf, -inf, 'fro', 'nuc'}, optional Order of the norm: ===== ============================ @@ -1360,6 +1360,7 @@ def cond(x, p=None): ===== ============================ None 2-norm, computed directly using the ``SVD`` 'fro' Frobenius norm + 'nuc' nuclear norm inf max(sum(abs(x), axis=1)) -inf min(sum(abs(x), axis=1)) 1 max(sum(abs(x), axis=0)) @@ -1368,8 +1369,9 @@ def cond(x, p=None): -2 smallest singular value ===== ============================ - inf means the numpy.inf object, and the Frobenius norm is - the root-of-sum-of-squares norm. + inf means the numpy.inf object, the Frobenius norm is + the root-of-sum-of-squares norm, and the nuclear norm is + the sum of singular values. Returns ------- @@ -1892,7 +1894,7 @@ def lstsq(a, b, rcond=-1): def _multi_svd_norm(x, row_axis, col_axis, op): - """Compute the extreme singular values of the 2-D matrices in `x`. + """Compute a function of the singular values of the 2-D matrices in `x`. This is a private utility function used by numpy.linalg.norm(). @@ -1902,16 +1904,16 @@ def _multi_svd_norm(x, row_axis, col_axis, op): row_axis, col_axis : int The axes of `x` that hold the 2-D matrices. op : callable - This should be either numpy.amin or numpy.amax. + This should be either numpy.amin or numpy.amax or numpy.sum. Returns ------- result : float or ndarray If `x` is 2-D, the return values is a float. Otherwise, it is an array with ``x.ndim - 2`` dimensions. - The return values are either the minimum or maximum of the + The return values are either the minimum or maximum or sum of the singular values of the matrices, depending on whether `op` - is `numpy.amin` or `numpy.amax`. + is `numpy.amin` or `numpy.amax` or `numpy.sum`. """ if row_axis > col_axis: @@ -1933,7 +1935,7 @@ def norm(x, ord=None, axis=None, keepdims=False): ---------- x : array_like Input array. If `axis` is None, `x` must be 1-D or 2-D. - ord : {non-zero int, inf, -inf, 'fro'}, optional + ord : {non-zero int, inf, -inf, 'fro', 'nuc'}, optional Order of the norm (see table under ``Notes``). inf means numpy's `inf` object. axis : {int, 2-tuple of ints, None}, optional @@ -1966,6 +1968,7 @@ def norm(x, ord=None, axis=None, keepdims=False): ===== ============================ ========================== None Frobenius norm 2-norm 'fro' Frobenius norm -- + 'nuc' nuclear norm -- inf max(sum(abs(x), axis=1)) max(abs(x)) -inf min(sum(abs(x), axis=1)) min(abs(x)) 0 -- sum(x != 0) @@ -1980,6 +1983,8 @@ def norm(x, ord=None, axis=None, keepdims=False): :math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}` + The nuclear norm is the sum of singular values. + References ---------- .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*, @@ -2143,6 +2148,8 @@ def norm(x, ord=None, axis=None, keepdims=False): ret = add.reduce(abs(x), axis=col_axis).min(axis=row_axis) elif ord in [None, 'fro', 'f']: ret = sqrt(add.reduce((x.conj() * x).real, axis=axis)) + elif ord == 'nuc': + ret = _multi_svd_norm(x, row_axis, col_axis, sum) else: raise ValueError("Invalid norm order for matrices.") if keepdims: |