estimate_smallest_singular_value(U)

Notes

The procedure is based on and is done in two steps. First, it finds a vector e with components selected from {+1, -1} such that the solution w from the system U.T w = e is as large as possible. Next it estimate U v = w . The smallest singular value is close to norm(w)/norm(v) and the right singular vector is close to v/norm(v) .

The estimation will be better more ill-conditioned is the matrix.

Parameters

U : ndarray

Square upper triangular matrix.

Returns

s_min : float

Estimated smallest singular value of the provided matrix.

z_min : ndarray

Estimatied right singular vector.

Given upper triangular matrix U estimate the smallest singular value and the correspondent right singular vector in O(n**2) operations.

Examples

See :

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GitHub : /scipy/optimize/_trustregion_exact.py#44
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