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A very fast procedure to calculate the smallest singular value

The optimization problem of estimate a vector x such that minimize ∥Ax∥ subject to ∥x∥ = 1, where A is a m×n matrix, is frequently found in computer vision. The solution of this problem is the right singular vector associated to the the smallest singular value. This problem must be solved very fast, for example, in real time applications as augmented reality environments are.

It is show in this work that the old procedure to calculate directly the smallest singular value and to use one inverse iteration to calculate its associated singular vector is a faster procedure, compared with the state of the art algorithms to calculate the SVD, with relatively small square matrices.