  The inverse-free extreme learning machine (ELM) algorithm proposed in [4] was
based on an inverse-free algorithm to compute the regularized pseudo-inverse,
which was deduced from an inverse-free recursive algorithm to update the
inverse of a Hermitian matrix. Before that recursive algorithm was applied in
[4], its improved version had been utilized in previous literatures [9], [10].
Accordingly from the improved recursive algorithm [9], [10], we deduce a more
efficient inverse-free algorithm
  to update the regularized pseudo-inverse, from which we develop the proposed
inverse-free ELM algorithm 1. Moreover, the proposed ELM algorithm 2 further
reduces the computational complexity, which computes the output weights
directly from the updated inverse, and avoids computing the regularized
pseudoinverse. Lastly, instead of updating the inverse, the proposed ELM
algorithm 3 updates the LDLT factor of the inverse by the inverse LDLT
factorization [11], to avoid numerical instabilities after a very large number
of iterations [12]. With respect to the existing ELM algorithm, the proposed
ELM algorithms 1, 2 and 3 are expected to require only (8+3)/M , (8+1)/M and
(8+1)/M of complexities, respectively, where M is the output node number. In
the numerical experiments, the standard ELM, the existing inverse-free ELM
algorithm and the proposed ELM algorithms 1, 2 and 3 achieve the same
performance in regression and classification, while all the 3 proposed
algorithms significantly accelerate the existing inverse-free ELM algorithm
