The invariant operator method and the unitary transformation method will be adopted for this purpose.
In this paper, we present a novel method for solving the unitary Hessenberg eigenvalue problem.
We now apply different numerical methods for finding the unitary polar factors of many randomly generated rectangular matrices with complex entries.
As discussed in the preceding sections, the construction of the iterative methods for finding the unitary polar factor of a matrix mainly relies on the nonlinear equation solver which is going to be applied on the mapping (8).
This is known as one of the good ways in the literature for constructing an initial value to ensure the convergence of iterative Newton-type methods for finding the unitary polar factor of A. The other sections of this paper are organized as follows.
To construct some other new and useful iterative methods for finding the unitary polar factor of a matrix, we could again use the first sub-step of (9) along with different kinds of approximation for the newly appearing first derivative in the second sub-step.
The unitary transformation method which is very useful when treating time-dependent Hamiltonian systems in cases like this will be used.
Now, we are going to transform H ̂ into a simple form using the unitary transformation method, developed in [6] for a two-loop LC circuit, in order to simplify the problem.
If we consider the method of consecutive unitary transformation, the expectation value for an arbitrary operator O ̂ j in the original system can be evaluated from 〈 ψ s, n 1, n 2 ( t ) | O ̂ j | ψ s, n 1, n 2 ( t ) 〉 = 〈 ψ n 1, n 2 B ( 0 ) | Ŝ j ‡ D ̂ j ‡ T ̂ Bj ‡ Û B ‡ Û A ‡ O ̂ j Û A Û B T ̂ Bj D ̂ j Ŝ j | ψ n 1, n 2 B ( 0 ) 〉. (57).
Finally, the eigenvalues of H e can be obtained using the phase estimation algorithm, starting from trial states obtained with the previous method via unitary coupled-cluster techniques [24].
