HFB equationsSupermatrix standard representation:
Wasteful : 2Nx2N matrix diagonalization for N vectors
Equivalent formulation for real case:
Takagi factorization:
_Special case of SVD:
_ : positive eigenvalues only : quasi-particle energies
_ up to complex phases only
_Phase provided by phase-dependent equation *
HFB equations
Supermatrix standard representation:
Wasteful : 2Nx2N matrix diagonalization for N vectors
Equivalent formulation for real case:
Takagi factorization:
_Special case of SVD:
_ : positive eigenvalues only : quasi-particle energies
_ up to complex phases only
_Phase provided by phase-dependent equation *
Numerical results
Comparison of different methods (Householder + QL):
Conclusion:
Takagi factorization much faster than full diagonalization (gain ~ 3.5)
Speed comparable to NxN complex hermitian diagonalization
Divide-and-conquer or twisted factorization to be considered in the future
*
* Complex HFB equations Complex hamiltonians:
: complex symmetric (Berggren basis)
complex hermitian (rotating states)
previous method available for real symmetric
Generalization to complex case
Use of bicomplex numbers.
Not quaternions : commutative algebra (but not a field)
Generalization:
Takagi factorization:
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