3.8 Complex-Symmetric Operators and Matrices
133
followed by the QL method (stated as Lanczos-QL method in the following) is more
precise than the complex symmetric Householder-QL method, so that the former is
preferred to the latter when eigenvalues are well separated from one another. The
presented scheme in the Lanczos-QL method is, however, more expensive in terms
of storage, as one has to keep two matrices: the matrix to diagonalize and the matrix
of eigenvectors. On the contrary, the Householder-QL method is “in place,” that
is, the initial matrix to diagonalize will contain its eigenvectors at the end of the
calculation. This additional memory storage is however necessary in order to have
a numerically precise algorithm for Lanczos-QL.
One will study the precision of eigenvalues and eigenvectors obtained with the
Householder-QL, Lanczos-QL, and LAPACK routines. For this, for a given matrix
M, one considers the maximal value of ||MX − EX|| ∞ for all eigenvectors X, with
E the eigenvalue associated to X. The considered complex symmetric matrix has
off-diagonal matrix elements whose real part is uniformly distributed in [−0.01 :
0.01] and whose imaginary part is uniformly distributed in [−0.001 : 0.001] in
order to mimic the Hamiltonian interaction part in the continuum. Real numbers
uniformly varying from 1 to 200 have been added to the diagonal of the considered
matrix in order to simulate the one-body part of the Hamiltonian. Calculations have
shown that the proposed method is very fast and precise, as a numerical precision
of 10 −10 or less is typically obtained with dimensions smaller than 1000, while
the precision of the Householder-QL method is about 10 −9 and can reach 10 −8 .
The numerical precision of eigenvalues and eigenvectors becomes worse if their
imaginary parts are of the same order of magnitude as real parts, that is, if imaginary
part is uniformly distributed in [−0.01 : 0.01] instead of [−0.001 : 0.001]. The
precision of the Householder-QL method can become of the order of 10 −6 for
dimensions smaller than 1000, whereas the Lanczos-QL method which can be
more precise by two orders of magnitude, reach the precision as large as 10 −8 .
Maximal numerical precision, of about 10 −12 is provided by the LAPACK library
routine ZGEEV, whereas the overall precision of the Lanczos-QL method is close to
10 −10 for considered dimensions. The precision of about 10 −10 attained in LanczosQL method is sufficient in all practical cases, as the numerical error in calculated
observables which arises from the necessary discretization of the Berggren basis
contours is at best 10 −6 . Consequently, by using the Lanczos-QL method, the
diagonalization of a dense complex symmetric Hamiltonian matrix as obtained in
Gamow shell model, poses no problem, and the only numerical problem therein
is the multiplication operation: Hamiltonian times vector, as in the standard shell
model.
An example of the time taken for diagonalization for a few complex-symmetric
matrices is shown in Fig. 3.8. The calculation time for all considered methods is
comparable, as can be seen in Fig. 3.8. The simplicity of the Lanczos-QL method
allows it to be easily parallelized, as the Lanczos method is the combination of
matrix multiplications and classical Gram-Schmidt orthogonalizations. This is a
very convenient aspect of the method, especially when one has to consider many
complex-symmetric matrices of sizable dimension. On the contrary, the LAPACK
library provides no parallel routine for the diagonalization of complex matrices.
133
followed by the QL method (stated as Lanczos-QL method in the following) is more
precise than the complex symmetric Householder-QL method, so that the former is
preferred to the latter when eigenvalues are well separated from one another. The
presented scheme in the Lanczos-QL method is, however, more expensive in terms
of storage, as one has to keep two matrices: the matrix to diagonalize and the matrix
of eigenvectors. On the contrary, the Householder-QL method is “in place,” that
is, the initial matrix to diagonalize will contain its eigenvectors at the end of the
calculation. This additional memory storage is however necessary in order to have
a numerically precise algorithm for Lanczos-QL.
One will study the precision of eigenvalues and eigenvectors obtained with the
Householder-QL, Lanczos-QL, and LAPACK routines. For this, for a given matrix
M, one considers the maximal value of ||MX − EX|| ∞ for all eigenvectors X, with
E the eigenvalue associated to X. The considered complex symmetric matrix has
off-diagonal matrix elements whose real part is uniformly distributed in [−0.01 :
0.01] and whose imaginary part is uniformly distributed in [−0.001 : 0.001] in
order to mimic the Hamiltonian interaction part in the continuum. Real numbers
uniformly varying from 1 to 200 have been added to the diagonal of the considered
matrix in order to simulate the one-body part of the Hamiltonian. Calculations have
shown that the proposed method is very fast and precise, as a numerical precision
of 10 −10 or less is typically obtained with dimensions smaller than 1000, while
the precision of the Householder-QL method is about 10 −9 and can reach 10 −8 .
The numerical precision of eigenvalues and eigenvectors becomes worse if their
imaginary parts are of the same order of magnitude as real parts, that is, if imaginary
part is uniformly distributed in [−0.01 : 0.01] instead of [−0.001 : 0.001]. The
precision of the Householder-QL method can become of the order of 10 −6 for
dimensions smaller than 1000, whereas the Lanczos-QL method which can be
more precise by two orders of magnitude, reach the precision as large as 10 −8 .
Maximal numerical precision, of about 10 −12 is provided by the LAPACK library
routine ZGEEV, whereas the overall precision of the Lanczos-QL method is close to
10 −10 for considered dimensions. The precision of about 10 −10 attained in LanczosQL method is sufficient in all practical cases, as the numerical error in calculated
observables which arises from the necessary discretization of the Berggren basis
contours is at best 10 −6 . Consequently, by using the Lanczos-QL method, the
diagonalization of a dense complex symmetric Hamiltonian matrix as obtained in
Gamow shell model, poses no problem, and the only numerical problem therein
is the multiplication operation: Hamiltonian times vector, as in the standard shell
model.
An example of the time taken for diagonalization for a few complex-symmetric
matrices is shown in Fig. 3.8. The calculation time for all considered methods is
comparable, as can be seen in Fig. 3.8. The simplicity of the Lanczos-QL method
allows it to be easily parallelized, as the Lanczos method is the combination of
matrix multiplications and classical Gram-Schmidt orthogonalizations. This is a
very convenient aspect of the method, especially when one has to consider many
complex-symmetric matrices of sizable dimension. On the contrary, the LAPACK
library provides no parallel routine for the diagonalization of complex matrices.