128
3 Berggren Basis and Completeness Relations
precision is attained, but at the price of abandoning complex symmetry. Moreover,
no parallel version of the latter routine exists. Thus, it is the aim of this section
to devise a diagonalization routine for complex-symmetric matrices which is both
numerically precise and straightforward to parallelize.
The Hamiltonian matrix provided by a Berggren basis is a special type of
complex symmetric matrix, as it is diagonally dominant for most of its rows and
columns, and its imaginary parts are typically smaller than real parts. Consequently,
it turns out to be possible to devise a stable method to diagonalize complexsymmetric matrices, based on a generalization of already existing methods for real
symmetric matrices. Its first step consists to make the initial matrix tridiagonal,
for example with orthogonal transformations using the standard Householder
method [47]. The second step is to diagonalize the obtained tridiagonal matrix,
by successively representing the matrix to be diagonalized by a product of an
orthogonal matrix, commonly denoted as Q, times the matrix L, where only matrix
elements below the diagonal are nonzero. This method is then deemed as the QL
algorithm in numerical analysis [47].
Such an avenue has already been pursued in Refs. [73, 74], where the standard
Householder and QL methods were extended to the complex case. However, the
precision of eigenstates was assessed only for the largest eigenvalues in modulus.
Moreover, as will be shown in the following, the Householder method becomes
quickly imprecise when the dimension of the matrix increases. This phenomenon is
due to the non-unitary character of orthogonal transformations in the complex case.
Added to that, the parallelization of the Householder and QL methods were not
considered. In order to clarify the situation a numerical study of several different
methods to diagonalize complex-symmetric matrices will be effected in Sect. 3.8.3.
A simple and powerful method to transform a real symmetric matrix is the
Lanczos method. It only demands matrix-vector multiplications and hence is widely
used. However, it suffers from the well-known loss of orthogonality of Lanczos
vectors when the number of Lanczos vectors increases. A full reorthogonalization
of Lanczos vectors is an effective method to suppress this problem. Nevertheless,
it makes the tridiagonalization process twice longer, so that it has been abandoned
and replaced by the faster and stable real symmetric Householder method. However, as one will see in the following, the Lanczos method combined with full
reorthogonalization extended to the complex symmetric case greatly alleviates the
instabilities encountered in the complex symmetric Householder method. Moreover,
parallelization of the Lanczos method is straightforward to implement using the
open multiprocessing and message passing interface frameworks, contrary to that of
the Householder method, where only open multiprocessing parallelization has been
considered [74].
Nevertheless, the orthogonal transformation induced by the Lanczos method
is not completely devoid of numerical inaccuracy, so that one cannot reach the
full numerical precision of the LAPACK library routines diagonalizing the general
complex matrix therein. Moreover, in the complex symmetric case, the so-called
breakdown of Lanczos method can occur [73], which is the appearance of Lanczos
vectors of vanishing norm. Indeed, the Berggren metric is not the hermitian metric,
3 Berggren Basis and Completeness Relations
precision is attained, but at the price of abandoning complex symmetry. Moreover,
no parallel version of the latter routine exists. Thus, it is the aim of this section
to devise a diagonalization routine for complex-symmetric matrices which is both
numerically precise and straightforward to parallelize.
The Hamiltonian matrix provided by a Berggren basis is a special type of
complex symmetric matrix, as it is diagonally dominant for most of its rows and
columns, and its imaginary parts are typically smaller than real parts. Consequently,
it turns out to be possible to devise a stable method to diagonalize complexsymmetric matrices, based on a generalization of already existing methods for real
symmetric matrices. Its first step consists to make the initial matrix tridiagonal,
for example with orthogonal transformations using the standard Householder
method [47]. The second step is to diagonalize the obtained tridiagonal matrix,
by successively representing the matrix to be diagonalized by a product of an
orthogonal matrix, commonly denoted as Q, times the matrix L, where only matrix
elements below the diagonal are nonzero. This method is then deemed as the QL
algorithm in numerical analysis [47].
Such an avenue has already been pursued in Refs. [73, 74], where the standard
Householder and QL methods were extended to the complex case. However, the
precision of eigenstates was assessed only for the largest eigenvalues in modulus.
Moreover, as will be shown in the following, the Householder method becomes
quickly imprecise when the dimension of the matrix increases. This phenomenon is
due to the non-unitary character of orthogonal transformations in the complex case.
Added to that, the parallelization of the Householder and QL methods were not
considered. In order to clarify the situation a numerical study of several different
methods to diagonalize complex-symmetric matrices will be effected in Sect. 3.8.3.
A simple and powerful method to transform a real symmetric matrix is the
Lanczos method. It only demands matrix-vector multiplications and hence is widely
used. However, it suffers from the well-known loss of orthogonality of Lanczos
vectors when the number of Lanczos vectors increases. A full reorthogonalization
of Lanczos vectors is an effective method to suppress this problem. Nevertheless,
it makes the tridiagonalization process twice longer, so that it has been abandoned
and replaced by the faster and stable real symmetric Householder method. However, as one will see in the following, the Lanczos method combined with full
reorthogonalization extended to the complex symmetric case greatly alleviates the
instabilities encountered in the complex symmetric Householder method. Moreover,
parallelization of the Lanczos method is straightforward to implement using the
open multiprocessing and message passing interface frameworks, contrary to that of
the Householder method, where only open multiprocessing parallelization has been
considered [74].
Nevertheless, the orthogonal transformation induced by the Lanczos method
is not completely devoid of numerical inaccuracy, so that one cannot reach the
full numerical precision of the LAPACK library routines diagonalizing the general
complex matrix therein. Moreover, in the complex symmetric case, the so-called
breakdown of Lanczos method can occur [73], which is the appearance of Lanczos
vectors of vanishing norm. Indeed, the Berggren metric is not the hermitian metric,