solve GEVP for a real, symmetric correlator matrix
gevp(cf, Time, t0 = 1, element.order = 1:cf$nrObs, for.tsboot = TRUE,
sort.type = "vectors", sort.t0 = TRUE)correlation matrix preferably obtained with a call to
extrac.obs (or at leas with the same structure) or an already
averaged one.
cf is supposed to be an array of dim=c(N, n*(Time/2+1)), where
N is the number of observations and n is the number of single
correlators in the matrix. E.g. for a 2x2 matrix n would be 4.
time extent of the lattice.
initial time value of the GEVP, must be in between 0 and
Time/2-2. Default is 1.
specifies how to fit the n linearly ordered
single correlators into the correlator matrix.
element.order=c(1,2,3,4) leads to a matrix
matrix(cf[element.order], nrow=2).
for internal use of bootstrap.gevp. Alters
the returned values, see details.
Sort the eigenvalues either in descending order, or by using the scalar product of the eigenvectors with the eigenvectors at \(t=t_0+1\). Possible values are "values", "vectors" or "det".
if true (default), sort with respect to data at t0, otherwise with respect to t-1.
Returns a list with the sorted eigenvalues, sorted eigenvectors and sorted (reduced) amplitudes for all t > t0.
In case for.tsboot=TRUE the same is returned as one long vector with
first all eigenvalues concatenated, then all eigenvectors and then all
(reduced) amplitudes concatenated.
The generalised eigenvalue problem \( \)\( C(t) v(t,t_0) = C(t_0) \lambda(t,t_0) v(t,t_0) \)\( \) is solved by performing a Cholesky decomposition of \(C(t_0)=L^t \)\( L\) and transforming the GEVP into a standard eigenvalue problem for all values of \(t\). The matrices \(C\) are symmetrised for all \(t\). So we solve for \(\lambda\) \((L^t)^{-1} C(t) L^{-1} w = \lambda w\) with \(w = L v\) or the wanted \(v = L^{-1} w\).
The amplitudes can be computed from \( A_i^{(n)}(t) = \sum_{j}C_{ij}(t) v_j^{(n)}(t,t_0)/(\sqrt{(v^{(n)}, Cv^{(n)})(\exp(-mt)\pm \exp(-m(t-t)))}) \) and this is what the code returns up to the factor \( 1/\sqrt{\exp(-mt)\pm \exp(-m(t-t))} \) The states are sorted by their eigenvalues when "values" is chosen. If "vectors" is chosen, we take \( \max( \sum_i \langle v(t_0,i), v(t, j)\rangle) \) with \(v\) the eigenvectors. For sort type "det" we compute \( \max(...) \)
Michael, Christopher and Teasdale, I., Nucl.Phys.B215 (1983) 433, DOI: 10.1016/0550-3213(83)90674-0 Blossier, B. et al., JHEP 0904 (2009) 094, DOI: 10.1088/1126-6708/2009/04/094, arXiv:0902.1265
boostrap.gevp, extract.obs