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<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{L}(\mathbf{P}_{p,q}(2^{-k_{n}}
\mathbf{M}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r,} \quad </math> <ref name=″:24″ /> <ref name=":10"
where the matrices <math>\mathbf{M}_{n}, \quad \mathbf{L} \quad and \quad \mathbf{r}</math> are defined as
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* Hochbruck M. et al. (1998) introduce the LL scheme for ODEs based on Krylov subspace approximation. <ref name="″:22″">Hochbruck, M., Lubich, C., & Selhofer, H. (1998). Exponential integrators for large systems of differential equations. SIAM J. Scient. Comput. 19(5), 1552-1574.[[doi:10.1137/S1064827595295337|<small>doi:10.1137/S1064827595295337</small>]]</ref>
* Jimenez J.C. (2002) introduces the LL scheme for ODEs and SDEs based on rational Padé approximation. <ref name=″:23″>Jimenez, J. C. (2002). A simple algebraic expression to evaluate the local linearization schemes for stochastic differential equations. Applied Mathematics Letters, 15(6), 775-780.[[doi:10.1016/S0893-9659(02)00041-1|<small>doi:10.1016/S0893-9659(02)00041-1</small>]]</ref>
* Carbonell F.M. et al. (2005) introduce the LL method for RDEs. <ref name="″:24″">Carbonell, F., Jimenez, J. C., Biscay, R. J., & De La Cruz, H. (2005). The local linearization method for numerical integration of random differential equations. BIT
* Jimenez J.C. et al. (2006) introduce the LL method for DDEs. <ref name=":13" />
* De la Cruz H. et al. (2006,2007) and Tokman M. (2006) introduce the two classes of HOLL integrators for ODEs: the integrator-based <ref name=":1" /> and the quadrature-based.<ref name=":17" /><ref name=":2" />
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