Added references for symplectic algorithms.

docs/source/bibliography.bib

@@ -120,6 +120,18 @@
   publisher={SIAM}
 }
 
+@article{BBH2019,
+  author         = {Buchfink, Patrick and Bhatt, Ashish and Haasdonk, Bernard},
+  title          = {Symplectic Model Order Reduction with Non-Orthonormal Bases},
+  journal        = {Mathematical and Computational Applications},
+  year           = {2019},
+  volume         = {24},
+  number         = {2},
+  issn           = {2297--8747},
+  article-number = {43},
+  doi            = {10.3390/mca24020043},
+}
+
 @article{BEOR14,
   title={A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations},
   author={Buhr, Andreas and Engwer, Christian and Ohlberger, Mario and Rave, Stephan},
@@ -334,6 +346,17 @@
   doi =          {10.1109/31.1720}
 }
 
+@article{PM16,
+  author  = {Peng, L. and Mohseni, K.},
+  title   = {Symplectic Model Reduction of Hamiltonian Systems},
+  journal = {SIAM Journal on Scientific Computing},
+  year    = {2016},
+  volume  = {38},
+  number  = {1},
+  pages   = {A1--A27},
+  doi     = {10.1137/140978922},
+}
+
 @phdthesis{RL95,
   title={Analysis and implementation of an implicitly restarted Arnoldi iteration},
   author={Lehoucq, Richard Bruno},
@@ -364,6 +387,17 @@
   publisher={IEEE}
 }
 
+@article{S11,
+  title={Symplectic Gram-Schmidt Algorithm with Re-Orthogonalization},
+  author={Al-Aidarous, Eman Salem},
+  journal={Journal of King Abdulaziz University: Science},
+  volume={23},
+  number={1},
+  pages={11--20},
+  year = {2011},
+  doi = {10.4197/Sci.23-1.2},
+}
+
 @article{TRLBK14,
   author = {Tu, J. H. and Rowley, C. W. and Luchtenburg, D. M. and Brunton, S. L. and Kutz, J. N.},
   title = {On Dynamic Mode Decomposition: Theory and Applications},

src/pymor/algorithms/symplectic.py

@@ -236,6 +236,8 @@ class SymplecticBasis(BasicObject):
 def psd_svd_like_decomp(U, modes, balance=True):
     """Generates a |SymplecticBasis| with the PSD SVD-like decompostion.
 
+    This is an implementation of Algorithm 1 in :cite:`BBH19`.
+
     Parameters
     ----------
     U
@@ -287,6 +289,8 @@ def psd_svd_like_decomp(U, modes, balance=True):
 def psd_cotengent_lift(U, modes):
     """Generates a |SymplecticBasis| with the PSD cotangent lift.
 
+    This is an implementation of Algorithm 1 in :cite:`PM16`.
+
     Parameters
     ----------
     U
@@ -316,6 +320,8 @@ def psd_cotengent_lift(U, modes):
 def psd_complex_svd(U, modes):
     """Generates a |SymplecticBasis| with the PSD complex SVD.
 
+    This is an implementation of Algorithm 2 in :cite:`PM16`.
+
     Parameters
     ----------
     U
@@ -347,10 +353,7 @@ def symplectic_gram_schmidt(E, F, return_Lambda=False, atol=1e-13, rtol=1e-13, o
                             lmax=2, check=True, check_tol=1e-3, copy=True):
     """Symplectify a |VectorArray| using the modified symplectic Gram-Schmidt algorithm.
 
-    Reference::
-
-        Salam (2005), On theoretical and numerical aspects of symplectic Gram--Schmidt-like
-        algorithms
+    This is an implementation of Algorithm 3.2. in :cite:`S11`.
 
     Decomposition::