returns the multilevel lower-triangular Toeplitz matrix tt generated by the multi-dimensional input vector x in the QTT format If the size of the input vector is N, then the size of the output matrix is N x N. April 20, 2011 Vladimir Kazeev vladimir.kazeev@gmail.com INM RAS Moscow, Russia %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% For details please see the preprint http://www.mis.mpg.de/publications/preprints/2011/prepr2011-36.html Vladimir A. Kazeev, Boris N. Khoromskij and Eugene E. Tyrtyshnikov Multilevel Toeplitz matrices generated by QTT tensor-structured vectors and convolution with logarithmic complexity January 12, 2012 Vladimir Kazeev, Seminar for Applied Mathematics, ETH Zurich vladimir.kazeev@sam.math.ethz.ch %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0001 function [tt]=tt_qltrtoepl(x) 0002 0003 % returns the multilevel lower-triangular Toeplitz matrix tt generated by the multi-dimensional input vector x 0004 % in the QTT format 0005 % 0006 % If the size of the input vector is N, 0007 % then the size of the output matrix is N x N. 0008 % 0009 % April 20, 2011 0010 % Vladimir Kazeev 0011 % vladimir.kazeev@gmail.com 0012 % INM RAS 0013 % Moscow, Russia 0014 % 0015 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0016 % For details please see the preprint 0017 % http://www.mis.mpg.de/publications/preprints/2011/prepr2011-36.html 0018 % Vladimir A. Kazeev, Boris N. Khoromskij and Eugene E. Tyrtyshnikov 0019 % Multilevel Toeplitz matrices generated by QTT tensor-structured vectors and convolution with logarithmic complexity 0020 % January 12, 2012 0021 % Vladimir Kazeev, 0022 % Seminar for Applied Mathematics, ETH Zurich 0023 % vladimir.kazeev@sam.math.ethz.ch 0024 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0025 0026 d=size(x,1); 0027 tt=tt_qshiftstack_q(d); 0028 tt=tt_qreshape(tt,3,[4*ones(d,1),2*ones(d,1)]); 0029 tt=tt_mv(tt,x); 0030 tt=tt_qreshape(tt,1,2*ones(d,2)); 0031 0032 return 0033 end