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BetaThe matrix–vector multiplication can be done in () arithmetical operations where is the average number of nonzero elements in a row. The total complexity is thus O ( d m n ) {\displaystyle O(dmn)} , or O ( d n 2 ) {\displaystyle O(dn^{2})} if m = n {\displaystyle m=n} ; the Lanczos algorithm can be very fast for sparse matrices.
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