422 Matrix ToolsThe (two-sided) reduced singular-value decomposition from Definition 2.27leads toprocedureRSVD(n, m, r, M, U, Σ, V);{reduced SVD}input:M∈Kn×m;output:U∈Kn×r, V∈Km×rorthogonal withr= rank(M),Σ= diag{σ1, . . . , σr} ∈Rr×rwithσ1≥. . .≥σr>0.(2.28)Here the integersn, mmay also be replaced with index setsIandJ. For the costNSVD(n, m), see Corollary 2.24a.The left-sided reduced singular-value decomposition (cf. Remark 2.28) is de-noted byprocedureLSVD(n, m, r, M, U, Σ);{left-sided reduced SVD}input:M∈Kn×m;output:U, r, Σas in (2.28).(2.29)Its cost isNLSVD(n, m) :=12n(n+ 1)Nm+83n3,whereNmis the cost of the scalar product of rows ofM. In general,Nm= 2m-1holds, but it may be smaller for structured matrices (cf. Remark 7.16).In the procedures above,Mis a general matrix fromKn×m. MatricesM∈ Rr(cf. (2.6)) may be given in the formM=rXν=1rXμ=1cνμaνbHμ=ACBHaν∈Kn, A=[a1a2· · ·]∈Kn×r,bν∈Km, B=[b1b2· · ·]∈Km×r!.(2.30)Then the following approach has a cost proportional ton+mifrn, m(cf.[138, Alg. 2.17]), but also forrn, mit is cheaper than the direct computation18of the productM=ACBHfollowed by a singular-value decomposition.