Error analysis of ๐๐ updating with exponential windowing
G. W. Stewart ยท Mathematics of Computation ยท 1992
Exponential windowing is a widely used technique for suppressing the effects of old data as new data is added to a matrix. Specifically, given an n ร p n \times p matrix X n {X_n} and a "forgetting factor" ฮฒ โ ( 0 , 1 ) \beta \in (0,1) , one works with the matrix diag ( ฮฒ n โ 1 , ฮฒ n โ 2 , โฆ , 1 ) X n {\operatorname {diag}}({\beta ^{n - 1}},{\beta ^{n - 2}}, \ldots ,1){X_n} . In this paper we examine an updating algorithm for computing the QR factorization of diag ( ฮฒ n โ 1 , ฮฒ n โ 2 , โฆ , 1 ) X n {\operatorname {diag}}({\beta ^{n - 1}},{\beta ^{n - 2}}, \ldots ,1){X_n} and show that it is unconditionally stable in the presence of rounding errors.