Low memory and low complexity iterative schemes for a nonsymmetric algebraic Riccati equation arising from transport theory

A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory

“…Although it accelerates SDA ls for NARE (1), MSDA ls is still slower than the fast Newton's method proposed in [4]. We were not able to acquire implementation of the algorithms in [20], so we leave the comparison to other methods for future work.…”

Iterative methods for the quadratic bilinear equation arising from a class of quadratic dynamic systems

“…This is a standard Lyapunov equation and might be further accelerated by proper preconditioning. Here a cyclic Smith or ADI preconditioning as in [16,17] is employed. Specifically, by incorporating the ADI parameters p l > 0 for l 0 and rewriting the QBEH as…”

Structured Shamanskii methods for Chandrasekhar equation arising from radiation

“…So, the numerical experiments in Section 4 indicate the performance of 2-step structured Shamanskii method. (ii) The computation of two extremal eigenvalues of ∆ −1 − ∆ −1 (x (k) )η T and the determination of the J k optimal ADI parameters {p i } in rows 3-4 are totally same with [24], see also [16,22,23] for more details. (iii) Note that in row 8, the scalars [h (12) are not relevant to m as stated in Remark (iii), we omit the subscript in Algorithm 1.…”

“…Higher Shamanskii steps are omitted as they give poorer numerical results. In comparison, the structured Newton's method ("SN") in [20,24] is employed to highlight the efficiency of Algorithm 1.…”

“…Note that, by introducing x (k) = α∆X (k) e + e, the matrix B − X (k) CX (k) on the right hand side (RHS) is a product of two rank-two matrices (∆ −1 (e, x (k) − e))(∆ −1 (e, e − x (k) )) T ; and A − X (k) C on the left hand side (LHS) is a diagonal-plus-rank-one matrix ∆ −1 − α∆ −1 x (k) e T , then each X (k+1) in the above structured Newton step might be expressed implicitly and stored in low-rank form via the factor-alternating direction implicit (FADI) method [2,16]. The obvious advantage of using FADI method to solve subproblems, compared to those equipped with Krylov subspace methods [14], is that the complexity of solving the subproblem could be down to O(n) [16,20,24]. However, from the viewpoint of computational efficiency, the above structured Newton's method might still leave enough room for further improvement.…”