A Shifted Cyclic Reduction Algorithm for Quasi-Birth-Death Problems

He, Chunyang; Meini, Beatrice; Rhee, Noah H.

doi:10.1137/s0895479800371955

Cited by 54 publications

(35 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another approach is to transform the bilateral equation (1.1) into a unilateral equation and use methods based on cyclic reduction, including the Latouche-Ramaswami (LR) algorithm [23], in combination with a shift technique proposed in [19].…”

Section: A Z-matrix a Is Called An M -Matrix If S ≥ ρ(B) Where ρ(·mentioning

confidence: 99%

See 1 more Smart Citation

Iterative Solution of a Nonsymmetric Algebraic Riccati Equation

Guo¹,

Higham

2007

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

Abstract. We study the nonsymmetric algebraic Riccati equation whose four coefficient matrices are the blocks of a nonsingular M -matrix or an irreducible singular M -matrix M . The solution of practical interest is the minimal nonnegative solution. We show that Newton's method with zero initial guess can be used to find this solution without any further assumptions. We also present a qualitative perturbation analysis for the minimal solution, which is instructive in designing algorithms for finding more accurate approximations. For the most practically important case, in which M is an irreducible singular M -matrix with zero row sums, the minimal solution is either stochastic or substochastic and the Riccati equation can be transformed into a unilateral matrix equation by a procedure of Ramaswami. The minimal solution of the Riccati equation can then be found by computing the minimal nonnegative solution of the unilateral equation using the LatoucheRamaswami algorithm. When the minimal solution of the Riccati equation is stochastic, we show that the Latouche-Ramaswami algorithm, combined with a shift technique suggested by He, Meini, and Rhee, is breakdown-free and is able to find the minimal solution more efficiently and more accurately than the algorithm without a shift. When the minimal solution of the Riccati equation is substochastic, we show how the substochastic minimal solution can be found by computing the stochastic minimal solution of a related Riccati equation of the same type.

show abstract

Section: A Z-matrix a Is Called An M -Matrix If S ≥ ρ(B) Where ρ(·mentioning

confidence: 99%

“…The shift technique introduced in [19] is H = G − ev T , where v > 0 and v T e = 1. For our purposes here, we only require that v ≥ 0 and v T e = 1.…”

Section: Applicability Of the Shifted Lr Algorithmmentioning

confidence: 99%

Iterative Solution of a Nonsymmetric Algebraic Riccati Equation

Guo¹,

Higham

2007

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

show abstract

“…As can be seen from Table 2, the CPU times for the Newton method are much higher than the ones obtained in Table 1. One possibility to overcome the difficulty of the Newton method in this near critical case, is the use of Shift techniques [37,38].…”

Section: Examplementioning

confidence: 99%

On some Krylov subspace based methods for large-scale nonsymmetric algebraic Riccati problems

Bentbib

Jbilou²,

Sadek

2015

Computers & Mathematics with Applications

View full text Add to dashboard Cite

a b s t r a c tIn the present paper, we consider large scale nonsymmetric matrix Riccati equations with low rank right hand sides. These matrix equations appear in many applications such as transport theory, Wiener-Hopf factorization of Markov chains, applied probability and others. We show how to apply directly Krylov methods such as the extended block Arnoldi algorithm to get low rank approximate solutions. We also combine the Newton method and block Krylov subspace methods to get approximations of the desired minimal nonnegative solution. We give some theoretical results and report some numerical experiments for the well known transport equation.

show abstract

“…These drawbacks can be easily removed by means of the shift technique originally introduced be He, Meini and Rhee in [13] and applied to Riccati equations in [6] and [2].…”

Section: Shift Techniquementioning

confidence: 99%

“…The idea is to apply the shift technique originally introduced by C. He, B. Meini and N.H. Rhee [13] and used in the framework of Riccati equations by C.-H. Guo, B. Iannazzo, and B. Meini in [6]. With this technique, we replace the original Riccati equation with a new one having the same minimal solution as the original equation (1.1) but where the singularity of the Jacobian is completely removed.…”

mentioning

confidence: 99%

A Fast Newton's Method for a Nonsymmetric Algebraic Riccati Equation

Бини¹,

Iannazzo²,

Poloni³

2008

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

Abstract. A special instance of the algebraic Riccati equation XCX − XE − AX + B = 0 encountered in transport theory is considered where the n × n matrix coefficients A, B, C, E are rank structured matrices. Relying on the structural properties of Cauchy-like matrices, an algorithm is designed for performing the customary Newton iteration in O(n 2 ) arithmetic operations (ops). The same technique is used to reduce the cost of the algorithm proposed by L.-Z. Lu in [Numer. Linear Algebra Appl., 12 (2005), pp. 191-200] from O(n 3 ) to O(n 2 ) ops still preserving quadratic convergence in the generic case. As a byproduct we show that the latter algorithm is closely related to the customary Newton method by simple formal relations.In critical cases where the Jacobian is singular and quadratic convergence turns to linear, we provide an adaptation of the shift technique in order to get rid of the singularity. The original equation is transformed into an equivalent Riccati equation where singularity is removed and the matrix coefficients maintain the same structure as in the original equation. This leads to a quadratically convergent algorithm with complexity O(n 2 ) which provides approximations with the full precision.Numerical experiments and comparisons which confirm the effectiveness of the new approach are reported.

show abstract

A Shifted Cyclic Reduction Algorithm for Quasi-Birth-Death Problems

Cited by 54 publications

References 8 publications

Iterative Solution of a Nonsymmetric Algebraic Riccati Equation

Iterative Solution of a Nonsymmetric Algebraic Riccati Equation

On some Krylov subspace based methods for large-scale nonsymmetric algebraic Riccati problems

A Fast Newton's Method for a Nonsymmetric Algebraic Riccati Equation

Contact Info

Product

Resources

About