A Shamanskii-Like acceleration scheme for nonlinear equations at singular roots
Mathematics of Computation
Nonsymmetric Algebraic Riccati Equations and Hamiltonian-like Matrices
SIAM Journal on Matrix Analysis and Applications
Solution of the matrix equation AX + XB = C [F4]
Communications of the ACM
Nonsymmetric Algebraic Riccati Equations and Wiener--Hopf Factorization for M-Matrices
SIAM Journal on Matrix Analysis and Applications
On the Iterative Solution of a Class of Nonsymmetric Algebraic Riccati Equations
SIAM Journal on Matrix Analysis and Applications
A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation
Numerische Mathematik
Journal of Computational and Applied Mathematics
Iterative Solution of a Nonsymmetric Algebraic Riccati Equation
SIAM Journal on Matrix Analysis and Applications
On the Doubling Algorithm for a (Shifted) Nonsymmetric Algebraic Riccati Equation
SIAM Journal on Matrix Analysis and Applications
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
SIAM Journal on Matrix Analysis and Applications
Transforming algebraic Riccati equations into unilateral quadratic matrix equations
Numerische Mathematik
Accurate fourteenth-order methods for solving nonlinear equations
Numerical Algorithms
Numerical Solution of Algebraic Riccati Equations
Numerical Solution of Algebraic Riccati Equations
Accurate solutions of M-matrix algebraic Riccati equations
Numerische Mathematik
Alternating-directional Doubling Algorithm for $M$-Matrix Algebraic Riccati Equations
SIAM Journal on Matrix Analysis and Applications
Hi-index | 0.00 |
For the algebraic Riccati equation whose four coefficient matrices form a nonsingular M-matrix or an irreducible singular M-matrix K, the minimal nonnegative solution can be found by Newton's method and the doubling algorithm. When the two diagonal blocks of the matrix K have both large and small diagonal entries, the doubling algorithm often requires many more iterations than Newton's method. In those cases, Newton's method may be more efficient than the doubling algorithm. This has motivated us to study Newton-like methods that have higher-order convergence and are not much more expensive each iteration. We find that the Chebyshev method of order three and a two-step modified Chebyshev method of order four can be more efficient than Newton's method. For the Riccati equation, these two Newton-like methods are actually special cases of the Newton---Shamanskii method. We show that, starting with zero initial guess or some other suitable initial guess, the sequence generated by the Newton---Shamanskii method converges monotonically to the minimal nonnegative solution.We also explain that the Newton-like methods can be used to great advantage when solving some Riccati equations involving a parameter.