Convergence of pure and relaxed Newton methods for solving a matrix polynomial equation arising in stochastic models

Seo, Jong-Hyeon; Kim, Hyun-Min

doi:10.1016/j.laa.2013.10.043

Cited by 15 publications

(16 citation statements)

References 22 publications

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“…For the notation simplification, we introduce the recursion function : N × N × R m×m × R m×m → R m×m as defined in [38]:…”

Section: Normwise Condition Numbermentioning

confidence: 99%

“…Under certain conditions, usually satisfied in the applications, the matrix P is a nonsingular matrix as showed in [38]. We suppose that P is nonsingular in the remainder of this paper.…”

Section: Normwise Condition Numbermentioning

confidence: 99%

“…We use the Newton's method proposed in [38] to compute the minimal nonnegative solution S andS. Choose δ k = A k F , from Theorem 3.2 we get three kinds of local normwise perturbation bounds:…”

Section: Numerical Examplesmentioning

confidence: 99%

“…For k = 0, 1, 2, let A k = rand(m) • A k × 10 −s , where s is a positive integer, thenÃ k = A k + A k is the perturbed coefficient matrix of the corresponding perturbed equation. Similarly, the minimal positive solutionX of the perturbed matrix polynomial equation exists and can be obtained by using the Newton's method in [38]. Let Table 2 shows that the mixed and componentwise analysis give more tighter and revealing bounds than the normwise perturbation bounds.…”

Section: Example 62mentioning

confidence: 99%

“…Kratz and Stickel [26] proved that Newton's method could also be applied to solve this general case. Seo and Kim [38] studied the relaxed Newton's method for finding the minimal nonnegative solution of the MPE (1.1) and they also proved that the relaxed Newton's method could work more efficiently than the general Newton's method.…”

Section: Introductionmentioning

confidence: 99%

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Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

2018

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We consider a matrix polynomial equation (MPE) A n X n + A n−1 X n−1 + · · · + A 0 = 0, where A n , A n−1 , . . . , A 0 ∈ R m×m are the coefficient matrices, and X ∈ R m×m is the unknown matrix. A sufficient condition for the existence of the minimal nonnegative solution is derived, where minimal means that any other solution is componentwise no less than the minimal one. The explicit expressions of normwise, mixed and componentwise condition numbers of the matrix polynomial equation are obtained. A backward error of the approximated minimal nonnegative solution is defined and evaluated. Some numerical examples are given to show the sharpness of the three kinds of condition numbers.

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“…For the notation simplification, we introduce the recursion function : N × N × R m×m × R m×m → R m×m as defined in [38]:…”

Section: Normwise Condition Numbermentioning

confidence: 99%

“…Under certain conditions, usually satisfied in the applications, the matrix P is a nonsingular matrix as showed in [38]. We suppose that P is nonsingular in the remainder of this paper.…”

Section: Normwise Condition Numbermentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Example 62mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

2018

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Convergence of relaxed Newton method for order-convex matrix equations

Seo

2019

Comp. Appl. Math.

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In this paper, we suggest singular monotone Newton theorem and a relaxed Newton method for a nonlinear matrix equation F(X) = 0 where F is an order-convex matrix function. At first, we introduce monotone Newton theorem from Ortega and Rheinboldt (SIAM, 2000), and we show singular monotone Newton theorem which shows that the Newton sequence converges to a solution of F(X) = 0 even if the Fréchet derivative at the solution is singular. At last, we provide a relaxed Newton method for F(X) = 0 which is better than the pure Newton method and we give some numerical experiments for the relaxed Newton method. Keywords Order-convex function • Nonlinear matrix equation • Newton method • Relaxed Newton method Mathematics Subject Classification 65H10 n T for A = a 1 a 2 • • • a n , (1.1) Communicated by Carlos Conca.

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Two global quasi-Newton algorithms for solving matrix polynomial equations

Macías,

Pérez,

Martínez

2023

Comp. Appl. Math.

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In this article, we globalize the quasi-Newton algorithm proposed in Macías et al. (Appl Math Comput 441:127678, 2023) introducing an exact line search, we propose a polynomial approximation to the merit function and we deduce a sufficient condition for its minimization interval. We use the exact merit function and its approximation to propose two global quasi-Newton algorithms for solving matrix polynomial equations. For each algorithm, we prove that the exact line search does not affect the convergence of quasi-Newton method. In addition, we present comparative numerical tests of the algorithmic proposals in which we also compare with global Newton’s method.

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Convergence of pure and relaxed Newton methods for solving a matrix polynomial equation arising in stochastic models

Cited by 15 publications

References 22 publications

Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

Convergence of relaxed Newton method for order-convex matrix equations

Two global quasi-Newton algorithms for solving matrix polynomial equations

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