On the Newton Method for the Matrix <i>P</i>th Root

Iannazzo, Bruno

doi:10.1137/050624790

Cited by 66 publications

(67 citation statements)

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“…As in Iannazzo's paper [21] note that for any strictly accretive Hilbert space operator a, b = a 1 2 / a 1 2 is also strictly accretive by e.g. a result on p. 181 of [19], and has norm ≤ 1.…”

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confidence: 99%

Roots in Operator and Banach Algebras

Blecher

Wang

2015

Integr. Equ. Oper. Theory

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Abstract. We show that several known facts concerning roots of matrices generalize to operator algebras and Banach algebras. We show for example that the so-called Newton, binomial, Visser, and Halley iterative methods converge to the root in Banach and operator algebras under various mild hypotheses. We also show that the 'sign' and 'geometric mean' of matrices generalize to Banach and operator algebras, and we investigate their properties. We also establish some other facts about roots in this setting.In memoriam Charles Read-gentleman, brother, mathematical force of nature. IntroductionAn operator algebra is a closed subalgebra of B(H), for a complex Hilbert space H. In this paper we show that several known facts concerning roots of matrices generalize to operator algebras and Banach algebras. We begin by establishing some basic properties of roots that do not seem to be in the literature (although they may be known to some experts), as well as reviewing some that are. We then show that the 'sign' of a matrix generalizes to Banach algebras, and that Drury's variant of the 'geometric mean' of matrices generalizes to operators on a Hilbert space (we also generalize his definition slightly), and prove some basic facts about these. We also show that the so-called Newton (or Babylonian), binomial, Visser, and Halley iterative methods for the root converge to the root in Banach and operator algebras under various mild hypotheses inspired by the matrix theory literature. Some parts of our paper are fairly literal transfers of matrix results to the operator or Banach algebraic setting, using known tricks or standard theory, and here we will try to be brief. However we have not seen these in the literature and they seem quite useful. For example our results, particularly probably the geometric mean, should be applicable to our ongoing study of 'real positivity' in operator algebras (see e.g. [9,10,11,8,6] and references therein) initiated by the first author and Charles Read.Turning to background and notation, it is common when studying roots to make the assumption that the spectrum contains no strictly negative numbers. Note that a singular matrix with no strictly negative eigenvalues, may not have a square root (for example, E 12 in M 2 ), or may have a square root but not have a square root in {x} ′′ (for example, E 12 in M 3 , which has many square roots including E 13 + E 32 ), or may have infinitely many square roots in {x} ′′ (for example, 0 in an algebra with 1991 Mathematics Subject Classification. Primary 47A64, 47L10, 47L30, 47B44; Secondary 15A24, 15A60, 47A12, 47A60, 47A63, 49M15, 65F30.Key words and phrases. Roots, fractional powers, geometric mean, sign of operator, Newton method for roots, binomial method for square root, accretive operator, sectorial operator, nonselfadjoint operator algebra, numerical range, functional calculus.

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confidence: 99%

Roots in Operator and Banach Algebras

Blecher

Wang

2015

Integr. Equ. Oper. Theory

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“…In our previous work we have on several occasions been able to manipulate an unstable iteration into an equivalent stable iteration [18], [21], [22], [23], [24]. Inspired by these results, we were able to find a stable variant of the simplified Newton method, namely, The equivalence of (4.2) and (4.4) and the stability of (4.4) …”

Section: Using (21) We Obtainmentioning

confidence: 99%

“…4a. If T 11 is nonempty then compute X 11 = W k (T 11 ) as the limit of (4.4) with 22 is nonempty (which implies |k| 1) then compute X 22 = W k (T 22 ) as the limit of (4.4) with Z 0 = (−1) k (2eT 22 + 2I) 1/2 − I. 5.…”

Section: The Algorithmmentioning

confidence: 99%

An Algorithm for the Matrix Lambert $W$ Function

Fasi

Higham

Iannazzo³

2015

SIAM J. Matrix Anal. & Appl.

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Abstract. An algorithm is proposed for computing primary matrix Lambert W functions of a square matrix A, which are solutions of the matrix equation W e W = A. The algorithm employs the Schur decomposition and blocks the triangular form in such a way that Newton's method can be used on each diagonal block, with a starting matrix depending on the block. A natural simplification of Newton's method for the Lambert W function is shown to be numerically unstable. By reorganizing the iteration a new Newton variant is constructed that is proved to be numerically stable. Numerical experiments demonstrate that the algorithm is able to compute the branches of the matrix Lambert W function in a numerically reliable way.

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“…In order to have a general algorithm, some kind of preprocessing of the matrix A should be done. The first general and stable algorithm was given by Iannazzo [12] and some others have followed [3,4,13,16]. The computational cost of these algorithms is O(n 3 log 2 p) arithmetic operations (ops) and the storage required is O(n 2 log 2 p) real numbers.…”

Section: Introductionmentioning

confidence: 99%