An Algorithm for the Matrix Lambert $W$ Function

Fasi, Massimiliano; Higham, Nicholas J.; Iannazzo, Bruno

doi:10.1137/140997610

Cited by 20 publications

(16 citation statements)

References 29 publications

(40 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. , n, be as in (8). Then P [u] and Q [v] have the same block structure as T , and their off-diagonal blocks, for 1 ≤ i < j ≤ ν, are given by the formulae…”

Section: A Substitution Algorithm For Rational Equationsmentioning

confidence: 99%

“…where A, X ∈ C N ×N and f is a complex function applied to a matrix (in the sense of primary matrix functions, see Section 2). Remarkable examples of (1) are the matrix equations X k = A, e X = A, and Xe X = A, which define the matrix kth root [22,16], the matrix logarithm [1], and the matrix Lambert W function [8], respectively. Existence and finiteness of real and complex solutions to (1) are discussed, along with other properties of this matrix equation, in the excellent treatise by Evard and Uhlig [7].…”

Section: Introductionmentioning

confidence: 99%

“…ij= c m Y ij , which follows directly from the definition of P [m−1] in(8). For the inductive step, we have, for 1 < k ≤ m,P [m−k] ij…”

mentioning

confidence: 95%

See 2 more Smart Citations

Computing primary solutions of equations involving primary matrix functions

Fasi

Iannazzo

2019

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

The matrix equation f (X) = A, where f is an analytic function and A is a square matrix, is considered. Some results on the classification of solutions are provided. When f is rational, a numerical algorithm is proposed to compute all solutions that can be written as a polynomial of A. For real data, the algorithm yields the real solutions using only real arithmetic. Numerical experiments show that the algorithm performs in a stable fashion when run in finite precision arithmetic.

show abstract

“…. , n, be as in (8). Then P [u] and Q [v] have the same block structure as T , and their off-diagonal blocks, for 1 ≤ i < j ≤ ν, are given by the formulae…”

Section: A Substitution Algorithm For Rational Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Computing primary solutions of equations involving primary matrix functions

Fasi

Iannazzo

2019

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…A possible application of these techniques is the computation of more general matrix functions defined implicitly by equations of the form f (X) = A, where f is an analytic function. As an example, we develop an algorithm for computing the Lambert W function which, in a number of cases, is faster and more accurate than the reference algorithm [15,Alg. 1].…”

mentioning

confidence: 99%

Substitution algorithms for rational matrix equations

Fasi¹,

Iannazzo²

2020

etna

Self Cite

View full text Add to dashboard Cite

We study equations of the form r(X) = A, where r is a rational function and A and X are square matrices of the same size. We develop two techniques for solving these equations by inverting (through a substitution strategy) two schemes for the evaluation of rational functions of matrices. For triangular matrices, the new methods yield the same computational cost as the evaluation schemes from which they are obtained. A general equation can be reduced to upper triangular form by exploiting the Schur decomposition of the given matrix. For real data, the algorithms rely on the real Schur decomposition in order to compute real solutions using only real arithmetic. Numerical experiments show that our implementations are faster than existing alternatives without sacrificing accuracy.

show abstract

“…Estimating the forward error of algorithms for matrix functions requires a reference solution computed in higher precision, and an arbitrary precision algorithm for the matrix exponential can be used both for the exponential and for other types of functions as mentioned above. Furthermore, such an algorithm allows us to estimate the backward error of algorithms for evaluating matrix functions defined implicitly by equations involving the exponential, such as the logarithm [3], [12], the Lambert W function [13], and inverse trigonometric and hyperbolic functions [5].…”

mentioning

confidence: 99%

An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential

Fasi¹,

Higham²

2019

SIAM J. Matrix Anal. Appl.

Self Cite

View full text Add to dashboard Cite

The most popular algorithms for computing the matrix exponential are those based on the scaling and squaring technique. For optimal efficiency these are usually tuned to a particular precision of floating-point arithmetic. We design a new scaling and squaring algorithm that takes the unit roundoff of the arithmetic as input and chooses the algorithmic parameters in order to keep the forward error in the underlying Padé approximation below the unit roundoff. To do so, we derive an explicit expression for all the coefficients in an error expansion for Padé approximants to the exponential and use it to obtain a new bound for the truncation error. We also derive a new technique for selecting the internal parameters used by the algorithm, which at each step decides whether to scale or to increase the degree of the approximant. The algorithm can employ diagonal Padé approximants or Taylor approximants and can be used with a Schur decomposition or in transformation-free form. Our numerical experiments show that the new algorithm performs in a forward stable way for a wide range of precisions and that the most accurate of our implementations, the Taylor-based transformation-free variant, is superior to existing alternatives.

show abstract

An Algorithm for the Matrix Lambert $W$ Function

Cited by 20 publications

References 29 publications

Computing primary solutions of equations involving primary matrix functions

Computing primary solutions of equations involving primary matrix functions

Substitution algorithms for rational matrix equations

An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential

Contact Info

Product

Resources

About