Two Taylor Algorithms for Computing the Action of the Matrix Exponential on a Vector

Round 1

Reviewer 1 Report

The paper is concerned with the computation of F(A)v where A is a square matrix , v is a vector and F(x) is the exponential matrix.  Methods based on the scaled Taylor approximation of F(x) are considered. In particular, two strategies are suggested for computing the scaling parameter and the truncation level.

I  think the paper is interesting enough to deserve publication in revised form.  Specifically, I ask the authors to include comparisons in  terms of accuracy and complexity with some  methods exploiting the rational (Pade') approximation of F(x).  It would  also be interesting to know the behaviour of such methods in the difficult cases  described at page 6.   

Other minor changes are the following: 

a) at page 2 replace equality with inequality for s;

b) at the end of page 2 clarify  how the parameter m is chosen if  only   the first  term of the  error  is considered.

Author Response

Please, find our answers to reviewer 1 is attached in the file. 

Author Response File: Author Response.pdf

Reviewer 2 Report

Please see the attached for details.

Comments for author File: Comments.pdf

Author Response

Please, find our answers to reviewer 1 is attached in the file. 

Author Response File: Author Response.pdf

Reviewer 3 Report

In this paper, two algorithms devoted to the computation of the action of the matrix exponential on a vector have been described. Their numerical and computational performance has been evaluated in several experiments under a testbed composed of different state-of-the-art matrices. In general, these algorithms provided higher accuracy and a lower cost than state-of-the-art algorithms in the literature.
Both algorithms have been migrated in their implementation to be able to run and take advantage of a computational infrastructure based on GPUs or a traditional computer, making such execution configurable and fully transparent to the user from the MATLAB application itself.

Author Response

Please, find our answers to reviewer 1 is attached in the file. 

Author Response File: Author Response.pdf

Reviewer 4 Report

The paper proposes two algorithms to compute e^Av, where A is a square matrix with complex entires and v is a complex vector, making use of the scaling and recovering technique based on a backward or forward error analysis vector. The problem is a very important one for several branches, as mathematics, physics, engineering. Both algorithms improve
on those already existing in the literature, in terms of accuracy and response time.

The manuscript is well written, in a very pleasant and clear style, with a large discussion and presentation of the algorithms and their efficacy.

I consider that the paper deserves to be published in the current form.

Author Response

Please, find our answers to reviewer 1 is attached in the file. 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

I am not satisfied with this revision. 

a) Concerning comparisons with  rational-based methods  I think that for the significance of the resulting analysis  methods   computing the action of the exponential matrix function without computing  the matrix function should be considered.  Moreover, I would ask again the authors to provide  at least   one comparison on a difficult test.  I think that for a symmetric matrix we can consider E_1 as a  sufficiently accurate approximation  of the matrix exponential. It should be interesting to know what the behaviour of the proposed algorithms is. 

b) The sentence including (3) and (4) is not mathematicallly  clear. Please, rewrite this sentence. 

Author Response

The authors answers are attached.

Author Response File: Author Response.pdf

Reviewer 2 Report

After I have checked the revised manuscript, it noted that the authors considered all the comments from my previous report and I am very satisfied for this revised manuscript.

Author Response

Authors like to thank again the reviewer for his/her suggestions. 

Regards

Round 3

Reviewer 1 Report

I do not agree with the reply by the authors. I think the authors should consider numerical methods based on rational (Pade`) appproximation which compute the action of the exponential matrix on a vector without explicitly forming the exponential matrix.   For instance, such a method  is described in section 5 of MR3457698  Güttel, Stefan ;  Nakatsukasa, Yuji . Scaled and squared subdiagonal Padé approximation for the matrix exponential. SIAM J. Matrix Anal. Appl.  37  (2016),  no. 1, 145--170.