## 7 Conclusions  
We presented two novel spectrum-adapted polynomial methods for approximating \(f(\mathbf{A})\mathbf{b}\) for large sparse matrices: spectrum-adapted interpolation and spectrum-adapted weighted least squares/orthogonal polynomial expansion. These methods leverage an estimation of the cumulative spectral density of the matrix to build polynomials of a fixed order \(K\) that yield better approximations to \(f\) in the higher density regions of the matrix spectrum. In terms of approximation accuracy, numerical experiments showed that, relative to the state-of-the-art Lanczos and Chebyshev polynomial approximation techniques, the proposed methods often yield more accurate approximations at lower polynomial orders; however, the proposed spectrum-adapted interpolation method is not very stable at higher degrees (\(K>10\)) due to overfitting. The proposed spectrum-adapted weighted least squares method performs particularly well in terms of accuracy for matrices with many distinct interior eigenvalues, whereas the Lanczos method, e.g., is often more accurate when \(K\) is higher and/or the matrix \(\mathbf{A}\) has many extreme eigenvalues. One potential extension would be to investigate a hybrid method that combines the Lanczos and spectrum-adapted weighted least squares approaches. We did not notice consistent trends regarding relative approximation accuracy with respect to the shape of the function \(f\).  
In terms of computational complexity, the cost of our methods, like Chebyshev polynomial approximation, is \(\mathcal{O}(MK)\), where \(M=nnz(\mathbf{A})\). For very large, sparse matrices, this complexity reduces to \(\mathcal{O}(NK)\), where \(\mathbf{A}\) is an \(N\times N\) matrix. The Lanczos method, on the other hand, incurs an additional \(\mathcal{O}(NK^{2})\) cost due to the orthogonalization step, making it more expensive for large enough \(K\). Finally, the proposed spectrum-adapted methods, like the Chebyshev approximation, are amenable to efficient parallel and distributed computation via communication between neighboring nodes [33]. The inner products of the Lanczos method, on the other hand, may lead to additional communication expense or severe loss of efficiency in certain distributed computation environments (e.g., GPUs).  
Conceptualization, T.F., D.I.S., S.U., and Y.S.; data curation, T.F., D.I.S., and S.U.; formal analysis, T.F., D.I.S., and S.U.; funding acquisition, D.I.S.; investigation, T.F., D.I.S., and S.U.; methodology, T.F., D.I.S., S.U., and Y.S.; project administration, D.I.S.; software, T.F., D.I.S., and S.U.; supervision, D.I.S.; validation, T.F., D.I.S., and S.U.; visualization, T.F., D.I.S., and S.U.; writing--original draft, T.F., D.I.S., and S.U.; writing--review and editing, T.F., D.I.S., S.U., and Y.S. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
We thank the anonymous reviewers for their constructive comments on an earlier version of this article.  
The authors declare no conflict of interest.  
MATLAB code for all numerical experiments in this paper is available at [[http://www.macalester.edu/~dshuman1/publications.html](http://www.macalester.edu/~dshuman1/publications.html)]([http://www.macalester.edu/~dshuman1/publications.html](http://www.macalester.edu/~dshuman1/publications.html)). It leverages the open access GSPBox [57].