Numerische Mathematik Numerische https://doi.org/10.1007/s00211-021-01215-6 Mathematik Fiber product homotopy method for multiparameter eigenvalue problems Jose Israel Rodriguez1 · Jin-Hong Du2 · Yiling You3 · Lek-Heng Lim4 Author Proof Received: 30 June 2018 / Revised: 14 May 2021 / Accepted: 24 May 2021 © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021 1 Abstract 2 We develop a new homotopy method for solving multiparameter eigenvalue problems 3 (MEPs) called the fiber product homotopy method. For a k-parameter eigenvalue 4 problem with matrices of sizes n1,...,nk O(n), fiber product homotopy method = 5 requires deformation of O(1) linear equations, while existing homotopy methods for 6 MEPs require O(n) nonlinear equations. We show that the fiber product homotopy 7 method theoretically finds all eigenpairs of an MEP with probability one. It is especially 8 well-suited for a class of problems we call dimension-deficient singular problems that 9 are generic with respect to intrinsic dimension, as the fiber product homotopy method 10 is provably convergent with probability one for such problems as well, a fact borne 11 out by numerical experiments. More generally, our numerical experiments indicate 12 that the fiber product homotopy method significantly outperforms the standard Delta 16 13 method in terms of accuracy, with consistent backward errors on the order of 10− 14 without any use of extended precision. In terms of speed, it significantly outperforms 15 previous homotopy-based methods on all problems and outperforms the Delta method The work in this article is generously supported by DARPA D15AP00109, HR00112190040, and NSF IIS 1546413, DMS 1854831. LHL has also received support from the Eckhardt Faculty Fund. B Jose Israel Rodriguez [email protected] Jin-Hong Du [email protected] Yiling You [email protected] Lek-Heng Lim [email protected] 1 Department of Mathematics, University of Wisconsin, Madison, WI, USA 2 Department of Statistics, University of Chicago, Chicago, IL, USA 3 Department of Mathematics,uncorrected University of California, Berkeley, CA, USA proof 4 Computational and Applied Mathematics Initiative, University of Chicago, Chicago, IL, USA 123 SPI Journal: 211 Article No.: 1215 TYPESET DISK LE CP Disp.:2021/7/29 Pages: 36 Layout: Small-Ex J. I. Rodriguez et al. 1 16 on larger problems, and is also highly parallelizable. We show that the fiber product 17 MEP that we solve in the fiber product homotopy method, although mathematically 2 18 equivalent to a standard MEP, is typically a much better conditioned problem. 19 Mathematics Subject Classification 65H20 65H17 65H10 35P30 · · · 20 1 Introduction Author Proof 21 A multiparameter eigenvalue problem (MEP) is, in an appropriate sense, a system of 22 linear equations 23 a11x1 a12x2 a1k xk b1, + +···+ = 24 a21x1 a22x2 a2k xk b2, + +···+ = . 25 . 26 ak1x1 ak2x2 akkxk bk, (1) + +···+ = 27 where the coefficients aij’s and bi ’s are matrices, and where equality is interpreted 28 to mean on a point in a product of projective spaces (this will be made precise later). 29 These coefficients are square matrices but are of different dimensions in general, so 30 one may not usually regard (1) as a linear system over a matrix ring. There is a rich 31 mathematical theory behind MEP [2,3] that places it at the crossroad of linear and 32 multilinear algebra, ordinary and partial differential equations, spectral theory and 33 Sturm–Liouville theory, among other areas. The problem appeared as early as 1836 in 34 the works of Sturm and Liouville on periodic heat flow in a bar, and was studied over 35 the years by many: Klein, Lamé, Heine, Stieltjes, Pell, Carmichael, Bocher, Hilbert 36 among them (see [2, Preface] and [3, Chapter 1]). 37 An MEP encompasses many known types of eigenvalue problems: Standard eigen- 38 value problems Ax λx; generalized eigenvalue problems Ax λBx; quadratic = 2 = 39 eigenvalue problems (λ A λB C)x 0; polynomial eigenvalue problems m m 1 + + = 40 (λ Am λ − Am 1 A0)x 0; quadratic two-parameter eigenvalue problems + − +···+ = 2 2 41 (A00 λA10 µA01 λ A20 λµA11 µ A02)x1 0, + + + + + = 2 2 42 (B00 λB10 µB01 λ B20 λµB11 µ B02)x2 0 + + + + + = ; 43 may all be reduced to mathematically equivalent MEPs. 44 Nevertheless MEP remains in the blind spot of most modern mathematicians, 45 whether pure or applied. This is not for its lack of applications; as we pointed out, 46 the problem in fact originated from a study of heat flow, and we will see yet other 47 applications of MEP in Sect. 7 and that it contains eigenvalue problem and linear 48 system, both ubiquitous in science and engineering, as special cases. We think that a 49 main reason for the obscurityuncorrected of MEPs is that there are not many effective proof methods 50 for its computation and there is thus little to be gained from formulating a problem 123 SPI Journal: 211 Article No.: 1215 TYPESET DISK LE CP Disp.:2021/7/29 Pages: 36 Layout: Small-Ex Fiber product homotopy method... 51 as an MEP. It is with this in mind that we propose a new homotopy method based on 52 what we call fiber product homotopy for computing MEP solutions. 53 We will now formally define an MEP in more conventional notations. Instead of 54 having a single eigenvalue parameter λ, an MEP has multiple eigenvalue parameters 55 λ (λ1,...,λk). We will call = 56 H(λ) A0 λ1 A1 λk Ak, := − −···− 57 a linear polynomial matrix in k parameters λ1,...,λk with matrix coefficients Author Proof n n n 58 A0,...,Ak C . We will write P for the complex projective n-space. ∈ × n n 59 Definition 1 For a fixed k 2 and given matrices Aij C i i with j 0, 1,...,k, ≥ ∈ × = 60 i 1,...,k, consider the linear polynomial matrices = 61 Hi (λ) Ai0 λ1 Ai1 λ2 Ai2 λk Aik. := − − −···− 62 The multiparameter eigenvalue problem (MEP), or, more precisely, a k-parameter 63 eigenvalue problem, is to find λ1,...,λk C and corresponding (x1,...,xk) n 1 n 1 ∈ ∈ 64 P 1 P k such that − ×···× − 65 Hi (λ)xi 0, i 1,...,k. (2) = = k n1 1 nk 1 66 A solution (λ1,...,λk, x1,...,xk) C P − P − to the MEP is called ∈ ×n1 1 ×···×nk 1 67 an eigenpair,thek-tuple (x1,...,xk) P − P − an eigenvector, and the k ∈ ×···× 68 k-tuple λ (λ1,...,λk) C an eigenvalue. = ∈ 69 Written out in full, (2) takes the form 70 (λ1 A11 λ2 A12 λk A1k)x1 A10 x1, + +···+ = 71 (λ1 A21 λ2 A22 λk A2k)x2 A20 x2, + +···+ = . 72 . 73 (λ1 Ak1 λ2 Ak2 λk Akk)xk Ak0 xk. (3) + +···+ = 74 With λi ’s playing the role of xi ’s, Aij’s and Ai0’s playing the roles of aij’s and bi ’s 75 respectively in (1), and interpreting equality of the ith equation in (1) to mean equality n 1 76 on some xi P i , we may view (3) as an analogue of a linear system that we ∈ − 77 referred to at the beginning. The analogy is precise when n1 nk 1—(3)is =···= = 78 a linear system in the usual sense. 79 When k 1, (3) is a generalized eigenvalue problem. More generally, if Aij 0 = = 80 for all i j and j 0, then (3) is decoupled into k generalized eigenvalue problems. = = 81 Hence (3) contains both eigenvalue problems and linear systems as special cases. The 82 multiparameter eigenvalue problem is well studied and readers may refer to the books 83 [2,3,23] for a comprehensive treatment. 84 Since any scalar multiple of xi is also an eigenvector it is fitting to consider xi as an 85 Pni 1 element of the projective spaceuncorrected− although for practical reason oneproof might prefer 86 to simply normalize xi to have unit norm. 123 SPI Journal: 211 Article No.: 1215 TYPESET DISK LE CP Disp.:2021/7/29 Pages: 36 Layout: Small-Ex J. I. Rodriguez et al. 87 We develop a new homotopy method to solve a multiparameter eigenvalue problem 88 effectively, where effectiveness is measured by the following factors: 89 – Speed as measured by wall time. We record time per path, maximum time over 90 all paths, and total track time of all paths. Our algorithm is highly parallelizable 91 and the per-path times give good speed estimates when there are enough cores to 92 track all paths in parallel. 93 – Accuracy as measured by the backward error. We use the normwise backward error 94 in [14, Theorem 2] for an approximate eigenpair. Our homotopy method tracks Author Proof 95 several copies of the eigenvalue λ; they should all converge to the same value if 96 our method performs correctly and we include the difference between copies of 97 λ’s as another measure of accuracy. 98 – Certificates of quadratic convergence in terms of Shub–Smale α-theory. 99 – Number of divergent paths that fail to converge to the solutions. 100 The last two measures only apply to methods based on homotopy continuation. We 101 will compare our method to two existing methods: 102 (i) The Delta method [2], which is the de facto standard method for solving MEPs 103 by transforming them into a coupled system of generalized eigenvalue problems; 104 we use the MultiParEig package [17] in our experiments with this method. For 105 singular MEPs, we perform Delta method after extracting the common regular 106 parts of the Delta matrices with a staircase algorithm [17]. 107 (ii) The diagonal coefficient homotopy method recently proposed in [8] for solving 108 MEPs, where the start system is a random choice of diagonal matrices and the 109 homotopy is a straight-line homotopy that deforms n1 nk of n1 nk k +···+ +···+ + 110 equations.