Inexact rational Krylov method for evolution equations

Abstract

Linear and nonlinear evolution equations have been formulated to address problems in various fields of science and technology. Recently, methods using an exponential integrator for solving evolution equations, where matrix functions must be computed repeatedly, have been investigated and refined. In this paper, we propose a new method for computing these matrix functions which is called an inexact rational Krylov method. This is a more efficient version of the rational Krylov method with appropriate shifts, which was proposed by Hashimoto and Nodera (ANZIAM J 58:C149–C161, 2016). The advantage of the inexact rational Krylov method is that it computes linear equations that appear in the rational Krylov method efficiently while guaranteeing the accuracy of the final solution.

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References

  1. 1.

    Beckermann, B., Reichel, L.: Error estimates and evaluation of matrix functions via the Faber transform. SIAM J. Numer. Anal. 47(5), 3849–3883 (2009)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Benzi, M., Boito, P., Razouk, N.: Decay properties of spectral projectors with applications to electronic structure. SIAM Rev. 55(1), 3–64 (2013)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Berljafa, M., Güttel, S.: Parallelization of the rational Arnoldi algorithm. SIAM J. Sci. Comput. 39(5), S197–S221 (2017)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Börner, R., Ernst, O.G., Güttel, S.: Three-dimensional transient electromagnetic modelling using rational Krylov methods. Geophys. J. Int. 202(3), 2025–2043 (2015)

    Article  Google Scholar 

  5. 5.

    Botchev, M.A.: Krylov subspace exponential time domain solution of Maxwell’s equations in photonic crystal modeling. J. Comput. Appl. Math. 293, 20–34 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bouras, A., Frayssé, V., Giraud, L.: A relaxation strategy for inner-outer linear solvers in domain decomposition methods. Technical report TR/PA/00/17 (2000)

  7. 7.

    Crouzeix, M., Palencia, C.: The numerical range is a \((1+\sqrt{2})\)-spectral set. SIAM J. Matrix Anal. Appl. 38(2), 649–655 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dinh, K., Sidje, R.: An application of the Krylov-FSP-SSA method to parameter fitting with maximum likelihood. Phys. Biol. 14(6), 065001 (2017)

    Article  Google Scholar 

  9. 9.

    Druskin, V., Lieberman, C., Zaslavsky, M.: On adaptive choice of shifts in rational Krylov subspace reduction of evolutionary problems. SIAM J. Sci. Comput. 32(5), 2485–2496 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Elsworth, S., Güttel, S.: The block rational Arnoldi method. SIAM J. Matrix Anal. Appl. 41(2), 365–388 (2020)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Gallopoulos, E., Saad, Y.: Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Stat. Comput. 13(5), 1236–1264 (1992)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Göckler, T.: Rational Krylov Subspace Methods for ’\(\phi \)’-functions in Exponential Integrators. Karlsruher Instituts für Technologie, Karlsruhe (2014)

    Google Scholar 

  13. 13.

    Giraud, L., Langou, J., Rozložník, M., van den Eshof, J.: Rounding error analysis of the classical Gram-Schmidt orthogonalization process. Numer. Math. 101(1), 87–100 (2005)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Golub, G.H., Zhang, Z., Zha, H.: Large sparse symmetric eigenvalue problems with homogeneous linear constraints: the Lanczos process with inner-outer iterations. Linear Algebra Its Appl. 309(1), 289–306 (2000)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Grimm, V.: Resolvent Krylov subspace approximation to operator functions. BIT Numer. Math. 52, 639–659 (2012)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Güttel, S.: Rational Krylov Methods for Operator Functions. Technischen Universität Bergakademie Freiberg, Freiberg (2010)

    Google Scholar 

  17. 17.

    Güttel, S.: Rational Krylov approximation of matrix functions: numerical methods and optimal pole selection. GAMM-Mitteilungen 36(1), 8–31 (2013)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hashimoto, Y., Nodera, T.: Inexact shift-invert Arnoldi method for evolution equations. ANZIAM J. 58, E1–E27 (2016)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Hashimoto, Y., Nodera, T.: Shift-invert rational Krylov method for evolution equations. ANZIAM J. 58, C149–C161 (2016)

    Article  Google Scholar 

  20. 20.

    Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26(4), 1179–1193 (2005)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1998)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19(5), 1552–1574 (1997)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Hochbruck, M., Ostermann, A.: Exponential Runge–Kutta methods for parabolic problems. Appl. Numer. Math. 53(2–4), 323–339 (2005)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    MathSciNet  Google Scholar 

  25. 25.

    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, New York (1991)

    Google Scholar 

  26. 26.

    Lopez, L., Simoncini, V.: Analysis of projection methods for rational function approximation to the matrix exponential. SIAM J. Numer. Anal. 44(2), 613–635 (2006)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Moret, I.: On RD-rational Krylov approximations to the core-functions of exponential integrators. Numer. Linear Algebra Appl. 14(5), 445–457 (2007)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Moret, I., Novati, P.: RD-rational approximations of the matrix exponential. BIT Numer. Math. 44(3), 595–615 (2004)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Moret, I., Novati, P.: On the convergence of Krylov subspace methods for matrix Mittag-Leffler functions. SIAM J. Numer. Anal. 49(5), 2144–2164 (2011)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Moret, I., Popolizio, M.: The restarted shift-and-invert Krylov method for matrix functions. Numer. Linear Algebra Appl. 21(1), 68–80 (2014)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Novati, P.: Using the restricted-denominator rational Arnoldi method for exponential integrators. SIAM J. Matrix Anal. Appl. 32(4), 1537–1558 (2011)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Popolizio, M., Simoncini, V.: Acceleration techniques for approximating the matrix exponential operator. SIAM J. Matrix Anal. Appl. 30(2), 657–683 (2008)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Qiu, C., Güttel, S., Ren, X., Yin, C., Liu, Y., Zhang, B., Egbert, G.: A block rational Krylov method for 3-D time-domain marine controlled-source electromagnetic modelling. Geophys. J. Int. 218(1), 100–113 (2019)

    Article  Google Scholar 

  35. 35.

    Ruhe, A.: Rational Krylov: a practical algorithm for large sparse nonsymmetric matrix pencils. SIAM J. Sci. Comput. 19(5), 1535–1551 (1998)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Saff, E.B., Schönhage, A., Varga, R.S.: Geometric convergence to \(e^{-z}\) by rational functions with real poles. Numer. Math. 25(3), 307–322 (1975)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Sidje, R.B., Winkles, N.: Evaluation of the performance of inexact GMRES. J. Comput. Appl. Math. 235(8), 1956–1975 (2011)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Simoncini, V., Szyld, D.B.: Theory of inexact Krylov subspace methods and applications to scientific computing. SIAM J. Sci. Comput. 25(2), 454–477 (2003)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Skoogh, D.: A parallel rational Krylov algorithm for eigenvalue computations. In: Applied Parallel Computing Large Scale Scientific and Industrial Problems, pp. 521–526 (1998)

  41. 41.

    Svoboda, Z.: The convective-diffusion equation and its use in building physics. Int. J. Archit. Sci. 1(2), 68–79 (2000)

    Google Scholar 

  42. 42.

    Van den Eshof, J., Sleijpen, G.L.G.: Inexact Krylov subspace methods for linear systems. SIAM J. Matrix Anal. Appl. 26(1), 125–153 (2004)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Van den Eshof, J., Hochbruck, M.: Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Wu, G., Feng, T., Wei, Y.: An inexact shift-and-invert Arnoldi algorithm for Toeplitz matrix exponential. Numer. Linear Algebra Appl. 22(4), 777–792 (2015)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Wu, G., Feng, T., Zhang, L., Yang, M.: Inexact implementation using Krylov subspace methods for large scale exponential discriminant analysis with applications to high dimensionality reduction problems. Pattern Recognit. 66, 328–341 (2017)

    Article  Google Scholar 

  47. 47.

    Zhang, D.S., Wei, G.W., Kouri, D.J., Hoffman, D.K.: Burgers’ equation with high Reynolds number. Phys. Fluids 9(6), 1853–1855 (1997)

    MathSciNet  Article  Google Scholar 

  48. 48.

    Zhu, H., Shu, H., Ding, M.: Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method. Comput. Math. Appl. 60(3), 840–848 (2010)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

We would like to thank the anonymous referee and associate editor Michiel Hochstenbach, whose suggestions greatly improved the paper.

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Correspondence to Yuka Hashimoto.

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Communicated by Michiel E. Hochstenbach.

Appendices

A Proof of Proposition 3.1

We use the Householder reflectors for transforming the matrix \(\tilde{K}_m\) into an upper Hessenberg matrix. Let \(u_j:=(\tilde{k}_{j+1:m,j}-\eta _j e_1)/\Vert \tilde{k}_{j+1:m,j}-\eta _j e_1\Vert \) and let \(\tilde{Q}_{j+1}:=-2u_ju_j^*\). Then, \(I_{m-j}+\tilde{Q}_{j+1}\) is a unitary matrix and satisfies \((I_{m-j}+\tilde{Q}_{j+1})k_{j+1:m,j}=\eta _je_1\). Thus, the matrix \(Q_m\) which is defined as \(Q_m:=(I_m+\hat{Q}_{m-1})\cdots (I_m+\hat{Q}_2)\) is a unitary matrix, and there exists an upper Hessenberg matrix \(H_m\) such that \(Q_m\tilde{K}_mQ_m^*=H_m\). Here, \(\hat{Q}_{j+1}:={\text {diag}}\{O_j,\ \tilde{Q}_{j+1}\}\) and \(O_j\) is the \(j\times j\) zero matrix.

Our first goal is to evaluate the magnitude of each element of the matrix \(\tilde{Q}_{j+1}\). By the assumption (3.11), the inequality \(\Vert \tilde{k}_{j+1:m,j}-\eta _j e_1\Vert \ge \eta \) holds. In addition, since the vector \(\tilde{k}_{j+1:m,j}\) satisfies the condition (3.10), we have

$$\begin{aligned} \vert \eta _j\vert =\Vert \tilde{k}_{j+1:m,j}\Vert \le \sqrt{\sum _{k=1}^{\infty }(\hat{\alpha }\hat{\lambda }^k)^2}= \hat{\alpha }\sqrt{1/(1-\hat{\lambda }^2)}\hat{\lambda }. \end{aligned}$$

The first element of the vector \(\tilde{k}_{j+1:m,j}-\eta _je_1\) is equal to \(\tilde{k}_{j+1,j}-\eta _j\), and since the identity \(\vert \eta _j\vert =\Vert \tilde{k}_{j+1:m,j}\Vert \) holds, we obtain \(\vert \tilde{k}_{j+1,j}-\eta _j\vert \le 2\vert \eta _j\vert \). All the elements except for the first element of the vector \(\tilde{k}_{j+1:m,j}-\eta _je_1\) are the same as the element \(\tilde{k}_{j+1:m,j}\). For these reasons, on the basis of the assumption (3.10), \(|u_{i,j}|<\tilde{\alpha }\hat{\lambda }^i\) is satisfied for

$$\begin{aligned} \tilde{\alpha }:=\max (\hat{\alpha }/\eta ,2\hat{\alpha } \sqrt{1/(1-\hat{\lambda }^2)}\hat{\lambda }/\eta ), \end{aligned}$$

where \(u_{i,j}\) is the ith element of the vector \(u_j\in \mathbb {C}^{m-j}\). Thus, we have

$$\begin{aligned} |(\tilde{Q}_{j+1})_{k,l}|=2|(u_ju_j^*)_{k,l}|=2|u_{k,j}u_{l,j}|\le 2\tilde{\alpha }^2\hat{\lambda }^{k+l}. \end{aligned}$$

Next, we evaluate the magnitude of each element of the matrix \(Q_m\). Let \(\check{\alpha }:=2\tilde{\alpha }^2\). For \(l\ge 2\) and \(l<k_1<k_2<\cdots <k_r\), we have

$$\begin{aligned} |(\hat{Q}_{k_r}\cdots \hat{Q}_{k_1}\hat{Q}_l)_{i,j}|&\le \displaystyle \frac{\check{\alpha }^{r+1}\hat{\lambda }^{-2(l-r-1)}}{(1-\hat{\lambda }^2)^r} \hat{\lambda }^{i+j}=\check{\alpha }^{r+1}\alpha ''(l,r)\hat{\lambda }^{i+j}\quad (i,j\le k_r),\nonumber \\ |(\hat{Q}_{k_r}\cdots \hat{Q}_{k_1}\hat{Q}_l)_{i,j}|&=0\quad (i>k_r\ \text {or}\ j>k_r), s \end{aligned}$$
(A.1)

where \(\alpha ''(l,r):=\hat{\lambda }^{-2(l-r-1)}/(1-\hat{\lambda }^2)^r\). The inequality (A.1) is proved by the induction on r. For \(r=1\), we have

$$\begin{aligned}&|(\hat{Q}_{k_1}\hat{Q}_l)_{i,j}| \le \sum _{a=k_1}^m\check{\alpha }\hat{\lambda }^{i-k_1+1+a-k_1+1} \check{\alpha }\hat{\lambda }^{a-l+1+j-l+1}\\&\qquad =\check{\alpha }^2\hat{\lambda }^{i+j}\hat{\lambda }^{-2(k_1+l-2)} \sum _{a=k_1}^m\hat{\lambda }^{2a}\le \frac{\check{\alpha }^2 \hat{\lambda }^{-2(l-2)}}{1-\hat{\lambda }^2}\hat{\lambda }^{i+j} \quad (i,j\le k_1). \end{aligned}$$

For \(r\ge 2\), if the inequality (A.1) is satisfied with \(r-1\), then we obtain

$$\begin{aligned}&|(\hat{Q}_{k_r}\cdots \hat{Q}_{k_1}\hat{Q}_l)_{i,j}| \le \sum _{a=k_r}^{m}\check{\alpha }\hat{\lambda }^{i-k_r+1+a-k_r+1} \frac{\check{\alpha }^r\hat{\lambda }^{-2(l-r)}}{(1-\hat{\lambda }^2)^{r-1}} \hat{\lambda }^{a+j}\\&\qquad \le \hat{\lambda }^{i+j}\frac{\check{\alpha }^{r+1} \hat{\lambda }^{-2(k_r+l-r-1)}}{(1-\hat{\lambda }^2)^{r-1}}\sum _{a=k_r}^m \hat{\lambda }^{2a} =\frac{\check{\alpha }^{r+1}\hat{\lambda }^{-2(l-r-1)}}{(1-\hat{\lambda }^2)^r}\hat{\lambda }^{i+j} \quad (i,j\le k_r), \end{aligned}$$

and the inequality (A.1) is also satisfied with r. This is the proof of the inequality (A.1). In the inequality (A.1), if \(\hat{\lambda }\le 1/\sqrt{2}\) is satisfied, then the inequality \(\alpha ''(l,r+1)\le \alpha ''(l,r)\) holds for any l. This results in \(\alpha ''(l,r)\le \alpha ''(l,1)\) for any r and l. The matrix \(Q_m\) can be represented as

$$\begin{aligned} Q_m&=(I_m+\hat{Q}_{m-1})\cdots (I_m+\hat{Q}_2)\nonumber \\&=I_m+\sum _{k=3}^{m-1}\sum _{l=2}^{k-1}\sum _{(a_1,a_2,\ldots , a_{k-l-1})\in \{0,1\}^{k-l-1}}\hat{Q}_{k}\hat{Q}_{k-1}^{a_1} \hat{Q}_{k-2}^{a_2}\cdots \hat{Q}_{l+1}^{a_{k-l-1}} \hat{Q}_l+\sum _{k=2}^{m-1}\hat{Q}_k. \end{aligned}$$
(A.2)

As a result, for \(2\le \min {\{i,j\}}\le m-1\), the Eq. (A.2) and inequality (A.1) derive

$$\begin{aligned} |(Q_m-I_m)_{i,j}|&\le \check{\alpha }^2\sum _{k=3}^{\min {\{i,j\}}}\sum _{l=2}^{k-1} \left( 1+(k-l-1)\check{\alpha }+\left( \begin{array}{c}k-l-1\\ 2\end{array} \right) \check{\alpha }^2+\cdots +\check{\alpha }^{k-l-1}\right) \\&\qquad \cdot \alpha ''(l,1)\hat{\lambda }^{i+j} +\check{\alpha }^2\sum _{k=2}^{\min {\{i,j\}}}\alpha ''(k,1)\hat{\lambda }^{i+j}\\&\le \check{\alpha }^2\sum _{k=3}^{\min {\{i,j\}}}\sum _{l=2}^{k-1} (1+\check{\alpha })^{k-l-1} \alpha ''(l,1)\hat{\lambda }^{i+j}+\check{\alpha }^2 \sum _{k=2}^{\min {\{i,j\}}}\alpha ''(k,1)\hat{\lambda }^{i+j}. \end{aligned}$$

Therefore, for \(2\le \min {\{i,j\}}\le m-1\) and \(i\le j\), under the assumptions of (3.11) and (3.12), we have

$$\begin{aligned}&|(Q_m-I_m)_{i,j}|\\&\quad \le \sum _{k=3}^i\frac{(1+\check{\alpha })^{k-1}\check{\alpha }^2 \hat{\lambda }^4}{1-\hat{\lambda }^2} \hat{\lambda }^{i+j}\frac{((1+\check{\alpha })\hat{\lambda }^2)^{-2}}{((1+\check{\alpha })\hat{\lambda }^2)^{-1}-1}\left( ((1+\check{\alpha }) \hat{\lambda }^2)^{-k+2}-1\right) \\&\quad + \frac{\check{\alpha }^2\hat{\lambda }^4}{1-\hat{\lambda }^2}\hat{\lambda }^{i+j} \frac{\hat{\lambda }^{-4}}{\hat{\lambda }^{-2}-1}((\hat{\lambda }^{-2})^{i-1}-1)\\&\quad \le \frac{(1+\check{\alpha })\check{\alpha }^2\hat{\lambda }^4((1 +\check{\alpha })\hat{\lambda }^2)^{-1}}{(1-\hat{\lambda }^2)(1-(1+\check{\alpha })\hat{\lambda }^2)} \hat{\lambda }^{i+j}\sum _{k=3}^i(\hat{\lambda }^{-2})^{k-2} +\frac{\check{\alpha }^2\hat{\lambda }^4}{(1-\hat{\lambda }^2)^2} \hat{\lambda }^{i+j}\hat{\lambda }^{-2i}\\&\quad \le \frac{\check{\alpha }^2\hat{\lambda }^2}{(1-\hat{\lambda }^2)(1-(1+\check{\alpha })\hat{\lambda }^2)} \hat{\lambda }^{i+j}\frac{\hat{\lambda }^{-2}}{\hat{\lambda }^{-2}-1}(\hat{\lambda }^{-2})^{i-2} +\frac{\check{\alpha }^2\hat{\lambda }^4}{(1-\hat{\lambda }^2)^2} \hat{\lambda }^{i+j}\hat{\lambda }^{-2i}\\&\quad =\left( \frac{\check{\alpha }^2\hat{\lambda }^2}{(1-\hat{\lambda }^2) (1-(1+\check{\alpha })\hat{\lambda }^2)}\frac{\hat{\lambda }^4}{1-\hat{\lambda }^2} +\frac{\check{\alpha }^2\hat{\lambda }^4}{(1-\hat{\lambda }^2)^2}\right) \hat{\lambda }^{j-i}\\&\quad =:\alpha '\hat{\lambda }^{j-i}, \end{aligned}$$

where the sum \(\sum _{k=3}^i\) becomes 0 for \(i=2\), and we use the assumption (3.12) to derive the second inequality. In a similar manner, it can be deduced that \(|(Q_m-I_m)_{i,j}|\le \alpha '\hat{\lambda }^{i-j}\) for \(i>j\). In the case of \(\min {\{i,j\}}=m\), the identity \(i=j=m\) holds and we obtain

$$\begin{aligned} |(Q_m-I_m)_{m,m}|\le \check{\alpha }^2\sum _{k=3}^{m-1} \sum _{l=2}^{k-1}(1+\check{\alpha })^{k-l+1}\alpha ''(l,1) \hat{\lambda }^{i+j}+\check{\alpha }^2\sum _{k=2}^{m-1} \alpha ''(k,1)\hat{\lambda }^{i+j}\le \alpha '. \end{aligned}$$

For \(\min {\{i,j\}}=1\), by the definition of the matrix \(Q_m\), we have

$$\begin{aligned} \left\{ \begin{aligned} |(Q_m)_{1,1}|&=1\\ |(Q_m)_{i,j}|&=0\quad (i\ne 1\ \text {or}\ j\ne 1). \end{aligned} \right. \end{aligned}$$

Since the matrix \(I_m\) is a diagonal matrix, redefining the constant \(\alpha '\) as the sum of 1 and the previous \(\alpha '\) completes the proof.

B Proof of Proposition 3.2

First, we apply Lemma 3.1 and Proposition 3.1 to the matrix \(\phi _k\left( D_m-H_m^{-1}T_m\right) H_m^{-1}\). Since the matrix \(H_m\) is an upper Hessenberg matrix satisfying the condition (3.13), by setting \(f(z)=z^{-1}\) and applying Lemma 3.1, there exist constants \(\hat{\alpha }>0\) and \(0<\hat{\lambda }<1\) which satisfy the following inequality:

$$\begin{aligned} |(H_m^{-1})_{i,j}|\le \hat{\alpha }\hat{\lambda }^{i-j}\quad (i\ge j). \end{aligned}$$
(B.1)

Since the matrix \(D_m\) is a diagonal matrix and the matrix \(T_m\) is defined as the Eq. (2.5), redefining the constant \(\hat{\alpha }\) as the sum of \(\Vert D_m\Vert =N-h\) and the previous \(\hat{\alpha }\) leads to

$$\begin{aligned} |(D_m-H_m^{-1}T_m)_{i,j}|\le \hat{\alpha }\hat{\lambda }^{i-j}\quad (i\ge j), \end{aligned}$$

where the constants \(\hat{\alpha }\) and \(\hat{\lambda }\) do not depend on m. Let

$$\begin{aligned} \mathscr {G}^{{\text {exp}}}(\alpha ,\lambda ):=\{A:\text {a square matrix}\mid |(A)_{i,j}| \le \alpha \lambda ^{|i-j|}\quad (\forall i,j)\}. \end{aligned}$$

By Proposition 3.1, there exists a unitary matrix \(Q_m\) and an upper Hessenberg matrix \(\tilde{H}_m\) such that \(D_m-H_m^{-1}T_m=Q_m^*\tilde{H}_mQ_m\) and \(Q_m\in \mathscr {G}^{{\text {exp}}}(\alpha ',\hat{\lambda })\), where \(\alpha '>0\) is a constant which does not depend on m. Thus, it is deduced that there exists a matrix \(\hat{H}_m\in \mathscr {G}^{{\text {exp}}}(\hat{\alpha },\hat{\lambda })\) such that

$$\begin{aligned} \phi _k\left( D_m-H_m^{-1}T_m\right) H_m^{-1}e_1&=\phi _k(Q_m^*\tilde{H}_mQ_m)H_m^{-1}e_1 =Q_m^*\phi _k(\tilde{H}_m)Q_m\hat{H}_me_1. \end{aligned}$$

The second equality holds, since by the inequality (B.1), there exists a matrix \(\hat{H}_m\!\in \mathscr {G}^{{\text {exp}}}(\hat{\alpha },\hat{\lambda })\) which satisfies \(H_m^{-1}e_1=\hat{H}_me_1\).

Our task is now to derive an upper bound of the magnitude of each element of the matrix \(Q_m^*\phi _k(\tilde{H}_m)Q_m\hat{H}_m\), which is composed of the products of the matrices that have the decay properties. According to Benzi and Boito [2, Theorem 9.2], there exist constants \(\alpha ''>0\) and \(\lambda ''\) which do not depend on m and satisfy \(Q_m\hat{H}_m\in \mathscr {G}^{{\text {exp}}}(\alpha '',\lambda '')\). In addition, because the function \(\phi _k\) is an entire function, by setting \(f=\phi _k\) in Proposition 3.1, there exist constants \(\check{\alpha }>0\) and \(0<\check{\lambda }<1\) satisfying \(\vert (\phi _k(\tilde{H}_m))_{i,j}\vert \le \check{\alpha }\check{\lambda }^{i-j}\) for \(i\ge j\). Let \(\varSigma :=\bigcup _{m=1}^n W(D_m-H_m^{-1}T_m)\). Then, the set \(\varSigma \) is closed and bounded, and the matrix \(\tilde{H}_m\) satisfies

$$\begin{aligned} W(\tilde{H}_m)=W\left( Q_m(D_m-H_m^{-1}T_m)Q_m^*\right) =W(D_m-H_m^{-1}T_m) \subseteq \varSigma . \end{aligned}$$
(B.2)

Therefore, \(|e^z|\le C'\) for a constant \(C'>0\) is satisfied for \(z\in W(\tilde{H}_m)\), and we have

$$\begin{aligned} |\phi _k(z)|=\left| \int _0^1e^{sz}\frac{(1-s)^{k-1}}{(k-1)!}ds \right| \le |e^z|\left| \int _0^1\frac{(1-s)^{k-1}}{(k-1)!}ds \right| \le \frac{C'}{k!}\quad (z\in W(\tilde{H}_m)). \end{aligned}$$
(B.3)

Using the result by Crouzeix [7], the condition (B.2), and the inequality (B.3), there exists a constant \(2\le C\le 1+\sqrt{2}\) such that

$$\begin{aligned} \Vert \phi _k(\tilde{H}_m)\Vert \le C\sup _{z\in W(\tilde{H}_m)}|\phi _k(z)|\le \frac{CC'}{k!}. \end{aligned}$$

Redefining the constant \(\check{\alpha }\) as the sum of the constant \(CC'/(k!)\) and the previous \(\check{\alpha }\) results in

$$\begin{aligned}&\left| \left( \phi _k(\tilde{H}_m)\right) _{i,j}\right| \le \check{\alpha }\check{\lambda }^{i-j}\quad (i\ge j), \end{aligned}$$
(B.4)
$$\begin{aligned}&\left| \left( \phi _k(\tilde{H}_m)\right) _{i,j}\right| \le \Vert \phi _k(\tilde{H}_m)\Vert \le \check{\alpha }\quad (i<j). \end{aligned}$$
(B.5)

By the upper bounds (B.4) and (B.5), we obtain

$$\begin{aligned}&\left| \left( \phi _k(\tilde{H}_m)Q_m\hat{H}_m\right) _{i,1}\right| \le \sum _{k=1}^{i}\check{\alpha }\check{\lambda }^{i-k}\alpha ''\lambda ''^{k-1} +\sum _{k=i+1}^m\check{\alpha }\alpha '' \lambda ''^{k-1}\nonumber \\&\qquad \le i\check{\alpha }\alpha ''\bar{\lambda }^{i-1}+\check{\alpha }\alpha '' \frac{\lambda ''^i}{1-\lambda ''}\le i\check{\alpha }\alpha ''\left( 1+\frac{\lambda ''}{1-\lambda ''}\right) \bar{\lambda }^{i-1} =i\bar{\alpha }\bar{\lambda }^{i-1}, \end{aligned}$$
(B.6)

where \(\bar{\alpha }:=\check{\alpha }\alpha ''/(1-\lambda '')\) and \(\bar{\lambda }:=\max \{\check{\lambda },\lambda ''\}<1\). As a result, using the fact \(Q_m\in \mathscr {G}^{{\text {exp}}}(\alpha ',\hat{\lambda })\) and the upper bound (B.6), we have

$$\begin{aligned}&\left| \left( \phi _k(D_m-H_m^{-1}T_m)H_m^{-1}\right) _{i,1}\right| =\left| \left( Q_m^*\phi _k(\tilde{H}_m)Q_m\hat{H}_m\right) _{i,1}\right| \\&\quad \le \sum _{k=1}^{i}\alpha '\hat{\lambda }^{i-k}k\bar{\alpha }\bar{\lambda }^{k-1} +\sum _{k=i+1}^{m}\alpha '\hat{\lambda }^{k-i}k\bar{\alpha }\bar{\lambda }^{k-1}\\&\quad \le \frac{1}{2}(i+1)i\alpha '\bar{\alpha }\lambda ^{i-1}+\alpha '\bar{\alpha } \frac{i+1}{(1-\lambda ^2)^2}\lambda ^{i+1}\\&\quad \le \frac{1}{2}(i+1)i\alpha '\bar{\alpha }\left( 1+ \frac{2\lambda ^2}{(1-\lambda ^2)^2}\right) \lambda ^{i-1} =\frac{1}{2}(i+1)i\alpha \lambda ^{i-1}, \end{aligned}$$

where \(\alpha :=\alpha '\bar{\alpha }(1+2\lambda ^2/(1- \lambda ^2)^2)\) and \(\lambda :=\max \{\hat{\lambda },\bar{\lambda }\}<1\). This completes the proof of Proposition 3.2.

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Hashimoto, Y., Nodera, T. Inexact rational Krylov method for evolution equations. Bit Numer Math (2021). https://doi.org/10.1007/s10543-020-00829-w

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Keywords

  • Rational Krylov method
  • Inexact computation
  • Decay property
  • \(\phi \)-Function
  • Exponential integrator

Mathematics Subject Classification

  • 65F60
  • 65M22