Correction to: Inexact rational Krylov method for evolution equations

1 Correction to: BIT Numerical Mathematics (2021) 61:473–502 https://doi.org/10.1007/s10543-020-00829-w

1.1 Correction of the statements and proofs of Theorem 3.1 and Proposition 3.1

The proof of Proposition 3.1 is incorrect. First, the unitary matrix $Q_m$ in the proof does not transform ${\tilde{K}}_m$ into an upper Hessenberg matrix. For $j\ge 2$, the definition of $u_j$ has to be replaced by $u_j:=({\hat{k}}_{j+1:m,j}^j-\eta _je_1)/\Vert {\hat{k}}_{j+1:m,j}^j-\eta _je_1\Vert$, where ${\hat{k}}_{j+1:m,j}^j$ is the vector consisting of the elements in rows $j+1\sim m$ and column j of the matrix ${\hat{K}}_{m,j}:=(I_m+{\hat{Q}}_{j})\cdots (I_m+{\hat{Q}}_2){\tilde{K}}_m(I_m+{\hat{Q}}_{2})\cdots (I_m+{\hat{Q}}_j)$ and $\eta _j:=-{\text {sign}}({\hat{k}}_{j+1,j})\Vert {\hat{k}}_{j+1:m,j}\Vert$. In addition, the sum $\sum _{k=3}^{{\text {min}}\{i,j\}}$ in the formula after the equation (A.2) has to be replaced by $\sum _{k=3}^i$. Thus, the upper bound of $|(Q_m-I_m)_{i,j}|$ for $i>j$ is not the same as that for $i\le j$. Although we can correct the upper bound of $|(Q_m-I_m)_{i,j}|$ by deriving a similar upper bound for $i>j$ as $i\le j$, the error regarding the transformation into an upper Hessenberg matrix is crucial for deriving Proposition 3.1. However, we can derive the same conclusion as Theorem 3.1 by modifying the assumptions of Theorem 3.1 and Proposition 3.1. To show the modified version of Proposition 3.1, we do not need Householder reflectors. Instead, we use Cauchy’s integral formula to show it. Correct statements and proofs of Theorem 3.1 and Proposition 3.1 are as follows. Here, in addition, we reorganize the statements of Theorem 3.1 and Proposition 3.1 to clarify the dependence of each variable.

We define ${\tilde{H}}_m:=T_m-H_mD_m+\gamma _mH_m$ and $L_m(z):=(zH_m-{\tilde{H}}_m)^{-1}$ for $z\in {\mathbb {C}}$. Note that $zH_m-{\tilde{H}}_m$ is an upper Hessenberg matrix and thus, $L_m(z)$ is the inverse of an upper Hessenberg matrix. To show the theorem, we require the following assumption:

Assumption 3.1

Let ${\hat{\alpha }}>0$ and $0<{\lambda }<1$ be given constants and let $\Pi \subseteq {\mathbb {C}}$ be a given bounded open set. We assume for any $z\in \partial \Pi$

$$\begin{aligned}&|(L_m(z))_{i,j}|\le {\hat{\alpha }}{\lambda }^{i-j}\quad (i\ge j),\nonumber \\&\varLambda (K_m^{-1})\subseteq {\Pi }, \end{aligned}$$

where $\varLambda (K_m^{-1})$ is the spectrum of the matrix $K_m^{-1}$.

Theorem 3.1

Let ${\hat{\alpha }}>0$, $0<{\lambda }<1$, and $\delta >0$ be constants and let $\Pi \subseteq {\mathbb {C}}^+$ be a bounded open set whose boundary is a rectifiable Jordan curve oriented in positive sense. If the matrices $L_m(z)$ and $K_m$ satisfy Assumption 3.1 with ${\hat{\alpha }}$, $\lambda$, and $\Pi$, and if the residual of solving the linear equation $(\gamma _m I-A)x_m=V_mt_m$ satisfies $\Vert r_m^{{\text {sys}}}\Vert \le \delta$, then the first term of the Eq. (3.4) is bounded as

$$\begin{aligned}&\beta \left| h_{m+1,m}e_m^*\phi _k(D_m-H_m^{-1}T_m)H_m^{-1}e_1\right| \, \Vert (\gamma _m I-A)v_{m+1}\Vert \nonumber \\&\qquad \le {\beta }(1+\delta )\kappa (\gamma _m I-A) \alpha \lambda ^{m-1}, \end{aligned}$$

(3.5)

where $\alpha ={2\pi }^{-1}|\partial \Pi |\phi _k(N){\hat{\alpha }}$ and $|\partial \Pi |=\int _{\partial \Pi }|{d}z|$.

Moreover, for any tolerance ${\text {tol}}_{\phi }>0$ for approximating the vector $\phi _k(A)v$ and for any $m^{{\text {max}}}>0$, if $m\le m^{{\text {max}}}$ and if

$$\begin{aligned} \Vert r^{{\text {sys}}}_1\Vert&\le \frac{{\text {tol}}_{\phi }}{m^{{\text {max}}} \beta \Vert \phi _k(D_m-H_m^{-1}T_m)H_m^{-1}e_1\Vert }, \end{aligned}$$

(3.6)

$$\begin{aligned} \Vert r^{{\text {sys}}}_j\Vert&\le \frac{|g^m_{1,1}|\lambda }{|g^m_{j,1}|} \Vert r^{{\text {sys}}}_1\Vert \quad (j=2,\ldots ,m), \end{aligned}$$

(3.7)

then the second term of Eq. (3.4) can be evaluated as

$$\begin{aligned} \beta \Vert R^{{\text {sys}}}_m\phi _k(D_m-H_m^{-1}T_m)H_m^{-1}e_1 \Vert \le {\text {tol}}_{\phi }, \end{aligned}$$

(3.8)

where $g^m_{i,j}=(\phi _k(D_m-H_m^{-1}T_m)H_m^{-1})_{i,j}$.

Proposition 3.1

Let ${\hat{\alpha }}>0$, $0<{\lambda }<1$ be constants and let ${\Pi }\subseteq {\mathbb {C}}^{+}$ be a bounded open set whose boundary is a rectifiable Jordan curve oriented in positive sense. If the matrices $L_m(z)$ and $K_m$ satisfy Assumption 3.1 with ${\hat{\alpha }}$, $\lambda$, and $\Pi$, then we have

$$\begin{aligned} |{\big (\phi _k(D_m-H_m^{-1}T_m)H_m^{-1}\big )}_{i,j}|\le \frac{1}{2\pi }|\partial \Pi |\phi _k(N){\hat{\alpha }}{\lambda }^{i-j}\quad (i\ge j). \end{aligned}$$

(3.14)

Proof

Since $\phi _k$ is an entire function, by Cauchy’s integral formula, we have

$$\begin{aligned}&\phi _k(D_m-H_m^{-1}T_m)H_m^{-1} =H_m^{-1}\phi _k(\gamma _mI-K_m^{-1}) =\frac{1}{2\pi \mathrm {i}}H_m^{-1}\int _{\partial \Pi }\phi _k(\gamma _m-z)(zI-K_m^{-1})^{-1}\,dz\\&\qquad =\frac{1}{2\pi \mathrm {i}}H_m^{-1}\int _{\partial \Pi }\phi _k(\gamma _m-z)H_m(zH_m-{\tilde{H}}_m)^{-1}\,dz =\frac{1}{2\pi \mathrm {i}}\int _{\partial \Pi }\phi _k(\gamma _m-z)L_m(z)\,dz. \end{aligned}$$

Moreover, for $i\ge j$, we have

$$\begin{aligned}&\bigg |\bigg (\frac{1}{2\pi \mathrm {i}}\int _{\partial \Pi }\phi _k(\gamma _m-z)L_m(z)\,dz\bigg )_{i,j}\bigg | \le \frac{1}{2\pi }\int _{\partial \Pi }|\phi _k(\gamma _m-z)|\,|dz|{\hat{\alpha }}{\lambda }^{i-j}\\&\qquad \le \frac{1}{2\pi }|\partial \Pi |\max _{z\in \partial \Pi }\phi _k(\gamma _m-{\text {Re}} (z)){\hat{\alpha }}{\lambda }^{i-j} \le \frac{1}{2\pi }|\partial \Pi |\phi _k(N){\hat{\alpha }}{\lambda }^{i-j}, \end{aligned}$$

where ${\text {Re}}(z)$ is the real part of z. The second inequality holds since $\phi _k$ is represented as $\phi _k(z)=\int _0^1e^{(1-s)z}\frac{s^{k-1}}{(k-1)!}ds$ and the last inequality holds since $N\ge \gamma _m$ for any $m<N/h$. This completes the proof of Proposition 3.1. $\square$

The modified version of Theorem 3.1 is derived by Eq. (3.14) in the same manner as the proof in the original article.

Remark 3.2

If $zH_m-{\tilde{H}}_m$ is diagonalizable and invertible, there exist constants ${\hat{\alpha }}_m(z)>0$ and $0<\lambda _m(z)<1$ such that $|(L_m(z))_{i,j}|\le {\hat{\alpha }}_m(z)\lambda _m(z)^{i-j}$. The first assumption about $L_m(z)$ in Assumption 3.1 is about the uniformity of ${\hat{\alpha }}_m(z)$ and $\lambda _m(z)$. Indeed, let $P_m(z)\Delta _m(z)P_m(z)^{-1}$ be an eigenvalue decomposition of $zH_m-{\tilde{H}}_m$ and let $\Sigma _m(z)\subseteq {\mathbb {C}}\setminus \{0\}$ be a bounded open set whose boundary is a rectifiable Jordan curve oriented in positive sense such that $\Lambda (zH_m-{\tilde{H}}_m)\subseteq \Sigma _m(z)$. Since $zH_m-{\tilde{H}}_m$ is an upper Hessenberg matrix, for $i> j$, any polynomial $p\in {\mathcal {P}}_{i-j-1}$ satisfies

$$\begin{aligned} |(L_m(z))_{i,j}|&=|(L_m(z))_{i,j}-(p(zH_m-{\tilde{H}}_m))_{i,j}| \le \Vert L_m(z)-p(zH_m-{\tilde{H}}_m)\Vert \\&\le \Vert P_m(z)\Vert \sup _{w\in \Sigma _m(z)}|w^{-1}-p(w)|\Vert P_m(z)^{-1}\Vert . \end{aligned}$$

Let $f(w):=w^{-1}$. We set the polynomial p as the truncated Faber series of f [1, Section 2]. Then by Corollary 2.2 in Ellacott [1], there exist constants $C_m(z)>0$ and $0<\lambda _m(z)<1$ such that

$$\begin{aligned} \sup _{w\in \Sigma _m(z)}|w^{-1}-p(w)|\le C_m(z)\lambda _m(z)^{i-j}. \end{aligned}$$

Thus, we have $|(L_m(z))_{i,j}|\le \kappa (P_m(z)){C_m(z)}\lambda _m(z)^{i-j}$.

If $\Sigma _m(z)$ is independent of m and z, then $C_m(z)$ and $\lambda _m(z)$ are independent of m and z. Therefore, if in addition there exist a constant ${\tilde{\alpha }}>0$ such that $\kappa (P_m(z))\le {\tilde{\alpha }}$, then the first assumption about $L_m(z)$ in Assumption 3.1 is satisfied.

The second assumption about $K_m$ in Assumption 3.1 is satisfied if there exists a bounded open set $\Sigma \subseteq {\mathbb {C}}^+$ such that $W((\gamma _m I-A)^{-1})\subseteq \Sigma$ and if $f_j^{{\text {sys}}}={\mathbf {0}}$ for $j=1,\ldots ,m$, that is, the linear equations solved exactly. Indeed, by Eq. (3.2), the identity $K_m=V_m^*(\gamma _m I-A)^{-1}V_m$ holds in this case. Therefore, we have $\varLambda (K_m)\subseteq W(K_m)\subseteq W((\gamma _m I-A)^{-1})\subseteq \Sigma$.

1.2 Typos

• 1. Before Eq. (1.1), “$u(\cdot ,x)\in C(0,T)$ for all $x\in \Omega$” should read “$u\in C([0,T],L^2(\Omega ))$”.

• 2. In Eqs. (1.4) and (1.5) and the formula between them, g(y(s)), $g_l$, and $g_{i-1}$ should read $M^{-1}g(y(s))$, $M^{-1}g_l$, and $M^{-1}g_{i-1}$, respectively.

• 3. The sum $\sum _{k=1}^{r-1}$ in Eq. (1.7) should read $\sum _{k=0}^r$.

• 4. Eq. (2.5) should read $t_j=e_{\rho \lfloor (j-1)/\rho \rfloor +1}\in {\mathbb {R}}^j$.

• 5. In the middle of the proof of Theorem 3.1, $(\gamma _m I-A)^{-1}v_m-f_m^{{\text {sys}}}$ should read $(\gamma _m I-A)^{-1}V_mt_m-f_m^{{\text {sys}}}$.

• 6. Eq. (3.15) should read

  $$\begin{aligned}&\beta \Vert H_m^{-1}\phi _k((H_mD_m-T_m)H_m^{-1})e_1\Vert \approx \beta \Vert H_m^{-1}[\phi _k((H_mD_m-T_m)H_m^{-1})]_{1,1}e_1\Vert \nonumber \\&\qquad \approx \beta \Vert V_m^*(\gamma _1 I-A)V_m\phi _k((H_mD_m-T_m)H_m^{-1})e_1\Vert \nonumber \\&\qquad \approx \Vert (\gamma _1 I-A)y(t)\Vert \approx \Vert (\gamma _1 I-A)y(0)\Vert . \end{aligned}$$

  Moreover, the formula about ${\text {tol}}_1^{{\text {sys}}}$ in the last paragraph in Section 3 should read ${\text {tol}}_1^{{\text {sys}}}={\text {tol}}_{\phi }/[m^{{\text {max}}}\Vert (\gamma _1I-A)y(0)\Vert ]$.

• 7. In the last paragraph in Remark 3.1, the definition of $j_0$ should read $j_0:=\rho \lfloor (j-1)/\rho \rfloor +1$. Moreover, we need an additional assumption $|g_{j_0,1}^m|(j_0-1)/(j_0+1)\ge |g_{j,1}^m|(j-1)/(j+1)$ for deriving the last formula in Remark 3.1.

• 8. At the beginning of Example 2, $(-1.5,1.5)\times (-1,1)$ should read $\Omega =(-1.5,1.5)\times (-1,1)$.

• 9. Eq. (4.2) should read

  $$\begin{aligned} \left\{ \begin{aligned}&{\tilde{M}}\ddot{{\tilde{y}}}(t)={\tilde{L}}{\tilde{y}}(t)+{\tilde{b}}(t),\\&{\tilde{y}}(0)={\tilde{v}},\quad \dot{{\tilde{y}}}(0)={\mathbf {0}}. \end{aligned}\right. \end{aligned}$$

• 10. In Example 3, $\partial \Omega$ in the boundary condition “$u=0,\ v=0\ \text {on}\ (0,T]\times \partial \Omega$” should read $\partial \Omega _1$ and $\partial \Omega$ in “$\frac{\partial u}{\partial {\mathbf {n}}}=0,\ \frac{\partial v}{\partial {\mathbf {n}}}=0 \ \text {on}\ (0,T]\times \partial \Omega$” should read $\partial \Omega _2$, where $\partial \Omega _1=[-1.5,1.5]\times \{1,-1\}$ and $\partial \Omega _2=\partial \Omega \setminus \partial \Omega _1$.

• 11. In Example 3, the formula $g_i(y)=F(y)-L_{i-1}y=Q(y)y-Q(y_{i-1})y$ should read $g_i(y)=F(y)-L_{i}y=Q(y)y-Q(y_{i})y$ and the scheme of the exponential integrator should read

  $$\begin{aligned} y_{i+1}=y_i+\Delta t\phi _1(\Delta tM^{-1}L_i)M^{-1} F(y_i) -\Delta t\frac{2}{3}\phi _2(\Delta tM^{-1}L_i)M^{-1}(g_i(y_i)-g_i(y_{i-1})). \end{aligned}$$

  In addition, the formula $\phi _2(\Delta tM^{-1}L_i)(g_i(u_i)-g_i(u_{i-1}))$ written after the scheme should read $\phi _2(\Delta tM^{-1}L_i)M^{-1}(g_i(y_i)-g_i(y_{i-1}))$.