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The joint bidiagonalization process with partial reorthogonalization


The joint bidiagonalization (JBD) process is a useful algorithm for the computation of the generalized singular value decomposition (GSVD) of a matrix pair. However, it always suffers from rounding errors, which causes the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, we present a semiorthogonalization strategy. Our rounding error analysis shows that the JBD process with the semiorthogonalization strategy can ensure that the convergence of the computed quantities is not affected by rounding errors and the final accuracy is high enough. Based on the semiorthogonalization strategy, we develop the joint bidiagonalization process with partial reorthogonalization (JBDPRO). In the JBDPRO algorithm, reorthogonalizations occur only when necessary, which saves a big amount of reorthogonalization work, compared with the full reorthogonalization strategy. Numerical experiments illustrate our theory and algorithm.

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    Here, we use the result of an exercise from Higham’s book [9, Chapter 6, Problems 6.14], which gives the upper bound of the p-norm of a row/column sparse matrix.


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This work was supported in part by the National Science Foundation of China (No. 11771249).

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Correspondence to Zhongxiao Jia.

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Appendix: 1: Proofs of Lemma 3.1 and Lemma 3.2

Appendix: 1: Proofs of Lemma 3.1 and Lemma 3.2

Proof Proof of Lemma 3.1

We prove (3.11) by mathematical induction. For i = 1, from (3.9) and (3.10), we have

$$ \begin{array}{@{}rcl@{}} \hat{\alpha}_{1}{Q_{L}^{T}}\hat{u}_{1} & =& {Q_{L}^{T}}Q_{L}\hat{\nu}_{1} -{Q_{L}^{T}}\hat{f}_{1} \\ & =& (I_{n}-{Q_{A}^{T}}Q_{A})\hat{\nu}_{1} -{Q_{L}^{T}}\hat{f}_{1} \\ & =& \hat{\nu}_{1}-{Q_{A}^{T}}(\alpha_{1}u_{1}+\beta_{2}u_{2}+f_{1})-{Q_{L}^{T}}\hat{f}_{1} \\ & =& \hat{\nu}_{1}-\alpha_{1}(\alpha_{1} \nu_{1}+g_{1})- \beta_{2}(\alpha_{2}\nu_{2}+\beta_{2}\nu_{1}+g_{2}) -{Q_{A}^{T}}f_{1}-{Q_{L}^{T}}\hat{f}_{1} \\ & =& (1-{\alpha_{1}^{2}}-{\beta_{2}^{2}})\hat{\nu}_{1} + \alpha_{2}\beta_{2}\hat{v}_{2} + O(\bar{q}(m,n,p)\varepsilon) . \end{array} $$

Next, suppose (3.11) is true for the indices up to i. For i + 1, we have

$$\hat{\alpha}_{i+1}{Q_{L}^{T}}\hat{u}_{i+1} = {Q_{L}^{T}}Q_{L}\hat{\nu}_{i+1}-\hat{\beta}_{i}{Q_{L}^{T}}\hat{u}_{i}- {\sum}_{j=1}^{i-1}\hat{\xi}_{ji+1}{Q_{L}^{T}}\hat{u}_{j}-{Q_{L}^{T}}\hat{f}_{i+1} .$$

Since \((\hat {\beta }_{i}{Q_{L}^{T}}\hat {u}_{i}- {\sum }_{j=1}^{i-1}\hat {\xi }_{ji+1}{Q_{L}^{T}}\hat {u}_{j}) \in span\{\hat {\nu }_{1}, \dots , \hat {\nu }_{i+1}\}+ O(\bar {q}(m,n,p)\varepsilon )\), we only need to prove \({Q_{L}^{T}}Q_{L}\hat {\nu }_{i+1}\in span\{\hat {\nu }_{1}, \dots , \hat {\nu }_{i+2}\}+ O(\bar {q}(m,n,p)\varepsilon )\). Notice that

$$ \begin{array}{@{}rcl@{}} {Q_{L}^{T}}Q_{L}\hat{\nu}_{i+1} & =& (I_{n}-{Q_{A}^{T}}Q_{A})\hat{\nu}_{i+1} \\ & =& \hat{\nu}_{i+1}+(-1)^{i+1}{Q_{A}^{T}}(\alpha_{i+1}u_{i+1}+\beta_{i+1}u_{i+2}+{\sum}_{j=1}^{i}\xi_{ji+1}u_{j}+f_{i+1}) \end{array} $$
$$ \begin{array}{@{}rcl@{}} \phantom{{Q_{L}^{T}}Q_{L}\hat{\nu}_{i+1}}& =& \hat{\nu}_{i+1}+(-1)^{i+1}(\alpha_{i+1}{Q_{A}^{T}}u_{i+1}+\beta_{i+1}{Q_{A}^{T}}u_{i+2}+ {\sum}_{j=1}^{i}\xi_{ji+1}{Q_{A}^{T}}u_{j}) \\&&+ (-1)^{i+1}{Q_{A}^{T}}f_{i+1} .\end{array} $$

From (3.9), we have

$$ \begin{array}{@{}rcl@{}} (\alpha_{i+1}{Q_{A}^{T}}u_{i+1}+\beta_{i+1}{Q_{A}^{T}}u_{i+2}&+& {\sum}_{j=1}^{i}\xi_{ji+1}{Q_{A}^{T}}u_{j}) \in span\{\hat{\nu}_{1}, \dots, \hat{\nu}_{i+2}\}\\&+& O(\bar{q}(m,n,p)\varepsilon) , \end{array} $$

which completes the proof of the induction step.

By the mathematical induction principle, (3.11) holds for all \(i = 1, 2, \dots ,k\). □

Proof Proof of Lemma 3.2

By (3.8) and (3.9), the process of computing Uk+ 1 and Vk can be treated as the Lanczos bidiagonalization of QA with the semiorthogonalization strategy. Since the k-step Lanczos bidiagonalization process is equivalent to the (2k + 1)-step symmetric Lanczos process [2, §7.6.1], the bounds for Ck and Dk can be deduced from the property of the symmetric Lanczos process with the semiorthogonalization strategy; see [26, Lemma 4] and its proof.

Now, we give the bound of \(\widehat {C}_{k}\). At the (i − 1)-th step, from (3.10), we can write the reorthogonalization step of \(\hat {u}_{i}\) as

$$ \begin{array}{@{}rcl@{}} & \hat{\alpha}_{i}^{\prime}\hat{u}_{i}^{\prime} = Q_{L}\hat{\nu}_{i}-\hat{\beta}_{i-1}\hat{u}_{i-1}-\hat{f}_{i}^{\prime}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} & \hat{\alpha}_{i}\hat{u}_{i} = \hat{\alpha}_{i}^{\prime}\hat{u}_{i}^{\prime}- {\sum}_{j=1}^{i-2}\hat{\xi}_{ji}\hat{u}_{j}-\hat{f}_{i}^{\prime\prime} , \end{array} $$

where \(\|\hat {f}_{i}^{\prime }\|, \|\hat {f}_{i}^{\prime \prime }\| = O(q_{3}(p,n)\varepsilon )\). Thus, for \(l=1,\dots , i-2\), we have

$$\hat{\alpha}_{i}^{\prime}\hat{u}_{l}^{T}\hat{u}_{i}^{\prime} = \hat{u}_{l}^{T}Q_{L}\hat{\nu}_{i}-\hat{\beta}_{i-1}\hat{u}_{l}^{T}\hat{u}_{i-1}- \hat{u}_{l}^{T}\hat{f}_{i}^{\prime} .$$

From (3.11) and its proof, we know that

$$ {Q_{L}^{T}}\hat{u}_{l}={\sum}_{j=1}^{l+1}\lambda_{j}\hat{v}_{j} + O(\bar{q}(m,n,p)\varepsilon) $$

with modest constants λj for \(j=1,\dots , l+1\). Notice that \(\left |\hat {u}_{l}^{T}\hat {u}_{i-1}, \hat {\nu }_{j}^{T}\hat {\nu }_{i}\right | \leq \sqrt {\varepsilon /(2k+1)}\) for \(l=1,\dots , i-2\) and \(j=1,\dots , l+1\). We obtain

$$ \hat{\alpha}_{i}^{\prime}\hat{u}_{l}^{T}\hat{u}_{i}^{\prime}= {\sum}_{j=1}^{l+1}\lambda_{j}\hat{\nu}_{j}^{T}\hat{\nu}_{i}- \hat{\beta}_{i-1}\hat{u}_{l}^{T}\hat{u}_{i-1} + O(\bar{q}(m,n,p)\varepsilon) =O(\sqrt{\varepsilon}) .$$

Then, we prove \(M = \max \limits _{1\leq j \leq i-1}|\hat {\xi }_{ji}|=O(\sqrt {\varepsilon })\). Premultiplying (A.1) by \(\hat {u}_{l}^{T}\) and making some arrangement, we obtain

$$\hat{\xi}_{li} = \hat{\alpha}_{i}^{\prime}\hat{u}_{l}^{T}\hat{u}_{i}^{\prime}-\hat{\alpha}_{i}\hat{u}_{l}^{T}\hat{u}_{i}-{\sum}_{j=1,j\neq l}^{i-2}\hat{\xi}_{ji}\hat{u}_{l}^{T}\hat{u}_{j}-\hat{u}_{l}^{T}\hat{f}_{i}^{\prime\prime} .$$

Notice that \(\hat {u}_{l}^{T}\hat {u}_{i}=O(\sqrt {\varepsilon })\) and we have proved \(\hat {\alpha }_{i}^{\prime }\hat {u}_{l}^{T}\hat {u}_{i}^{\prime }=O(\sqrt {\varepsilon })\) for \(l=1,\dots , i-2\). We obtain

$$ |\hat{\xi}_{li}| \leq O(\sqrt{\varepsilon})+O(\sqrt{\varepsilon})+iM\sqrt{\varepsilon}+O(\bar{q}(m,n,p)\varepsilon) . $$

The above right-hand side does not depend on l anymore, and we finally obtain by taking the maximum on the the left-hand side:

$$ (1-i\sqrt{\varepsilon})M \leq O(\sqrt{\varepsilon}) + O(\bar{q}(m,n,p)\varepsilon) . $$

Therefore, we have \(M = O(\sqrt {\varepsilon })\). □

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Jia, Z., Li, H. The joint bidiagonalization process with partial reorthogonalization. Numer Algor 88, 965–992 (2021).

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  • Joint bidiagonalization
  • GSVD
  • Lanczos bidiagonalization
  • Orthogonality level
  • Semiorthogonalization
  • Partial reorthogonalization

Mathematics Subject Classification (2010)

  • 15A18
  • 65F15
  • 65F25
  • 65F50
  • 65G50