	\begin{algorithmic}[1]
		\Require $A$, $b$, $x_{0}=\mathbf{0}$
		\For{$k=1,2,\ldots,$}
		\State Compute $p_k$, $q_k$, $\alpha_k$, $\beta_k$ by the LBFRO  \algorithmiccomment{roundoff unit is $\mathbf{u}$}
		\State Compute $\rho_k$, $\theta_{k+1}$, $\bar{\rho}_{k+1}$, $\phi_{k}$, $\bar{\phi}_{k+1}$ by the updating procedure  \algorithmiccomment{double precision}
		\State Compute $x_k$, $w_{k+1}$ by the updating procedure  \algorithmiccomment{roundoff unit is $\mathbf{\bar{u}}$}
		\If{Early stopping criterion is satisfied}  \algorithmiccomment{DP or L-curve criterion}
		\State The semi-convergence point is estimated as $k_1$
		\State Terminate the iteration
		\EndIf
		\EndFor
		\Ensure Final regularized solution $\hat{x}_{k_1}$  \algorithmiccomment{Computed solution corresponding to $x_{k_1}$}
	\end{algorithmic}

\begin{algorithmic}
[1]
\Require $A$, $b$, $x_{0}=\mathbf{0}$
\For{$k=1,2,\ldots,$}
\State Compute $p_k$, $q_k$, $\alpha_k$, $\beta_k$ by the LBFRO \algorithmiccomment{roundoff unit is $\mathbf{u}$}
\State Compute $\rho_k$, $\theta_{k+1}$, $\bar{\rho}_{k+1}$, $\phi_{k}$, $\bar{\phi}_{k+1}$ by the updating procedure \algorithmiccomment{double precision}
\State Compute $x_k$, $w_{k+1}$ by the updating procedure \algorithmiccomment{roundoff unit is $\mathbf{\bar{u}}$}
\If{Early stopping criterion is satisfied} \algorithmiccomment{DP or L-curve criterion}
\State The semi-convergence point is estimated as $k_1$
\State Terminate the iteration
\EndIf
\EndFor
\Ensure Final regularized solution $\hat{x}_{k_1}$ \algorithmiccomment{Computed solution corresponding to $x_{k_1}$}
\end{algorithmic}
{
    "title": "double precision is not necessary for lsqr for solving discrete linear   ill-posed problems",
    "id": "2210.11025",
    "abstract": "the growing availability and usage of low precision foating point formats has attracts many interests of developing lower or mixed precision algorithms for scientific computing problems. in this paper we investigate the possibility of exploiting lower precision computing in lsqr for solving discrete linear ill-posed problems. we analyze the choice of proper computing precisions in the two main parts of lsqr, including the construction of lanczos vectors and updating procedure of iterative solutions. we show that, under some mild conditions, the lanczos vectors can be computed using single precision without loss of any accuracy of final regularized solutions as long as the noise level is not extremely small. we also show that the most time consuming part for updating iterative solutions can be performed using single precision without sacrificing any accuracy. the results indicate that the most time consuming parts of the algorithm can be implemented using single precision, and thus the performance of lsqr for solving discrete linear ill-posed problems can be significantly enhanced. numerical experiments are made for testing the single precision variants of lsqr and confirming our results.",
    "categories": "math.na cs.na",
    "doi": "10.1007/s10915-023-02447-4",
    "created": "2022-10-20",
    "updated": "2024-02-12",
    "authors": [
        "haibo li"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2210.11025"
}
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