### Abstract

We present a randomized iterative algorithm that exponentially converges in the mean square to the minimum ℓ_{2}-norm least squares solution of a given linear system of equations. The expected number of arithmetic operations required to obtain an estimate of given accuracy is proportional to the squared condition number of the system multiplied by the number of nonzero entries of the input matrix. The proposed algorithm is an extension of the randomized Kaczmarz method that was analyzed by Strohmer and Vershynin.

Original language | English (US) |
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Pages (from-to) | 773-793 |

Number of pages | 21 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 34 |

Issue number | 2 |

DOIs | |

State | Published - Jul 29 2013 |

### Fingerprint

### Keywords

- Iterative method
- LAPACK
- Linear least squares
- Minimum-length solution
- Overdetermined system
- Random sampling
- Randomized algorithms
- Sparse matrix
- Underdetermined system

### ASJC Scopus subject areas

- Analysis

### Cite this

*SIAM Journal on Matrix Analysis and Applications*,

*34*(2), 773-793. https://doi.org/10.1137/120889897

**Randomized extended Kaczmarz for solving least squares.** / Zouzias, Anastasios; Freris, Nikolaos.

Research output: Contribution to journal › Article

*SIAM Journal on Matrix Analysis and Applications*, vol. 34, no. 2, pp. 773-793. https://doi.org/10.1137/120889897

}

TY - JOUR

T1 - Randomized extended Kaczmarz for solving least squares

AU - Zouzias, Anastasios

AU - Freris, Nikolaos

PY - 2013/7/29

Y1 - 2013/7/29

N2 - We present a randomized iterative algorithm that exponentially converges in the mean square to the minimum ℓ2-norm least squares solution of a given linear system of equations. The expected number of arithmetic operations required to obtain an estimate of given accuracy is proportional to the squared condition number of the system multiplied by the number of nonzero entries of the input matrix. The proposed algorithm is an extension of the randomized Kaczmarz method that was analyzed by Strohmer and Vershynin.

AB - We present a randomized iterative algorithm that exponentially converges in the mean square to the minimum ℓ2-norm least squares solution of a given linear system of equations. The expected number of arithmetic operations required to obtain an estimate of given accuracy is proportional to the squared condition number of the system multiplied by the number of nonzero entries of the input matrix. The proposed algorithm is an extension of the randomized Kaczmarz method that was analyzed by Strohmer and Vershynin.

KW - Iterative method

KW - LAPACK

KW - Linear least squares

KW - Minimum-length solution

KW - Overdetermined system

KW - Random sampling

KW - Randomized algorithms

KW - Sparse matrix

KW - Underdetermined system

UR - http://www.scopus.com/inward/record.url?scp=84880534082&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880534082&partnerID=8YFLogxK

U2 - 10.1137/120889897

DO - 10.1137/120889897

M3 - Article

AN - SCOPUS:84880534082

VL - 34

SP - 773

EP - 793

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 2

ER -