LSQR

Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems. An iterative method is given for solving Ax = b and min|| Ax - b||2 , where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerical properties. Reliable stopping criteria are derived, along with estimates of standard errors for x and the condition number of A. These are used in the FORTRAN implementation of the method, subroutine LSQR. Numerical tests are described comparing LSQR with several other conjugate-gradient algorithms, indicating that LSQR is the most reliable algorithm when A is ill-conditioned.

This software is also peer reviewed by journal TOMS.


References in zbMATH (referenced in 343 articles , 1 standard article )

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  1. Dodani, Mahesh H.; Babu, A. J. G.: Karmarkar’s projective method for linear programming: A computational survey (1990)
  2. Duff, Iain S.: The solution of large-scale least-squares problems on supercomputers (1990)
  3. Forrest, J. J. H.; Tomlin, J. A.: Vector processing in simplex and interior methods for linear programming (1990)
  4. Kaufman, Linda: Solving emission tomography problems on vector machines (1990)
  5. Schaefer, Mark J.: A polynomial based iterative method for linear parabolic equations (1990)
  6. van der Sluis, A.; van der Vorst, H. A.: SIRT- and CG-type methods for the iterative solution of sparse linear least-squares problems (1990)
  7. Björck, Åke: A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations (1988)
  8. Goldfarb, Donald; Mehrotra, Sanjay: A relaxed version of Karmarkar’s method (1988)
  9. Goldfarb, Donald; Mehrotra, Sanjay: Relaxed variants of Karmarkar’s algorithm for linear programs with unknown optimal objective value (1988)
  10. Saad, Youcef: Preconditioning techniques for nonsymmetric and indefinite linear systems (1988)
  11. Van der Vorst, Henk A.; Dekker, Kees: Conjugate gradient type methods and preconditioning (1988)
  12. Cavalier, Tom M.; Schall, Kenneth C.: Implementing an affine scaling algorithm for linear programming (1987)
  13. de Pillis, John: Tensor equivalents for solution of linear systems: a parallel algorithm (1987)
  14. Martínez, J. M.: An algorithm for solving sparse nonlinear least squares problems (1987)
  15. van der Vorst, Henk A.: Large tridiagonal and block tridiagonal linear systems on vector and parallel computers (1987)
  16. Zlatev, Zahari: A survey of the advances in the exploitation of the sparsity in the solution of large problems (1987)
  17. Foster, Leslie V.: Rank and null space calculations using matrix decomposition without column interchanges (1986)
  18. Gill, Philip E.; Murray, Walter; Saunders, Michael A.; Tomlin, J. A.; Wright, Margaret H.: On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method (1986)
  19. Zielke, G.: Report on test matrices for generalized inverses (1986)
  20. Nolet, Guust: Solving or resolving inadequate and noisy tomographic systems (1985)

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