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. Hansen, Per Christian: Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems (1994)
  2. Heinkenschloss, M.: On the solution of a two ball trust region subproblem (1994)
  3. Lukšan, L.: Inexact trust region method for large sparse systems of nonlinear equations (1994)
  4. McQuain, W. D.; Ribbens, C. J.; Watson, L. T.; Melville, R. C.: Preconditioned iterative methods for sparse linear algebra problems arising in circuit simulation (1994)
  5. Van der Vorst, H. A.; Vuik, C.: GMRESR: A family of nested GMRES methods (1994)
  6. Barlow, Jesse L.: Numerical aspects of solving linear least squares problems (1993)
  7. Barrett, Richard; Berry, Michael; Chan, Tony F.; Demmel, James; Donato, June; Dongarra, Jack; Eijkhout, Victor; Pozo, Roldan; Romine, Charles; van der Vorst, Henk: Templates for the solution of linear systems: building blocks for iterative methods (1993)
  8. Chan, Raymond H.; Nagy, James G.; Plemmons, Robert J.: FFT-based preconditioners for Toeplitz-block least squares problems (1993)
  9. Dierckx, Paul: Curve and surface fitting with splines (1993)
  10. Lukšan, Ladislav: Inexact trust region method for large sparse nonlinear least squares (1993)
  11. Freund, Roland W.; Golub, Gene H.; Nachtigal, Noël M.: Iterative solution of linear systems (1992)
  12. Hansen, Per Christian: Analysis of discrete ill-posed problems by means of the L-curve (1992)
  13. Kratzer, Steven G.: Sparse QR factorization on a massively parallel computer (1992)
  14. Nachtigal, Noël M.; Reddy, Satish C.; Trefethen, Lloyd N.: How fast are nonsymmetric matrix iterations? (1992)
  15. Bierlaire, M.; Toint, Ph. L.; Tuyttens, D.: On iterative algorithms for linear least squares problems with bound constraints (1991)
  16. Christiansen, E.; Kortanek, K. O.: Computation of the collapse state in limit analysis using the LP primal affine scaling algorithm (1991)
  17. Golub, Gene H.; von Matt, Urs: Quadratically constrained least squares and quadratic problems (1991)
  18. Gomes, Herminio Simões; Martínez, José Mario: A numerically stable reduced-gradient type algorithm for solving large- scale linearly constrained minimization problems (1991)
  19. Karmarkar, N. K.; Ramakrishnan, K. G.: Computational results of an interior point algorithm for large scale linear programming (1991)
  20. Strakoš, Z.: On the real convergence rate of the conjugate gradient method (1991)

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