A New Iterative Solution Method for Solving Multiple Linear Systems

ABSTRACT

In this paper, a new iterative solution method is proposed for solving multiple linear systems A^{(i)}x^{(i)}=b^{(i)}, for 1≤ i ≤ s, where the coefficient matrices A^{(i)} and the right-hand sides b^{(i)} are arbitrary in general. The proposed method is based on the global least squares (GL-LSQR) method. A linear operator is defined to connect all the linear systems together. To approximate all numerical solutions of the multiple linear systems simultaneously, the GL-LSQR method is applied for the operator and the approximate solutions are obtained recursively. The presented method is compared with the well-known LSQR method. Finally, numerical experiments on test matrices are presented to show the efficiency of the new method.

In this paper, a new iterative solution method is proposed for solving multiple linear systems A

Cite this paper

S. Karimi, "A New Iterative Solution Method for Solving Multiple Linear Systems,"*Advances in Linear Algebra & Matrix Theory*, Vol. 2 No. 3, 2012, pp. 25-30. doi: 10.4236/alamt.2012.23004.

S. Karimi, "A New Iterative Solution Method for Solving Multiple Linear Systems,"

References

[1] C. C. Paige and M. A. Saunders, “LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares,” ACM Transactions on Mathematical, Vol. 8, No. 1, 1982, pp. 43-71. doi:10.1145/355984.355989

[2] R. Plemmons, “FFT-Based RLS in Signal Processing,” Proceeding of the IEEE International Conference on Acoustics, Speech and Signal Processing, Minneapolis, 27-30 April 1993, pp. 571-574.

[3] W. E. Boyse and A. A. Seidl, “A Block QMR Method for Computing Multiple Simultaneous Solutions to Compex Symmetric Systems,” SIAM Journal on Scientific Computing, Vol. 17, No. 1, 1996, pp. 263-274.

[4] R. Kress, “Linear Integral Equations,” Springer-Verlag, New York, 1989. doi:10.1007/978-3-642-97146-4

[5] D. O’Leary, “The Block Conjugate Gradient Algorithm and Related Methods,” Linear Algebra and Its Applications, Vol. 29, 1980, pp. 293-322. doi:10.1016/0024-3795(80)90247-5

[6] A. Jain, “Fundamentals of Digital Image Processing,” Prentice-Hall, Englewwood Cliffs, 1989.

[7] S. Karimi and F. Toutounian, “The block Least Squares Method for Solving Nonsymmetric Linear Systems with Multiple Right-Hand Sides,” Applied Mathematics and Computation, Vol. 177, No. 2, 2006, pp. 852-862. doi:10.1016/j.amc.2005.11.038

[8] G. Golub and C. Loan, “Matrix Computations,” 2nd Edition, Johns Hopkins Press, Bultimore, 1989.

[9] A. El Guennouni, K. Jbilou and H. Sadok, “The Block Lanczos Method for Linear Systems with Multiple Right-Hand Sides,” Applied Numerical Mathematics, Vol. 51, No. 2-3, 2004, pp. 243-256.

[10] A. El Guennouni, K. Jbilou and H. Sadok, “A Block Bi-CGSTAB Algorithm for Mutiple Linear Systems,” Electronic Transactions on Numerical Analysis, Vol. 16, 2003, pp. 129-142.

[11] S. Karimi and F. Toutounian, “On the Convergence of the BL-LSQR Algorithm for Solving Matrix Equations,” International Journal of Computer Mathematics, Vol. 88, No. 1, 2011, pp. 171-182. doi:10.1080/00207160903365883

[12] T. F. Chan and M. K. Ng, “Galerkin Projection Methods for Solving Multiple Linear Systems,” SIAM Journal on Scientific Computing, Vol. 21, No. 3, 2012, pp. 836-850. doi:10.1137/S1064827598310227

[13] F. Toutounian and S. Karimi, “Global Least Squares (GLLSQR) Method for Solving General Linear Systems with Several Right-Hand Sides,” Applied Mathematics and Computation, Vol. 178, No. 2, 2006, pp. 452-460. doi:10.1016/j.amc.2005.11.065

[1] C. C. Paige and M. A. Saunders, “LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares,” ACM Transactions on Mathematical, Vol. 8, No. 1, 1982, pp. 43-71. doi:10.1145/355984.355989

[2] R. Plemmons, “FFT-Based RLS in Signal Processing,” Proceeding of the IEEE International Conference on Acoustics, Speech and Signal Processing, Minneapolis, 27-30 April 1993, pp. 571-574.

[3] W. E. Boyse and A. A. Seidl, “A Block QMR Method for Computing Multiple Simultaneous Solutions to Compex Symmetric Systems,” SIAM Journal on Scientific Computing, Vol. 17, No. 1, 1996, pp. 263-274.

[4] R. Kress, “Linear Integral Equations,” Springer-Verlag, New York, 1989. doi:10.1007/978-3-642-97146-4

[5] D. O’Leary, “The Block Conjugate Gradient Algorithm and Related Methods,” Linear Algebra and Its Applications, Vol. 29, 1980, pp. 293-322. doi:10.1016/0024-3795(80)90247-5

[6] A. Jain, “Fundamentals of Digital Image Processing,” Prentice-Hall, Englewwood Cliffs, 1989.

[7] S. Karimi and F. Toutounian, “The block Least Squares Method for Solving Nonsymmetric Linear Systems with Multiple Right-Hand Sides,” Applied Mathematics and Computation, Vol. 177, No. 2, 2006, pp. 852-862. doi:10.1016/j.amc.2005.11.038

[8] G. Golub and C. Loan, “Matrix Computations,” 2nd Edition, Johns Hopkins Press, Bultimore, 1989.

[9] A. El Guennouni, K. Jbilou and H. Sadok, “The Block Lanczos Method for Linear Systems with Multiple Right-Hand Sides,” Applied Numerical Mathematics, Vol. 51, No. 2-3, 2004, pp. 243-256.

[10] A. El Guennouni, K. Jbilou and H. Sadok, “A Block Bi-CGSTAB Algorithm for Mutiple Linear Systems,” Electronic Transactions on Numerical Analysis, Vol. 16, 2003, pp. 129-142.

[11] S. Karimi and F. Toutounian, “On the Convergence of the BL-LSQR Algorithm for Solving Matrix Equations,” International Journal of Computer Mathematics, Vol. 88, No. 1, 2011, pp. 171-182. doi:10.1080/00207160903365883

[12] T. F. Chan and M. K. Ng, “Galerkin Projection Methods for Solving Multiple Linear Systems,” SIAM Journal on Scientific Computing, Vol. 21, No. 3, 2012, pp. 836-850. doi:10.1137/S1064827598310227

[13] F. Toutounian and S. Karimi, “Global Least Squares (GLLSQR) Method for Solving General Linear Systems with Several Right-Hand Sides,” Applied Mathematics and Computation, Vol. 178, No. 2, 2006, pp. 452-460. doi:10.1016/j.amc.2005.11.065