Abstract
In this paper, an alternating direction method (ADM) is proposed for nonnegative solutions of the matrix equation \(AX+YB=C\). In addition, the preliminary convergence of the proposed method is given and proved. Numerical experiments illustrate the effectiveness of the method.
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References
Alfred MB, Elad M, Zibulevsky M (2008) On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations. IEEE Trans Inform Theory 54:4813–4820
Baksalary JK, Kala R (1979) The matrix equation \(AX+YB = C\). Linear Algebra Appl 25:41–43
Hartwig RE (1987) A note on light matrices. Linear Algebra Appl 97:153–169
Hestenes MR (1969) Multiplier and gradient methods. J Optim Theory App 4:303–320
Huang JP (2004) The solution of the quaternion matrix equation \(AX-YB=C\) and its optimal approximation. Math Theory Appl 24:1–4
Ke YF, Ma CF (2014) A preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation. Comput Math Appl 68:1409–1420
Peng ZY, Wang L, Peng JJ (2012) The solution of matrix equation \(AX=B\) over a matrix inequality constraint. SIAM J Matrix Anal Appl 33:554–568
Powell MJD (1969) A method for nonlinear constraints in minimization problems. In: Fletcher R (ed) Optimization. pp 283–298
Rockafellar RT (1973) The multiplier method of Hestenes and Powell applied to convex programming. J Optim Theory App 12:555–562
Xu YY, Yin WT, Wen ZW, Zhang Y (2012) An alternating direction algorithm for matrix completion with nonnegative factors. Front Math China 7:365–384
Yong HL (2006) Ranks of solutions of the linear matrix equation \(AX+YB=C\). Comput Math Appl 52:861–872
Zak SH (1985) On the polynomial matrix equation \(AX+YB=C\). IEEE Trans Automat Contr 30:1240–1242
Ziȩtak K (1984) The \(l_p\)-solution of the linear matrix equation \(AX+YB=C\). Computing 32:153–162
Ziȩtak K (1988) Properties of the strict Chebyshev solutions of the linear matrix equation \(AX+YB=C\). J Approx Theory 55:140–149
Acknowledgments
The project was supported by the National Natural Science Foundation of China (11071041, 11201074) and the Fujian Natural Science Foundation (2013J01006).
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Communicated by Jinyun Yuan.
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Ke, Y., Ma, C. An alternating direction method for nonnegative solutions of the matrix equation \(AX+YB=C\) . Comp. Appl. Math. 36, 359–365 (2017). https://doi.org/10.1007/s40314-015-0232-5
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DOI: https://doi.org/10.1007/s40314-015-0232-5