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Vladimir O.

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Vladimir O.

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DOI: 10.17586/2226-1494-2018-18-3-437-446

DOI: 10.17586/2226-1494-2018-18-3-437-446

# MATRIX-ITERATIVE SOLUTION METHOD FOR SYSTEM OF LINEAR EQUATIONS AND ITS APPLICATION IN SPACE TOMOGRAPHY SCANNING USING RADAR

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**Article in**Russian

**For citation:**Samoilenko M.V. Matrix-iterative solution method for system of linear equations and its application in space tomography scanning using radar.

*Scientific and Technical Journal of Information Technologies, Mechanics and Optics*, 2018, vol. 18, no. 3, pp. 437–446 (in Russian). doi: 10.17586/2226-1494-2018-18-3-437-446

**Abstract**

A new solution method for a system of linear algebraic equations (SLAE), called matrix-iterative, is proposed. The method was obtained while analyzing the problem of space tomography scanning – one of the methods of multichannel tomography, which is being developed by the author. Practically all the methods of multichannel tomography are reduced to the restoration of the searched distribution of certain physical quantity by solving a SLAE, which is often underdetermined. The searched distribution is represented by vector-original. In tomography scanning the point sources power distribution over the area review sector is restored. The task determines certain a priori data, which made it possible to develop a new method of solving a SLAE: the most components of a searched vector are equal to the background value, for example, zero, and exceeding values represent the d-shaped rare values. The resulting vector-matrix equation in the considered problem, identical to SLAE, is underdetermined and often includes a matrix of incomplete rank or badly conditioned matrix. In contrast to the known methods of solving SLAE with partial a priori information about the restored vector, the matrix-iterative method implies no transformation of the original equation preceding the solution. The method is simple in implementation and contains no such ambiguous quantities as, for example, the regularization parameter. In essence, this is a software method quite easy to be realized. The alternative possibility to solve the SLAE without its preliminary transformations is the pseudo-inverse method, which needs no a priori information and gives either the solution with minimum norm (if it is the joint system), or the solution providing a minimum quadratic deviation. This method has served as a basis for matrix-iterative method, being, in particular, its first iteration. On the following iterations the columns are excluded from the system matrix, the ones corresponding to those values of the solution on the previous iteration, that had been identified as the background. The process of pseudo-inverse solutions with changing matrix is being repeated until all the results are more than the background value. The matrix-iterative method makes it possible to receive practically exact solution if vector-original comes in countless solutions of SLAE. If the vector-original is not included in the number of solutions of SLAE, because of the deviations of the task parameters, then, in this case, the method gives a fairly good result, confirming its stability.

**Keywords:**multichannel tomography, matrix-iterative method, space tomography scanning, underdetermined SLAE, matrix conditioning coefficient

**References**

1. Samoilenko M.V.

*Signal Processing in Problems of Location Measurements and Estimation*. Moscow, Spectr Publ., 2016, 260 p. (In Russian)
2. Samojlenko M.V.

*Method for Definition of Directions to Radiation Sources and Angular Discrimination of Sources*. Patent RU2392634, 2010.
3. Kir'yanov D.V., Kir'yanova E.N.

*Computational Physics*. Moscow, Polibuk Mul'timedia Publ., 2006, 352 p. (In Russian)
4. Tikhonov A.N. On Ill-Posed Problems in Linear Algebra and a Stable Method for Their Solution.

*Doklady AN USSR*, 1965, vol. 163, no. 6, pp. 591–595. (In Russian)
5. Sevast'yanov L.A., Lovetskii K.P., Laneev E.B.

*Regular Methods and Algorithms for Calculating Ill-Posed Problems in Models of Optical Structures*. Moscow, RUDN Publ., 2008, 155 p. (In Russian)
6. Tikhonov A.N., Arsenin V.Ya.

*Solutions of Ill-Posed Problems*. New York, Wiley, 1977.
7. Butler J.P., Reeds J.A., Dawson S.V. Estimating solutions of first kind integral equations with nonnegative constraints and optimal smoothing.

*SIAM Journal on Numerical Analysis*,1981,vol. 18, no. 3, pp. 410-421. doi: 10.1137/0718025
8. Antyufeev V.S. Finding the most probable nonnegative solutions of systems of linear algebraic equations by the maximum likelihood method.

*Numerical Analysis and Applications*, 2013, vol. 6, no. 3, pp. 187–196. doi: 10.1134/S1995423913030026
9. Turchin V.F., Kozlov V.P., Malkevich M.S. The use of mathematical-statistics methods in the solution of incorrectly posed problems.

*Soviet Physics Uspekhi*, 1971, vol. 13, no. 6, pp. 681–703. doi: 10.1070/PU1971v013n06ABEH004273
10. Mosegaard K., Tarantola A. Monte Carlo sampling of solutions to inverse problem.

*Journal of Geophysical Research*, 1995, vol. 100, no. B7, pp. 12431-12447.
11. Voskoboinikov Yu.E., Mitsel' A.A.

*Modern Problems of Applied Pathematics*. Part 1. Lecture Course. Tomsk, TUSUR Publ., 2015, 136 p. (In Russian)
12. Voskoboinikov Yu.E., Vtyurin K.A., Litasov V.A. Descriptive algorithm for restoring input signals of optical systems.

*Optoelectronics, Instrumentation and Data Processing*, 2005, vol. 41, no. 3, pp. 3–10. (In Russian)
13. Vasin V.V., Ageev A.L.

*Ill-Posed Problems with A Priori Information*. Ekaterinburg, UIF Nauka Publ., 1993, 264 p. (in Russian)
14. Gantmakher F.R.

*Matrix Theory*. Moscow, Nauka Publ., 1988, 552 p. (In Russian)
Samoilenko V.I., Puzyrev V.A., Grubrin I.V.

*Technical Cybernetics: Textbook*. Moscow, MAI Publ., 1994, 280 p. (In Russian)