Mmimo detection preprocessing and parameter pre-training method for correlated channels

Abstract

The application provides an mMIMO detection preprocessing and parameter pre-training method for a correlated channel, and the preprocessing method comprises the following steps: a1, modeling uplink of a large-scale MIMO system; a2, calculating a Gram matrix W; a3, obtaining a preprocessing matrix P according to the Gram matrix W; a4, preprocessing the Gram matrix W by using the preprocessing matrix P to obtain a new Gram matrix W1: W1=PW; and a5, processing the matrix W1 by using an iterative algorithm instead of a matrix inversion algorithm. The method can be applied to various iterative algorithms instead of matrix inversion, thereby accelerating the convergence of the algorithms and improving the detection performance by more than 2dB under the correlated channel.

Description

Technical Field

[0001] This invention relates to a method for mMIMO detection preprocessing and parameter pretraining for associated channels. Background Technology

[0002] Massive Multiple-Input Multiple-Output (mMIMO) technology has become one of the key technologies for 5G wireless communication systems due to its excellent data transmission rate. By deploying dozens or even hundreds of antennas at base stations, mMIMO systems can significantly improve data transmission rate, spectral efficiency, and energy efficiency. However, the increase in the number of antennas also leads to an increase in computational complexity during signal detection. Current signal detection methods are mainly divided into linear detection and nonlinear detection. Linear detection is relatively simple to compute, with the main complexity concentrated in matrix inversion operations, such as the minimum mean square error (MMSE) (Reference 1) and zero-forcing (Reference 2) detection algorithms. Nonlinear detection has higher complexity but usually has better detection performance. Among them, the expectation propagation (EP) (Reference 3) algorithm has received widespread attention in recent years, but the EP algorithm also requires a large number of matrix inversion operations, resulting in extremely high complexity.

[0003] To overcome the complexity bottleneck of matrix inversion, some iterative approximation algorithms that replace inversion have been applied to detection algorithms such as MMSE and EP, taking advantage of channel hardening characteristics. These include Jacobi (Reference 4), Gauss-Seidel (GS) (Reference 5), Richardson (RI) (Reference 6), Second-Order Richardson Iteration (SORI) (References 7, 8), and weighted Neumann series approximation (wNSA) (Reference 9), effectively reducing computational complexity. However, these methods are only applicable to ideal propagation conditions. In real channels (correlated channels), due to mutual interference between antennas, the channel hardening characteristics decrease or even disappear, and the performance of the previous iterative inversion algorithms drops sharply.

[0004] Professor Zhang Chuan's team at Southeast University proposed improved algorithms for the MMSE algorithm (TL-BD-INSA, Reference 10) and the EP algorithm (BD-NS-EPA, Reference 11) in 2021 and 2023, respectively, for correlated channels. However, they only focused on interference between antennas of the same user and between antennas of the base station, ignoring interference between different users. The special correlated channel they considered is referred to as a block-like correlated channel. Under generalized correlated channels, due to the weakening or even disappearance of channel hardening characteristics, the performance of a series of iterative algorithms replacing matrix inversion deteriorates, and no suitable solution has yet been provided. Furthermore, these iterative algorithms replacing matrix inversion require calculating the scaling factor needed for iteration based on eigenvalues. However, the method proposed by Zhang Chuan et al. regarding eigenvalue parameters leads to some frames failing to converge, resulting in significant performance loss. The team of He Guanghui at Shanghai Jiao Tong University, in Reference 7, utilizes channel hardening to select the optimal eigenvalue parameters; therefore, this method is no longer applicable under correlated channels, and detection performance deteriorates sharply.

[0005] Reference 1: N.Kim, Y.Lee, and H.Park, "Performance analysis of mimo system with linear mmse receiver," IEEE Trans.Wireless Commun., vol.7, no.11, pp.4474–4478, 2008.

[0006] Document 2: L.Bai and J.Choi, Low complexity MIMO detection, New York, NY, USA: Springer-Verlag, 2012.

[0007] Reference 3: J.Cespedes, PMOlmos, MS′anchez-Fern′andez, and F.Perez-Cruz, “Expectation propagation detection for high-order high-dimensional mimosystems,” IEEE Trans.Commun., vol.62, no.8, pp.2840–2849, Aug.2014.

[0008] Reference 4: W. Song, X. Chen, L. Wang, and X. Lu, “Joint conjugate gradient and jacobi iteration based low complexity precoding for massive mimo systems,” in Proc. IEEE / CIC Int. Conf. Commun. China (ICCC), Jul. 2016, pp. 1–5.

[0009] Reference 5: L. Dai, X. Gao, X. Su, S. Han, I. Chih-Lin, and Z. Wang, “Low-complexity soft-output signal detection based on gauss–seidel method for uplink multiuser large-scale mimo systems,” IEEE Trans. Veh. Technol., vol. 64, no. 10, pp. 4839–4845, Oct. 2015.

[0010] Reference 6: X. Gao, L. Dai, Y. Ma, and Z. Wang, “Low-complexity near-optimal signal detection for uplink large-scale mimo systems,” Electronics letters, vol. 50, no. 18, pp. 1326–1328, 2014.

[0011] Reference 7: J. Tu, M. Lou, J. Jiang, D. Shu, and G. He, “An efficient massive mimo detector based on second-order richardson iteration: From algorithm to flexible architecture,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 67, no. 11, pp. 4015–4028, Nov. 2020.

[0012] Reference 8: Y. Chen, S. Song, Z. Wang, and J. Lin, “An efficient massive mimo detector based on approximate expectation propagation,” IEEE Trans. Very Large Scale Integr. (VLSI) Syst., vol. 31, no. 05, pp. 696–700, 2023.

[0013] Reference 9: X. Tan, H. Han, M. Li, K. Sun, Y. Huang, X. You, and C. Zhang, “Approximate expectation propagation massive mimo detector with weighted neumann-series,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 68, no. 2, pp. 662–666, Feb. 2021.

[0014] Reference 10: H. Wang, Y. Ji, Y. Shen, W. Song, M. Li, X. You, and C. Zhang, “An efficient detector for massive mimo based on improved matrix partition,” IEEE Trans. Signal Process., vol. 69, pp. 2971–2986, 2021.

[0015] Reference 11: H. Wang, B. Cheng, X. Tan, X. You, and C. Zhang, “An efficient approximate expectation propagation detector with block-diagonal neumann series,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 70, no. 3, pp. 1403–1416, March 2023. SUMMARY OF THE INVENTION

[0016] Purpose of the invention: The technical problem to be solved by the present invention is to address the shortcomings of the prior art by providing a method for mMIMO detection preprocessing and parameter pretraining for correlated channels. This method can be applied to various iterative matrix inversion algorithms in signal detection of large-scale multiple-input multiple-output antenna systems, thereby accelerating the convergence speed of these algorithms.

[0017] The mMIMO detection preprocessing method for correlated channels includes the following steps:

[0018] Step a1: Model the uplink of the large-scale MIMO system;

[0019] Step a2: Calculate the Gram matrix W;

[0020] Step a3: Obtain the preprocessing matrix P based on the Gram matrix W;

[0021] Step a4: Preprocess the Gram matrix W using the preprocessing matrix P to obtain a new Gram matrix W1: W1 = PW;

[0022] Step a5: The matrix W1 is processed using an iterative method instead of the matrix inversion algorithm.

[0023] Step a1 includes: In the uplink of a massive MIMO system, the number of antennas at the transmitting end is K, and the number of antennas at the receiving end is N;

[0024] Transmitted signal vector x = [x1, x2, ..., x] K The received signal vector is y = [y1, y2, ..., y]. N ], where x K y represents the transmitted signal of the Kth antenna. N Let represent the received signal of the Nth receiving antenna. Then the system model is expressed as y = Hx + n, where H is the channel matrix; n represents a signal with a mean of 0 and a variance of 0. Additive white Gaussian noise; where, considering interference between antennas in the actual channel, the Kronecker channel model is used for modeling, expressed as:

[0025] H=R 1 / 2H0T 1 / 2 ,

[0026] in, Indicates a Rayleigh fading channel. Represents an N x K matrix; The correlation matrix representing the receiving antenna. This represents the correlation matrix of the transmitting antenna.

[0027] In step a1, matrices R and T are represented as follows:

[0028]

[0029] Among them, R u，v represents the element in the \(u\)-th row and \(v\)-th column of matrix \(R\), and \(T u，v represents the element in the \(u\)-th row and \(v\)-th column of matrix \(T\), and \(\zeta r and \(\zeta t represent the correlation coefficient, represents the phase factor, represents \(R u，v 's conjugate transpose, represents \(T u，v 's conjugate transpose, and \(e\) is the natural constant, and \(j\) is the imaginary unit.

[0030] In step a2, the Gram matrix \(W = H H H\) is calculated, where \((·) H represents the conjugate transpose of the matrix, and the Gram matrix \(W\) is a \(K\times K\) matrix.

[0031] Step a3 includes:

[0032] In step a3-1, extract the other diagonals of the Gram matrix \(W\) except the main diagonal, and give the scaling factor \(\varepsilon = [\varepsilon_1, \varepsilon_2, \ldots, \varepsilon k-1 \), where \(-1 < \varepsilon i \leq 0\), \(i\) takes values from 1 to \(k - 1\), and \(\varepsilon i represents the scaling factor of the \(i\)-th level diagonal;

[0033] In step a3-2, obtain the preprocessing matrix \(P\) through the scaling factor and the extracted diagonal values.

[0034] In step a3-2, the preprocessing matrix \(P\) is obtained according to the following formula:

[0035]

[0036] where \(1\leq i < K\), \(S_1\) and represent the sub-diagonal of \(W\), \(S_2\) and represent the third diagonal, and so on, \(S i and represent the \((i + 1)\)-th diagonal; \(I K represents the \(K\times K\) identity matrix; represents the mapping relationship, that is, the expression on the right side of the equal sign.

[0037] The present invention also provides a pre-training method for mMIMO detection parameters for an associated channel, including the following steps:

[0038] In step b1, according to the number of antennas \(N\) at the actual receiving end and the number of antennas \(K\) at the transmitting end, and the given correlation coefficient \(\zetar and ζ t The following formula is used to obtain n frames of data, and each data point corresponds to the H matrix of the n frames of data:

[0039] H=R 1 / 2 H0T 1 / 2 ,

[0040]

[0041] Step b2: Based on the H matrix of n frames of data, n new Gram matrices W1 are obtained using steps a1 to a4. The maximum and minimum eigenvalues ​​are then obtained from these n new Gram matrices to obtain the set of maximum eigenvalues. and the set of minimum eigenvalues Let these represent the largest and smallest eigenvalues ​​of the nth matrix, respectively.

[0042] Step b3, based on the set of largest eigenvalues ​​λ max and the set of minimum eigenvalues ​​λ min The estimates λ′ for the largest eigenvalue are obtained respectively. max and the estimate of the minimum eigenvalue λ′ min :

[0043] λ′ max =max(λ) max ),

[0044] λ′ min =αmax(λ min )+βmin(λ min ),

[0045] Where α and β are constants, and α≥0, β≥0;

[0046] Step b4, based on the obtained estimate λ′ of the largest eigenvalue max and the estimate of the minimum eigenvalue λ′ min The scaling factor is calculated by replacing the matrix inversion algorithm with different iterations.

[0047] Beneficial Effects: This invention proposes a novel diagonal preprocessing (DP) method by studying the characteristics of correlated channels. This method can be applied to various algorithms that replace matrix inversion through iterative methods, thereby accelerating the convergence of these algorithms and improving their detection performance by more than 2dB under correlated channels. In iterative matrix inversion algorithms, scaling factors are often used, which are calculated based on the maximum and minimum eigenvalues. Existing eigenvalue parameter selection schemes lead to some performance loss under correlated channels. Therefore, this invention proposes a pre-training method, providing an effective eigenvalue parameter pre-training scheme to calculate the optimal eigenvalue parameters for iterative matrix inversion in different MIMO systems. Compared to existing technologies, this pre-training device ensures convergence for all frames, enabling the detection algorithm to achieve optimal detection performance. This invention applies the two proposed methods to the SORI algorithm, resulting in the DP-SORI algorithm, whose convergence speed is more than twice that of the original SORI algorithm. Attached Figure Description

[0048] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments, and the advantages of the present invention in the above and / or other aspects will become clearer.

[0049] Figure 1 This is a diagram of the method architecture of the present invention.

[0050] Figure 2 This is a schematic diagram illustrating the generation of the preprocessing matrix P.

[0051] Figure 3 This is a diagram showing the comparison of simulation results. Detailed Implementation

[0052] like Figure 1 As shown, this invention provides an mMIMO detection preprocessing method for correlated channels, including the following steps:

[0053] Step a1: In the uplink of a large-scale MIMO system, the number of antennas at the transmitting end is K, and the number of antennas at the receiving end is N.

[0054] The transmitted signal vector is x = [x1, x2, ..., x]. K The received signal vector is y = [y1, y2, ..., y]. N The system model is then represented as y = Hx + n, where H is the channel matrix, n represents a mean of 0, and a variance of . Additive white Gaussian noise. In practical channels, interference between antennas is considered, and the Kronecker channel model is used for modeling, expressed as:

[0055] H=R 1 / 2 H0T1 / 2 ,

[0056] in, This indicates a Rayleigh fading channel. The correlation matrix representing the receiving antenna. The correlation matrix representing the transmitting antenna is shown as follows:

[0057]

[0058] Where, ζ r and ζ t Represents the correlation coefficient. x represents the phase factor. * This represents the conjugate transpose of x.

[0059] Step a2, calculate the Gram matrix W = H H H. Among them, (·) H This represents the conjugate transpose of a matrix.

[0060] Step a3: Obtain the preprocessing matrix P based on the Gram matrix W.

[0061] Step a3 includes the following steps:

[0062] Step a3-1: Extract the larger diagonals from the Gram matrix W (excluding the main diagonal) and assign scaling factors ε = [ε1, ε2, ..., ε3]. i ], where -1 < ε i ≤0.

[0063] Step a3-2, using the scaling factor and the extracted diagonal values, and according to the following formula:

[0064]

[0065] The preprocessing matrix P is obtained, where S1 and Denotes the second diagonal of W, S2 and This represents the third diagonal, and so on. I K Let K represent the identity matrix. Figure 2 A schematic diagram of generating the preprocessing matrix P is given.

[0066] Step a4: Preprocess matrix W using matrix P: W1 = PW, thereby obtaining a new Gram matrix W1.

[0067] Step a5: The obtained matrix W1 is processed using an iterative algorithm instead of a matrix inversion algorithm.

[0068] This invention also provides a method for pre-training mMIMO detection parameters for correlated channels, comprising the following steps:

[0069] Step b1, based on the actual number of antennas N and K at the receiving and transmitting ends, and the given correlation coefficient ζ r and ζ t ,

[0070] The following formula is used to obtain n frames of data, and each data point corresponds to the H matrix of the n frames of data:

[0071] H=R 1 / 2 H0T 1 / 2 ,

[0072]

[0073] Step b2: Based on the H matrix of n frames of data, n new Gram matrices W1 are obtained using steps a1 to a4. The maximum and minimum eigenvalues ​​are then extracted from these n new Gram matrices to obtain two sets of eigenvalues. and

[0074] Step b3, based on the set of maximum and minimum eigenvalues ​​λ min and λ max The estimates λ′ for the largest eigenvalue are obtained respectively. max and the estimate of the minimum eigenvalue λ′ min :

[0075] λ′ max =max(λ) max ),

[0076] λ′ min =αmax(λ min )+βmin(λ min )

[0077] Where α≥0, β≥0. Recommended values ​​are α=1 / 4, β=3 / 4.

[0078] Step b4, based on the obtained estimate λ′ of the largest eigenvalue max and the estimate of the minimum eigenvalue λ′ min The scaling factor is calculated by replacing the matrix inversion algorithm with different iterations.

[0079] Example

[0080] In this embodiment, the novel matrix preprocessing method and pre-training method for selecting optimal eigenvalue parameters proposed in this invention, applicable to correlated channels, are applied to the SORI algorithm, resulting in a new iterative matrix inversion algorithm, DP-SORI. Furthermore, the obtained DP-SORI algorithm is applied to the EPA algorithm, resulting in a simplified EPA-DP-SORI algorithm. The specific implementation process is as follows:

[0081] The SORI detection algorithm, proposed by He Guanghui's team at Shanghai Jiao Tong University in reference 7, is a second-order detection algorithm based on Richardson iteration, used to achieve minimum mean square error detection in large-scale MIMO systems. The formula is as follows:

[0082]

[0083] Where l1 represents the number of iterations, l1≥1, and μ represents the estimate of the received number. This represents the estimate of the received signal during the l1-th iteration. B = H H y, ω, and δ are scaling factors.

[0084] Under correlated channels, the convergence speed of the SORI algorithm drops sharply. Therefore, the novel matrix preprocessing method proposed in this invention, suitable for correlated channels, can be used to preprocess matrices W and B to obtain W1 = PW and B1 = PB. Substituting these into the iterative formula of the SORI algorithm, we get:

[0085]

[0086] The values ​​of scaling factors ω and δ are calculated as follows:

[0087] δ=2 / (λ min +λ max )

[0088]

[0089] Where, λ min and λ max These refer to the minimum and maximum eigenvalues ​​of matrix W1, respectively. Further, using the matrix eigenvalue parameter pre-training scheme proposed in this invention, n frames of data are pre-trained to obtain n Gram matrices W1, and their maximum and minimum eigenvalues ​​are calculated, resulting in two eigenvalue sets. and Based on the set of maximum and minimum eigenvalues ​​λ min and λ max This yields an estimate of the largest eigenvalue, λ′. max and the estimate of the minimum eigenvalue λ′ min :

[0090] λ′ max =max(λ) max ),

[0091] λ′ min ==(max(λ) min )+3min(λ min )) / 4

[0092] This leads to the new iterative matrix inversion algorithm DP-SORI, which features fast convergence speed in correlated channels. This algorithm is essentially an approximation of the MMSE algorithm and can also be applied to other MIMO signal detection algorithms involving matrix inversion operations, such as the EPA algorithm. A detailed explanation follows.

[0093] In the EP algorithm, the posterior probability distribution of the transmitted symbol vector x is approximated as a Gaussian distribution for MIMO signal detection, expressed as:

[0094]

[0095] in, The mean is Hx and the variance is... The Gaussian distribution is denoted by Diag(), where Λ = [Λ1, Λ2, ..., Λ]. K ] represents the variance introduced by each dimension. This represents the mean introduced from each dimension. Let represent a diagonal matrix composed of vectors. This formula conforms to a joint Gaussian distribution, and its covariance matrix and mean are expressed as follows:

[0096] and

[0097] Where, initialized to In each iteration, Λ and γ need to be updated.

[0098] Then update μ. The EPA algorithm eliminates the matrix inversion operation in each iteration compared to the EP algorithm, but the initial matrix inversion still exists:

[0099] and

[0100] To further reduce the complexity of the EPA algorithm, the method of this invention is used to approximate the matrix inversion calculation in the above equation, thus obtaining the EPA-DP-SORI algorithm. That is, μ in the above equation... 0 Approximating through iteration, it can be expressed as:

[0101]

[0102] Where l1≥1,

[0103] The proposed EPA-DP-SORI algorithm is compared with the EPA-SORI algorithm that directly uses the SORI algorithm to eliminate matrix inversion (Reference 8) and the EPA-DP-SORI algorithm with some replacements of other schemes. The simulation uses a large-scale MIMO system with K=32 and N=128, the modulation scheme is 64-QAM, and the coefficients of the associated channels are selected as (0.2, 0.4).

[0104] The number of iterations for DP-SORI and SORI was set to 6, and the number of iterations for EPA was set to 3. The simulation results are as follows. Figure 3 As shown in the figure (7 refers to reference 7; 10 and 11 refer to references 10 and 11), simulation results show that compared to the algorithm without preprocessing, the method of this invention can achieve algorithm convergence and significantly improve detection performance. Regarding the eigenvalue parameter selection scheme, the eigenvalue parameter pre-training scheme used in this invention has performance essentially the same as the scheme using precise eigenvalue calculation, with only a small performance loss. However, the eigenvalue parameter selection schemes in references 10, 11, and 7 all lead to performance losses. In particular, the eigenvalues ​​obtained by the pre-training scheme in references 10 and 11, which use the expected value of the eigenvalue set as an estimate, can cause some data to fail to converge, resulting in severe performance loss. The obtained EPA-DP-SORI algorithm avoids complex eigenvalue calculations and achieves the best balance between performance and complexity.

[0105] This invention provides a method for mMIMO detection preprocessing and parameter pretraining for correlated channels. Many methods and approaches exist for implementing this technical solution; the above description is merely a preferred embodiment of the invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications should also be considered within the scope of protection of this invention. All components not explicitly stated in this embodiment can be implemented using existing technologies.

Claims

1. A method for mMIMO detection preprocessing for correlated channels, characterized in that, Includes the following steps: Step a1: Model the uplink of the large-scale MIMO system; Step a2: Calculate the Gram matrix W; Step a3: Obtain the preprocessing matrix P based on the Gram matrix W; Step a4: Preprocess the Gram matrix W using the preprocessing matrix P to obtain a new Gram matrix. : =PW; Step a5, convert the matrix Iteration is used instead of matrix inversion algorithm for processing; Step a3 includes: Step a3-1, extract the Gram matrix For the diagonals other than the main diagonal, give the scaling factor. ],in The value of i ranges from 1 to k-1. This represents the scaling factor for the i-th order diagonal; Step a3-2: Obtain the preprocessing matrix using the scaling factor and the extracted diagonal values. ; In step a3-2, the preprocessing matrix is ​​obtained according to the following formula. : ， in , Denotes the second diagonal of W. This represents the third diagonal, and so on. and Represents the (i+1)th diagonal; express The identity matrix; This indicates a mapping relationship, i.e., the expression on the right side of the equals sign.

2. The method according to claim 1, characterized in that, Step a1 includes: In the uplink of a massive MIMO system, the number of transmitting antennas is K and the number of receiving antennas is N.

3. The method according to claim 2, characterized in that, In step a1, the signal vector is transmitted. Receive signal vector ,in This represents the transmitted signal of the Kth antenna. Let represent the received signal of the Nth receiving antenna, then the system model is expressed as: ,in The channel matrix; This indicates that the mean is 0 and the variance is 0. Additive white Gaussian noise; where, considering interference between antennas in the actual channel, the Kronecker channel model is used for modeling, expressed as: ， in, Indicates a Rayleigh fading channel. Represents an N x K matrix; The correlation matrix represents the receiving antenna. This represents the correlation matrix of the transmitting antenna.

4. The method according to claim 3, characterized in that, In step a1, the matrix and They are represented as follows: ， in, Representation matrix The Middle Line number Column elements, Representation matrix The Middle Line number Column elements, and Represents the correlation coefficient. Represents the phase factor. express The conjugate transpose of . express The conjugate transpose of , where e is the natural constant and j is the imaginary unit.

5. The method according to claim 4, characterized in that, In step a2, the Gram matrix is ​​calculated. ,in, Represents the conjugate transpose of a matrix, Gram matrix It is a K×K matrix.

6. A method for pre-training mMIMO detection parameters for correlated channels, characterized in that, Includes the following steps: Step b1, based on the actual number of antennas N at the receiving end and the number of antennas K at the transmitting end, and the given correlation coefficient... and The following formula is used to obtain n frames of data, and each data point corresponds to the H matrix of the n frames of data: ， ； in, Indicates a Rayleigh fading channel. Represents an N x K matrix; The correlation matrix represents the receiving antenna. The correlation matrix representing the transmitting antenna; Representation matrix The Middle Line number Column elements, Representation matrix The Middle Line number Column elements, and Represents the correlation coefficient. Represents the phase factor. express The conjugate transpose of . express The conjugate transpose of , where e is the natural constant and j is the imaginary unit; Step b2: Based on the H matrix of the n frames of data, obtain n new Gram matrices using steps a1 to a4 of the method described in any one of claims 1 to 5. The maximum and minimum eigenvalues ​​are obtained from n new Gram matrices to obtain the set of maximum eigenvalues. and the set of minimum eigenvalues ; Let these represent the largest and smallest eigenvalues ​​of the nth matrix, respectively. Step b3, based on the set of largest eigenvalues and the set of minimum eigenvalues Estimates of the largest eigenvalue were obtained respectively. and estimation of the minimum eigenvalue : ， ， in, and It is a constant, and ; Step b4, based on the obtained estimate of the largest eigenvalue and estimation of the minimum eigenvalue The scaling factor is calculated by replacing the matrix inversion algorithm with different iterations.

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