A Weighted Gauss-Seidel Iterative Algorithm with Fast Convergence
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Abstract:
Unacceptable amounts of computation can be generated in massive multiple-input multiple-output (MIMO) systems with minimum mean squared error (MMSE) detection at the received side. A weighted Gauss-Seidel iterative algorithm with fast convergence is launched. The proposed algorithm uses a mixture of Conjugate Gradient and Jacobi iterations to select the optimal search direction. Then weighting factor is used to accelerate the traditional Gauss-Seidel iterative algorithm. The results show that the detection capability of the scheme has been improved. The theoretical analysis verifies that the proposed algorithm has a lower computational complexity compared to the MMSE algorithm. After simulation analysis, the recommended algorithm can gain better convergence speed and BER function with fewer iterations. If user antennas setting values is similar to base station antennas, the proposed algorithm is significantly better.Keywords:
Gauss–Seidel method
Using the Gauss-Seidel iterative method for the solution of the linear equations,the convergence of the Gauss-Seidel iterative method is discussed under a type of preconditioned matrix.With a more general splitting,we compare the convergence of the preconditioned Gauss-Seidel iterative method and the corresponding Gauss-Seidel iterative method.Furthermore,we get some comparison theorems.Finally,a numerical example is given to illustrate the validity of the conclusions.
Gauss–Seidel method
Matrix (chemical analysis)
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Matrix (chemical analysis)
Sylvester matrix
Linear equation
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บทคดยอ วธทำซำสำหรบหาผลเฉลยระบบเชงเสนมอยดวยกน 2 แบบใหญ ๆ คอ วธทำซำอยางนงกบวธปรภมยอยไครลอฟ บทความวชาการนอภปรายแนวคดทวไปและเทคนคพนฐานของวธทำซำอยางนง ไดแก วธจาโคบ วธเกาส-ไซเดล และวธผอนปรนเกนสบเนอง นอกจากนนยงพจารณาวธทำซำทพฒนาตอยอดจากวธดงกลาว ไดแก วธผอนปรนเกนแบบเรง วธทำซำทมฐานจากเกรเดยนต เและวธทำซำกำลงสองนอยสด สำหรบวธปรภมยอยไครลอฟนนมตนแบบมาจากวธคอนจเกตเกรเดยนต วธดงกลาวจะสรางฐานหลกเชงตงฉากของปรภมแบบยคลดจากเมทรกซสมประสทธโดยพจารณาจากเกรเดยนตของฟงกชนกำลงสองทสอดคลอง ฐานหลกดงกลาวประกอบดวยเวกเตอรทมทศทางททำใหผลเฉลยคาประมาณเขาใกลผลเฉลยจรงไดเรวทสด กลาวโดยสรปไดวา วธทำซำอยางนง 4 วธแรกทกลาวมานนจะการนตการลเขาของลำดบของผลเฉลยโดยประมาณสผลเฉลยจรงเมอใชกบระบบทมเมทรกซสมประสทธอยในรปแบบเฉพาะ เชน เมทรกซแนวทแยงมมขมแท เมทรกซลดทอนไมได และเมทรกซแบบแอล โดยตองกำหนดตวแปรเสรมทเหมาะสม สวนวธทำซำทมฐานจากเกรเดยนตและวธทำซำกำลงสองนอยสดใชไดกบระบบทมเมทรกซสมประสทธมคาลำดบชนเตม สำหรบวธคอนจเกตเกรเดยนตใชไดกบระบบทเมทรกซสมประสทธเปนเมทรกซสมมาตรทเปนบวกแนนอน คำสำคญ: ระบบเชงเสน วธผอนปรนเกนสบเนอง วธผอนปรนเกนแบบเรง วธทำซำทมฐานจากเกรเดยนต วธคอนจเกตเกรเดยนต ABSTRACT There are two major types of iterative methods for solving linear systems, namely, stationary iterative methods and Krylov subspace methods. This survey article discusses general ideas and elementary techniques for stationary iterative methods such as Jacobi method, Gauss-Seidel method, and the successive over-relaxation method. Moreover, we investigate further developed methods, namely, the accelerated over-relaxation method, the gradient based iterative method, and the least squares iterative method. On the other hand, Krylov subspace methods have prototypes from the conjugate gradient method. The latter method constructs an orthogonal basis for the Euclidean space from the gradient of the associated quadratic function. Such basis consists of vectors in directions so that the approximated solutions fastest approach to the exact solution. In conclusions, all 1st-4th mentioned stationary iterative methods guarantee the convergence of the sequence of approximated solutions to the exact solution when applying to the system with specific coefficient matrices such as strictly diagonally dominant matrices, irreducible matrices, and L-matrices. Here, the parameters in the methods must be appropriate. The gradient based iterative method and the least squares iterative method can be applied to systems with full-column rank coefficient matrices. The conjugate gradient method is applicable for the system whose coefficient matrix is a positive definite symmetric matrix. Keywords: linear system, successive over-relaxation method, accelerated over-relaxation method, gradient based iterative method, conjugate gradient method
Coefficient matrix
Krylov subspace
Successive over-relaxation
Matrix (chemical analysis)
Biconjugate gradient method
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The preconditioned Gauss-Seidel iterative method is introduced. It is proved that if the coefficient matrix is an irreducible Z-matrix, H-matrix or positive definite matrix, then the preconditioned Gauss-Seidel iterative method is convergent. Thus it expands the applicable scope of the method. Finally, a simple numerical example shows the validity of the conclusions.
Gauss–Seidel method
Matrix (chemical analysis)
Coefficient matrix
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This paper discusses the Extrapolated Gauss-Seidel method with semi-iterative techniques to solve large sparse linear systems of the form Ax = b, when A is a generalized consistently ordered (GCO (p– 1,1)) matrix, (see [3], [2]) and the pth power of the Jacobi iterative matrix B possesses non-negative eigenvalues. The theoretical results demonstrate that the combination of the extrapolated Gauss-Seidel method with the semi-iterative strategy is competitive with the SOR method, and the larger p is, the better the method becomes.
Gauss–Seidel method
Matrix (chemical analysis)
M-matrix
Gaussian elimination
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Combining the iterative method and the column pivoting,two improved iterative methods for linear equations is given,based on the Gauss-Seidel iterative method and SOR iterative method.The improved method expands the range to the Gauss-Seidel iterative method and SOR iterative method,which brings practical application value.By writing MATLAB program,the convergence of Gauss-Seidel iterative and the SOR iterative methods are verified.Meanwhile,through to comparison of the convergence speed of the classical Gauss-Seidel iterative,SOR iterative method,the improved Gauss-Seidel iterative method,and SOR iterative method,a conclusion is given: the two new iterative methods has a wider scope,a faster convergence speed.
Gauss–Seidel method
Successive over-relaxation
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In this papert,he convergence analysis for a new preconditioned Gauss-Seidel iterative method was discussed.If the matrix is the strictly dominant L-matrixt,he convergence rate of the preconditioned Gauss-Seidel iterative method is faster than one of the original one.Furthermore,the spectral radius of the reconditioned Gauss-Seidel iterative method is monotonically decreasing.At last,an example was given to confirm the fact that the preconditioned Gauss-Seidel iterative method is better than the original one.
Gauss–Seidel method
Spectral Radius
Matrix (chemical analysis)
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First, the convergence conditions of Gauss - Seidel method and Jacobi method are shown. And then the iterative times of the two methods are discussed and the formula of estimate iterative times is obtained. Finally, a numerical example is calculated by the two methods and the results show the actual iterative times and the estimate iterative times are basically equal.
Gauss–Seidel method
Successive over-relaxation
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A new preconditioned matrix was introduced for solving linear systems with Gauss-Seidel iterative method.Under the condition that the coefficient matrix being an M-matrix,which is widely used,the comparison theorem between preconditioned Gauss-Seidel iterative method and classic Gauss-Seidel iterative method was given.The preconditioned Gauss-Seidel iterative method is convergent and it accelerates the convergent speed of classic Gauss-Seidel iterative method.It showed that newly proposed preconditioned Gauss-Seidel iterative method is superior to the Gauss-Seidel iterative method mentioned.A numerical example was given to verify the effectiveness of the conclusions.
Gauss–Seidel method
Matrix (chemical analysis)
M-matrix
Coefficient matrix
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