 Gauss–Seidel method

With $A=D+L+U$, the Gauss-Seidel method solves $(D+L)x^{(k+1)}=b-U{x}^{(k)}$. It uses each newly computed coordinate immediately in the remaining triangular solve. It is successive over-relaxation with parameter one, and converges from every start when $A$ is a symmetric positive-definite matrix.

To solve a linear system $Ax=b$, split its matrix into the diagonal matrix $D$ and the strictly lower and upper parts $L,U$, retaining their actual signs. Every diagonal entry must be nonzero. A sweep updates coordinates in order:The Jacobi method uses the old vector throughout; the Gauss-Seidel method reuses each fresh coordinate. Thus the update order is part of the method.^[ref]

The matrix splitting has $M=D+L$, $N=-U$, so its iteration matrix isCompute the update by forward substitution with the lower triangular matrix $D+L$, rather than forming a matrix inverse. For a sparse matrix, a sweep only visits stored entries and can overwrite the working vector. Dependencies on fresh coordinates constrain parallel execution. This makes the Gauss-Seidel method an inexpensive solver component, including in a multigrid method.^[ref]

The Gauss-Seidel method is an iterative technique used to solve a system of linear equations of the form \(Ax = b\), where \(A\) is a matrix, \(x\) is the vector of unknowns, and \(b\) is the output vector. This method is particularly useful for large systems where direct methods like Gaussian elimination might be computationally expensive.