basic_GS: Gauss-Seidel method

In kisungyou/Rlinsolve: Iterative Solvers for (Sparse) Linear System of Equations

Gauss-Seidel method

Description

Gauss-Seidel(GS) method is an iterative algorithm for solving a system of linear equations, with a decomposition A = D+L+U where D is a diagonal matrix and L and U are strictly lower/upper triangular matrix respectively. For a square matrix A, it is required to be diagonally dominant or symmetric and positive definite. For an overdetermined system where nrow(A)>ncol(A), it is automatically transformed to the normal equation. Underdetermined system - nrow(A)<ncol(A) - is not supported.

Usage

lsolve.gs(
  A,
  B,
  xinit = NA,
  reltol = 1e-05,
  maxiter = 1000,
  adjsym = TRUE,
  verbose = TRUE
)

Arguments

A	an (m\times n) dense or sparse matrix. See also sparseMatrix.
B	a vector of length m or an (m\times k) matrix (dense or sparse) for solving k systems simultaneously.
xinit	a length-n vector for initial starting point. NA to start from a random initial point near 0.
reltol	tolerance level for stopping iterations.
maxiter	maximum number of iterations allowed.
adjsym	a logical; TRUE to symmetrize the system by transforming the system into normal equation, FALSE otherwise.
verbose	a logical; TRUE to show progress of computation.

Value

a named list containing

x

solution; a vector of length n or a matrix of size (n\times k).

iter

the number of iterations required.

errors

a vector of errors for stopping criterion.

References

\insertRef

demmel_applied_1997Rlinsolve

Examples

## Overdetermined System
set.seed(100)
A = matrix(rnorm(10*5),nrow=10)
x = rnorm(5)
b = A%*%x

out = lsolve.gs(A,b)
matout = cbind(matrix(x),out$x); colnames(matout) = c("true x","est from GS")
print(matout)

kisungyou/Rlinsolve documentation built on Aug. 21, 2023, 4:52 a.m.