basic_JACOBI: Jacobi method
In SolveLS: Iterative Methods for Solving Linear System of Equations

Description Usage Arguments Value References Examples

Jacobi method is an iterative algorithm for solving a system of linear equations, with a decomposition A = D+R where D is a diagonal matrix. For a square matrix A, it is required to be diagonally dominant. 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.

1 2	lsolve.jacobi(A, B, xinit = NA, reltol = 1e-05, maxiter = 1000, weight = 2/3, adjsym = TRUE, verbose = TRUE)

`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.
`weight`	a real number in (0,1]; 1 for native Jacobi.
`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.

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.

\insertRef

demmel_applied_1997SolveLS

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

out1 = lsolve.jacobi(A,b,weight=1,verbose=FALSE)   # unweighted
out2 = lsolve.jacobi(A,b,verbose=FALSE)            # weight of 0.66
out3 = lsolve.jacobi(A,b,weight=0.5,verbose=FALSE) # weight of 0.50
print("* lsolve.jacobi : overdetermined case example")
print(paste("*   error for unweighted    Jacobi case : ",norm(out1$x-x)))
print(paste("*   error for 0.66 weighted Jacobi case : ",norm(out2$x-x)))
print(paste("*   error for 0.50 weighted Jacobi case : ",norm(out3$x-x)))