Nothing
#' Jacobi method
#'
#' Jacobi method is an iterative algorithm for solving a system of linear equations,
#' with a decomposition \eqn{A = D+R} where \eqn{D} is a diagonal matrix.
#' For a square matrix \eqn{A}, it is required to be diagonally dominant. For an overdetermined system where \code{nrow(A)>ncol(A)},
#' it is automatically transformed to the normal equation. Underdetermined system -
#' \code{nrow(A)<ncol(A)} - is not supported.
#'
#' @param A an \eqn{(m\times n)} dense or sparse matrix. See also \code{\link[Matrix]{sparseMatrix}}.
#' @param B a vector of length \eqn{m} or an \eqn{(m\times k)} matrix (dense or sparse) for solving \eqn{k} systems simultaneously.
#' @param xinit a length-\eqn{n} vector for initial starting point. \code{NA} to start from a random initial point near 0.
#' @param reltol tolerance level for stopping iterations.
#' @param maxiter maximum number of iterations allowed.
#' @param weight a real number in \eqn{(0,1]}; 1 for native Jacobi.
#' @param adjsym a logical; \code{TRUE} to symmetrize the system by transforming the system into normal equation, \code{FALSE} otherwise.
#' @param verbose a logical; \code{TRUE} to show progress of computation.
#'
#' @return a named list containing \describe{
#' \item{x}{solution; a vector of length \eqn{n} or a matrix of size \eqn{(n\times k)}.}
#' \item{iter}{the number of iterations required.}
#' \item{errors}{a vector of errors for stopping criterion.}
#' }
#'
#' @examples
#' ## Overdetermined System
#' set.seed(100)
#' 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)))
#'
#' @references
#' \insertRef{demmel_applied_1997}{Rlinsolve}
#'
#' @rdname basic_JACOBI
#' @export
lsolve.jacobi <- function(A,B,xinit=NA,reltol=1e-5,
maxiter=1000,weight=2/3,adjsym=TRUE,verbose=TRUE){
if (verbose){
message("* lsolve.jacobi : Initialiszed.")
}
if (any(is.na(A))||any(is.infinite(A))||any(is.na(B))||any(is.infinite(B))){
stop("* lsolve.jacobi : no NA or Inf values allowed.")
}
# Preprocessing : sparsity
# https://dirk.eddelbuettel.com/tmp/RcppArmadillo-sparseMatrix.pdf
sparseformats = c("dgCMatrix","dtCMatrix","dsCMatrix")
if (aux.is.sparse(A)||aux.is.sparse(B)){
A = Matrix(A,sparse=TRUE)
B = Matrix(B,sparse=TRUE)
sparseflag = TRUE
} else {
A = matrix(A,nrow=nrow(A))
if (is.vector(B)){
B = matrix(B)
} else {
B = matrix(B,nrow=nrow(B))
}
sparseflag = FALSE
}
# xinit
if (length(xinit)==1){
if (is.na(xinit)){
xinit = matrix(rnorm(ncol(A)))
} else {
stop("* lsolve.jacobi : please use a valid 'xinit'.")
}
} else {
if (length(xinit)!=ncol(A)){
stop("* lsolve.jacobi: 'xinit' has invalid size.")
}
xinit = matrix(xinit)
}
# Preprocessing : symmetricity warning
if (nrow(A)==ncol(A)){
if (norm(abs(t(A)-A),"f")>1e-10){
if (verbose){
message("* lsolve.jacobi : A may not be symmetric.")
}
if (adjsym){
B = t(A)%*%B
A = t(A)%*%A
if (verbose){
message("* lsolve.jacobi : making it normal equation form via 'adjsym' flag.")
}
}
}
}
# Preprocessing : JACOBI ONLY : weight should be (0,1]
if ((weight<=0)||(weight>1)){
stop("* lsolve.jacobi : weight should be a positive real number in (0,1].")
}
# Preprocessing : no NA or Inf
if (any(is.infinite(A))||any(is.na(A))||any(is.infinite(B))||any(is.na(B))){
stop("* lsolve.jacobi : no NA, Inf, -Inf values are allowed.")
}
# Preprocessing : size argument : A and B
m = nrow(A)
if (is.vector(B)){
mB = length(B)
if (m!=mB){
stop("* lsolve.jacobi : a vector B should have a length of nrow(A).")
}
} else {
mB = nrow(B)
if (m!=mB){
stop("* lsolve.jacobi : an input matrix B should have the same number of rows from A.")
}
}
if (is.vector(B)){
B = as.matrix(B)
}
# Preprocessing : size argument : A case
# Overdetermined - A'Ax = A'b
# Underdetermined - not supporting this case.
n = ncol(A)
if (m<n){
stop("* lsolve.jacobi : underdetermined case is not supported.")
} else if (m>n){
B = (t(A)%*%B)
A = (t(A)%*%A)
if (verbose){
message("* lsolve.jacobi : overdetermined case : turning into normal equation.")
}
}
# Preprocessing : aux.is.dd
if (aux.is.dd(A)==FALSE){
if (verbose){
message("* lsolve.jacobi : LHS matrix A is not diagonally dominant.")
message("* : solution from Jacobi method is not guaranteed.")
}
}
# # Preprocessing : aux.is.dd
# if (aux.is.dd(A)==FALSE){
# if (verbose){
# message("* lsolve.jacobi : LHS matrix A is not diagonally dominant.")
# message("* : solution from Jacobi method is not guaranteed.")
# }
# }
# Preprocessing : adjust diagonal entries for A
if (any(diag(A)==0)){
cvec = rnorm(10)
adjconst = cvec[sample(which(cvec!=0),1)]/(1e+5)
diag(A) = diag(A)+adjconst
}
# Main Computation
ncolB = ncol(B)
if (ncolB==1){
if (!sparseflag){
vecB = as.vector(B)
res = linsolve.jacobi.single(A,vecB,xinit,reltol,maxiter,weight)
} else {
vecB = B
res = linsolve.jacobi.single.sparse(A,vecB,xinit,reltol,maxiter,weight)
}
} else {
x = array(0,c(n,ncolB))
iter = array(0,c(1,ncolB))
errors = list()
for (i in 1:ncolB){
if (!sparseflag){
vecB = as.vector(B[,i])
tmpres = linsolve.jacobi.single(A,vecB,xinit,reltol,maxiter,weight)
} else {
vecB = Matrix(B[,i],sparse=TRUE)
tmpres = linsolve.jacobi.single.sparse(A,vecB,xinit,reltol,maxiter,weight)
}
x[,i] = tmpres$x
iter[i] = tmpres$iter
errors[[i]] = tmpres$errors
if (verbose){
message(paste("* lsolve.jacobi : B's column.",i,"being processed.."))
}
}
res = list("x"=x,"iter"=iter,"errors"=errors)
}
# Return
if (verbose){
message("* lsolve.jacobi : computations finished.")
}
return(res)
}
Any scripts or data that you put into this service are public.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.