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#' Biconjugate Gradient method
#'
#' Biconjugate Gradient(BiCG) method is a modification of Conjugate Gradient for nonsymmetric systems using
#' evaluations with respect to \eqn{A^T} as well as \eqn{A} in matrix-vector multiplications.
#' 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. Preconditioning matrix \eqn{M}, in theory, should be symmetric and positive definite
#' with fast computability for inverse, though it is not limited until the solver level.
#'
#' @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 preconditioner an \eqn{(n\times n)} preconditioning matrix; default is an identity matrix.
#' @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
#' A = matrix(rnorm(10*5),nrow=10)
#' x = rnorm(5)
#' b = A%*%x
#'
#' out1 = lsolve.cg(A,b)
#' out2 = lsolve.bicg(A,b)
#' matout = cbind(matrix(x),out1$x, out2$x);
#' colnames(matout) = c("true x","CG result", "BiCG result")
#' print(matout)
#'
#' @references
#' \insertRef{watson_conjugate_1976}{SolveLS}
#'
#' \insertRef{voevodin_question_1983}{SolveLS}
#'
#' @rdname krylov_BICG
#' @export
lsolve.bicg <- function(A,B,xinit=NA,reltol=1e-5,maxiter=10000,
preconditioner=diag(ncol(A)),verbose=TRUE){
###########################################################################
# Step 0. Initialization
if (verbose){
message("* lsolve.bicg : Initialiszed.")
}
if (any(is.na(A))||any(is.infinite(A))||any(is.na(B))||any(is.infinite(B))){
stop("* lsolve.bicg : no NA or Inf values allowed.")
}
sparseformats = c("dgCMatrix","dtCMatrix","dsCMatrix")
if ((class(A)%in%sparseformats)||(class(B)%in%sparseformats)||(class(preconditioner)%in%sparseformats)){
A = Matrix(A,sparse=TRUE)
B = Matrix(B,sparse=TRUE)
preconditioner = Matrix(preconditioner,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))
}
preconditioner = matrix(preconditioner,nrow=nrow(preconditioner))
sparseflag = FALSE
}
# xinit
if (is.na(xinit)){
xinit = matrix(rnorm(ncol(A)))
} else {
if (length(xinit)!=ncol(A)){
stop("* lsolve.bicg : 'xinit' has invalid size.")
}
xinit = matrix(xinit)
}
###########################################################################
# Step 1. Preprocessing
# 1-1. Neither NA nor Inf allowed.
if (any(is.infinite(A))||any(is.na(A))||any(is.infinite(B))||any(is.na(B))){
stop("* lsolve.bicg : no NA, Inf, -Inf values are allowed.")
}
# 1-2. Size Argument
m = nrow(A)
if (is.vector(B)){
mB = length(B)
if (m!=mB){
stop("* lsolve.bicg : a vector B should have a length of nrow(A).")
}
} else {
mB = nrow(B)
if (m!=mB){
stop("* lsolve.bicg : an input matrix B should have the same number of rows from A.")
}
}
if (is.vector(B)){
B = as.matrix(B)
}
# 1-3. Adjusting Case
if (m > ncol(A)){ ## Case 1. Overdetermined
B = t(A)%*%B
A = t(A)%*%A
} else if (m < ncol(A)){ ## Case 2. Underdetermined
stop("* lsolve.bicg : underdetermined case is not supported.")
}
# 1-4. Preconditioner : only valid for square case
if (!all.equal(dim(A),dim(preconditioner))){
stop("* lsolve.bicg : Preconditioner is a size-matching.")
}
if (verbose){message("* lsolve.bicg : preprocessing finished ...")}
###########################################################################
# Step 2. Main Computation
ncolB = ncol(B)
if (ncolB==1){
if (!sparseflag){
vecB = as.vector(B)
res = linsolve.bicg.single(A,vecB,xinit,reltol,maxiter,preconditioner)
} else {
vecB = B
res = linsolve.bicg.single.sparse(A,vecB,xinit,reltol,maxiter,preconditioner)
}
} else {
x = array(0,c(ncol(A),ncolB))
iter = array(0,c(1,ncolB))
errors1 = list()
errors2 = list()
for (i in 1:ncolB){
if (!sparseflag){
vecB = as.vector(B[,i])
tmpres = linsolve.bicg.single(A,vecB,xinit,reltol,maxiter,preconditioner)
} else {
vecB = Matrix(B[,i],sparse=TRUE)
tmpres = linsolve.bicg.single.sparse(A,vecB,xinit,reltol,maxiter,preconditioner)
}
x[,i] = tmpres$x
iter[i] = tmpres$iter
errors1[[i]] = tmpres$errors1
errors2[[i]] = tmpres$errors2
if (verbose){
message(paste("* lsolve.bicg : B's column.",i,"being processed.."))
}
}
res = list("x"=x,"iter"=iter,"errors1"=errors1,"errors2"=errors2)
}
###########################################################################
# Step 3. Finalize
if ("flag" %in% names(res)){
flagval = res$flag
if (flagval==0){
if (verbose){
message("* lsolve.bicg : convergence well achieved.")
}
} else if (flagval==1){
if (verbose){
message("* lsolve.bicg : convergence not achieved within maxiter.")
}
} else {
if (verbose){
message("* lsolve.bicg : breakdown.")
}
}
}
if (verbose){
message("* lsolve.bicg : computations finished.")
}
return(res)
}
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