R/lsolve_BICG.R

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

#' 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
#' set.seed(100)
#' 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}{Rlinsolve}
#'
#' \insertRef{voevodin_question_1983}{Rlinsolve}
#'
#' @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 (aux.is.sparse(A)||aux.is.sparse(B)||aux.is.sparse(preconditioner)){
    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 (length(xinit)==1){
    if (is.na(xinit)){
      xinit = matrix(rnorm(ncol(A)))
    } else {
      stop("* lsolve.bicg: please use a valid 'xinit'.")
    }
  } 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)
}