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solv-eq.R
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solv-eq.R
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#############################################################################
# Copyright (c) 2012 Christophe Dutang
#
# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program; if not, write to the
# Free Software Foundation, Inc.,
# 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA
#
#############################################################################
### equation solver in GNE
###
### R functions
###
eqsolve <- function(xinit, f, jac,
method=c("Newton", "Levenberg-Marquardt", "Broyden"),
global=c("line search", "none"), control=list())
{
method <- match.arg(method, c("Newton", "Levenberg-Marquardt","Broyden"))
global <- match.arg(global, c("line search", "none"))
#default control parameters
con <- list(tol=1e-6, maxit=100, echo=0, sigma=1/2, btol=1e-2,
echofile=NULL, echograph="NULL", delta=2, initlnsrch=0, minstep=1e-5)
namc <- names(con)
con[namc <- names(control)] <- control
xk <- xinit
fk <- f(xinit)
if(method == "Broyden")
{
# Wk <- sqrt(sum(fk^2)) * diag(length(xk))
Bk <- jac(xinit)
}
if(method != "Broyden")
Jacfk <- jac(xinit)
iter <- 0
if(class(con$echo) == "character")
stop("wrong echo parameter.")
if(!is.numeric(con$tol) || !is.numeric(con$maxit) || !is.numeric(con$sigma) || !is.numeric(con$btol))
stop("wrong control parameters.")
if(!is.null(con$echofile))
{
dirbefore <- getwd()
dirname <- sub(":", "m", sub(":", "h", Sys.time()))
dirname <- paste(dirname, " - example", sep="")
dir.create(dirname, showWarnings=FALSE)
setwd(paste("./", dirname, sep=""))
}else
{
dirname <- NULL
}
#traces in R console
if(con$echo >= 1 && is.null(con$echofile))
{
cat("**** k", iter, "\n")
cat("x_k", xk, "\n")
}
#traces in text file
if(con$echo >= 2 && !is.null(con$echofile))
cat("**** k", iter, "\n", file=con$echofile, append=TRUE)
if(con$echo >= 1 && !is.null(con$echofile))
cat("x_k", xk, "\n", file=con$echofile, append=TRUE)
while(sqrt( sum( fk^2 ) ) > con$tol && iter < con$maxit)
{
if(method == "Newton")
xkp1 <- try( NewtonNext(xk, fk, Jacfk, TRUE) )
if(method == "Levenberg-Marquardt")
xkp1 <- try( LevenMarqNext(xk, fk, Jacfk, TRUE, con$delta) )
if(method == "Broyden")
xkp1 <- try( QuasiNewtonNext(xk, fk, Bk, silent=TRUE, inv=FALSE) )
if(class(xkp1) == "try-error")
break
if(con$echo >= 1 && is.null(con$echofile))
cat("**** k", iter+1, "\n")
if(con$echo >= 2 && !is.null(con$echofile))
cat("**** k", iter+1, "\n", file=con$echofile, append=TRUE)
dk <- xkp1 - xk
if(con$echograph == "line")
{
if(!is.null(con$echofile))
png(paste("line search - ", iter, ".png", sep=""), 1600, 800)
par(mfrow=c(1, 2))
graphlinesearch(f, xk, dk, iter)
graphContourOneIter(f, xk, dk, iter, delta=.2)
if(!is.null(con$echofile))
dev.off()
}
if(global == "line search")
{
dk <- xkp1 - xk
normfk <- crossprod(fk)/2
i <- con$initlnsrch
if(method == "Broyden")
slopek <- crossprod(Bk %*% dk, fk)
if(method != "Broyden")
slopek <- crossprod(Jacfk %*% dk, fk)
while(i < 32)
{
stepk <- con$sigma^i
normfp <- crossprod(f(xk + stepk*dk))/2
#traces in R console
if(con$echo >= 3 && is.null(con$echofile))
cat("i", i, "\tlambda", stepk, "\tright term\t", normfk + con$btol * stepk * slopek, "\tleft term", normfp, "\n")
#traces in text file
if(con$echo >= 3 && !is.null(con$echofile))
cat("i", i, "\tlambda", stepk, "\txk + stepk*dk\t", xk + stepk*dk, "\n", file=con$echofile, append=TRUE)
cat("largest\t", max(abs(f(xk + stepk*dk))), "\n")
#check Armijo condition
if(normfp <= normfk + con$btol * stepk * slopek)
{
cat("Armijo satisfied\n")
break
}
cat("\tnorm stepk*dk\t", sqrt(sum(stepk*dk^2)), "\n\n")
i <- i+1
}
xkp1 <- xk + stepk*dk
}
xk_1 <- xk
xk <- xkp1
#control step size
if(sqrt( sum( (xk - xk_1)^2 ) ) <= con$minstep)
{
cat("too small step\n")
break
}
fk_1 <- fk
fk <- f(xk)
if(method == "Broyden")
{
sk <- xk - xk_1
yk <- fk - fk_1
# Wkyk <- Wk %*% yk
# Wk <- Wk + (sk - Wkyk) %*% t(Wkyk) / as.numeric(crossprod(sk, Wkyk))
Bk <- Bk + (yk - Bk %*% sk) %*% t(sk) / as.numeric(crossprod(sk))
if(any(is.nan(Bk)))
break
}
if(method != "Broyden")
{
Jacfk <- jac(xk)
if(any(is.nan(Jacfk)))
break
}
iter <- iter+1
#traces in R console
if(con$echo >= 1 && is.null(con$echofile))
cat("x_k", xk, "\n")
if(con$echo >= 2 && is.null(con$echofile))
cat(" ||f(x_k)||", sqrt( sum(fk^2) ), "\n")
if(con$echo >= 4 && is.null(con$echofile))
{
if(method != "Broyden")
{
cat(" Jac Phi(x_k)\n")
print(Jacfk)
}
if(method == "Broyden")
{
cat(" B_k\n")
print(Bk)
}
}
if(con$echo >= 1 && is.null(con$echofile))
cat("\n")
#traces in text file
if(con$echo >= 1 && !is.null(con$echofile))
cat("x_k", xk, "\n", file=con$echofile, append=TRUE)
if(con$echo >= 2 && !is.null(con$echofile))
cat(" ||f(x_k)||", sqrt( sum(fk^2) ), "\n", file=con$echofile, append=TRUE)
if(con$echo >= 4&& !is.null(con$echofile))
{
if(method != "Broyden")
{
cat(" Jac Phi(x_k)\n", file=con$echofile, append=TRUE)
capture.output(print(Jacfk), file=con$echofile, append=TRUE)
}
if(method == "Broyden")
{
cat(" B_k\n", file=con$echofile, append=TRUE)
capture.output(print(Bk), file=con$echofile, append=TRUE)
}
}
if(con$echo >= 2 && !is.null(con$echofile))
cat("\n", file=con$echofile, append=TRUE)
}
if(!is.null(con$echofile))
setwd(dirbefore)
if(method == "Broyden")
exitcode <- (iter >= con$maxit)*1 + (sqrt( sum( fk^2 ) ) > con$tol)*10 + any(is.nan(Bk))*100
if(method != "Broyden")
exitcode <- (iter >= con$maxit)*1 + (sqrt( sum( fk^2 ) ) > con$tol)*10 + any(is.nan(Jacfk))*100
list(par= as.vector(xk), value=sqrt( sum( fk^2 ) ), counts=iter, iter=iter,
code= exitcode, message=NULL, dirname=dirname, echofile=con$echofile)
}
graphlinesearch <- function(f, xk, dk, k, tbound=c(0, 2))
{
phi <- function(t) sapply(t, function(u) crossprod(f(xk + u* dk)) /2)
times <- seq(tbound[1], tbound[2], length=201)
yval <- phi(times)
plot(times, yval, type="l", main=paste("phi_k(t) (k=", k, ")", sep=""), ylim=c(min(yval), yval[1]), xlab="t", ylab="phi_k(t)")
abline(h=phi(0), lty=2, col="grey")
legend("topright", legend=paste("xk", as.vector(xk)))
}
graphContourOneIter <- function(f, xk, dk, k, delta=.2, tols=c(1e-1, 1e-3, 1e-6),
cex=2, line=TRUE, tbound=c(0, 1/2, 1, 2))
{
if(length(xk) == 2)
{
xkp1 <- xk + dk
xlim <- cbind( min = (1-delta)*pmin(xk, xkp1)*(pmin(xk, xkp1) >0) + (1+delta)*pmin(xk, xkp1)*(pmin(xk, xkp1) <0) - delta*(pmin(xk, xkp1) == 0),
max = (1+delta)*pmax(xk, xkp1)*(pmax(xk, xkp1) >0) + (1-delta)*pmax(xk, xkp1)*(pmax(xk, xkp1) <0) + delta*(pmin(xk, xkp1) == 0) )
x <- seq(xlim[1, "min"], xlim[1, "max"], length=101)
y <- seq(xlim[2, "min"], xlim[2, "max"], length=101)
normf <- function(x, y)
{
xy <- cbind(x, y)
sapply(1:NROW(xy), function(i) crossprod(f(xy[i, ])/2) )
}
z <- outer(x, y, normf )
colnames(z) <- paste("y", y)
rownames(z) <- paste("x", x)
contour(x, y, z, nlevels=20, main="Contour of norm(F(x)), with 'zeros'", cex=cex, col="grey50")
idzero <- which(abs(z) < tols[1], arr.ind=TRUE)
points(cbind( x[idzero[, "row"]], y[idzero[, "col"]] ), col="yellow", pch=".", cex=1.5)
idzero <- which(abs(z) < tols[2], arr.ind=TRUE)
points(cbind( x[idzero[, "row"]], y[idzero[, "col"]] ), col="orange", pch=20)
idzero <- which(abs(z) < tols[3], arr.ind=TRUE)
points(cbind( x[idzero[, "row"]], y[idzero[, "col"]] ), col="red", pch=19)
if(line)
{
trials <- function(t) sapply(t, function(u) xk + u* dk)
times <- c(tbound[1], tbound[2], tbound[3], tbound[4])
vals <- t(trials(times))
lines(vals[3:4,1], vals[3:4,2], col="black", lty="dotted")
lines(vals[2:3,1], vals[2:3,2], col="black", lty="dotdash")
lines(vals[1:2,1], vals[1:2,2], col="black", lty="solid")
points(vals[1,1], vals[1,2], pch=4)
legend("topright", legend=c("t<=1/2", "t<=1", "t<=2"), lty=c("dotted","dotdash","solid"), col="black")
}
}
}
graphContourMultIter <- function(f, xks, xlim,
option, nbgrid=101, tols=c(1e-1, 1e-3, 1e-6),
cex=1, lwd=1, itercol=c("blue", "green"), zerocol=c("yellow", "orange","red"),
title=NULL, posleg=NULL)
{
if(missing(xlim))
xlim <- apply(xks, 2, range)
if(is.null(colnames(xlim)))
colnames(xlim) <- c("min", "max")
if(any( colnames(xlim) != c("min", "max") ) )
colnames(xlim) <- c("min", "max")
option <- match.arg(option, c("none", "zeros", "colored", "numbered"), several.ok=TRUE)
if(!is.null(title))
{
mytitle <- paste("Contour of ||F(x)||", title)
}else if(length(option) == 1)
{
mytitle <- switch(option,
none= "Contour of norm(F(x))",
zeros = "Contour of norm(F(x)), with 'zeros'",
colored = "Contour of norm(F(x)), with colored iterates",
numbered = "Contour of norm(F(x)), with numbered iterates")
}else
mytitle <- "Contour of norm(F(x)), with iterates and zeros"
x <- seq(xlim[1, "min"], xlim[1, "max"], length= nbgrid)
y <- seq(xlim[2, "min"], xlim[2, "max"], length= nbgrid)
normf <- function(x, y)
{
xy <- cbind(x, y)
sapply(1:NROW(xy), function(i) crossprod(f(xy[i, ])/2) )
}
z <- outer(x, y, normf )
colnames(z) <- paste("y", y)
rownames(z) <- paste("x", x)
contour(x, y, z, nlevels=20, main=mytitle, cex=cex, col="grey50", xlab="x_1", ylab="x_2")
if("zeros" %in% option)
{
idzero <- which(abs(z) < tols[1], arr.ind=TRUE)
points(cbind( x[idzero[, "row"]], y[idzero[, "col"]] ), col=zerocol[1], pch=".", cex=cex)
idzero <- which(abs(z) < tols[2], arr.ind=TRUE)
points(cbind( x[idzero[, "row"]], y[idzero[, "col"]] ), col=zerocol[2], pch=20)
idzero <- which(abs(z) < tols[3], arr.ind=TRUE)
points(cbind( x[idzero[, "row"]], y[idzero[, "col"]] ), col=zerocol[3], pch=19)
if(!is.null(posleg))
legend(posleg, legend=c(paste("<", rev(tols), sep=""), paste(">", tols[1], sep="")),
fill=c(rev(zerocol), "white"), bg ="white")
}
if("colored" %in% option)
{
mycol <- colorRampPalette( itercol )(NROW(xks))
points(xks[,1], xks[,2], col=mycol, pch=3, cex=cex, lwd=lwd)
}
if("numbered" %in% option)
{
n <- NROW(xks)
mycol <- colorRampPalette( itercol )(n)
text(xks[,1], xks[,2], 0:(n-1), col=mycol, cex=cex, lwd=lwd)
}
}