| # Copyright (C) 1997-2009, 2017 The R Core Team |
| |
| ### Helical Valley Function |
| ### Page 362 Dennis + Schnabel |
| |
| require(stats); require(graphics); require(utils) |
| |
| theta <- function(x1,x2) (atan(x2/x1) + (if(x1 <= 0) pi else 0))/ (2*pi) |
| ## but this is easier : |
| theta <- function(x1,x2) atan2(x2, x1)/(2*pi) |
| |
| f <- function(x) { |
| f1 <- 10*(x[3] - 10*theta(x[1],x[2])) |
| f2 <- 10*(sqrt(x[1]^2+x[2]^2)-1) |
| f3 <- x[3] |
| return(f1^2 + f2^2 + f3^2) |
| } |
| |
| ## explore surface {at x3 = 0} |
| x <- seq(-1, 2, length.out=50) |
| y <- seq(-1, 1, length.out=50) |
| z <- apply(as.matrix(expand.grid(x, y)), 1, function(x) f(c(x, 0))) |
| contour(x, y, matrix(log10(z), 50, 50)) |
| str(nlm.f <- nlm(f, c(-1,0,0), hessian = TRUE)) |
| points(rbind(nlm.f$estim[1:2]), col = "red", pch = 20) |
| stopifnot(all.equal(nlm.f$estimate, c(1, 0, 0))) |
| |
| ### the Rosenbrock banana valley function |
| |
| fR <- function(x) |
| { |
| x1 <- x[1]; x2 <- x[2] |
| 100*(x2 - x1*x1)^2 + (1-x1)^2 |
| } |
| |
| ## explore surface |
| fx <- function(x) |
| { ## `vectorized' version of fR() |
| x1 <- x[,1]; x2 <- x[,2] |
| 100*(x2 - x1*x1)^2 + (1-x1)^2 |
| } |
| x <- seq(-2, 2, length.out=100) |
| y <- seq(-0.5, 1.5, length.out=100) |
| z <- fx(expand.grid(x, y)) |
| op <- par(mfrow = c(2,1), mar = 0.1 + c(3,3,0,0)) |
| contour(x, y, matrix(log10(z), length(x))) |
| |
| str(nlm.f2 <- nlm(fR, c(-1.2, 1), hessian = TRUE)) |
| points(rbind(nlm.f2$estim[1:2]), col = "red", pch = 20) |
| |
| ## Zoom in : |
| rect(0.9, 0.9, 1.1, 1.1, border = "orange", lwd = 2) |
| x <- y <- seq(0.9, 1.1, length.out=100) |
| z <- fx(expand.grid(x, y)) |
| contour(x, y, matrix(log10(z), length(x))) |
| mtext("zoomed in");box(col = "orange") |
| points(rbind(nlm.f2$estim[1:2]), col = "red", pch = 20) |
| par(op) |
| with(nlm.f2, |
| stopifnot(all.equal(estimate, c(1,1), tol = 1e-5), |
| minimum < 1e-11, abs(gradient) < 1e-6, code %in% 1:2)) |
| |
| fg <- function(x) |
| { |
| gr <- function(x1, x2) |
| c(-400*x1*(x2 - x1*x1)-2*(1-x1), 200*(x2 - x1*x1)) |
| x1 <- x[1]; x2 <- x[2] |
| structure(100*(x2 - x1*x1)^2 + (1-x1)^2, |
| gradient = gr(x1, x2)) |
| } |
| |
| nfg <- nlm(fg, c(-1.2, 1), hessian = TRUE) |
| str(nfg) |
| with(nfg, |
| stopifnot(minimum < 1e-17, all.equal(estimate, c(1,1)), |
| abs(gradient) < 1e-7, code %in% 1:2)) |
| |
| ## or use deriv to find the derivatives |
| |
| fd <- deriv(~ 100*(x2 - x1*x1)^2 + (1-x1)^2, c("x1", "x2")) |
| fdd <- function(x1, x2) {} |
| body(fdd) <- fd |
| nlfd <- nlm(function(x) fdd(x[1], x[2]), c(-1.2,1), hessian = TRUE) |
| str(nlfd) |
| with(nlfd, |
| stopifnot(minimum < 1e-17, all.equal(estimate, c(1,1)), |
| abs(gradient) < 1e-7, code %in% 1:2)) |
| |
| fgh <- function(x) |
| { |
| gr <- function(x1, x2) |
| c(-400*x1*(x2 - x1*x1) - 2*(1-x1), 200*(x2 - x1*x1)) |
| h <- function(x1, x2) { |
| a11 <- 2 - 400*x2 + 1200*x1*x1 |
| a21 <- -400*x1 |
| matrix(c(a11, a21, a21, 200), 2, 2) |
| } |
| x1 <- x[1]; x2 <- x[2] |
| structure(100*(x2 - x1*x1)^2 + (1-x1)^2, |
| gradient = gr(x1, x2), |
| hessian = h(x1, x2)) |
| } |
| |
| nlfgh <- nlm(fgh, c(-1.2,1), hessian = TRUE) |
| str(nlfgh) |
| ## NB: This did _NOT_ converge for R version <= 3.4.0 |
| with(nlfgh, |
| stopifnot(minimum < 1e-15, # see 1.13e-17 .. slightly worse than above |
| all.equal(estimate, c(1,1), tol=9e-9), # see 1.236e-9 |
| abs(gradient) < 7e-7, code %in% 1:2)) # g[1] = 1.3e-7 |