Nothing
# Chebyquad functions (no solution for n=8)
library("nleqslv")
chebyquad <- function(x) {
n <- length(x)
y <- numeric(n)
for(j in 1:n) {
t1 <- 1.0
t2 <- 2.0*x[j] - 1.0
tmp <- 2.0*t2
for(i in 1:n) {
y[i] <- y[i] + t2
t3 <- tmp * t2 - t1
t1 <- t2
t2 <- t3
}
}
y = y / n
for(i in 1:n) {
if ( i%%2 == 0 ) {
y[i] = y[i] + 1.0 / (i * i - 1)
}
}
y
}
chebyinit <- function(n) {
x <- (1:n) / (n + 1)
}
for (n in c(1:7,9)) { # exclude n=8 since there is no solution
xstart <- chebyinit(n)
fstart <- chebyquad(xstart)
zz <- nleqslv(xstart, chebyquad, global="dbldog",
control=list(ftol=1e-8,xtol=1e-8,trace=0,btol=.01,delta=-2))
print(all(abs(zz$fvec)<=1e-8))
}
for (n in c(1:7,9)) { # exclude n=8 since there is no solution
xstart <- chebyinit(n)
fstart <- chebyquad(xstart)
zz <- nleqslv(xstart, chebyquad, global="dbldog", method="Newton",
control=list(ftol=1e-8,xtol=1e-8,trace=0,btol=.01,delta=-2))
print(all(abs(zz$fvec)<=1e-8))
}
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