[R] How to solve a non-linear system of equations using R

[Ravi Varadhan]

Jorge,

You can use the package "BB" to try and solve this problem.

I have re-written your functions a little bit.

 # ------------------------------
# Constants
# ------------------------------

l=1
m=0.4795
s=0.4795

# ------------------------------
# Functions to estimate f_i-k_i
# ------------------------------

myfn <- function(d){
d1 <- d[1]
d2 <- d[2]
d3 <- d[3]
d4 <- d[4]
res <- rep(NA, 4)
res[1] <-
2*d1+2*sqrt(2)*d1*d2+2*sqrt(3)*d2*d3+4*d3*d4-l*m*(1+d1^2+d2^2+d3^2+d4^2)
res[2] <-
2*sqrt(2)*d2+2*d1^2+2*sqrt(6)*d1*d3+4*d2^2+4*sqrt(3)*d2*d4+6*d3^2+8*d4^2-l*(
m^2+m^3*s^(-1))*(1+d1^2+d2^2+d3^2+d4^2)
res[3] <-
6*d1+12*sqrt(2)*d1*d2+18*sqrt(3)*d2*d3+48*d3*d4+2*sqrt(6)*d3+4*sqrt(6)*d1*d4
-l*(m^3+3*m^4*s^(-1)+3*m^6*s^(-2))*(1+d1^2+d2^2+d3^2+d4^2)
res[4] <-
12*sqrt(2)*d2+12*d1^2+36*d2^2+72*d3^2+120*d4^2+20*sqrt(6)*d1*d3+56*sqrt(3)*d
2*d4+4*sqrt(6)*d4-l*((m^4+6*m^6*s^(-1)+15*m^6*s^(-2)+15*m^7*s^(-3))-3*l^(2)*
(m^2+m^3*s^(-1))^2)*(1+d1^2+d2^2+d3^2+d4^2)
res
}

myfn.opt <- function(d){
# re-writing "myfn" to be used for minimization
d1 <- d[1]
d2 <- d[2]
d3 <- d[3]
d4 <- d[4]
res <- rep(NA, 4)
res[1] <-
2*d1+2*sqrt(2)*d1*d2+2*sqrt(3)*d2*d3+4*d3*d4-l*m*(1+d1^2+d2^2+d3^2+d4^2)
res[2] <-
2*sqrt(2)*d2+2*d1^2+2*sqrt(6)*d1*d3+4*d2^2+4*sqrt(3)*d2*d4+6*d3^2+8*d4^2-l*(
m^2+m^3*s^(-1))*(1+d1^2+d2^2+d3^2+d4^2)
res[3] <-
6*d1+12*sqrt(2)*d1*d2+18*sqrt(3)*d2*d3+48*d3*d4+2*sqrt(6)*d3+4*sqrt(6)*d1*d4
-l*(m^3+3*m^4*s^(-1)+3*m^6*s^(-2))*(1+d1^2+d2^2+d3^2+d4^2)
res[4] <-
12*sqrt(2)*d2+12*d1^2+36*d2^2+72*d3^2+120*d4^2+20*sqrt(6)*d1*d3+56*sqrt(3)*d
2*d4+4*sqrt(6)*d4-l*((m^4+6*m^6*s^(-1)+15*m^6*s^(-2)+15*m^7*s^(-3))-3*l^(2)*
(m^2+m^3*s^(-1))^2)*(1+d1^2+d2^2+d3^2+d4^2)
sum(res^2)
}

library(BB)
 
p0 <- runif(4, -1,0)
ans1 <- dfsane(par=p0, fn=myfn)
ans2 <- spg(par=p0, fn=myfn.opt)
ans1
ans2


Note that the above does not produce a redual of zero, so the system can't
be solved exactly.  I tried a large number of random starting values without
improving upon the solution provided by "spg".  So, you may want to check
your system for its correctness.

Hope this helps,
Ravi.

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On
Behalf Of Jorge Ivan Velez
Sent: Tuesday, June 03, 2008 5:09 PM
To: R mailing list
Subject: [R] How to solve a non-linear system of equations using R

Dear R-list members,

I've had a hard time trying to solve a non-linear system (nls) of equations
which structure for the equation i, i=1,...,4, is as follows:


                f_i(d_1,d_2,d_3,d_4)-k_i(l,m,s) = 0    (1)


In the expression above, both f_i and k_i are known functions and l, m and s
are known constants. I would like to estimate the vector d=(d_1,d_2,d_3,d_4)
which is solution of (1). Functions in R to estimate f_i-k_i are at the end
of this message.

Any help/suggestions/comments would be greatly appreciated.

Thanks in advance,

Jorge


# ------------------------------
# Constants
# ------------------------------

l=1
m=0.4795
s=0.4795

# ------------------------------
# Functions to estimate f_i-k_i
# ------------------------------

f1=function(d){
d1=d[1]
d2=d[2]
d3=d[3]
d4=d[4]
res=2*d1+2*sqrt(2)*d1*d2+2*sqrt(3)*d2*d3+4*d3*d4-l*m*(1+d1^2+d2^2+d3^2+d4^2)
res
}

f2=function(d){
d1=d[1]
d2=d[2]
d3=d[3]
d4=d[4]
res=2*sqrt(2)*d2+2*d1^2+2*sqrt(6)*d1*d3+4*d2^2+4*sqrt(3)*d2*d4+6*d3^2+8*d4^2
-l*(m^2+m^3*s^(-1))*(1+d1^2+d2^2+d3^2+d4^2)
res
}

f3=function(d){
d1=d[1]
d2=d[2]
d3=d[3]
d4=d[4]
res=6*d1+12*sqrt(2)*d1*d2+18*sqrt(3)*d2*d3+48*d3*d4+2*sqrt(6)*d3+4*sqrt(6)*d
1*d4-l*(m^3+3*m^4*s^(-1)+3*m^6*s^(-2))*(1+d1^2+d2^2+d3^2+d4^2)
res
}


f4=function(d){
d1=d[1]
d2=d[2]
d3=d[3]
d4=d[4]
res=12*sqrt(2)*d2+12*d1^2+36*d2^2+72*d3^2+120*d4^2+20*sqrt(6)*d1*d3+56*sqrt(
3)*d2*d4+4*sqrt(6)*d4-l*((m^4+6*m^6*s^(-1)+15*m^6*s^(-2)+15*m^7*s^(-3))-3*l^
(2)*(m^2+m^3*s^(-1))^2)*(1+d1^2+d2^2+d3^2+d4^2)
res
}

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