R help - Aug 2007 - Solve equations

Hello,

 I have a system of five equations to solve with five 'unknows'(V, W, X,
Y and Z) and  constraints. The equations are:
0.007= 2VZ
0.03= W(Y+Z)
0.034= X(y+Z)
0.013 = (X+W)Y +(X-W)Z
X = W+V
Constraints:
0<V<W<X
0<Y<Z<1

Does anyone know a R-package to solve this system?

 Thanks,

E-mail: sebastien.puechmaille at ucd.ie

Try brute force:
> f <- function(x) {
+ V <- x[1]; W <- x[2]; X <- x[3]; Y <- x[4]; Z <- x[5]
+ if (!(V < W && W < X && Y < Z)) return(Inf)
+ ((0.007 - 2*V*Z)^2 +
+ (0.03 - W*(Y+Z))^2 +
+ (0.034 - X*(Y+Z))^2 +
+ (X - W+V)^2 +
+ (0.013 - (X+W)*Y +(X-W)*Z))^2
+ }
> s <- 1:10/10
> g <- expand.grid(V = s, W = s, X = s, Y = s, Z = s)
> idx <- which.min(apply(g, 1, f))
> idx
[1] 10541
> g[idx,]
        V   W   X   Y   Z
10541 0.1 0.5 0.6 0.1 0.2
> as.vector(f(g[idx,]))
[1] 3.8025e-08


On 8/6/07, sebastien puechmaille <sebastien.puechmaille at ucd.ie>
wrote:
> Hello,
>
>  I have a system of five equations to solve with five 'unknows'(V,
W, X,
> Y and Z) and  constraints. The equations are:
> 0.007= 2VZ
> 0.03= W(Y+Z)
> 0.034= X(y+Z)
> 0.013 = (X+W)Y +(X-W)Z
> X = W+V
> Constraints:
> 0<V<W<X
> 0<Y<Z<1
>
> Does anyone know a R-package to solve this system?
>
>  Thanks,
>
> E-mail: sebastien.puechmaille at ucd.ie
>
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

Hi Sebastian,

Your equations can be easily solved - no programming
is required!
Let's number you equations:

(1) 0.007= 2VZ
(2) 0.03= W(Y+Z)
(3) 0.034= X(y+Z)
(4) 0.013 = (X+W)Y +(X-W)Z
(5) X = W+V

Substitute (5) into (3) and then divide (2) by (3) to
get

(W+V)/W = 0.034/0.03 
so that 
(6) V = (2/15)*W
(7) X = V+W = (17/15)*W

Substitute (7) into (3) and (4) to get:
(3') (17/15)*(WY + WZ) = 0.034
(4') (32/15)WY + (2/15)WZ = 0.013

Solving (3') and (4') for two new unknowns WY and WZ
yields

(8) WY = 0.0045
(9) WZ = 0.0255

So that 

(10) Z = (17/3)*Y

Substitute (10) into (1) to get
2*Y*(17/3)*Y = 0.07

So (since Y > 0), Y = sqrt(0.21/34) = 0.07859052
>From (10), Z = (17/3)*Y = 0.4453463
>From (8), W = 0.0045/Y = 0.05725881
>From (6), V = (2/15)*W = 0.007634508
Finally, X = V + W = 0.06489332

So there is a unique solution (which luckily satisfies
your constraints!).

Regards,

Moshe.


--- sebastien puechmaille
<sebastien.puechmaille at ucd.ie> wrote:
> Hello,
> 
>  I have a system of five equations to solve with
> five 'unknows'(V, W, X,
> Y and Z) and  constraints. The equations are:
> 0.007= 2VZ
> 0.03= W(Y+Z)
> 0.034= X(y+Z)
> 0.013 = (X+W)Y +(X-W)Z
> X = W+V
> Constraints:
> 0<V<W<X
> 0<Y<Z<1
> 
> Does anyone know a R-package to solve this system?
> 
>  Thanks,
> 
> E-mail: sebastien.puechmaille at ucd.ie
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained,
> reproducible code.
>