Nonlinear regression I have some functions of $x$, in the form of $d\sqrt{x}$ or $d\log(x)$ where $d$ is known. I would like to rewrite (approximate is fine) them in the form $a/(1 + bx^c)$, where $a$, $b$ and $c$ are arbitrary.
I don't think there are $a$, $b$ and $c$ such that the two curves will match exactly, so I think maybe try to fit a nonlinear regression and find the closest $a$, $b$ and $c$. I tried the following R code, and I get the singular gradient error. 
x     = seq(1, 50000, by=1000)
y     = 50*sqrt(x)
model = nls(y ~ a/(1 + b*x^c), start=list(a=1, b=-0.01, c=0.01))

 A: You have several problems there. The biggest problems are easiest to see if you reparameterize your fitted function from
$y = a/(1 + b x^{-c})$
to
$y = 1/(\frac{1}{a} + \frac{b}{a} x^{-c})$
$\quad = 1/(a_1 + b_1 x^{-c})$
(this gives the same model fit, just some of the parameters are different from your expression of the model)

Now let's look at your data:
$y = 50 x^\frac{1}{2}$
$\quad = 1/( \frac{1}{50} x^{-\frac{1}{2}})$
$\quad = 1/(0 + \frac{1}{50} x^{-\frac{1}{2}})$
That is, your model exactly fits your data if $a_1=0$, $b_1=\frac{1}{50}$ and $c=-\frac{1}{2}$.
Therefore, taking that back to the original form you tried to fit, $b/a = \frac{1}{50}$ or $a=50 b$ and $1/a = 0$. 
So three problems:
(i) there's a ridge along $a=50b$  
(ii) the bigger $a$ is, the better the fit (the sums of squares of error will be minimized as $a\to\infty$.
(iii) as $a\to\infty$, there's no error in the fit. This causes some difficulties with the fitting algorithm - it doesn't terminate nicely, but it can still find the fit if you solve the first two problems. (If you turn trace=TRUE on after the first two problems are fixed, and start in a reasonable place, it does locate the parameter values I mention - 1.662634e-22 :   0.02 -0.50 are the values trace gives for the SSE, b1 and c. If you play with the convergence criteria, you may be able to get it to store the results in model.)
(Well, Alexis correctly points out your starting values are no good, so maybe that's four problems.)
