@acer Yes, I was busy trying to get the numerical ODE to solve and hadn't payed attention to the actual IVP I had sent. I'm not sure if you're interested in the problem, but its a physics one.
I have the function V, which is a potential. It contains three parts V_0, VCWc and V_TDD. Because of the complications of V_TDD, I have given it default values of lambda and mu (defined at the start of the sheet). The other two, I have used more accurate coefficients for lambda and mu, namely the solutions of the fsolve argument where I've set rho=250 and 125^2. Func is essentially V_C + VCW (if mu and lambda were undefined variables). Lets call this "Func". I take the first derivative of of Func wrt to rho and at the point rho=246 this derivative should be zero. Similarly the second derivative should be 125^2. These conditions give me values of mu and lambda as a function of kappa.
The physical problem is that I am writting a modified potential to the Higgs field. Its derivative at the known minimum (electroweak minumum 246 Gev) should be zero and the second derivative should be the higgs mass, 125^2.
In regards to accuracy, I'm trying to solve this with what ever accuracy I can get. The ODE I'm solving is like a ball rolling down a hill with friction. The potential looks like
where the blue line is the negative potential. I need to find the initial rho value which gives a result such that it rolls down the potential and lands onto the second bump. I do this via a bisection. Essentially I pick some initial point p(0.001)=190, solve my ODE see if it goes over or under the bump. Then repeat with a higher/lower value until I narrow down to some degree of accuracy where it sits on the second hill for a little bit before falling back to its minimum (in this case around 0). I'm using the Student[NumericalAnalysis]-Roots procedure to do this.
This means, I need to repetitively solve this IVP and require speed. I spose I dont need a rediculous level of accuracy for each iteration.
I will have a play with some polynomial fitting functions to approximate this derivative.