I am a little confused about this question; it has been many years since I studied numerical methods. The first part is fine. Why must we define a sequence of successive approximations? What do we use them for? Surely to use the trapezium rule we would need successive values of the function INSIDE the integral, since we are trying to solve the integral? Or are we looking for some kind of convergence? Also, does anybody else think these course notes are really full of errors? Thanks! Hope someone can help.
Hi Nina79, Sometimes, we don't have to define these successive approximations, especially in the exam questions where the maths will simplify rather more quickly than in real life. The point about the equation is that each probability has another probability inside the integral so we end up with a set of recursive equations that don't necessarily help us. However, if we constrain the probabilities to involve n jumps then a computer can step in and help us out. You can get from state i to state j in n steps by staying in state i for a bit, jumping to state k and then going from state k to state j in n-1 steps. Eventually n, n-1, .... becomes 1 and we have a probability we can assign a number too. Then the computer steps in and sorts the rest out. In terms of CT4, I wouldn't worry too much about this, you've only ever been forced to go through the integral forms once (Apr 06 A6), I see them more as a (very useful) mathematical tool to have in your locker. Set up a HSD model with constant transition rates and no return to H once sick. Try and use the integral equation to find pHS(t) - it's quicker than going down the other route where you need intergrating factor. Good luck! John
