yr2 chap 10 numerical methods

• the root of a function is where it crosses the x axis
• the phrase you should always write when finding roots or proving there is a root is: because f(x) is continuous and there is a change of sign in the interval [a, b] then there must be at least one real root in the interval
• an interval with a change of sign can have multiple real roots as long as there is an odd number
• an interval with no change of sign can have multiple real roots as long as there is an even number
• if an asymptote is present in a function, it is not continuous, so there can be a sign change but no real root
• an iterative formula is a formula which can be used to find the roots of a function, it will be in the form $x_{(n+1)} =$$f(x_{n})$
• the starting point for an iterative formula makes a big difference as it can cause it to be either converging or diverging
• graphically, a converging iteration would be shown using a staircase diagram or a cobweb diagram
• questions usually ask you to rearrange the function into a specific form for the iterative formula to prove they are equal, this should be possible using simple algebra
• a staircase diagram is when successive iterations approach the root from the same direction
• a cobweb diagram is when successive iterations alternate being above and below the root
• to draw a cobweb or spider diagram you need the curve and the line y=x drawn, then draw a line vertically from the first x value until it reaches the curve, then draw a line horizontally from that point until it reaches the line y=x, then repeat the process of vertical and horizontal lines
• the newton-raphson method is a way of finding roots using a formula which is given to you in the formula book, the question will specify if you should use this method
• to use the newton-raphson procedure, substitue values into the formula to get the required terms and make sure to use the correct number of decimal places
• normally the question gives you the starting value for a newton-raphson method, but if you are asked to choose your own then you should choose one that is quite far away from the root so it converges more quickly, and definitely do not choose one that is a turning point as the derivative will be zero and it is impossible to divide by zero