Media not available in the Digital Archive
Description
This programme analyses the sequence of iterates obtained from the formula Xn+1=f(x,n) for various functions f and various starting values x1. It looks in detail at calculating values of iterates a...nd scale factors.
Module code and title: MS283, An introduction to calculus MS283; 06 02-05-1979 1979 Rights owned or controlled by The Open University This material can be used in accordance with The Open University conditions of use. A link to the conditions can be found at the bottom of all OU Digital Archive web pages. 00:24:00 + Show more... Jack Koumi Colin Rourke; Allan I.,1936-2013 Solomon BBC Open University Diagrams; Staircase Allan Solomon introduces the programme. Solomon looks at the function f(x)=x and its solution by formula iteration starting with x =.5' The sequence converges to 0.I. Colin Rourke manipulates a mapping diagram for the function f(x)=x to illustrate how successive iterations converge on a solution. An animated mapping diagram is also used. Rourke goes on to demonstrate, again using a mapping diagram, why the solution can never be calculated exactly but rather converges on a solution to any desired accuracy. Allan Solomon points out that the formula iteration method works only if the first approximation is close to the solution. He goes on to explain how one can, with the aid of a graph, determine an approximate solution for the function. An animated graph shows what happens to the actual iterations during the process. The result is a "staircase diagram". Allan Solomon and Colin Rourke, with the aid of a mapping diagram and a staircase diagram, examine a case where the sequence diverges away from the solution. They point out that any starting values above 0.9 are bound to make the sequence diverge and that values below 0.9 will make the sequence converge on the solution. Allan Solomon, with the aid of animated staircase diagrams, looks at various functions with various slopes at the point of solution to see what happens as the slopes become shallower at the solution. Allan Solomon, together with Colin Rourke, goes on to look at the case of negative slopes. Animated diagrams and still graphs illustrate the iterative sequence in a case like this. Another animated diagram illustrates the changes which take place when the negative slope of a function changes. Solomon, with the aid of several still graphs, sums up the programme. 6HT/73037 00525_4315 466 no