Page 1Zoom in Page 1 2 3 4 5 Next Level: AS and A Level Subject: Maths Word count: 1974 Save View my saved documents Submit similar document Share this Decimal search. Download this essay Print Save AS AND A LEVEL CORE & PURE MATHEMATICS An extract from this document… Decimal search The equation f (x) = 0, where f (x) = x3-x2+x-2, has only one real root, but there is no simple analytical method of finding it. Therefore, a spreadsheet had been used to solve the equations numerically using decimal search. x f(x) -3 -41 -2 -16 -1 -5 0 -2 1 -1 2 4 3 19
The table and graph above illustrated the first approximations to the roots of the equation x3-x2+x-2=0. As the curve crosses the x-axis, f (x) changes sign, so provided that f (x) is a continuous function, once you have located an interval in which f(x) changes sign, you know that that interval must contain a root. In the table, you first take increments in x of size 1 within the interval 1<x<2, working out the value of the function x5-5x+3 for each one. You do this until you find a change of sign of f(x) between the value of x = 1 and 2 , so you should know that there is a root lying in the interval 1<x<2 on the graph.
This method is based on the iteration. xn+1 = xn – , n = 0,1,2,3,……. with initial approximation x0. The Newton – Raphson iterative formula is based on evaluating the gradient of the tangent to the curve y = f(x) at x = x0 The gradient of the tangent at (x0, f (x0) is f ? (x) f ? (x0) = Rearranging this, ==> x1 = x0 – To solve the equation ex-7x-3, f (x) = ex-7x-3==> f ? (x) = ex-7. This will give rise to the Newton Raphson iteration formula, that is, xn+1 = xn – You will then be able to find the upper root ? , let the initial approximation x0 = 4 etc The sequence converge rapidly towards the upper root to give ? =3. 2478 to 5 s.
f. In this case, Newton-Raphson Iteration gives an extremely rapid rate of convergence. This is the case for examples, even when the first approximation is not particularly good. For manual caluations it is almost always the most efficient method. You may try to solve the equation using spreadsheet, it gives a Iteration: xn+1 = x0 – (ex-7x-3)/(ex-7) Convergence towards upper root n xn 0 4 1 3. 504221277 2 3. 286134403 3 3. 248830171 4 3. 247850646 5 3. 247849987 6 3. 247849987 7 3. 247849987 8 3. 247849987 9 3. 247849987 10 3. 247849987 Error Bounds for this root can be established for 5 s. f. by evaluating f(x) for x = 3. 2478 ?
0. 00005. f (3. 24775) = e3. 24775 – 7 (3. 24775) -3 = -0. 001873116 < 0 f (3. 24785) = e3. 24785 – 7 (3. 24785) – 3 = 0. 000000251 > 0 Since there is a change of sign in f(x) over the interval 3. 24775<x<3. 24785, and the function is continuous, there must be a root in this interval, i. e. ?=3. 2478 to 5 s. f. Fixed Point Iteration The equation f(x) = 0 is rearranged into the form x = g(x).
Roots of the equation x = g(x) are therefore roots of the equation f(x) = 0. This gives rise to the iterative formula x n+1 = g (x n), where n = 0,1,2,3… with initial approximation x0. The sequence x0, x1, x2, x3… will converge to a root of the equation x = g(x) provided a suitable starting value x0 is chosen and -1<g? (? )<1. The equation y = x3-6x+1, f (x)=0 is plotted on the graph below f (x) = x3-6x+1 = 0 may be re-arranged in a form g(x) as follows: ==> ==> The x-intercept of the graphs of y = x and g (x) = represent roots of the original equation x3+4×2-3x-7 = 0 The re-arrangement leads to the iteration To find the middle root ? , let initial approximation x0 = 2 x1 = x03+1 = 23 + 1 = 1. 5 6 6 x2 = x13+1 = 1. 53 + 1 = 0. 7291666667 6 6 x3 = x23+1 = 0. 72916666673 + 3 = 0. 231281045 6 6 x4 = x33+1 = 0. 2312810453 + 3 = 0. 1687285727 6 6 x5 = x43+1 = 0.
Decimal Search Fixed Point Iteration Newton-Raphson Iteration Graphical Calculator It is especially useful to solve an equation numerically by searching for a change of sign. Necessarily the technique involves much repetition of calculations. Each step is illustrated by a screen dump from the graphical calculator, but the techniques may be readily adapted for use with other model of graphical calculators. It allows zooming in the area you wish to enlarge and trace to find a better approximation to the required root. Alternatively, a more systematic approach, which can be illustrated by careful use of the table options.
The decimal search technique progressively refines the interval in which the root lies, until a desired degree of accuracy is achieved. Before carry out the Fixed Point iteration, you are required to work out and consider a suitable rearrangement for an equation. It is important to decide an appropriate starting value is chosen, so that the iterative sequence converges, and find the root to the required degree of accuracy. It allows you to enter an initial approximation which close to the root, you could try out and decide which rearrangement best suit with the root you required.
If an appropriate starting value or rearrangement is chosen, this will lead the sequence diverges to infinity. Some equations may involve complicated functions to generate the next approximation from the first one. These will be a risk of enter the formula incorrectly in the calculator, so you should pay more attention when entering the formula. It requires the same skills as you use in Fixed Point Iteration. Spreadsheet It allows searching for a change of sign within an interval with different size of increment. Graphs could be set scales for any intervals (e. g. the scales may be set within a interval between 1. 35<x<1. 35, 1. 353<x< 1.
354 or even smaller interval) illustrates where the graphs of a function cut the x-axis. Give indication where there is a change of sign. Copy and paste is particularly useful as you will not need to perform repeated calculations for each value of f (x), as this correspond to the value of x. It may be involves the risk errors if you carry out your calculation manually, e. g. entering wrong number. You may find monotonous when doing the repeat calculations. If you found the number you have entered was wrong, you could possibly re-enter the correct number in the spreadsheet, and you would find the dependent number change automatically.
It does not involves a lot of programming skills, but you must unsure the formula you have entered is correct, because if you enter a wrong formula generates a incorrect result, then all the remains will be affected. When the formula for a rearrangement of an equation g(x) is entered correctly with an input of initial approximation(i. e. x0 in this case). Then the approximation would converge towards the root required by simply copy and paste the formula until there is no any changing result. Sometimes, you have to try which rearrangement is suitable for a particular roots.
Plot the line y =x and the curve g (x), then find the gradient near the root, if the gradient is greater than 1 (i. e. the gradient of the line y = x ), illustrated that this is a unsuitable rearrangement of finding the root, and vice versa. It requires the same techniques as you used in Fixed Point Iteration. With the first approximation, it then converges gradually to the root required. Sometimes, there is a problem of plotting a graph by using spreadsheet. For example, +5 , it is impossible to plot the point where x = 0, it ends up with a continuous function which is not the expected discontinuous function.