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# numerical methods revision sheet

By Gene Lee,2014-07-05 09:45
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numerical methods revision sheet

Trapezium rule Simpson’s rule

bSimpson’s rule is a more accurate way of To approximate an integral, such as , fxdx()estimating integrals. Suppose the area is split into an strips (n must be even), then we can divide the area into n strips. Then bhba(){4(...)，？？？？？fxdxyyyyy0131nn is the width of each strip. h3 an

The formula for the trapezium rule is: ？？？？2(...)}yyy242nb hh. ()2(...)，？？？？？fxdxyyyyy;；OR IN WORDS . {ends + 4dds + 2vens}0121Numerical solution of equations nn23aThe equation f(x) = 0 has a solution (or root) 1.2122between x = a and x = b if f(x) changes sign in xExample: Find an approximation for sin()xdxExample: Find an approximation for edxthis interval. 01 using Simpson’s rule with 6 strips. using the trapezium rule with 6 ordinates. x ex，？62 Example: Show that the equation (NB: 6 ordinates means 5 strips). 1.2(0) has a root in the interval 1 < x < 2. Solution: h，，0.21(1)？？ 6Solution: h，，0.4 Solution: Rearrange to the form f(x) = 0: 0 0.2 0.4 0.6 0.8 1 1.2 5x xx0 0.040 0.159 0.352 0.597 0.841 0.991 -1 -0.6 -0.2 0.2 0.6 1 exex，？(？？，62260 . y x

2.718 1.433 1.041 1.041 1.433 2.718 y Substitute in x = 1:

x1So, exe？？，？？，？26261.28... bSo, 0.2Substitute in x = 2: bfxdx，？？？？(){00.9914(0.0400.3520.841)0.4x2 fxdx，？？？？？()2.7182.7182(1.4331.0411.0411.433);；3 exe？？，？？，26465.39... a2aAs there is a change in sign, there is a root in ？？，2(0.1590.597)}0.496the interval 1 < x < 2.

Sol’ns can be found by programming a calculator: Example: The sequence defined by the Iterative methods for finding roots

A root of the equation x = F(x) can sometimes iterative formula 1 = 0 . 5 × ln ( 2 × ANS + 3 ) = = etc be found using the iterative formula xxx，？，0.5ln(23),1The values you get converge to 0.753 (to 3 d.p.) nn11x = F(x) you should show all the values you get in the exam. n+1nconverges to α. starting with some initial value x. 1a) Find α correct to 3 d.p. xxandThis method only works if the sequence of b) To get the equation solved, replace nn1b) State an equation of which α is a root. values converges. in the iterative formula by x:

x，！？，0.5ln(213)0.8047Solution: a) xx，？0.5ln(23) 2

2xx，！？，0.5ln(20.80473)0.7641 23xe？，This is equivalent to 3

etc

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