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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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