Trapezoid Method and Iteration & Acceleration

Trapezoid Method The trapezoid method is a way to approximate an integral. It approximates the curve by a polygonal line and computes the area formed by the trapezoids where lines are dropped perpendicular down to the x-axis from each of the connecting points. f(x) f(x) b b Area approximated using 2 trapezoids. Area: Area approximated using 4 trapezoids. Area: This is A2 for [0,b] This is A3 for [0,b]

In general any set of points that partition the interval [a,b] can be used as a trapezoidal approximation. We are going to use certain types of partitions that will make the calculation easier. In the partitions we say on the previous slide all had the points equally spaced. This results in the following convention for the trapezoid approximation with n trapezoids. Add up all the function values in the partition multiplied by 2 except for the a and the b, the multiply by the width of each partition and ½. This will calculate the areas of the trapezoid under the curves. Iteration An iterative method generates a sequence of approximations to an answer. Each time you generate an approximation this is called an iteration.

Acceleration The term acceleration is a term used in numerical analysis that refers to how you can improve the results of an iterative algorithm by applying another algorithm to it. In the example we use here we can show how we can use the results of the trapezoid method to improve (give a more accurate estimate of the integral) how quickly it arrives at the answer in terms of the number of calculations that need to be performed. The example of acceleration I will show you is called an extrapolation method (specific case of Richardson extrapolation). Define a sequence of values Ai where: A1 = Trapezoid method with 1 partition (i.e. ½(b-a)(f(a)+f(b)) A2 = Trapezoid method with 2 partitions A3 = Trapezoid method with 4 partitions A4 = Trapezoid method with 8 partitions Each Ai calculation is considered an iteration.

Now we define sequences of values (B1,B2,B3,…),(C1,C2,C3,…),…