Engineering Analysis ENG 3420 Fall 2009 Dan C Marinescu Office HEC 439 B Office hours Tu Th 11 00 12 00 1 Lecture 25 Attention The last homework HW5 and the last project are due on Tuesday November 24 Last time Cubic splines Today Searching and sorting Numerical integration chapter 17 Next Time Numerical integration of functions chapter 18 Lecture 25 2 Search algorithms Find an element of a set based upon some search criteria Linear search Compare each element of the set with the target Requires O n operations if the set of n elements is not sorted Binary search Can be done only when the list is sorted Requires O log n comparisons Algorithm Check the middle element If the middle element is equal to the sought value then the position has been found Otherwise the upper half or lower half is chosen for search based on whether the element is greater than or less than the middle element 3 Sorting algorithms Algorithms that puts elements of a list in a certain order e g numerical order and lexicographical order Input a list of n unsorted elements Output the list sorted in increasing order Bubble sort complexity average O n2 worst case O n2 Compare each pair of elements swap them if they are in the wrong order Go again through the list until no swaps are necessary Quick sort complexity average O n log n worst case O n2 Pick an element called a pivot from the list Reorder the list so that all elements which are less than the pivot come before the pivot and all elements greater than the pivot come after it equal values can go either way After this partitioning the pivot is in its final position Recursively sort the sub list of lesser elements and the sub list of greater elements 4 Sorting algorithms cont d Merge sort invented by John von Neumann 1 2 3 4 5 Complexity average O n log n worst case O n log n If the list is of length 0 or 1 then it is already sorted Otherwise Divide the unsorted list into two sublists of about half the size Sort each sublist recursively by re applying merge sort Merge the two sublists back into one sorted list Tournament sort Complexity average O n log n worst case O n log n It imitates conducting a tournament in which two players play with each other Compare numbers in pairs then form a temporary array with the winning elements Repeat this process until you get the greatest or smallest element based on your choice 5 Integration Integration I f x dx b a is the total value or summation of f x dx over the range from a to b 6 Newton Cotes formulas Replace a function difficult or impossible to integrate analytically or tabulated data with a polynomial easy to integrate I f x dx f x dx b b a a n fn x is an nth order interpolating polynomial The integrating function can be polynomials for any order for example a straight lines or b parabolas The integral can be approximated in one step or in a series of steps to improve accuracy 7 Trapezoidal rule The trapezoidal rule is the first of the Newton Cotes closed integration formulas it uses a straight line approximation for the function I f x dx b a n f b f a I a f a x a dx b a b f a f b I b a 2 8 Error of the trapezoidal rule An estimate for the local truncation error of a single application of the trapezoidal rule is 1 3 Et f b a 12 where is somewhere between a and b This formula indicates that the error is dependent upon the curvature of the actual function as well as the distance between the points Error can thus be reduced by breaking the curve into parts 9 Composite trapezoidal rule Assuming n 1 data points are evenly spaced there will be n intervals over which to integrate The total integral can be calculated by integrating each subinterval and then adding them together I xn x0 fn x dx I x1 x0 x1 x0 fn x dx f x0 f x1 2 x2 x1 x2 x1 fn x dx L f x1 f x2 2 xn x n 1 fn x dx L xn xn 1 f xn 1 f xn 2 n 1 h I f x0 2 f xi f xn 2 i 1 10 11 Simpson s rules The error of the trapezoidal rule is related to the second derivative of the function To improve the accuracy use a 2nd and b 3rd order polynomials the results are called Simpson s rules 12 Simpson s 1 3 rule Simpson s 1 3 rule corresponds to using second order polynomials Using the Lagrange form for a quadratic fit of three points fn x x x1 x x2 f x x x0 x x2 f x x x0 x x1 f x x0 x1 x0 x2 0 x1 x0 x1 x2 1 x2 x0 x2 x1 2 Integration over the three points simplifies to I I h f x0 4 f x1 f x2 3 x2 x0 fn x dx 13 Error of Simpson s 1 3 rule An estimate for the local truncation error of a single application of Simpson s 1 3 rule is 1 5 4 Et 2880 f b a where again is somewhere between a and b This formula indicates that the error is dependent upon the fourth derivative of the actual function as well as the distance between the points Note that the error is dependent on the fifth power of the step size rather than the third for the trapezoidal rule Error can thus be reduced by breaking the curve into parts 14 Composite Simpson s 1 3 rule Simpson s 1 3 rule can be used on a set of subintervals in much the same way the trapezoidal rule was except there must be an odd number of points Because of the heavy weighting of the internal points the formula is a little more complicated than for the trapezoidal rule I xn x0 fn x dx x2 x0 fn x dx x4 x2 fn x dx L xn x n 2 fn x dx h h h f x 4 f x f x f x 4 f x f x L f xn 2 4 f xn 1 f xn 0 1 2 2 3 4 3 3 3 n 1 n 2 h I f x0 4 f xi 2 f xi f xn 3 i 1 j 2 i odd j even I 15 Simpson s 3 8 rule Simpson s 3 8 rule corresponds to using third order polynomials to fit four points Integration over the four points simplifies to x3 I x0 fn x dx 3h I f x0 3 f x1 3 f x2 f x3 8 Simpson s 3 8 rule is generally used in concert with Simpson s 1 3 rule when the number of segments …