Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: - - PowerPoint PPT Presentation

1

Engineering Analysis ENG 3420 Fall 2009

Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00

2 Lecture 25

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).

3

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.

4

Sorting algorithms

• Algorithms that puts elements of a list in a certain order, e.g., numerical
• rder 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.

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Sorting algorithms (cont’d)

• Merge sort – invented by John von Neumann:

1.

Complexity: average O(n log(n)); worst case O(n log(n));

2.

If the list is of length 0 or 1, then it is already sorted. Otherwise:

3.

Divide the unsorted list into two sublists of about half the size.

4.

Sort each sublist recursively by re-applying merge sort.

5.

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.

6

Integration

• Integration:

is the total value, or summation, of f(x) dx over the range from a to b:

I = f x

( )

a b

∫

dx

7

Newton-Cotes formulas

• Replace a function difficult or

impossible to integrate analytically or tabulated data with a polynomial easy to integrate:

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
• ne step or in a series of steps to

improve accuracy.

I = f x

( )

a b

∫

dx ≅ fn x

( )

a b

∫

dx

8

Trapezoidal rule

• The trapezoidal rule is the first
• f the Newton-Cotes closed

integration formulas; it uses a straight-line approximation for the function:

I = fn x

( )

a b

∫

dx I = f (a)+ f b

( )− f a ( )

b − a x − a

( )

⎡ ⎣ ⎢ ⎤ ⎦ ⎥

a b

∫

dx I = b − a

( ) f a ( )+ f b ( )

2

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Error of the trapezoidal rule

• An estimate for the local truncation

error of a single application of the trapezoidal rule is: where ξ 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.

Et = − 1 12 ′ ′ f ξ

( ) b − a ( )

3

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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 = fn x

( )

x0 xn

∫

dx = fn x

( )

x0 x

1

∫

dx+ fn x

( )

x1 x2

∫

dx+L+ fn x

( )

xn−1 xn

∫

dx I = x

1− x0

( ) f x0 ( )+ f x

1

( )

2 + x2 − x

1

( ) f x

1

( )+ f x2 ( )

2 +L+ xn − xn−1

( ) f xn−1 ( )+ f xn ( )

2 I = h 2 f x0

( )+ 2

f xi

( )

i=1 n−1

∑

+ f xn

( )

⎡ ⎣ ⎢ ⎤ ⎦ ⎥

11

12

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.

13

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:

Integration over the three points simplifies to:

I = fn x

( )

x0 x2

∫

dx I = h 3 f x0

( )+ 4 f x1 ( )+ f x2 ( )

[ ]

fn x

( )=

x − x1

( )

x0 − x1

( )

x − x2

( )

x0 − x2

( )

f x0

( )+ x − x0 ( )

x1 − x0

( )

x − x2

( )

x1 − x2

( )

f x1

( )+ x − x0 ( )

x2 − x0

( )

x − x1

( )

x2 − x1

( )

f x2

( )

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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: where again ξ is somewhere between a and b.

• This formula indicates that the error is dependent upon the fourth-derivative
• f 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.

Et = − 1 2880 f

4

( ) ξ

( ) b − a ( )

5

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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 = fn x

( )

x0 xn

∫

dx = fn x

( )

x0 x2

∫

dx + fn x

( )

x2 x4

∫

dx +L+ fn x

( )

xn−2 xn

∫

dx I = h 3 f x0

( )+ 4 f x1 ( )+ f x2 ( )

[ ]+ h

3 f x2

( )+ 4 f x3 ( )+ f x4 ( )

[ ]+L+ h

3 f xn−2

( )+ 4 f xn−1 ( )+ f xn ( )

[ ]

I = h 3 f x0

( )+ 4

f xi

( )

i=1 i, odd n−1

∑

+ 2 f xi

( )

j=2 j, even n−2

∑

+ f xn

( )

⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥

16

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:

• Simpson’s 3/8 rule is generally used in

concert with Simpson’s 1/3 rule when the number of segments is odd.

I = fn x

( )

x0 x3

∫

dx I = 3h 8 f x0

( )+ 3f x1 ( )+ 3f x2 ( )+ f x3 ( )

[ ]

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Higher-order formulas

Higher-order Newton-Cotes formulas may also be used - in general,

the higher the order of the polynomial used, the higher the derivative

• f the function in the error estimate and the higher the power of the

step size.

As in Simpson’s 1/3 and 3/8 rule, the even-segment-odd-point

formulas have truncation errors that are the same order as formulas adding one more point. For this reason, the even-segment-odd- point formulas are usually the methods of preference.

18

Integration with unequal segments

The trapezoidal rule with data containing unequal segments:

I = fn x

( )

x0 xn

∫

dx = fn x

( )

x0 x1

∫

dx + fn x

( )

x1 x2

∫

dx +L+ fn x

( )

xn−1 xn

∫

dx I = x1 − x0

( ) f x0 ( )+ f x1 ( )

2 + x2 − x1

( ) f x1 ( )+ f x2 ( )

2 +L+ xn − xn−1

( ) f xn−1 ( )+ f xn ( )

2

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Integration with unequal segments

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Built-in functions

MATLAB has built-in functions to evaluate integrals based on the

trapezoidal rule z = trapz(y) z = trapz(x, y) produces the integral of y with respect to x. If x is omitted, the program assumes h=1. z = cumtrapz(y) z = cumtrapz(x, y) produces the cumulative integral of y with respect to x. If x is

• mitted, the program assumes h=1.

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Multiple Integrals

• Multiple integrals can be determined

numerically by first integrating in one dimension, then a second, and so on for all dimensions of the problem.