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Engineering Analysis ENG 3420 Fall 2009
Dan C. Marinescu
Office: HEC 439 B
Office hours: Tu-Th 11:00-12:00
Lecture 24
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Attention: The last homework HW5 and the last project are
due on Tuesday November 24!!
Last time:
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Today
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Lagrange interpolating polynomials
Splines
Cubic splines
Searching and sorting
Numerical integration (chapter 17)
Next Time
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Numerical integration
Lecture 24
2
Cubic splines
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Cubic splines the simplest representation with the appearance of
smoothness and without the problems of higher order polynomials.

Linear splines have discontinuous first derivatives
 Quadratic splines have discontinuous second derivatives and require setting
the second derivative at some point to a pre-determined value
 Quartic or higher-order splines tend to exhibit ill-conditioning or oscillations.

The cubic spline function for the ith interval can be written as:
si x  ai  bi x  xi  ci x  xi   di x  xi 
2
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3
For n data points, there are (n-1) intervals and thus 4(n-1) unknowns to
evaluate to solve all the spline function coefficients.

3

Conditions to determine the spline coefficients
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The first condition  the spline function goes through the first and last
point of the interval; this leads to 2(n-1) equations:
s i xi   fi  ai  fi
s i xi1   fi s i xi1   ai  bi xi1  xi  ci xi1  xi   di xi1  xi   fi

2
3
The second condition  the first derivative should be continuous at
each interior point; this leads to (n-2) equations:
'
si' xi1   si1
xi1   bi  2ci xi1  xi  3di xi1  xi   bi1
2
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The third condition  the second derivative should be continuous at
each interior point; this leads to (n-2) equations:
''
si'' xi1  si1
xi1  2ci  6di xi1  xi   2ci1
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So far we have (4n-6) equations; we need (4n-4) equations!
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Two additional equations
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There are several options for the final two equations:
Natural end conditions  the second derivative at the end knots are zero.
 Clamped end conditions  the first derivatives at the first and last knots are known.
 “Not-a-knot” end conditions  force continuity of the third derivative at the second
and penultimate points (results in the first two intervals having the same spline
function and the last two intervals having the same spline function)
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Built-in functions for piecewise interpolation
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MATLAB has several built-in functions to implement piecewise interpolation.
spline yy=spline(x, y, xx)
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Performs cubic spline interpolation, generally using not-a-knot conditions.
If y contains two more values than x has entries, then the first and last value in y are
used as the derivatives at the end points (i.e. clamped)
Example:
 Generate data:
x = linspace(-1, 1, 9);
y = 1./(1+25*x.^2);
 Calculate 100 model points and
determine not-a-knot interpolation
xx = linspace(-1, 1);
yy = spline(x, y, xx);
 Calculate actual function values
at model points and data points, the
9-point not-a-knot interpolation (solid),
and the actual function (dashed),
yr = 1./(1+25*xx.^2)
plot(x, y, ‘o’, xx, yy, ‘-’, xx, yr, ‘--’)
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Clamped example
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Generate data w/ first derivative information:
x = linspace(-1, 1, 9);
y = 1./(1+25*x.^2);
yc = [1 y -4]
Calculate 100 model points and
determine not-a-knot interpolation
xx = linspace(-1, 1);
yyc = spline(x, yc, xx);
Calculate actual function values
at model points and data points, the
9-point clamped interpolation (solid),
and the actual function (dashed),
yr = 1./(1+25*xx.^2)
plot(x, y, ‘o’, xx, yyc, ‘-’, xx, yr, ‘--’)
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interp1 built-in function
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interp1 function performs several different kinds of interpolation:
yi = interp1(x, y, xi, ‘method’)
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x & y contain the original data
 xi contains the points at which to interpolate
 ‘method’ is a string containing the desired method:
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‘nearest’ - nearest neighbor interpolation
‘linear’ - connects the points with straight lines
‘spline’ - not-a-knot cubic spline interpolation
‘pchip’ or ‘cubic’ - piecewise cubic Hermite interpolation
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Piecewise Polynomial Comparisons
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Multidimensional Interpolation
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The interpolation methods for onedimensional problems can be
extended to multidimensional
interpolation.
Example - bilinear interpolation using
Lagrange-form equations:
f xi , yi  
xi  x2 yi  y2
f x1 , y1  
x1  x2 y1  y2
xi  x1 yi  y2
f x2 , y1  
x2  x1 y1  y2
xi  x2 yi  y1
f x1 , y2  
x1  x2 y2  y1
xi  x1 yi  y1
f x2 , y2 
x2  x1 y2  y1
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Built-in functions for two- and three-dimensional
piecewise interpolation
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2-D interpolation: the inputs are vectors or same-size matrices.
zi = interp2(x, y, z, xi, yi, ‘method’)
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3-D interpolation: the inputs are vectors or same-size 3-D arrays.
vi = interp3(x, y, z, v, xi, yi, zi, ‘method’)
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‘method’ is a string containing the desired method:
‘nearest’,
‘linear’,
‘spline’,
‘pchip’,
‘cubic’
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Search algorithms
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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
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Binary search:
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Can be done only when the list is sorted.
 Requires O(log(n)) comparisons.
 Algorithm:
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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.
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Sorting algorithms
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Algorithms that puts elements of a list in a certain order, e.g., numerical
order and lexicographical order.
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Input: a list of n unsorted elements.
 Output: the list sorted in increasing order.
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Bubble sort  complexity: average O(n2); )); worst case O(n2).
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Compare each pair of elements; swap them if they are in the wrong order.
 Go again through the list until no swaps are necessary.
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Quick sort  complexity: average O(n log(n)); worst case O(n2).
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Pick an element, called a pivot, from the list.
 Reorder the list so that
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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)
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Merge sort – invented by John von Neumann:
1.
2.
3.
4.
5.
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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:
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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.
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Integration
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I
Integration:
 f x dx
b
a
is the total value, or summation, of f(x) dx over the range from a to b:

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Newton-Cotes formulas
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The most common numerical integration schemes.
Based on replacing a complicated function or tabulated data with a
polynomial that is easy to integrate:
I
 f x dx   f x dx
b
b
a
a n
where fn(x) is an nth order interpolating polynomial.

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Newton-Cotes Examples
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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.
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The 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   f (a) 
x  adx
a
ba


b
f a  f b
I  b  a
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:
1
3
Et   f  b  a
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
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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.
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Composite Trapezoidal Rule
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I
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:

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 

f x1  f x2 
2

xn
x n 1
fn x dx
 xn  xn1
f xn1  f xn 
2
n1

h 
I  f x0   2 f xi   f xn 
2 

i 1
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MATLAB Program
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Simpson’s Rules
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One drawback of the trapezoidal rule is that the error is related to the
second derivative of the function.
More complicated approximation formulas can improve the accuracy for
curves - these include using (a) 2nd and (b) 3rd order polynomials.
The formulas that result from taking the integrals under these polynomials
are called Simpson’s rules.
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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
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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:
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.
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Composite Simpson’s 1/3 rule
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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 


xn
x n2
fn x dx
h
h
f
x

4
f
x

f
x

 0
 1   2   f x2  4 f x3  f x4 

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

h
f xn2   4 f xn1   f xn 

3
25
Simpson’s 3/8 rule
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
Simpson’s 3/8 rule corresponds to using
third-order polynomials to fit four points.
Integration over the four points simplifies
to: x3
I

I
3h
f x0  3 f x1  3 f x2  f x3 

8
x0
fn x dx
Simpson’s 3/8 rule is generally used in
concert with Simpson’s 1/3 rule when the
number of segments is odd.
26
Higher-order formulas

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Higher-order Newton-Cotes formulas may also be used - in general,
the higher the order of the polynomial used, the higher the derivative
of 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-oddpoint formulas are usually the methods of preference.
27
Integration with unequal segments

The trapezoidal rule with data containing unequal segments:
I

xn
x0
fn x dx 

x1
x0
fn x  dx 

x2
x1
fn x  dx 


f x0   f x1 
f x1   f x2 
I  x1  x0 
 x2  x1 

2
2
xn
x n1
fn x dx
f xn1   f xn 
 xn  xn1 
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
omitted, 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.
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