Polynomials and Curve Fitting in MATLAB

11
Polynomials and Curve Fitting
Lecture 07
by
Shameer A Koya

Polynomial
 Degree : 3
 For help abut polynomials in matlab, type help polyfun
2

Polynomials in MATLAB
 MATLAB provides a number of functions for the
manipulation of polynomials. These include,
 Evaluation of polynomials
 Finding roots of polynomials
 Addition, subtraction, multiplication, and division of polynomials
 Dealing with rational expressions of polynomials
 Curve fitting
Polynomials are defined in MATLAB as row vectors made up of the
coefficients of the polynomial, whose dimension is n+1, n being the degree
of the polynomial
p = [1 -12 0 25 116] represents x4 - 12x3 + 25x + 116
poly2sym(p)
sym2poly(x^4-12*x^3+25*x+116)
3

Polynomials in MATLAB…
Vector must include all polynomial
coefficients, even those that are zero
4

Value of a Polynomial
 MATLAB provides the function polyval to evaluate
polynomials. To use polyval you need to provide the
polynomial to evaluate and the range of values where the
polynomial is to be evaluated. Consider,
>>p = [1 4 -7 -10];
To evaluate p at x=5, use
>> polyval(p,5)
To evaluate for a set of values,
>>x = linspace(-1, 3, 100);
>>y = polyval(p,x);
>>plot(x,y)
>>title(‘Plot of x^3 + 4*x^2 – 7*x – 10’)
>>xlabel(‘x’)
5

Roots
 >>p = [1, -12, 0, 25, 116]; % 4th order polynomial
(x-r1)(x-r2)(x-r3)(x-r4)
>>r = roots(p)
r =
11.7473
2.7028
-1.2251 + 1.4672i
-1.2251 - 1.4672i
 From a set of roots we can also determine the polynomial
>>pp = poly(r)
r =
1 -12 -1.7764e-014 25 116
6

Addition and Subtraction
 MATLAB does not provide a direct function for adding or subtracting
polynomials unless they are of the same order, when they are of the
same order, normal matrix addition and subtraction applies, d = a + b
and e = a – b are defined when a and b are of the same order.
 When they are not of the same order, the lesser order polynomial must
be padded with leading zeroes before adding or subtracting the two
polynomials.
>>p1=[3 15 0 -10 -3 15 -40];
>>p2 = [3 0 -2 -6];
>>p = p1 + [0 0 0 p2];
>>p =
3 15 0 -7 -3 13 -46
The lesser polynomial is padded and then added or subtracted as
appropriate.
7

Multiplication
 Polynomial multiplication is supported by the conv function. For the
two polynomials
a(x) = x3 + 2x2 + 3x + 4
b(x) = x3 + 4x2 + 9x + 16
>>a = [1 2 3 4];
>>b = [1 4 9 16];
>>c = conv(a,b)
c =
1 6 20 50 75 84 64
or c(x) = x6 + 6x5 + 20x4 + 50x3 + 75x2 + 84x + 64
8

Multiplication II
 Couple observations,
 Multiplication of more than two polynomials requires repeated
use of the conv function.
 Polynomials need not be of the same order to use the conv
function.
 Remember that functions can be nested so conv(conv(a,b),c)
makes sense.
9

Division
 Division takes care of the case where we want to divide one polynomial
by another, in MATLAB we use the deconv function. In general
polynomial division yields a quotient polynomial and a remainder
polynomial. Let’s look at two cases;
 Case 1: suppose f(x)/g(x) has no remainder;
>>f=[2 9 7 -6];
>>g=[1 3];
>>[q,r] = deconv(f,g)
q =
2 3 -2 q(x) = 2x2 + 3x -2
r =
0 0 0 0 r(x) = 0
10

Division II
 The representation of r looks strange, but MATLAB outputs
r padding it with leading zeros so that length(r) = length of
f, or length(f).
 Case 2: now suppose f(x)/g(x) has a remainder,
>>f=[2 -13 0 75 2 0 -60];
>>g=[1 0 -5];
>>[q,r] = deconv(f,g)
q =
2 -13 10 10 52 q(x) = 2x4 - 13x3 + 10x2 + 10x + 52
r =
0 0 0 0 0 50 200 r(x) = -2x2 – 6x -12
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Derivatives
 Single polynomial
 k = polyder(p)
 Product of polynomials
 k = polyder(a,b)
 Quotient of two polynomials
 [n d] = polyder(u,v)
12

Curve Fitting
 Curve fitting is the process of adjusting a mathematical
function so that it lays as closely as possible to a set of data
points
 MATLAB provides a number of ways to fit a curve to a set of
measured data. One of these methods uses the “least
squares” curve fit. This technique minimizes the squared
errors between the curve and the set of measured data.
 The function polyfit solves the least squares polynomial
curve fitting problem.
13

Curve Fitting ....
 Consider 11 measurements of Voltage for Field current change from 0
to 1 A, spaced 0.1 A.
>>x = [0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1];
>>y = [-.447 1.978 3.28 6.16 7.08 7.34 7.66 9.56 9.48 9.30 11.2];
>>n = 2;
>>p = polyfit( x, y, n )
p =
-9.8108 20.1293 -0.317
or p(x) = -9.8108x2 + 20.1293 – 0.317
>>xi = linspace(0,1,100);
>>yi = polyval(p, xi);
>>plot(x,y,’-o’, xi, yi, ‘-’)
>>xlabel(‘x’), ylabel(‘y = f(x)’)
>>title(‘Second Order Curve Fitting Example’)
15

Curve Fitting – Order of Polynomial
 The order of polynomial relates to the number of turning
points (maxima and minima) that can be accommodated
 Given n data points (xi,yi), can make a polynomial of degree
n-1 that will pass through all n points.
 Given a set of n data points (xi,yi), can often make a
polynomial of degree less than n-1 that may not pass through
any of the points but still approximates the set overall.
 For an nth order polynomial normally n-1 turning points (sometimes less
when maxima & minima coincide).
16

Curve Fitting – 2nd Order Polynomial
 Consider the ‘cosine’ function
 x = [0:10]; y = [1 0.54 -0.42 -0.99 -0.65 0.28 0.96 0.75 -0.15 -0.91 -0.83];
 Start with polynomial of degree 2 (i.e. quadratic):
p=polyfit(x,y,2)
p =
-0.0040 -0.0427 0.3152
 So the polynomial is
-0.0040x2 - 0.0427x + 0.3152
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Curve Fitting – 5th Order Polynomial
p=polyfit(x,y,5)
p =
0.0026 -0.0627 0.4931 -1.3452 0.4348 1.0098
 So a polynomial 0.0026x5 - 0.0627x4 + 0.4931x3 -1.3452x2
+ 0.4348x + 1.0098
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 Not bad. But can it be
improved by increasing the
polynomial order?

Curve Fitting – 8th Order Polynomial
19

Summary of Polynomial Functions
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Function Description
conv Multiply polynomials
deconv Divide polynomials
poly Polynomial with specified roots
polyder Polynomial derivative
polyfit Polynomial curve fitting
polyval Polynomial evaluation
polyvalm Matrix polynomial evaluation
residue Partial-fraction expansion (residues)
roots Find polynomial roots

Polynomials and Curve Fitting in MATLAB

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