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POLYNOMIAL MODELS WITH PYTHON
José M. Garrido
Department of Computer Science
January 2016
College of Computing and Software Engineering
Kennesaw State University
c 2015 J. M. Garrido
Polynomial Models with Python
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General Forms of Polynomial Functions
Linear and quadratic equations are special cases of polynomial functions. The degree
of a polynomial function is the highest degree among those in its terms. A linear
function such as: y = 3x + 8, is a polynomial equation of degree 1 and a quadratic
equation, such as: y = 4.8x2 + 3x + 7, is a polynomial function of degree 2; thus the
term “second degree equation”.
A function such as: y = 2x4 + 5x3 − 3x2 + 7x − 10.5, is a polynomial function
of degree 4 because 4 is the highest exponent of the independent variable x. A
polynomial function has the general form:
y = p1 xn + p2 xn−1 + p3 xn−2 + . . . pk−1 x + pk
This function is a polynomial equation of degree n, and p1 , p2 , p3 , . . . pk are the
coefficients of the equation and are constant values. Another general form of a
polynomial function P(x) is:
P (x) = c0 + c1 x + c2 x2 + . . . + cn−1 xn−1 + cn xn
In this second form of a polynomial, c0 , c1 , c2 , c3 , . . . cn are the coefficients of the
equation and are constant values. In addition to the algebraic form of a polynomial
function, the graphical form is also important; a polynomial function is represented
by a graph. As mentioned previously, Gnuplot is used for plotting data.
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The polynomial Module of the numpy Package
The polynomial module of the Numpy package provides several functions for manipulating polynomial series. A polynomial function is represented by a vector of
coefficients in ascending order. The following is a list of the relevant functions for
dealing with polynomials.
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polyval(x,c[,tensor])Evaluate a polynomial at points x.
polyval2d(x,y,c)
Evaluate a 2-D polynomial at points (x,
y).
polyval3d(x,y,z,c)
Evaluate a 3-D polynomial at points (x, y,
z).
polygrid2d(x,y,c)
Evaluate a 2-D polynomial on the Cartesian product of x and y.
polygrid3d(x,y,z,c) Evaluate a 3-D polynomial on the Cartesian product of x, y and z.
polyroots(c)
Compute the roots of a polynomial.
polyfromroots(roots) Generate a monic polynomial with given
roots.
polyfit(x,y,deg
Least-squares fit of a polynomial to data.
[,rcond,full,w])
polyvander(x,deg)
Vandermonde matrix of given degree.
polyvander2d(x,y,deg)Pseudo-Vandermonde matrix of given degrees.
polyvander3d(x,y,z, Pseudo-Vandermonde matrix of given dedeg)
grees.
polyder(c[,m,scl,
Differentiate a polynomial.
axis])
polyint(c[,m,k,lbnd, Integrate a polynomial.
scl,axis])
polyadd(c1,c2)
Add one polynomial to another.
polysub(c1,c2)
Subtract one polynomial from another.
polymul(c1,c2)
Multiply one polynomial by another.
polymulx(c)
Multiply a polynomial by x.
polydiv(c1,c2)
Divide one polynomial by another.
polypow(c,pow[,
Raise a polynomial to a power.
maxpower])
polyline(off,scl)
Returns an array representing a linear
polynomial.
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Evaluation of Polynomial Functions
With a computer program, a relatively large number of values of x can be used to
evaluate the polynomial function for every value of x. The set of values of x that
are used to evaluate a polynomial function are taken from an interval a ≤ x ≤ b.
Where a is the lower bound of the interval and b is the upper bound. The interval is
known as the domain of the polynomial function. In a similar manner, the interval
of the values of the function y is known as the range of the polynomial function.
A polynomial function is evaluated by using various values of the independent
variable x and computing the value of the dependent variable y. In a general mathematical sense, a polynomial function defines y as a function of x. With the appropriate expression, several values of polynomial can be computed and graphs can be
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produced.
The functions in the polynomial module of the Numpy package evaluate a polynomial, P(x), using Horner’s method for stability. Using the general form of the
polynomial of degree n:
P (x) = c0 + c1 x + c2 x2 + . . . + cn−1 xn−1 + cn xn
Function polyval evaluates polynomials with real coefficients for the real variable
x. The function returns the values of P(x) for the given values in list x. The following
assignment statement has a call to function polyval that evaluates a polynomial at
the values in list x with the coefficients in list c .
y = polyval (x, c)
The function computes the values of the polynomial (values of y) for all the
given values of x. The following example evaluates the polynomial function y =
2x3 − 3x2 − 36x + 14 and the values in the two arrays x and y are computed by the
program in file polyval1.py. Only 20 values of x are evaluated in this example.
The following listing is the output produced by the program.
$ python polyval1.py
Coefficient list
[ 14. -36. -3.
2.]
Evaluating a polynomial
-6.0 -310.0
-5.31578947368 -179.827525878
-4.63157894737 -82.3265782184
-3.94736842105 -13.6534480245
-3.26315789474 30.0355736988
-2.57894736842 52.5841959469
-1.89473684211 57.8361277154
-1.21052631579 49.6350779997
-0.526315789474 31.8247557953
0.157894736842 8.24887009768
0.842105263158 -17.2488700977
1.52631578947 -40.8247557953
2.21052631579 -58.6350779997
2.89473684211 -66.8361277154
3.57894736842 -61.5841959469
4.26315789474 -39.0355736988
4.94736842105 4.65344802449
5.63157894737 73.3265782184
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6.31578947368 170.827525878
7.0 301.0
Figure 1: Graph of the equation y = 14 − 36x − 3x2 + 2x3 .
Listing 1 shows the source code of the Python program in file polyval1.py.
The program evaluates the polynomial and computes the values in vector y. Line
18 defines the array with the polynomial coefficients: a, b, c, and d. Line 10 defines
array x with 20 different values from the interval −6.0 ≤ x ≤ 7.0. Line 14 calls
function polyval that computes the value of the function for every value in array x.
Listing 1: A program that computes the values of a polynomial.
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# This program evaluates a polynomial given
# the coefficients in list c
# at points in list x
# J M Garrido, August 2, 2014
import numpy as np
from numpy.polynomial.polynomial import polyval
M = 20
# number of data points
xi = -6.0 # first value of x
xf = 7.0
# final value
x = np.linspace(xi, xf, M)
c = np.array([14.0, -36.0, -3.0, 2.0])
print "Coefficient list"
print c
y = polyval(x, c)
print "Evaluating a polynomial"
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16 for j in range(M):
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print x[j], y[j]
A graph is easily produced with Gnuplot using the values in arrays of x and y
computed. Figure 1 shows the graph of the polynomial function y = 14 − 36x −
3x2 +2x3 with the data computed previously. The graph is produced with the script
files polyval1.cgp.
To evaluate the polynomial equation y = −2 + 3x5 , the corresponding coefficient
vector is [−2.0, 0, 0, 0, 0, 3.0]. The program calls function polyval to evaluate the
polynomial in the interval −2.5 ≤ x ≤ 2.5. The program is basically the same as
the previous example. The following listing shows the Python interpreter running
the program.
$ python polyval2.py
Coefficient list
[-2. 0. 0. 0. 0. 3.]
Evaluating polynomial
-2.5 -294.96875
-2.23684210526 -169.995597296
-1.97368421053 -91.8482429544
-1.71052631579 -45.9308953696
-1.44736842105 -21.0553407421
-1.18421052632 -8.98659937214
-0.921052631579 -3.98858195139
-0.657894736842 -2.36974585578
-0.394736842105 -2.02875143775
-0.131578947368 -2.00011831867
0.131578947368 -1.99988168133
0.394736842105 -1.97124856225
0.657894736842 -1.63025414422
0.921052631579 -0.011418048612
1.18421052632 4.98659937214
1.44736842105 17.0553407421
1.71052631579 41.9308953696
1.97368421053 87.8482429544
2.23684210526 165.995597296
2.5 290.96875
In a similar manner, executing the command file polyval2.cgp with Gnuplot
produces a graph of polynomial equation y = −2 + 3x5 . Figure 2 shows the graph
of this polynomial function.
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Figure 2: Graph of the equation y = −2 + 3x5 .
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Solving Polynomial Functions
The solution to quadratic equation, which is a second degree equation, is relatively
straight forward. The solution to this equation involves finding two values of x that
give y value zero. These two values of x are known as the roots of the function.
For higher order polynomial functions, the degree of the polynomial determines the
number of roots of the function. A polynomial function of degree 7 will have 7 roots.
It is not generally feasible to find the roots of polynomial equations of degree 4
or higher by analytical methods.
Function polyroots of the polynomial module computes the complex roots of a
polynomial function. This function takes the coefficients vector of the polynomial
function as the argument, and returns a vector with the roots. This vector is known
as the roots vector. The function implements an algorithm that uses an iterative
method to find the approximate locations of roots of polynomials. The only parameter of this function is the array of coefficients of the polynomial function.
Listing 2 shows the Python source code of a program that computes the roots
of the polynomial function 3 + 3x − 14x2 − 17x3 + 23x4 . The program is stored in
file polyrootsp.py.
Listing 2: A program that computes the roots of a polynomial function.
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# This program computes the roots of a polynomial
# given the coefficients in list c
# J M Garrido, Aug 2, 2014. Program: polyrootsp.py
import numpy as np
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import numpy.polynomial.polynomial as poly
c = np.array([3.0, 3.0, -14.0, -17.0, 23.0])
print "Solving a polynomial"
print "Coefficient list"
print c
r = poly.polyroots(c)
print "Roots of the polynomial"
print r
The following output was produced after starting the Python interpreter and
running the program. The solution vector, r, has the values of the (complex) roots
of the polynomial function.
$ python polyrootsp.py
Solving a polynomial
Coefficient list
[ 3.
3. -14. -17. 23.]
Roots of the polynomial
[-0.43665695-0.19425443j -0.43665695+0.19425443j
0.52528824+0.j 1.08715609+0.j ]
For the polynomial function: y = 2x3 − 3x2 − 36x + 14, the following output
was produced after starting the Python interpreter and running the program. The
solution vector, r, has the values of the (real) roots of the polynomial function. The
individual values of vector r are: r0 = −3.7688, r1 = 0.3799, and r2 = 4.8889.
$ python polyroots2.py
Solving a polynomial
Coefficient list
[ 14 -36 -3
2]
Roots of the polynomial
[-3.76883165 0.37990763
4.88892403]
The solution to the polynomial equation y = −2 + 3x5 is computed in a similar
manner. Four of the roots computed are complex values, which are known as complex
roots.
$ python polyroots3.py
Solving a polynomial
Coefficient list
[-2. 0. 0. 0. 0. 3.]
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Polynomial Models with Python
Roots of the polynomial
[-0.74600097-0.54200143j -0.74600097+0.54200143j
0.28494702-0.87697674j 0.28494702+0.87697674j
0.92210791+0.j
]
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