Polynomials

1. Polynomials
Jordi Cortadella
Department of Computer Science
Outline
• How to represent a polynomial?
• Evaluation of a polynomial.
• Basic operations: sum, product and division.
• Greatest common divisor.
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Representation of polynomials
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A list of coefficients
Properties of the representation:
• 𝑎𝑛 ≠ 0 (the topmost element is not zero).
• 𝑃 𝑥 = 0 is represented with an empty vector.
𝑃 𝑥 = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ + 𝑎1𝑥 + 𝑎0
𝑛 𝑛 − 1 ⋯ 1 0
𝑎𝑛 𝑎𝑛−1 ⋯ 𝑎1 𝑎0
Polynomial evaluation (Horner’s scheme)
• Design a function that evaluates the value of a
polynomial.
• A polynomial of degree 𝑛 can be efficiently
evaluated using Horner’s algorithm:
𝑃 𝑥 = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ + 𝑎1𝑥 + 𝑎0 =
(⋯ ((𝑎𝑛𝑥 + 𝑎𝑛−1)𝑥 + 𝑎𝑛−2)𝑥 + ⋯ )𝑥 + 𝑎0
• Example:
3𝑥3
− 2𝑥2
+ 𝑥 − 4 = 3𝑥 − 2 𝑥 + 1 𝑥 − 4
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2. Polynomial evaluation (Horner’s scheme)
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3𝑥3
− 2𝑥2
+ 𝑥 − 4
3 𝑥 − 2 𝑥 + 1
3 𝑥 − 2 𝑥 + 1 𝑥 − 4
3 𝑥 − 2
3
Polynomial evaluation (Horner’s scheme)
def poly_eval(P, x):
"""Returns the evaluation of P(x)"""
r = 0
# Evaluates the polynomial using Horner's scheme
# The coefficients are visited from highest to lowest
for i in range(len(P)-1, -1, -1):
r = r*x + P[i]
return r
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Product of polynomials
Example:
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def poly_mul(P, Q):
"""Returns P*Q"""
𝑃 𝑥 = 2𝑥3 + 𝑥2 − 4
𝑄 𝑥 = 𝑥2
− 2𝑥 + 3
𝑃 ⋅ 𝑄 𝑥 = 2𝑥5 + −4 + 1 𝑥4 + 6 − 2 𝑥3 + −4 + 3 𝑥2 + 8𝑥 − 12
𝑃 ⋅ 𝑄 𝑥 = 2𝑥5
− 3𝑥4
+ 4𝑥3
− 𝑥2
+ 8𝑥 − 12
Product of polynomials
• Key point:
Given 𝑃 𝑥 = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ + 𝑎1𝑥 + 𝑎0
and 𝑄 𝑥 = 𝑏𝑚𝑥𝑚 + 𝑏𝑚−1𝑥𝑚−1 + ⋯ + 𝑏1𝑥 + 𝑏0,
what is the coefficient 𝑐𝑖 of 𝑥𝑖
in (𝑃 ⋅ 𝑄)(𝑥)?
• The product 𝑎𝑖𝑥𝑖
⋅ 𝑏𝑗𝑥𝑗
must be added to the term
with 𝑥𝑖+𝑗
.
• Idea: for every 𝑖 and 𝑗, add 𝑎𝑖 ⋅ 𝑏𝑗 to the (𝑖 + 𝑗)-th
coefficient of the product.
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3. Product of polynomials
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2 -1 9 -5 10 -3
𝑥5
𝑥4
𝑥3
𝑥2
𝑥1
𝑥0
𝑥3 𝑥2 𝑥1 𝑥0
2 -1 3
1 0 3 -1
×
2 6
-1
0 -2
0 -3 1
3 0 9 -3
+
Product of polynomials
def poly_mul(P, Q):
"""Returns P*Q"""
# Special case for a polynomial of size 0
if len(P) == 0 or len(Q) == 0:
return []
R = [0]*(len(P)+len(Q)-1) # list of zeros
for i in range(len(P)):
for j in range(len(Q)):
R[i+j] += P[i]*Q[j]
return R
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Sum of polynomials
• Note that over the real numbers,
degree 𝑃 ⋅ 𝑄 = degree 𝑃 + degree(𝑄)
(except if 𝑃 = 0 or 𝑄 = 0).
So we know the size of the result vector a priori.
• This is not true for the polynomial sum, e.g.,
degree 𝑥 + 5 + −𝑥 − 1 = 0
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Sum of polynomials
A function to normalize a polynomial might be useful in
some algorithms, i.e., remove the leading zeros to
guarantee the most significant coefficient is not zero.
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def poly_normalize(P):
"""Resizes the polynomial to guarantee that the
leading coefficient is not zero."""
while len(P) > 0 and P[-1] == 0:
P.pop(-1)

4. Sum of polynomials
def poly_add(P, Q):
"""Returns P+Q"""
if len(Q) > len(P): # guarantees len(P) >= len(Q)
P, Q = Q, P
# Adds the coefficients up to len(Q)
R = []
for i in range(len(Q)):
R.append(P[i]+Q[i])
R += P[i+1:] # appends the remaining coefficients of P
poly_normalize(R)
return R
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P
Q
Euclidean division of polynomials
• For every pair of polynomials 𝐴 and 𝐵, such that
𝐵 ≠ 0, find 𝑄 and 𝑅 such that
𝐴 = 𝐵 ∙ 𝑄 + 𝑅
and degree 𝑅 < degree(𝐵).
𝑄 and 𝑅 are the only polynomials satisfying this
property.
• Classical algorithm: Polynomial long division.
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Polynomial long division
def poly_div(A, B):
"""Returns the quotient (Q) and the remainder (R)
of the Euclidean division A/B. They are returned
as a tuple (Q, R)"""
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A B R
Q A B R
Q
Polynomial long division
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def poly_div(A, B):
"""Returns the quotient (Q) and the remainder (R)
of the Euclidean division A/B. They are returned
as a tuple (Q, R)"""

5. A B R
Q
Polynomial long division
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def poly_div(A, B):
"""Returns the quotient (Q) and the remainder (R)
of the Euclidean division A/B. They are returned
as a tuple (Q, R)"""
A B R
Q
Polynomial long division
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def poly_div(A, B):
"""Returns the quotient (Q) and the remainder (R)
of the Euclidean division A/B. They are returned
as a tuple (Q, R)"""
A B R
Q
Polynomial long division
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def poly_div(A, B):
"""Returns the quotient (Q) and the remainder (R)
of the Euclidean division A/B. They are returned
as a tuple (Q, R)"""
A B R
Q
Polynomial long division
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def poly_div(A, B):
"""Returns the quotient (Q) and the remainder (R)
of the Euclidean division A/B. They are returned
as a tuple (Q, R)"""

6. A B R
Q
Polynomial long division
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def poly_div(A, B):
"""Returns the quotient (Q) and the remainder (R)
of the Euclidean division A/B. They are returned
as a tuple (Q, R)"""
Polynomial long division
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A B R
Q
def poly_div(A, B):
"""Returns the quotient (Q) and the remainder (R)
of the Euclidean division A/B. They are returned
as a tuple (Q, R)"""
Polynomial long division
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2 6 -3 1 0 -1 2 0 1 -1
1 3 -2
5 4 3 2 1 0 3 2 1 0
2 1 0
R: B:
Q:
6 -4 2 0 -1
-4 -1 3 -1
-1 5 -3
A
Invariant: A = BQ+R
Polynomial long division
def poly_div(A, B):
"""Returns the quotient (Q) and the remainder (R) of the
Euclidean division A/B. They are returned as a tuple (Q, R)"""
R = A[:] # copy of A
if len(A) < len(B):
return [], R
Q = [0]*(len(A)-len(B)+1)
# For each digit of Q
for iQ in range(len(Q)-1, -1, -1):
Q[iQ] = R[-1]/B[-1]
R.pop(-1)
for k in range(len(B)-1):
R[k+iQ] -= Q[iQ]*B[k]
poly_normalize(R)
return Q, R
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iQ
k
k+iQ

7. GCD of two polynomials
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Example:
Re-visiting Euclidean algorithm for gcd
# gcd(a, 0) = a
# gcd(a, b) = gcd(b, a%b)
def gcd(a, b):
"""Returns gcd(a, b)"""
while b > 0:
a, b = b, a%b
return a
For polynomials:
• a and b are polynomials.
• a%b is the remainder of
the Euclidean division.
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Euclidean algorithm for gcd
def poly_gcd(A, B):
"""Returns gcd(A, B)"""
while len(B) > 0:
Q, R = poly_div(A, B)
A, B = B, R
# Converts to monic polynomial (e.g., 3x-6  x-2)
c = A[–1]
for i in range(len(A)):
A[i] /= c
return A
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Euclidean algorithm for gcd
monic polynomial
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8. Conclusions
• Polynomials are used very often in numerical
analysis.
• Polynomial functions have nice properties:
continuous and simple derivatives and
antiderivatives.
• There are simple root-finding algorithms to
find approximations of the roots.
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