Polynomial Addition

Polynomial addition

PROBLEM

We wish to perform calculations such as

(2 + 3*X + 6*X**2) + (2 - X**2) = (4 + 3*X + 5*X**2)

When polynomials are represented by the arrays, then following procedure is a straightforward implementation of polynomial addition:

IMPLEMENTATION

Unit: nonrecursive internal subroutine

Global variables: an array A. of N+1 coefficients, an array B. of N+1 coefficients.

Parameter: a positive integer N - a degree of the A. and B. polynomials.

Result: The A. array includes coefficients of polynomial addition.

POLYADD: procedure expose A. B.
parse arg N
do J = 0 to N; A.J = A.J + B.J; end
return

EXAMPLE

(12*X**54 + 65*X**80 + 3*X*10000) +
(3*X**12 - 13*X**54 + 13*X**98 + 7*X**10000) =
3*X**12 - X**54 + 65*X*80 + 13*X*98 + 10*X**10000

N = 10000
A. = 0; A.54 = 12; A.80 = 65; A.10000 = 3
B. = 0; B.12 = 3; B.54 = -13
B.98 = 13; B.10000 = 7
call POLYADD N
do I = 1 to N
if A.I <> 0 then say I A.I
end
exit
POLYADD: procedure expose A. B.
...

displays on the screen

12 3
54 -1
80 65
98 13
10000 10

SPARSE POLYNOMIALS

For processing "sparse" polynomials with many zero coefficients, we can have array nodes represent only the nonzero terms. A sparse polynomial

A.N*X**N+...+A.J*X**J+...+A.1*X+A.0

can be represented by an ascending sequence of pairs J A.J only for nonzero coefficients A.J.

IMPLEMENTATION

Unit: nonrecursive internal subroutine

Global variables: an ascending sequence A. of M couples coefficient power, an ascending sequence B. of N couples coefficient power, and an output array C. as result of polynomial addition

Parameters: positive integers M, N - size of the A. and B. arrays.

Result: Returns size of the C. array. The C. array includes couples represent of polynomial addition.

SPARSE_POLYADD: procedure expose A. B. C.
parse arg M, N
K = 1; L = 1; J = 0
do while K <= M & L <= N
  parse var A.K Apower Acoef
  parse var B.L Bpower Bcoef
  select
    when Apower < Bpower
      then do; J = J + 1; C.J = A.K; K = K + 1; end
    when Apower > Bpower
      then do; J = J + 1; C.J = B.L; L = L + 1; end
    otherwise
      Sum = Acoef + Bcoef
      if Sum <> 0
        then do; J = J + 1; C.J = Apower Sum; end
      K = K + 1; L = L + 1
  end
end
do while K <= M; J = J + 1; C.J = A.K; K = K + 1; end
do while L <= N; J = J + 1; C.J = B.L; L = L + 1; end
return J

EXAMPLE

The following fragment of program

M = 3
A.1 = 54 12; A.2 = 80 65; A.3 = 10000 3
N = 4
B.1 = 12 3; B.2 = 54 "-13"; B.3 = 98 13
B.4 = 10000 7
J = SPARSE_POLYADD(M, N)
do I = 1 to J; say C.I; end
exit
SPARSE_POLYADD: procedure expose A. B. C.
...

displays on the screen

12 3
54 -1
80 65
98 13
10000 10

CONNECTIONS

Evaluating the Polynomials
Evaluation of polynomial and its derivatives (simultaneously)
Division of polynomials
Merging