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H. T. Kung

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Carnegie Mellon University
Research Showcase @ CMU
Computer Science Department School of Computer Science
1973
Fast evaluation and interpolation
H. T. Kung
Carnegie Mellon University
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The copyright law of the United States (title 17, U.S. Code) governs the making 
of photocopies or other reproductions of copyrighted material. Any copying of this 
document without permission of its author may be prohibited by law. FAST EVALUATION AND INTERPOLATION 
H. T. Kung 
Department of Computer Science 
Carnegie-Mellon University 
Pittsburgh, Pa. 
January, 1973 
This work was supported in part by the National Science 
Foundation under grant GJ-32111 and the Office of Naval 
Research under Contract N00014-67-A-0314-0010, NR 044-422. ABSTRACT 
A method for dividing a polynomial of degree (2n-l) by a precomputed 
nth degree polynomial in 0(n log n) arithmetic operations is given. This 
is used to prove that the evaluation of an nth degree polynomial at n+1 
arbitrary points can be done in 0(n log^ n) arithmetic operations, and con-
sequently, its dual problem, interpolation of an nth degree polynomial from 
2 
n+1 arbitrary points can be performed in 0(n log n) arithmetic operations. 
The best previously known algorithms required 0(n log^ n) arithmetic opera-
tions . 1• INTRODUCTION 
Given (x^y^) (0 £ i £ n) , the interpolation problem is the determina-
tion of the coefficients {c^} (0 £ i £ n) of the unique polynomial 
P(x) « Z c.x
1 of degree £ n such that P(x.) = y. (0 £ i £ n). If a 
O^i^n
 1 1 1 
classical method such as the Lagrange or Newton formula is used, inter-
2 
polation takes 0(n ) operations. (In this paper all arithmetic operations 
will be counted. We simply write operations to denote arithmetic operations.) 
3 
However, Horowitz (1972) has shown that interpolation can be done in 0(n log r 
operations by using the Fast Fourier Transform (FFT), and he has shown that 
interpolation is reducible to evaluation of an nth degree polynomial at nH-l 
points. Moenck and Borodin (1972) have shown that the evaluation problem 
is reducible to the division problem, and they have shown that both evalua-
3 
tion and interpolation can be done in 0(n log n) operations, and precomputed 
2 
interpolation (knowing the x^ in advance) can be performed in 0(n log n) 
operations. The purpose of this paper is to show that, without using any 
2 
precomputation, both evaluation and interpolation can be done in 0(n log n) 
operations. As a corollary we show that an nth degree polynomial and all 
2 
its derivatives can be evaluated at any point in 0(n log n) operations. 
We shall use the same approach as used by Moenck and Borodin (1972). 
2 
But we shall first precompute all necessary divisors in 0(n log n) operations 
so that each division can be done in 0(n log n) operations. This results in 
faster evaluation and faster interpolation. 
After the work reported here was completed, the author received a re-
port from V. Strassen, entitled, "Die Berechnungskomplexitat von elementar-
symmetrischen Funktionen und von Interpolationskoeffizienten
1'. Using -2-
different techniques Strassen proves that interpolation can be done in 
0(n log n) multiplications or divisions and he states that his techniques 
2 
can be used to prove that interpolation can be done in 0(n log n) arithmetic  
operations. -3-
2. PRELIMINARIES 
We shall work over the field of complex numbers. 
Theorem 2.1. (Fast Polynomial Multiplication) 
Let A(x) = 2 ax
1 and B(x) « 2 b.x
1 be any two polynomials. Let 
O^i^n O^i^n-1
1 
A(x)-B(x) = S c.x
1. Then [c. } (0 £ i £ 2n-1) can be obtained in 0(n log n) 
0=^2n-l
 1 
operations. 
Theorem 2.2. 
Let {ai} (0 ^ i £ n) and {V} (0 £ i £ n-1) be any two sequences of 
numbers. Then 
(2.1) 
an Vl
 ai 
n-1 
n 
'n-2 
Vl 
can be computed in 0(n log n) operations. 
Proof. 
Let A(x) = £ ax
1 and B(x) « £ b.x
1. Suppose that 
0<a^n . O^i^n-1
 1 
A(x)*B(x) = E c x . It is clear that the computation of (2.1) is 
0£i^2n-l
 1 
equivalent to the computation of {c } (n £ i £ 2n-l). Thus the proof fol-
lows from Theorem 2.1. QED _4_ 
Theorem 2.3. 
For any sequence [a^ (0 £ i ^ n) of numbers with
 a
n ^ °»
 le
t 
T, a.x
1 be the unique polynomial q(x) such that 
O^n-1
1 
(2.2) x
2n_
1
 = q(x)-( E ax
1) + r(x), deg r < n. 
O^i^n 
Then, we have that 
(2.3) 
n n-1 1 
a  n-1 
n 
a a  n-1 n-2 
an-i 
*n-l 
and the sequence {£.} (0 £ i £ n-1) can be obtained in 0(n log n) operations. 
Proof. 
- i>  Let ( E axV< S a x^) = J ex.- Then £ c.x = 
1 O^n-1 0£i£2n-l O^i^n-l 
2n-l  - r(x), 
0£i**i 
Since deg r < n, = 1 and c. = 0 for i - n,n+l,...,2n-2. Therefore, 
(2.4) 
a a - ... a. 
n n-1 1 
• • • 
n 
*n-2 
0 
1 
Furthermore, from (2.4), one can easily show that, for any i (1 ^ i * n) , -5-
a  n-1 
0 
1 
0 
0 
ith component 
This proves (2.3), By using the fast division algorithm given by Moenck 
and Borodin (1972), the unique polynomial q(x) (i.e., the sequence {a^} 
2 
(0 ^ i ^ n-1)) can be computed in 0(n log n) operations. QED 
Definition 2. 4.(Precomputing) 
Given any polynomial P(x) = £ a.x
1 with a ^ 0, by precomputing P(x), 
0£i£n_
 1 n 
we shall mean the computation of the {
a^} (0 ^ i £ n-1) which are defined by 
(2.2) or (2.3). That is, precomputing P(x) is just the division of x^
n ^ by 
P(x). 
Hence, by Theorem 2.3, we can precompute an nth degree in 0(n log n) 
operations. Since this bound will be sufficient to prove the results in 
this paper, no attempt has been made to improve it. -6-
3. FAST DIVISION USING PRECOMPILED DIVISOR 
Theorem 3.1. 
Let U(x) = £ u.x
1 and V(x) = 2 v.x
1 (v £ 0). Suppose that 
0£i£2n-l
 1 Ofi^n
 1 n 
V(x) has already been precomputed, i.e., {v } (0 ^ i ^ n-1) are available 
with no associated cost. Then we can compute the unique polynomials Q(x) 
and R(x) such that 
(3.1) U(x) « Q(x)-V(x) + R(x), deg R < n 
in 0(n log n) operations. 
Proof. 
It suffices to show that to compute Q(x) we only require 0(n log n) 
operations, since R(x) = U(x) - Q(x)*V(x) and Q(x)*V(x) can be computed in 
0(n log n) operations by Theorem 2.1. Let Q(x) = E q.*
1*
 and let 
Q(x)*V(x) = T. c.x
1. From (3.1), it is clear that u. = c# for 
0^i<2n-1
 1 1 X 
i = n,...,2n-l. Therefore, 
v v - n n-1 
n-1 
n 
— — 
• 
• 
un 
• 
• 
•  • 
^n-2 
- V
1- >1 
and hence, by (2.3), 
V2 
The theorem then follows from Theorem 2.2. 
n-1  'n-2 
n-2 
v -,  n-1 
u  n 
*>n-
*2n-
QED -7-
4. FAST EVALUATION 
Moenck and Borodin (1972) have shown that evaluation is reducible to 
division and have proved the following theorem: 
Theorem 4.1. (Moenck and Borodin (1972)) 
polynomial in 0(g(n)) operations and multiply two nth degree polynomials 
in 0(f(n)) operations. 
This fast evaluation algorithm requires certain divisions. The divisors 
are exactly the members of the following family except the polynomial at 
level r+1. 
X"
XQ> x-x.|, X-X2> x-x^, ••• »
 x~
x
n 1 >
 x"
Xn Level 1 
Let U(x) be a polynomial of degree n = 2
r-1. Then we can evaluate U(x) 
at nH-1 arbitrary points x0,Xl,...,xn in O(g(n)log n+f(n)log n) operations, 
provided that we can divide a polynomial of degree (2n-l) by an nth degree 
(x-xQ)(x-x1), (x-x2)(x-x3),  (x-x  )(x-x ) 
n 
Level 2 
n (x-x.) 
i=n-3
 1 
n 
Level 3 
(4.1) 
n (x-x ) 
i«0
 x 
n (x-x.) 
r-1
 1 
Level r 
i=2 
n (x-x.) 
i=0 
Level r+1 
Theorem 4.2. 
All polynomials in (4.1) can be precomputed in 0(n log
2 n) operati -8-
Proof. 
We first convert all polynomials in (4.1) into the form £ h.x
1. This 
2
 i 
can be done in 0(n log n) operations (see Horowitz (1972)). Then we shall 
precompute the polynomials at level j from the precomputed polynomials at 
level j+1, for j = r,r-l,...,l. By Theorem 2.3, we can precompute the poly-
2 
nomial at level r+1 in 0(n log n) operations. Suppose that all polynomials 
at level j+1 have been precomputed. Let D(x) = £ d.x^ be a polynomial 
0£i£2J
 1 
at level j+1, and let E(x) = E. . e.x
1 and F(x) = S. , f.x be those 
0^is2
J"
 1 0^2
J"
 1 
two polynomials such that D(x) = E(x)«F(x). By (2.2), we know that 
2
j+1-1 - i i  x = ( S. d.x
L).D(x) + r (x), deg r < 2
J. 
0£i=S2
J -1
 1 
Since D(x) = E(x)«F(x), it follows that 
2J-1 <
 Si
 5 xL),F(x
)
 r (x) 
xf _ _ 0^i^2
J-1
 1
 +
 rD
W 
E(x) " 2j 2j 
x x E(x) • 
But, by (2.2), 
x
2J_
1
 - i
 rE
(x
)
 1-1 
Hence, if ( 2. d XVF(X) = S , then 
0^i^2J-1 0sa.S23+2J"'-l 
Si
 Si
X r (x) • r„(x) 
By the uniqueness of the partial fraction expansion, it is easy to see that -9-
e. = g.+2j for all i = 0,1,...,2
J-1-1. 
Therefore, we can precompute E(x) by computing ( T, . d.x
1)»F(x), which 
0<a^2
 J-1
 1 
can be performed in 0(j»2^) operations by Theorem 2.1. Similarly, we can 
i 2
r 
precompute F(x) in 0(j»2
J) operations. Since there are — 1 polynomials 
2
J 
at level j, all polynomials at level j can be precomputed in 
0( . «j «2^) c= 0(j»2
r ) operations. Hence, all polynomials in (4.1) can 
2
J 
r+1 2 r 2  be precomputed in 0( S j»2 ) « 0(r •! ) = 0(n log n) operations. QED 
l£j£r 
Theorem 4.3. 
Let U(x) be a polynomial of degree n « 2 -1, Then we can evaluate U(x) 
2 
at n+1 arbitrary points XQ,x^,...,xn in 0(n log n) operations. 
Proof. 
We first precompute all divisors needed for the algorithm of Theorem 
2 
4.1. By Theorem 4.2, this takes 0(n log n) operations. Then by Theorem 3.1, 
all divisions used in the algorithm of Theorem 4.1 can be performed in 
0(n log n) operations. The proof follows from Theorem 4.1 by letting 
g(n) = f(n) = n log n. OFD 5. FAST INTERPOLATION 
Horowitz (1972) has shown that interpolation is reducible to fast 
evaluation. 
Theorem 5.1. (Horowitz (1972)) 
Given n+1 = 2 pairs of numbers (x±,y^) (0 ^ i ^ n), the coefficients 
of the unique polynomial P(x) of degree ^ n such that y^ = P(x^) (0 ^ i ^ n) 
can be obtained in 0(h(n) + f(n)log n) operations, provided that evaluation 
at n+1 point is 0(h(n)) operations and multiplication is 0(f(n)) operations. 
Theorem 5.2. 
Given n+1 = 2 pairs of points (x^,y^) (0 ^ i ^ n), the coefficients 
of the unique polynomial P(x) of degree ^ n such that y^ = P(x^) (0 ^ i ^ n) 
2 
can be obtained in 0(n log n) operations. 
Proof. 
Apply the result of Theorem 4.3 to Theorem 5.1. QED 
Corollary 5.3. 
An nth degree polynomial and all its derivatives can be evaluated at 
any point in 0(n log n) operations. 
Proof. 
Suppose that we want to evaluate the nth degree polynomial P(x) and all 
its derivatives at some point Then it suffices to show that 0*^} (0 ^ i ^ 
• 2 
such that P(x) = £ d. (X-Q0 , can be obtained in 0(n log n) operations. 
O^i^n
 1 -11-
First, we evaluate P(x) at n+1 arbitrary distinct points xn,x1,...,x . 
U I n 
2 
This takes 0(n log n) operations by Theorem 4.3. Next, we determine 
n 
[d } (0 < i £ n) such that E d y
X = P(x.), y. = x.-r* for j « 0,1,...,n. 
i=0
 1 J J J
 2
J 
This is an interpolation problem and takes 0(n log n) operations by 
Theorem 5.2. QED 
ACKNOWLEDGMENT 
The author wishes to express his appreciation to his advisor, Professor 
J. F. Traub, for providing information on the subject and for his comments 
on this paper. -12-
6. BIBLIOGRAPHY 
E. Horowitz (1972), "A Fast Method for Interpolation Using Preconditioning,
1
1 
 
Information Processing Letters 1, pp. 157-163. 
R. Moenck and A. Borodin (1972), "Fast Modular Transform Via Division,
1
1 
 
Proceedings of 13th Symposium on Switching and Automata Theory, pp. 90-96. 

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