C. Domb, "On Iterative Solutions of Algebraic Equations,". Proc Camb. Phil. Soc, 4 5 ( 1 ... of Equations," Phil. Trans. Roy. ... McGraw-Hill, (1953), 125. [15]. I. Kiss ...
On a Class of Iteration Formulas and Some Historical Notes by J. F. Traub Bell Telephone Laboratories, Incorporated Murray Hill, New Jersey
The class of iteration formulas obtainable by rational approximations of "Euler's formula" will be derived with the corresponding error estimates.
Section I contains
some historical notes on Iterative procedures.
Section II
f
gives a derivation of Euler s formula with the associated error estimate in a new notation which simplifies the error estimate and suggests generalizations.
Section III considers
the Pade approximants to the "Euler polynomial" and shows how a number of known formulas may be derived from this unified approach.
There is a short discussion of the "best" formula. I.
SOME HISTORICAL NOTES
In the author's earlier paper, "Comparison of Iterative Methods for the Calculation of N-th Roots," Comm. ACM 4, No. 3 (1961), henceforth referred to as [I], formula (5) was incorrectly ascribed to Bailey.
This formula has been
independently rediscovered by many authors.
In the recent
literature it has been ascribed to Richmond [20],[28], Wall [24],[11], Frame [8],[11], Schroder [22],[13], and Konig [19].
As has been pointed out by a number of
authors [ 9 L [ 2 7 h the formula may be traced back to
- 2 -
Halley ( 1 6 9 4 ) , [ 1 0 ] , [ 2 ] .
Similarly/ formula (7) of I, a
special case of Halley's formula as applied to the calculation of N-th roots, which has generally been ascribed to Bailey ( 1 9 4 1 ) , [l],[l4],
dates back to Lambert ( 1 7 7 0 ) , [ 1 7 ] , [ 1 8 ] , [ 1 5 ] . A comment of Rubbert ( 1 9 4 8 ) , [ 2 1 ] demonstrates in
striking fashion how the advent of computers forces us to re-evaluate our ideas.
r
Rubbert concludes that Halley s method
is far preferable to the logarithm-exponential method for calculating N-th roots while [I], written thirteen years later, reports the opposite conclusion. Many papers have recently appeared concerning higher Bodewig [4]
order iterations involving higher derivatives.
gives an algorithm for generating iteration methods of arbitrary order and points out that the results may also be derived by a method due to Euler [7.]. simplified by Curry ['5]. results.
1
Bodewig s proof is
See [ 2 6 ] , [ 2 7 ] , [19] , [ 3 ] for analogous
Similar results may be already found in Schroder (1870)
[ 2 2 ] , which contains a goldmine of information.
Schroder knew
that Newton's method was of second order for single roots but only first order for multiple roots, he was aware of the modification in Newton's method which makes it second order for multiple roots, and he rederived Halley'-s method as a special case.
- 3 -
II.
1
EULER S FORMULA
We give a short derivation of the algorithm for 1
generating iteration formulas of arbitrary order by Euler s method and give an estimate of the errpr.
The notation will
be the same as in [I] except that higher derivatives of f(x) will be denoted by f ^ as f,
and that f(x ) = f ±
±
will be abbreviated
We will assume that f(x) is regular in the neighborhood
of the root although this condition may be modified In an obvious manner.
We will assume that we are in the neighborhood
of a root of multiplicity one and will denote the inverse of f(x) as g(y). Thus we have y - f(x) ,
x = g(y) .
Then a * g(0) s. g(y, -y. )
(1)
and developing the right side of (1) in a Taylor series yields
a =
Let u = f/f
(2)
and define j+l (j+1)
- 4 -
Then
(3)
a = x -u^u ±
^
This suggests the iteration formula m
(4)
x
1 + 1
= xj-u^y xj j=o m
Defining the Euler polynomial as Y(u) * to write (4) as (5)
u^Yj enables us ^
0
= x -uY(u) . 1
Define (6)
Dj = f(J)/f' .
We have the following theorem:
Yj is a polynomial in
Differentiating (2) w.r.t. x yields
(7)
Tj. = ( j D ^
Y
0=
1
- —jji-yj+l ,
for
J >0
- 5 -
Differentiating (6) w.r.t. x yields (8)
Dj = D g D j ^ + £
(Dj^) ,
J > 1
for
Using (7) and (8), a simple induction completes the proof of the theorem. We have
Y
0=
Y
1
- D /2 - D /6 2
2
3
This formulation permits an easy estimate of the error.
Subtracting (3) from (4) we have
e
(9)
i l = +
Noting that u'^ ~
u
Y
)_/ j m+1
and expanding Yj in a power series
about a, we have
(10)
e
1 + 1
~ Y (a)( r m+1
ei
2
,
M
It is easy to show that tJie • %ultl"term formula of [I] may be derived from (4) with the corresponding error estimate derived from (10). III.
RATIONAL APPROXIMATIONS
THE ETJLER FORMULA
We may write (3) as (II)
a =
where E ~ Y , ( e . )
m + 2
.
- uY(u) - E
XJJ
We will consider iteration formulas
of the form
(12)
x
X
u
i l = I " +
| ©
where
>
P(u) = ^ V
?o = i
1=0
Q(u) =
^ 1=0
Subtracting ( 1 1 ) from (12) yields
(13)
e
1 + 1
~ - .»
- Y(u)]
+
E - - u.
+
where (14)
H(u) = P(u) - Y(u)Q(u) = ^ ^ u
1
.
- 7 -
If the lowest term of Q(u) is a constant and the lowest term , then (12) defines iterative
of H(u) is proportional to u formulas of order m+2. p+q+1 parameters
'Therefore, we will choose the
P-JJQJ
SO
that E
±
= 0, for i = 0 , 1 , . . . , p + q
with p+q = m. Let co^j - 1
for
i < j
co j = 0
for
i > j
±
and let k = min[>t, q] . Equating coefficients in (14) yields k (15)
-
Q J Y ^ J
=
O
for
I = 0,l,...,p+q
J-l Equation (15) may be used to calculate and Q , Q , . . . , Q Q
given by
1
q
recursively.
P
1
, P
2
, . . . , P
The corresponding error Is
P
- (WW-™*
2
The rational function P(u)/Q(u) is the Pade' approximant [25] of Y(u).'
Defining elements I
= x - uP(u)/Q(u), for
XT'
p = 0 , 1 , . . . , m and q = 0 , 1 , . . . ,m, with p+q - m, permits a conx
venient classification of the iteration formulas
= ^pq(' j_)
generated by this method. A two-dimensional array of rational fraction approximations to a given function is called a Pade table. The arrangement of the I
into such a table is a slight modifica-
tion of the usual procedure. We will give some of the lower order formulas and their associated error estimates explicitly. 0 :
^0
1:
I Z
1 Q
=
x
2 Q
u
= x - u(l+Y ) lU
01 "
I
"
X
u ~ 1-Y u
= x - u(l Y +
(Newton)
e
i+l
£
i+l
=
£
i+l
=
£
I 1
=
ll "
X
U
"
i l
=
2
3
£ i
(Halley)
2
l U +
Y u ) 2
+
T
x ( )
Y L
-
Y
u
U 2
£
Y
e
)
3* l '*
+
Y
2Y
Y
+Y
e
= ( - 1 2 ?)( J 3
-
9
-
For m fixed, which of the m+1 iteration formulas generated by this process is preferable?
No general theorem
[9],[16],
is available, but there is indications
that the
formulas I which are the most commonly used are often not po the best while there are some grounds for believing that the formulas I are preferable, pp Consider, for example, f(x) = x -A and define C pq n
b y
e
i+l
=
C
e
pq( i)
m + 2 ,
F
small as possible.
o
r
m
f l x e d
w
>
e
w
o
u
l
d
l
i
k
e
c
a s p
q
We have
m = 1:
m = 2:
C
1 0
- (n-1)(2n-l)/6a
C
Q 1
= (n -l)/l2a
G
2 Q
= (n-l)(2n-l)(3n-l)/24a
C
n
C
Q 2
2
2
2
2
= (n -l)(2n-l)/72a
2
= n(n -l)/24a
3
3
3
and
11m n —oo
m °oi
4 ,
lim n —
2^2
oo ° n
=
9
,
lim n-*oo
= § . °11
- 10
-
It should be noted that the condition that a formula have the form I does not lead to a unique formula. Consider pq the fourth order formula of Kiss [15] which in our notation is (1 - u D / 2 ) g
x
i+l
This has the form of I ^ with p+q = 2.
X
2
i "
u
x _ uD -hD u /6 * 2
2
3
but has the error term of a formula
It may be derived from I
the numerator and denominator of I
Q 2
by multiplying
by 1 - u D / 2 and neglect2
Q 2
ing terms in u . Snyder [23J derives Halley s formula by a "method of f
replacement" and he derives a fourth order formula by a "method of double replacement".
When the fourth order formula Is
translated into our notation, It is seen to be identical with the formula that Kiss derived by entirely different methods.
REFERENCES
A limited bibliography follows.
A more complete
bibliography will appear in a later paper. [1]
V. A. Bailey, "Prodigious Calculation," Australian J, of Sei., 3 ( 1 9 ^ 1 ) , 78-80.
[2]
H. Bateman, "Halley's Methods of Solving Equations," Am. Math. Mo., 45 (1938),
11-17.
- 11
[3]
D. R. Blaskett and H. Schwerdtfeger, "A Formula for the Solution of an Arbitrary Analytic Equation," Quart. Appl. Math., 3 (1945), 78-80.
[4]
E. Bodewig, "On Types of Convergence and on the Behavior of Approximations in the Neighborhood of a Multiple Root of an Equation," Quart. Appl. Math.,
[5]
( 1 9 4 9 ) ,
3 2 5 - 3 3 .
H. B. Curry, "Note on Iterations with Convergence of Higher Degree,"Quart. Appl. Math.,
[6]
7
9
( 1 9 5 1 ) *
204-5.
C. Domb, "On Iterative Solutions of Algebraic Equations," Proc Camb. Phil. S o c ,
45
( 1 9 4 9 ) ,
237-40.
[7]
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[8]
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Mo., 51 (1944), 36-8. [9]
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[10]
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( 1 9 5 3 ) .
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[11]
( 1 9 5 0 ) ,
5 1 7 - 5 2 2 .
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[13]
( 1 6 9 4 ) ,
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[12]
18
45
( 1 9 4 9 ) ,
2 3 0 - 6 .
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1 2
( 1 9 5 8 ) ,
5 8 - 6 0 .
1 3 6 .
- 12
[14]
-
A. Householder, Principles of Numerical Analysis, McGraw-Hill, (1953), 125.
[15]
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[16]
Z. Kopal, "Operational Methods in Numerical Analysis Based on Rational Approximations," Symposium on Numerical Approximation at U. of Wisconsin edited by R. E. Langer, (1959),
[17]
25-44.
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[18]
M. Muller, Zeit. Angew. Math. Mech,, 29 (1949), 160.
[19]
D. Pham et M. Ghinea, "Sur une Methode d Iteration dans
1
la theorie des Equations," Comptes Rendus, Paris, 249
[20]
(1959), 2262-4.
H. W. Richmond, "On Certain Formulae for Numerical Approximation," J. London Math. Soc., 19 (1944), 3 1 - 8 .
[21]
P. K. Rubbert, "Zur Radizierung Mit der Rechenmaschine," Zeit. Angew. Math. Mech., 28 (1948),
[22]
190-1.
E. Schroder, "Ueber Unendlich Viele Algorithmen zur Anflosung der Gleichungen," Math. Ann., 2 (1870),
[23]
317-365.
R. W. Snyder, "One More Correction Formula," Am. Math. Mo., 62 ( 1 9 5 5 ) , 7 2 2 - 5 .
-
[24]
-
H. S. Wall, "A Modification of Newton's Method/' Am. Math. Mo.,
[25]
1 3
55
(1948),
90-4.
H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand,
( 1 9 4 8 ) ,
Chap. XX.
[26]
F. A. Willers, Practical Analysis, Dover,
£27]
J. M. Wolfe, "A Determinant Formula for Higher Order Approximations of Roots," Math. Mag.,
[28]
31-
1
(1948),
(1958},
2 2 2 - 3 .
1 9 7 - 9 .
P. Wynn, "On a Cubically Convergent Process for Determining the Zeros of Certain Functions," MTAC, 9 7 - 1 0 0 .
10
( 1 9 5 6 ) ,
