An Algorithm for the Evaluation of Finite Trigonometric Series
Author(s): Gerald Goertzel
Reviewed work(s):
Source:
The American Mathematical Monthly,
Vol. 65, No. 1 (Jan., 1958), pp. 34-35
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2310304 .
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34
MATHEMATICAL
NOTES
[January
3) If
G
has
nzo
elements
whose orders
divide n2
-
n
or if
G has no elements
whose
orders
divide
n
-
I
when
X
is
an
automorphism,
then
G
is
Abelian.
It should
be
noted
that
if G is the
direct
product
of
two groups
A and
B,
where
A
(n
-1)
=-(1)
and
B
(n) (1),
then
n
leaves
A
elementwise
fixed and maps
B
into
(1).
Hence
any group
of
this
type
admits
Ft
as an
endomorphism,
and
some
such
restriction
as
in
3)
is
necessary
if
G is to be
Abelian.
The
proof
of the
proposition
is
as
follows.
Since
in
is
an endomorphism
anbn
=
(ab)
n. If
a
is
cancelled
on
the left and
b on the
right,
then
an-'bn-
I(ba)n-'.
It follows
that bl-nal-n
(ba)1-n
and
1-n
is an
endomorphism
(cf.
[1]).
Then
(aan-lbn)(a1-nb-nb)
=
(ab)
n(ab)1n
=ab,
and
1
=
a-lbnal-b-n,
or
an-1bn
=bnan-1.
This means
that nth
powers
commute
with
(tp
-
l)st
powers,
whence
G(n2-n)
is
Abelian
(cf.
[2]
p.
29
Ex.
4).
Now
the
product
of the two
endomorphisms
nt
by 1-n
is an
endomorphism
of G onto
the Abelian
group
G(n2-n)
with kernel G{n2
-n
}.
This
proves
the
first statement
of the
proposition.
Statement
2)
follows
from
the fact
that when
n
is an
automorphism,
every
element
is an
nth
power,
and
therefore
the equation
an-lbn
bna'-I implies
that
G(n
-1)
is in the center
of the
group.
It follows
that
1-n
is an endomorphism,
mapping
G onto
the Abelian
group
G(n
-1)
with kernel G {
n-1
.
Statement 3)
follows
immediately
from
1)
and
2).
We
are indebted
to the referee
for the referenlces
to the
literature.
References
1.
J.
W.
Young,
On
the
holomorphisms
of
a
group,
Trans.
Amer.
Math.
Soc.,
vol. 3, 1902,
pp. 186-191.
2.
Hans
Zassenhaus, Group
Theory (English
Edition),
New
York,
1949.
AN ALGORITHM
FOR
THE
EVALUATION
OF
FINITE
TRIGONOMETRIC
SERIES
GERALD
GOERTZEL,
Nuclear
Development
Corporation
of
America,
White
Plains,
N.
Y.
The algorithm
described
below
enables
one to
obtain the simultaneous
numerical
evaluation
of
C_
EO
ak
cos kx and
S_
El
ak
sin
kx for
given ak,
cos x,
and
sin
x.
Tables
for
sin
kx and
cos kx are not needed
and
only
about
N
multiplications
and
about
2N additions
or subtractions
are
required,
so
the
method
may
be
of
interest
to
programmers
of
digital computers.
The
algorithm
is
defined
by
UN+
2-
=
UN+
1
=
0;
Uk
=
ak+
2
COS
XUk+1
-
Uk+2,
k
=
N, N
-
1,
...
,
1'
C
=
ao
+
U1 cos
x-
U2,
S
=
U1
sin x.
To
establish
this
result,
consider
1958]
CLASSROOM NOTES
35
N
Vk=
E
aj
sin
(j
-
k
+
1)x;
k
=
1,
** *,
N,
j=k
VN+1
=
VN+2
0 .
Then
ak
sin x
+
2
COS
XVk+1
-
Vk+2
N
=
ak
sin
x
+
E
aj[2
cos
x
sin
(j-
k)x-
sin
(j-
k
-
1)x]
j2k+1
N
=
ak
sinx
+
E
aj
sin
(j-k
+
l)x Vk,
j=k+1
whence
Vk
=
Uk
sin
x
and,
in
particular, S-Vi
=
U
sin x.
Furthermore
N
ao
sin
x
+
V,
cos
x
-V2
=
ao
sin
x
+
E
aA[cos
x
sin
jx
-
sin
(j
-
1)x]
j=1
N
ao
sin
x
+ E
a1
cosjx
sin
x
=
C sin
x,
jl1
whence
C=ao+
U1 cos
x-
U2.
CLASSROOM NOTES
EDITED
BY C.
0.
OAKLEY, Haverford College
All material for this department should be sent
to
C. 0. Oakley, Department of
Mathe-
mlatics, Haverford
College,
Haverford,
Pa.
A
DIRECT
PROOF FOR THE LEAST
SQUARES SOLUTION
OF
A
SET
OF
CONDITION
EQUATIONS
ERWIN
SCHMID,
Coast and Geodetic
Survey, Washington,
D. C.
The problem of finding
the solution
of a
set
of
m
independent
"condition"
equations,
linear
in
the
n
variables
v1
,*
Vn,
n>m
n
(1)
~E
aiv,-
ajo=OX
i
=
1,,
m
j=1
such
that
1 v;
be
a
minimum is
generally solved, following Lagrange, by
mini-
mizing instead,
an
equivalent
function
involving
the
so-called
Lagrangian multi-
pliers.
The following approach
seems more
direct,
and
generalizes
a
basic
theorem
in
analytic geometry
to
n dimensions.
Multiplying
in
turn
each
of
equations (1) by
one of the
m
parameters