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NTIA Report 86208
A Method
Univariate Interpolation
That Has the Accuracy
a
Third Degree Polynomial
Hiroshi Akima
u
EP RTMENT
OF
OMM R
Malcolm Baldrige Secretary
Alfred C Sikes Assistant Secretary
for Communications and Information
November 1986
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TABLE OF CONTENTS
ABSTRACT
INTRODUCTION
2 DESIRED
PROPERTIES OF INTERPOLATION
3 THE INTERIM
METHOD
4.
THE
METHOD
5
USE
OF A HIGHER-DEGREE POLYNOMIAL A VARIATION
6 EXAMPLES
7. CONCLUSIONS
8
ACKNOWLEDGMENTS
9
REFERENCES
APPENDIX
A:
THE
UVIPIA
SUBROUTINE SUBPROGRAM
APPENDIX
B: INTERPOLATING FUNCTIONS
IN
A UNIT INTERVAL
Page
3
7
7
35
7
38
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LIST
OF
FIGURES
Page
Figure
1
Deflected line data Case
Figure
2.
Deflected line data
Case 2.
22
Figure
3.
Straight line plus cubic curve
y = 0 and y = x
2
/2 - 5x/6.
24
Figure 4.
Cubic curve y = x
3
21x /20.
25
Figure
5.
Sine
curve
y = s i n ~ x .
26
Figure
6. Akima
data
J.ACM
1970 .
27
Figure
7.
Akima
data
Modification A.
28
Figure
8.
Akima data Modification
B.
30
Figure
9.
Aklma
data
Modification
C.
3
Figure
10.
Akima
data
Modification
D.
33
Figure
Akima
data
Modification
E.
34
Figure
B-1. Function based on an nth degree polynomial
with n
3.
55
Figure
B-2. Function based
on
an nth degree polynomial
with n
4.
56
Figure
B-3.
Function based on an nth degree polynomial
with n
5.
57
Figur.e
B-4.
Function based on
an
nth degree polynomial
with
n 6.
58
Figure
B-5. Function
based on
an nth degree polynomial
with n
8.
59
Figure B-6. Function based
on an nth degree polynomial
with n
10. 60
Figure
B-7.
Function based
on
an nth degree and second-degree
polynomials
with
n
=
6.
6
Figure B-8.
Function based
on
an nth degree and
second-degree
polynomials
with
n =
10.
62
Figure
B-9.
Function based
on an
exponential
function
exp ax
with a = 1.
63
Figure
B-10. Function based
on
an exponential function
exp ax
with a
= 2. 64
Figure
B 11
Function based
on an
exponential
function
exp ax
with a =
5.
65
Figure
B-12. Function
based
on
an exponential function
exp ax
with
a
=
10.
66
Figure
B-13.
Piecewise
function composed
of two
second-degree
polynomials.
67
iv
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A METHOD OF
UNIVARIATE INTERPOLATION
mAT HAS THE ACCURACY OF
A THIRD-DEGREE POLYN OMIAL
Hiroshi
Akima
A method of
interpolation
th at accuratelyl n terpolates data
values
that
satisfy
a
function is said to
have
the accuracy
of
that
function.
The
desired
or
required propert ies
for a
uni val iate
interpolation
method
are
reviewed, and
the accuracy of
a
third-degree
polynomial
is
found
to
be one
of the
desirable propert ies . A method
of uni variate
interpolation
having
the accuracy of
a
third-degree
polynomial
while
retaining
the
des irab le propert ies
of th e
method
developed
earlier
by Akima(
M
17, PP. 589-602,
1970)
has been
developed.
The
newly de veloped method
is
based
on a
piecewise
function composed
of
a
set of
polynomials, each of degree three, at
most and
applicable
to
successi
ve intervals of the gi ven data
points
The
method
estimates the
f i r s t
derivative
of
the
interpolating
function
(or the slope of the curve) at
each given
data
point from
the
coordinates
of
seven
data
points. The resultant curve
looks
natural
in many cases when th e m ~ t h o
is
applied to curve
fi t t ing. The method
is presented with
examples.
Possible
use
of
a
higher-degree
polynomial
in
each interval is also examined.
Key
words: curve fi t t ing; interpolation; polynomial; second-degree polynomial;
third-degree polynomial; univariate
interpolation
1.
INTRODUCTION
Interpolation is a mathematical procedure for
supplying intermediate
terms
in
a given
series
of terms. In
this
report,
we consider
only interpolation of
univariate (one-var iable) single-val ued funct
ions.
We
seek
a method
of
interpolation that
will
produce a
natural-looking
curve
when
i t
is
applied to
curve fi t t ing. (When
there
is no risk
of
confusion, two terms
interpolation
and
curve fit t ing
will be used synonymously in this
report.)
The
author is
with
the Institute fo r
Telecommunication
Sciences, Nat iona l
Telecommunications and
Information
Administration,
U.S. Department of
Commerce Boulder Colorado 80303-3328.
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Some time ago, Akima 1970, 1972) developed a method of interpolation and
smooth curve
fitting hereinafter r f rr
to
as the original
A method)
that
produced
natural-looking curves
in many cases..
The original A method
emphasizes natural appearance of
the resultant
curve,
by suppressing excessive
undulations
or
wiggles) of
the
curves.
t
is
described in
a textbook
Carnahan and Wilkes, 1973) and
is included
in
the
IMSL
International
Mathematical and
Statistical Libraries , Inc .)
Library under
the
routine name of
IQHSCU
IMSL
1979). Examinations of
the
method
with some hostile
examples,
however, have
revealed
that
the method needs further improvement.
There
r several
aspects of
interpolation.
A method
emphasizes
an
aspect.
Another method emphasizes
another
aspect.
The method developed by
Fritsch and Carlson 1980) or the improved version of the method by Fritsch
1982 or by
Fr i
tach
and Butl and 1984)
hereinafter r f rr
to
collecti vely
as
the
F-C-B method) outperforms the
original
method
when
monotonici ty
of the
data
must
be preserved. The
method developed by
Roulier
1980) or the method
by
McAllister
and
Roulier
1981a; 1981b)
hereinafter r f rr
to
collectively
as
the
M-Rmethod) preserves the convexity of
the
data in addition to the
m ono ton icl ty. There r also many
other
shape-preserving methods as
described
by Gregory 1985).
In the primi t ve stage of development, a method can be superior to
all
other methods
in
most cases.
In
the advanced stage of development with many
methods
exist ing,
t
becomes
apparent
that
a method
is
superior to other
methods only
in
some applications and
another
method is
superior
in other
applications. Desired
or
required properties for interpolation methods differ
widely from case
to case,
and
some
properties
are
mutually
incompatible.
In this
report ,
we identify desired or required
properties for an
interpolation
method, discuss their mutual
compatibility, establish
our goals,
and derive
the
guidelines for developing an improved method.
I t turns
ou t
that
one
of our goals
is
to
develop a method
that
has
the
accuracy
of thi rd-degree
cubic)
polynomial,
1.e.,
a method
that accurately interpolates the
given data
when the given set of data points l les on a cubic curve. We have developed a
method
hereinafter referred to
as
the
improved A method)
that
meets
the goals.
Like
the
original A method,
the
improved A method does not always
preserve
monotonici ty
or convexi ty;
we have not intended to
preserve
t
in
developing
the improved A method.
We
propose the improved A method a s a replacement for
the
original A method when
natural
appearance
of the resultant
curve
is
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important;
we do no t propose
i t
as a replacement
for
th e F-C-B method or the
M R
method
when monotonicity
or convexity must be preserved.
In this
report,
desired
properties
of
interpolation ar e discussed and
the
goals for development are
established in S ec tion
2. An interim method
that
has
th e
accuracy
of
a
second-degree
polynomial and
some
other desirable properties
is
developed
as
th e interim
step
of improvement of th e original A method in
Section
3.
A method that has th e accuracy
of
a third-degree polynomial and
meets the goals is developed
as
the improved A method in Section 4. Possible
use of higher-degree polynomials is c on si de re d a s a
variation
of
th e improved A
method in Section
5
wi th th e details l e f t
to
Appendi x
B.
Both
methods
developed in S ec tio ns 3 and 4 us e third-degree
polynomials.)
Some examples are
shown in Section
6.
A brief summary and so:me remarks for the use of the
/ improved A method ar e given
in
Section
7.
A
Fortran subroutine
subprogram
that
implements th e method developed in Sections
4
and
5
is listed
in
Appendix A.
Throughout this
report
we use
some conventions. We
assume
o
that
th e x and y
variables
represent th e independent
variable
and th e
f un ct io n v al ue ,
o
that
th e x and y
variables
also represent th e abscissa and ordinate
of a two-dimensional
Ca rte sia n c oo rdina te,
o that th e
f i r s t
derivative
of
the function or the slope
of
th e curve
is
represented by
y ,
o
that
th e
x, y, and
y values
at
data
point
Pi
a re r ep re se nt ed
by
x i
Yi and YI and
o that th e sequence {xi}
is in
an
increasing
order.
2. DESIRED PROPERTIES O INTERPOLATION
There are several
desired
properties
of i n t erp o l at i o n
depending on
particular
purposes of the user. Since some desired prop ert ies are mutually
incompatible,
as
discussed
l at er,
anyone method cannot possess a l l
the
desired
properties
simultaneously. In this section we w ill i de ntif y major
properties
desired for
interpolation,
discuss their mutual compati
b i l i
ty, and
se t
ou r
goals
by selecting
some
of
them.
In order for
th e
curve
to
be smooth
when interpolation
is applied
to curve
f i t t i ng, we require that th e interpolating function and i ts f irs t derivative be
continuous, i e in mathematical terms, t he f un ct io n be C continuous.
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Since
t
is desired in many
cases that th e curve is
affected
only
in
a
small
neighborhood of the data point when a
data
point is added, deleted, or
moved,
we
require that
the
method be based on local
procedures confined
to a
small neighborhood
of
th e
point a t which in te r p o la tio n of the value
r equir ed.
This
requirement
was
already
recognized
before the turn of th e
century; a method based
on
local procedures was developed by Karup (1899) and
l a t e r
improved
by
Ackland
1915).
do
not consider the so-called global
interpolation
methods such as the spline-function method
O re v i l l e , 1967;
Cline, 1974). Although some traditional methods such a s L ag ra ng e s or Newton s
method Hildebrand, 1956; Davis, 1975)
are
based on
local procedures,
we also
do
not
consider these methods since these methods fai l
to
produce a C
1
continuous
curve.
P re se rv in g t he shape of th e data such as monoton1city and convexity
is
a
desired property in some cases.
Monotonic
ty
must be preserved,
fo r
example,
when the
s e t
of
input data points represents
a probability distr tbution
function.
The property
of preserVing
monotonicity dictates
that
the
p ortio n o f
th e curve between a pair of successive data po ints must
l ie
between the two
points
n t
ordinates
and that th e portion of th e curve between a pair of
successive d ata p oin ts having an identical ordinate value must be a
horizontal
line segment. The property of preservin g convexity
dictates that th e
portion
of the curve that connects
three
collinear data points must be a
line
segment.
Invariance
under
certain
types o f c oo rd in at e t ra ns fo rm at io n
is
a
desired
p ro per ty i n
some applications.
Desirability
of
invariance under a
linear-scale
transformation
x
=
a u
Y b v,
where a and b are nonzero
constants, is
obvious.
In
some c as es , in va ria nc e
under
another
type of linear
coordinate
transformation
x = a u
2
Y b u
C
v,
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where a, b,
and c
are
nonzero constants, is desirable.
In studying the
fluctuation of a clock, for example one m y plot ei ther the reading of the
clock
against
the correct time as an almost
0
slope curve or the error
against
the correct
time
a s
an
almost
horizontal curve , and
both
plottings
should
represent physically th e same phenomenon.
Symmetry of
the
method
is another desired
property
in
most
applications.
Symmetry is
described
here
as
the property that the method p ro du ce s a sym metric
curve when the data points
are symmetric .
Pro du cing a curve
that looks
natural is another
important
property desired
for the method.
Since
a
skilled
draftsman draws a
natural-looking
curve
with
French
curves, i t is
desirable
that the
method
simulate
a
skilled
draftsman.
Since
a natural-looking curve has
no t
been defined
mathematically,
we
will
consider here
what
behavior
of
a
curve
makes
the
curve look
natural.
t
seems
that
a good w y
of describing
a natural-looking
curve
is to
describe the
behavior of the curve in terms of curva ture
that is
the
rate
of deviation of a
curve from a s tr aigh t l in e. believe that a curve does not look natural i f
i ts curvature changes
excessively
and
too
frequently.
believe that
change
of
curvature must be suppressed
as
much
as possible
to m ke
the curve look
na
ur
ale
rom this
general
requirement, several
specifi
c requirements
are
deri ved. Excessi ve undulations
that
are accompanied by excessi ve curvature
changes
should
be
avoided. Since the curve
cha nge s from convex
to
concave
or
the reverse and therefore changes i ts
curvature at
an
inflection
point, the
number
of inflection points
must
be kept
to a
small
number.
Since
a
line
segment embedded in a generally curved l ine exhibit s large changes of curva ture
and additional inflection
points near
the
endpoints
of
the
line segment in m ny
cases, producing
such a
l ine
segment must be a voi ded i f possible.
In
addition
to these descriptions terms of curva tu re ,
we also
describe
the
property of
producing
a natural-looking curve in terms of
accuracy of
a
simple
mathematical function.
say
that
an
interpolation
method has the
accuracy
of
a
function
if the
method
accurately interpolates
the data
when
the
data
points l ie on
a curve of the function.
know intuitively
that curves of
some
simple mathematical functions such as a low-degree polynomial or a sine
or cosine
function
look good and natural. Therefore, accurate or close
interpolation
of
data points
given on
sueh curves i s also
desi red
Particularly, a
third-degree
polynomial is a polynomi al of the
lowest degree
that can have an inflection point. t can be
used
at
l eas t
local ly to
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approximate a curve that has inflection points. Therefore, w require
th e
accuracy
of
a third-degree polynomial for our method.
Both
th e
method developed by Karup
and the original
A method have
th e
accuracy
of
a s ec on d-d eg re e polynomial
conditionally,
i .
e they interpolate
given
data
accurately when
th e
given
data
points are
equally
spaced
in their
abscissas. The osculatory method by Ackland nas the accuracy of a second
degree polynomial unconditionally,
i e
i t
interpolates
given d at a a cc ur at el y
even
when the gi
ven
data
points are
unequally
spaced.
will l a t e r
review
these
methods in more detail and develop an interim method that has
the
unconditional) accuracy of a second-degree polynomial and other
de s i rabl e
properties.
n
interpolation
method is desired to be continuous in the
sense
that th e
resultant
curve changes very
l i t t le
when
a
small
change
is
made
in the input
data. The
curve
resulting
from
the original
A method
will
change
abruptly ~ n
three data points,
P
i
-
2
, P
i
-
1
, and
Pi,
are
collinear
and three
data points, P i
Pi+1 and P
i
+
2
, are changed from almost collinear to exactly collinear. If
such
discontinuous
b eh av io r cannot be eliminated entirely, i t is desired to
reduce th e chance of occurrence of such
behavior.
L in ea ri ty o f th e
method
is
another
desired property in some applications.
Linearity is described
here
as th e property that th e interpolated values
satisfy y x) a
y 1) x)
+ b
y 2) x) { ~
a
y 1) xi)
+ b
y 2) Xi)
fo r
a ll
i
,
where a and b ar e nonzero constants.
Unfortunately, some of t he se d es ir ed properties are mutually incompatible.
One of the
most
serious
problems
is
incompatibility
of the
pro perty
of
preserving
monotonici
ty
wi th
some
other
properties. Its incompatibili ty wi th
the property of invariance under
the l i n e a r
c oo rd in at e t ra n sf o rm a ti o n
represented by 2)
is
obvious. The requirement for preserving monotoniclty or
convexity
sometimes produces embedded
line
segments and an
excessive
number of
inflection points. The m onot onl cit y or c on ve xi ty
requirement is
incompatible
with the
requirement
of
accuracy
of
a
third-degree
polynomial.
I t
may
no t
be a
good
idea to require preserving
monotonicity
or convexity when
such a
property
is no t
really necessary.
Since
we
already have
th e
F-C-B
or M R
method as a
good i nt e r pol a t i on method, we wi l l drop th e requirements fo r preserving
monotonicity
and
convexity.
Even within th e same general desirability
for producing a natural-looking
curve, re qui re me nt s fo r
some d es i rab l e
pr ope r t i e s need adjustment
and
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compromise. Suppression of excessi ve undulations and embedded l ine s egments
need a mutual adjustment.
When several
successive
data points are
on a
straig ht lin e such as th e
x
axis)
and
other
data points
are
elsewhere, th e
portion
of the
curve
that connects the collinear data
points
is generally
desired to
be a
l ine
segment.
The original
A method and
the M R
method produce
a
l ine
segment
when three data points are
collinear, and
the
F-C-B method doe s
the same
when
three data points
are
on a horizontal l ine.
This
property tends
to produce unnatural-looking l ine segments. We feel
that a
deviation
from the
line segment
should
be
allowed when
only three data
points
are collinear. In
the
method
developed
in this report,
we
require a line segment
when
four
data
points or more are col l inear ,
but
not when only
three
data
points
are
collinear.
The
requirement
of
l ine
segment
for severa l
collinear
data
points,
regard
less of the
number
of collinear data points,
is
incompatible with requirements
for other properties
such as continuity and l inearity of
the
method as
in
the
original
A method. Although
the
discontinuity
of the
method
is not entirely
eliminated,
the increase
in
th e number of collinear data
points for
a line
segment from
three to
four is
expected to reduce
the chance
of
occurrence
of
discontinuous
behavior. This
can be
accounted
for
by
th e
fact
that
the
probability of having
four
collinear data points by
chance
is much
less
than
the probability of
having
three collinear data point s by chance.
3. THE INTERIM
M THO
s
a
preliminary step in developing
an
i n t ~ r p o l a t i o n
method that meets
our
goals
including
having
the accuracy of a third-degree
polynomial, we try
to
develop,
in this
section,
an
interim
method that
has the
accuracy
of
a
second
degree
polynomial
and
other
des irab le proper ties .
For
this
purpose,
we
r s t
review basic procedures
and majo r
properties
of the osculatory
method Ackland,
1915) that has th e accuracy of a s e c o n d d e g r e E ~ polynomial and th e original A
method Akima, 1970, 1972)
that has some of the desirable propert ies .
In
both
the
osculatory
and
original
A methods, th e
interpolating
function
is a piecewise function composed
of
a set
of thi rd-degree cubic)
polynomials,
applicable
to
successive intervals
of
th e given
data points.
Function value y
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corresponding to an x
value
in
th e interval
between
xi
and
xi 1
is
calculated
by
where aO a1 a2 and
a3
are the coefficients of the polynomial
fo r that
inter
val. These coefficients ar e determined by th e given y values and t he e st im at ed
y
values i .e.
th e f i r s t derivatives)
at t he e nd po in ts
of the interval as
4)
and
where mi is th e slope of th e line segment
connecting
Pi and P
i
1
and is
represented
by
5)
The only difference
between
th e two
methods
is in th e
procedure
of
estimating
th e f i r s t
derivative of th e interpolating function at
each given data
point.
In th e
osculatory
method
th e
f i r s t derivative of the
interpolating
function at data point Pi
is
estimated with a set of three data points, P
i
-
1
Pi and P
i
1
I t is estimated as th e f i r s t
derivative
of the second-degree
polynomial
f i tted to th e three data
points.
I t is clear
from this procedure
that
th e f i r s t
derivative
is
accurately estimated when th e three
data p oin ts
ar e
on a curve
of
a second-degree polynomial.
In th e original
A method
th e
f i r s t
deri vati ve of th e function at
data
point Pi is
estimated
with a
set of
five points, P
i
-
2
P
i
-
1
P i
P
i
1
and
Pi 2. Two
line-segment
slopes,mi_1 the slope
of
th e
line
segment connecting
Pi-1 and
Pi )
and
mi th e
slope of the
line
segment
connecting
Pi and P
i
+,),
are
used
as the
primary es tim ate s of
th e
f i r s t
derivative,
and
th e final
estimate
is calculated
as
th e weighted
mean of
th e primary estimates,
i .e .
8
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(6)
The
weight
for Mi
is
th e reciprocal of the absolute value of the
difference
between M
i
1
and
mi-2
and
th e
weight
for
mi
is
the
reciprocal
of the absolu te
value of
th e
difference between m
i
1
and
mi
i .e . ,
(7)
where abs{ } stands for the absolute value
of.
The basic concept behind the
selection
of the
weight
is
that
the
primary
estimate
based
on
the
data points
on th e l e f t
(or
right)
side
of the point
in
question should be given a small
weight
i f
the
data
points on the lef t (or
right)
side are
volati le (or
far
from being collinear). The f i rs t
derivative
is accurately estimated
when
the
five
data
po in ts are
on a
curve
of a second-degree polynomial and are equally
spaced in
their
abscissas.
Before developing th e interim
method
w present the expression of the
f i rs t
derivative, at data point Pi
of
a second-degree polynomial fit ted to a
set of three data
points, Pi
P
j
and P
k
I f
we denote
th e f i rs t
derivative
by
F i , j ,k ,
i t
is
given
by
F(i , j ,k)
(8)
Note that the f i rs t index in th e expression o f F(i , j ,k) must be i , which is the
point number of the point
i n questi on ,
and that th e
remaining indices
can be
given in any order.
With this
notation,
the estimate of the f i rs t derivative in th e oscula
tory method is
represented
by
F(i,i-1 ,1 1 .
9
(9)
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The
set of
three successive data points that can be used fo r e st im at in g
th e fi rst derivative of
th e
interpolating
function at
data
point
Pi by f i t t i ng
a second-degree polynomial is not limited to
th e set of
three points used in
th e osculatory
method,
i e th e set of t hr ee p oi nt s, Pi-1 Pi
and P
i
+
1
Any
set of ,consecutive three data
points can
be
used
i f the set includes
Pi. There
r
three qua l i f i e d s e t s , i e P
i
-
2
, P
i
-
1
, P i ) P
i
-
1
, Pi P
i
+
1
), and
Pi P
i
+
1
, P
i
+
2
).
can, therefore,
f i t three
second-degree polynomials to
the
three
sets of three data points and
calculate
three primary
estimates.
The
estimate 9)
used
in the osculatory
method can be
considered
one
of the three
primary estimates.
The three
primary estimates
of the
f i r s t
derivative of the
interpolating function at
Pi
r
Ylm
F i , i -2, i -1),
and
YI0
F i,i-1,i+1),
yIp
F i,i+1,i+2).
10
Since
a ll these three
primary
estimates
for
the f i r s t derivative r accurate
when
a ll th e five data points,
P
i
-
2
through P
i
+
2
,
ar e on
a curve
of
a second
degree polynomial, use
of
a weighted
mean of th e three
primary
estimates with
any
set o f w ei gh ts , i e
11
as th e
final
es tim ate fo r th e f i r s t
deri
vati
ve
of the
interpolating
function
will yield
a method
that
has
th e
accuracy
of
a second-degree polynomial.
The
osculatory method can be considered a special case where th e two weights
im
and
w
ip
equal zero.
The
basic
concept behind
th e
original
A method
dictates that
a small
weight be
assigned to
a primary estimate
i f th e
primary estimate is
calculated
from a set of volatile data points.
In
th e interim method, we could
take
the
absolute
value
of
the
second
derivative of the
second-degree polynomial
fit ted
to a
set
of three d ata p oi nts as a measure
of vol a t i l i t y,
use
the
reciprocal
of
th e measure of vol a t i l i t y
as th e
weight , and assign
th e
weight
to the
primary
estimate
calculated
from
th e set of
data
points. Use of the
second-degree
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polynomial as a measure of
v o l at i l i t y ,
however:,
is applicable
only to a
se t
of
three
data points. For
future
development, w e need a measure of vol a t i l i ty
t h a t
is
independent
of the num er of
data points. s
such a measure of
vol a t i l i t y,
we
take
the sum of squares of deviations from a straight l ine
of
the least-square f i t Then, th e measure of
v ol at il it y is
represented y
V i,j,k)
12
where b
O
and b, are the coefficients of the first-degree
linear)
polynomial of
the
least-square
f i t to
th e
data
points
and
are
represented y
In 12 and 13), symbol
represents
a summation over three data points, Pi
P
j
, and P
k
Note that
the
three
indices in the
expression
of
V i,j,k)
in 12
can be given
in
any
order.
In addition
to the
v o l at i l i t y
of the
data point set, we
include
another
factor
in
th e weight. We consider that the primary estimate should be given a
small weight
i f
the data point
set includes
a data point
or data
points far
distant from the data point
in
question. We define the
distance
factor
y
14 )
Note that th e
f irs t
index
in
the
expression
of D i,j,k) must be
i
which
is
th e
point num er
of
the point in question, and that th e remaining
indices
can be
given
in
any
order.
We
use
the
reciprocal
of
the product of
V i,j,k)
and
D i,j,k)
as the
weight and assi gn the weight
to
the primary estimate calculated from th e
set of
data
points,
Pi P
and P
k
Then, th e t h r ~ weights corresponding
to the
three
primary estimates 10
are
represented by
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W
im
1 /
[V i,i-2,1-1) 0 i, i-2,1-1)J,
[V i,1-1,i+1) 0 i ,1-1,i+1)J,
wip 1 /
[V 1,i+1,i+2) 0 I,i+1,i+2)J.
5
When a set of three data poInts is collinear, th e V
value
equals zero, and th e
corresponding weight becomes i n fi n i t e . When any weight becomes i n fi n i t e , we
reset infinite
weights
to
unity and f in ite weights to
zero before using
11 to
calculate th e final
estimate
of the
f i r s t
derivative.
Note that
th e
interim method
uses
a to ta l
of five
data
points
P
i
_
2
through
P
i
2
to
estimate
th e rst derivative
at data
point Pi.
Because
of the
use
of these
weights
1 5 ) ,
the interim
method has
th e
property t h a t , when a s e t
of
three data p o in ts, P
i
-
1
through P
i
+
1
, is
collinear,
the estimate of the f i r s t
derivative
at Pi
equais
the slope
of
th e
s tr ai gh t l in e passing through th e set of data points. The method also has th e
property that,
when
a set of three data points, Pi through P
i
+
2
, is
collinear,
the estimate of the rs t deri vati ve at P
equals the slope of th e straight
line passing
through th e
set
of data
points
unless
another set
of t h r e a d a t a
points, P
i
-
2
through P i 1s also
collinear.
Note that th e inte rim method has
inherited these
properties
from th e original A method.
When the data point in question is th e f i r s t
or
th e
l as t
data point, only
one set of three data points
is
available and therefore only one pr
imary
estimate can be
calculated. When th e
data point in question is
th e
second
or
th e second l as t data point,
only
two primary
estimates
can be calculated. In
these
cases,
we use
only
th e
available primary estimate
or
estimates for
calculating th e final
estimate
of th e
f i r s t
derivative.
Like th e osculatory and
original
A methods, th e
interim
method also uses a
third-degree polynomial
fo r
th e
interval
between each pair of s uc ce ss iv e d at a
points.
The
interim
method
interpolates
th e
y
value
with
3), 4),
and
5).
Since the interIm method retains some of th e desirable
properties
of th e
original A
method with an a dd it io na l d es ir ab le property of the accuracy
of
a
second-degree polynomial, we
also
call th e
interim
method th e improved A method
of the second-degree polynomial version.
will examine th e performance of this interim method with some examples
in Section
6.
2
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4.
TH
M THO
In the preceding
section, we
have developed an
interim
method that
ha s th e
accuracy
of a second-degree polynomial and retains some of the
desirable
properties of
th e original A method. In
this
section we
will develop
a method
that
ha s
th e accuracy of a third-degree polynomial by modifying th e
interim
method
to
a
third-degree polynomial version. We
also
modify
th e osculatory
method which has th e accuracy of a second-degree polynomial} i n su ch a,way,
that the
modified
method
ha s th e accuracy of
a
third-degree polynomial.
Before we proceed, we present the expression of the f i rs t derivative, at
da
t a point P i of
a
third-degree
polynomial
f i tted to
a se t of four data
points,
P i P
j
P
k
and
Pl.
If we denote th e f i rs t
derivative
by
F i , j ,k , l ,
i t
is
represented
by
F i , j ,k , l
6
Note that
th e f i rs t
index
in
the
e xp re ss io n o f F i , j ,k , l
must be
i ,
which
i s
the point number of the point in question, and that the
remaining indices
can
be
given
in any
order.
We also
present
th e
sum of squares of
deviations
from a
s tr aig ht lin e of
th e
least-square
f i t as th e measure
of
voIatili ty. The measure of
v o l a t i l i ty
is
represented
by
V i , j , k , l )
7
where b
O
and b
1
are
th e coefficients of the f ir s t- d eg r ee l in e ar )
polynomial
of
th e least-square
f i t
to th e
data
points
and
are
represented
as
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and
18)
In 17) and
18), symbol
represents
a summation over four
data points, Pi
P
j
P
k
, and
Pl.
Note that the four
indices in
t he e xp re ss io n of V i,j ,k,l) in
17) can be given n any order.
The
distance
factor
can be represented as
19
Note
that
th e
f i r s t
index in
th e expression of
D i , j , k, l )
must
be i ,
which
is
th e point
number
of the point in Q uestion, and
that
th e remaining
indices
can
be
given
in
any
order.
There ar e
four
sets of four consecutive data
points
that
include Pi 1 e
P
i
-
3
, P
i
-
2
P
i
-
1
, P i ) P
i
-
2
, P
i
-
1
, Pi Pi+1)
P
i
1
, Pi Pi+1 Pi+2) and
Pi P
i
+
1
, P
i
+
2
P
i
+
3
). Seven
data points,
P
i
-
3
through
P i ~ ar e involved.
cal cu\l at e f our
primary estimates of th e f i r s t deri
vati
ve of th e i nt e r-
polating
function,
each as the
f i r s t deri
vati ve of a third-degree polynomial
fit ted to a
set
of
fou r c on se cu tive
data points. These primary estimates are
yImm F i , i- 3 ,i- 2 ,i- 1 ) ,
yIm F i , i-2, i-1 , i+1),
20
yIp
F 1,i-1,i+1,i+2),
and
yI
pp
= F 1,i+1,i+2,i+3).
Since
these
primary
e sti ma te s a re
a ll
accurate
when all
th e
seven
data p oin ts
are on a curve
of
a third-degree polynom ial, use of any combination
of
these
primary estimates, i e
2
)
1
4
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as the final
estimate
of the f i rs t derivative
of
the interpolating
function
yields
a method
that
has the accuracy of a third-degree
polynomial.
With
these
notations, developing the final method by modifying the interim
method to the
third-degree
polynomial
version is
rather
straightforward.
Like
the
interim
method,
the
final
method
uses
al l
primary
estimates,
i e four
primary
estimates
calculated by
20 in this
case. I t uses
the
reciprocal
of
the product of V(i, j ,k, l ) and O(i, j ,k, l ) calculated by 17 and 19 as
the
weight for the
primary
estimate calculated froIn the set of data,. p.o.tnt.s,. Pi P
j
P
k
, and
Pl.
Then, the four weights correspondi,ng to the four
primary
estimates
20
are represented
by
w
imm
1 /
[V(i , i-3,i-2,i-1) 0(i , i -3, i -2, i -1)J ,
w
im
1 / [V(i,i-2,i-1,i+1)
0(i , i-2,i-1,i+1)J,
22
W
ip
1 / [V(i,i-1,i+1,i+2) 0(I,i-1,i+1,i+2)J,
and
ipp
=
1 /
[V(i,i+1,i+2,i+3) 0(i,i+1,i+2,i+3)J.
hen a set of four data points is collinear,
the
V value equals zero , and
the
corresponding weight becomes inf inite. hen any weight becomes infini te , we
reset
infinite
weights
to
unity
and
finite
weights
to
zero
before
using
21
to
calculate
the final estimate of the
f i rs t
derivative.
Note that the final method uses a total
of
seven
data
points, P
i
3
through
P
i
+
3
, to estimate the
f i rs t derivative
at data point
Pi.
Because of the use of these weights shown in (22), the final method
has
the
property
that, when a set of four data points, P
i
-
1
through P
i
+
2
, is
collinear, the
estimate of
the f i rs t derivative
at
Pi equals
the
slope of the
s tr aigh t l in e passing
through
the set of data points. he method also has the
property that, when a set of four data points, Pi through P
i
+
3
,
is collinear,
the
estimate
of the
f i rs t
deri vati ve at Pi
equals
the slope of the straight
line passing
through
the set
of
data points unless another set of
four
data
points, P
i
-
3
through
Pi is also collinear. I t clear from these
properties
of the method that,
when
a set of four data points or more
is
collinear, the
method
will
produce
a
line segment across the set of data
points.
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When
the data
point in
question
is
th e
f i rs t or the
last
data
point, only
one set of four data points
is available
and
therefore only
one primary
est i
mate can be calculated. When th e
data point in
question is
the
second or the
second last data
point,
only two primary estimates can be
calculated.
When
the
data
point
in
question is the
third or the third
last data
point, only
three
primary estimates can be
calculated.
In these
cases, the
method uses only th e
available
primary estimate or est imates for calculating
the
final estimate of
the f i rs t derivative.
Like the interim method as well as
the
osculatory and original A methods,
th e final
method
also
uses a
third-degree
polynomial for
th e interval
between
each pair
of successive data
points. This method interpolates th e y
value
with
3 ,
4 ,
and 5).
Since
the
final
method
is
expected
to
retain
the
desirable propert ies of
the original
A method
with addit iona l des irab le
property o f
th e accuracy
of a
third-degree polynomial, we also call
the
final method
the
improved A method of
the third-degree polynomial version
or
simply the improved A method.
For completeness in la tel comparisons, we develop
another
method by
modifying t he osculato ry method to i ts third-degree polynomial version. he
osculatory method
of the
original
version or
the second-degree polynomial
version)
uses only one primary estimate based on the
set of
three data points
that includes the data point in question as
the
center
point. I t therefore
uses no weights based on
the
volati l i ty
of
the
data
points. o
satisfy
the
symmetry requirements , t he third-degree polynomial version uses two primary
estimates, each based
of
the
set
of
four data
points that includes the data
point in question near the center of the set. I t
uses
no weights based on the
volati l i ty
of
the
data points; i t uses a simple mean of the second and third
primary es timate s in 20). If we use 21) to calculate the final
es timate of
the f i rs t derivative, we can represent the weights as
W
imm
0,
wim
wip
and
w
ipp
o
1
6
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Note
th a t t h i s
method uses only
five
data
points,
P
i
-
2
through
Pi+2
to
e st im at e th e final estimate of the f irs t
derivative.
In this
osculatory
method of the
third-degree
polynomial version, we also
use a
third-degree
polynomial
fo r
the interval between each pair
of
successive
data
points.
This method
interpolates
th e
y
val ue with
3), 4),
and
5).
We will examine the performances of these two methods with some examples
in
Section 6.
5. USE O A HIGHER-DEGREE
POLYNOMI L
A VARIATION
So
far
in th e
preceding
two
s e c t i o n s ,
we
have,
concentr ated
on
the
procedure f or e st im at in g
th e
f i r s t derivative f
th e
interpolating
function at
each
data
point. We have assumed a third-degree polynomial to be
applied
to
th e
interval of each
successive
pair
of data
points.
A third-degree polynomial
is not, however,
th e only
function that is d1etermined by
th e
va l ues
of the
function and f i r s t derivative at
two points.
A
hyperbolic
function r a
combination of
exponential
functions),
rational function
i .e .
a quotient of
two polynomials), a com bi nati on of
two
second-degree polynomials, and higher
degree polynomial
are
some of the examples
of such functions.
In
this
section,
we present
an
interpolating
function
based
on
an
nth-degree
polynomial, with n
being
equal
to three
or greater.
See Appendix B fo r
th e
behavior, in an
inter val, of some interpolating
functions
inclUding
the
polynomials
of various
degrees.)
Fo r simplicity, we consider an
interpolc:iting
function y y x) in an,
interval between Pi and Pi+1
in
a
new coordinate
system
in
which
the
abscissa
values
equal
0 and 1
at
Pi and Pi +1 respecti
vely,
and
th e
o rd in at e, v al ue s
equal to 0 at these two
points.
We call th e
new
coordinate system th e u-v
coor dinate
system.
The
l i n ear
c oo rd in at e t ra ns fo rm at io n
between
the
u-v
coordinate system and th e x-y coordinate system
is
represented by
24)
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The
irs t
derivatives
in the
two coordinate systems,
y
ar e related by
dy/dx and v
dV/du,
25
where
t 1s clear from
24)
through
26)
that
26)
u = 0,
v
= 0,
v
0
2 7
at P
i
+
1
,
where v and v are the v values at
u
and u
=
1. The set o f eq ua tion s
27
indicates that
th e u-v coordinate sys tem h as
the
p ~ p r t y described in th e
beginning
of this
paragraph.
As t he v u) func t ion, we present here an nth-degree polynomial in u
represented by
v u) 28)
The coefficients A
O
and A
1
are given by
A
O
[v
O
+
n
1)vi
J
/ [n n - 2)J,
A
1
=
[ n - 1)v6
v,J
/
[n n
-
2)J.
29)
When
th e y and y values are given at Pi and P
i
1
, we can calculate the v
values
at
these points by 27) and 26), and the, A
O
and coefficients by
29). For a given x value, we can calculate th e corresponding u value by
30)
which is equivalent to the irs t equation in 24), the
v u)
value by 28), and
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finally the y value by
31
which
is equ iva lent to
the second equation
in
(24).
I t is
easy
to show that
31 with supplementary relations 26 through 30 reduces
to
3 with 4 and
(5)
when
n equals
three.
Use of a
higher-degree
polynomial has an
advantage.
Undulations in the
resultant
curve
will
be reduced when the Inte rpolat ion method
is
applied
to
curve
f i t t ing. Use
of a higher-degree polynomilal, however, has a disadvantage,
also. The
resultant curve sometimes is too
tight, i . e .
th e
portion
of the
curve between a
pair
of successive data points
is
so
close
to
the
line
segment
connecting the pair of data points that the whole curve looks as if i t were
deflected. Use
of a higher-degree polynomial has
another
disadvantage.
The
interpolation method will
not
achieve
the
accuracy of a third-degree
polynomial.
have
implemented
the
variation i .e .
the use of a higher-degree
polynomial
descri
bed
in
this
section)
in the improved A method
as
a user
option.
In the
Fortran subprogram subroutine listed in Appendix A selection
of the degree of the polynomial
is
left to the user. Depending on the
user's
situation, the user has an option:
the
user
will
either use a
value greater
than
three
as the degree of the polynomial
to
reduce undulations while giving
up
the
accuracy
of
a tnird-degree polynomial,
or
the
user
will use a value
of
three
retaining the accuracy of a third degree polynomial. Dependence of the
performance of the curve of v u in
28
on the degree of the polynomial n is
shown graphical ly in
Appendix
B
Examples presented
in
Section
6
include curve
fitted with n = 6. I t is expected that the
user
of the improved Amethod will
develop a general idea
on
the
selection
of the degree
of
the polynomial from
the information in
AppendixB and
the
examples
in Sect ion
6.
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6. X MPL S
This section i l lustrates performances of the methods developed in this
report
wi
th
examples in
Figures
1 through 11. In each figure
presented in
this
section,
curves
resulting
from
two
existing
methods
are
also p lo tted
fo r
comparison. Six curves
in
each figure re from the top to
the
bottom,
1) the original osculatory method
or
the osculato ry method of the
second-degree polynomial version developed
by
Ackland, 1915)
2) the
modified osculatory
method
o r
the osculato ry method
of
the
third-degree polynomial version developed in Section 4
3 the original A method developed
by
Akima 1970)
4 the
interim
method o r the improved A method of the second-degree
polynomial version developed
in
Section 3
5 the improved A method
wi
th n = or
the
improved A method
of
the
third-degree polynomial version developed in Section 4 without the
variation described in Section 5)
6
the improved A method wi th n
=
6 or the improved A method of the
third-degree polynomial version developed in
Section
4, wi
th
the
variation described
in
Section 5, with the degree
of
polynomial set
to six .
Each data point is
plotted
with an x symbol. The x and y
coordinate values
of the data
points
are
tabulated
above
the caption of
each
figure.
In Figure 1,
data points are
taken from a deflected
line. The
top two
curves
resulting
from the
two
osculatory
methods) exhibit
overshoots
in
the
horizontal portions of the
curve,
while the
overshoots
are nonexistent
in
other
curves resulting from the original A method, interim method, and improved A
method . The
bottom
two curves result ing from the improved A method)
i l lustrate the effect of the degree of polynomials , n = 3 versus n = 6.
In Figure 2, data
points
are taken also from a
deflected
line; they
consist
of
all
data
points
in Figure
1
plus
a
data point
at
the
center
of the
s loping r eg ion. The top four
curves
resulting from the two osculatory
methods,
original
A method, and in,terim method) exhibit overshoots
in
the
horizontal portions of the
curve,
while
the overshoots
are nonexistent in the
bottom two curves
resulting
from
the
improved A method).
The
bottom
two
curves again i l lustrate the
effect
of the degree of polynomials, n
=
3 versus
n
6.
20
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DEFLE TED
LINE
SE
5
4
3
2
2
5
4
3 2
1 2
3
4
5
X
x
=
4
3
2 1
y
=
1
4
1 1 1
Figure
1.
Deflected l ine data Case
1.
21
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DEFLECTED
L N CASE
2
7
6
5
4
3
2
1
0
1
2
5 4 3
2
1
1
2
3 4
5
X
x
= 4 3 2 1 0
y
=
1
4
1 1 1
Figure
2
Deflected l ine data
Case 2
22
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In
Figure
3, the
f i rs t
four data points
are
on a horizontal straight
line
and the last six data
points are
on a curve of a
third-degree
polynomial, with
the
third and
fourth points
overlapping on bloth
lines.
The top two curves
result ing from the two
osculatory
methods exhi
bi
t overshoots around x
=
0,
while
the
other
four
curves
resulting
from
the
original
A method,
interim
method, and improved A method look good.
Figure demonstrates how the improved A method interpolates the data
points given on the curves
of
a simple known
function.
The
data points
are
on
a cubic
curve
at unequal intervals .
As is expected, the
second curve
resulting from the modified
osculatory
method and
the
second curve from
the
bottom resulting from the improved A method with n
= 3
look good, while the
f i rs t third, and fourth curves resulting from the original
osculatory
method,
original A method, and
interim
method, respecti vely exhibi t irregulari
t ies.
In each of the two intervals around the cen te r point , the portion of
the
f i rs t
curve from the original osculatory method has an inflection point.
In
the
two
intervals around the c.enter point, the portions of the
third
and fourth
curves from the original A method and
interim
method, respectively are line
segments.
The
disadvantage
of the
use
of
a
higher-degree
polynomial is demon-
strated in
the bottom curve resulting from the improved A method
with
n
6 .
Figure
also demonstrates how the improved A method interpolates
the
data
points
given
on the curves of a simple known
function.
The data points
are
on
a
sine
curve
at
unequal
intervals.
In
this
rather
contri
ved example,
the
general
trends
of the curves in Figure
are
even more pronounced. Figures
and indicate that higher-degree polynomials should be used sparingly.
The data
points for
Figure 6 are taken from Akima 1970 .
The
top
two
curves resulting from the two
osculatory
methods exhibit undulations
in
the
interval between x =
7
and 8, while
all
other
curves
resulting from the
original A method,
interim
method, and improved A method look good. will
modify this data point set in several ways and see how
the
curves resulting
from various methods behave for each of the modified data
point
sets in the
figures
that follow.
The data point set for Figure 7 is Modif:ication A of
the
original data
point set fo r Figure 6. Two
leftmost
data point.s are removed from the original
set
and
the
remaining
data points are
moved horizontally. As is
expected,
removal of the
two
points has
no
effect on the curves resulting from ll
methods. The undula t ions in the top
two curves
resulting from the two
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STRAIGHT LINE
CUBIC
CURVE
9
8
7
6
5
4
3
2
0
1
2
4
3
2
1
0
2
3
X
x = 3 2 1 1 2 2 5 3
y 0 0 0
1 1 2
Figure 3 Straight line plus cubic curve Y = 0 and Y =
x
3
/3 x
2
2
5x/6
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U I URVE
Y
=
{X 3
21X 20
9...... r ..... .. r r ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .........
8
7
6
5
4
3
>
o
1
2
3
4
5 ~ a ............ .... a... ...Io ...........a. Ioo
6
5 4 3 2
3 4 5 6
X
x =
5 4
2 4 5
y
1 1
1 7
1 7
1 1
Figure
4. Cubic curve y = x
3
21x /20.
5
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SINE
URV
Y
SIN PI*X
4 5
4
3 5
3
2 5
2
1 5
1
5
.0
5
1 0
1 5
2 0
5 1
1 5
2
X
x = 0 05 0 10 0.20 0.40 1.00 1.60 1.80 1.90 1 95
Y
1564 3090 5878 9511 0000 9511 .5878 .3090
.1564
Figure
5 Sine curve Y = s i n ~ x .
26
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KIM
J ACM 1970
2
20
19
18
17
6
5
4
13
2
>
10
9
8
7
6
5
4
3
2
1
0
a
I I I
0
1 2
3
4 5
6
7 8
9
10
12
X
x
=
1
2
3
4
5
6
7
8
9
1
Y
0
0 0 0 0 0
O
1 1
8
15
igure
6. Akima
d t J.ACM,1970 .
27
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AKIMA
MODIFICATION
A
22
............
............
....... .....-..-...-...........-............
21
2
9
8
7
6
5
4
3
12
11
> 10
. . . . . . . . . ~ I
o ~
9
8 ~ ~ p - - - - - . j l i - - - - M - ~
7
6 . . . . . . . . . t ~ I f t I M . . . r
5
4
t o i ~ I I _ _ _ _ _ i ~
... r
3
~ ~ ~ ~ ~ v ~ ~
1
o ~ ~ j J l ~ ~ ~ :
1
2 ~ ~ a a
o 1 2 3 4 5 6 7 8 9 1112 3 4 5
X
x =
1 2 4 6.5
8
10
13
4
Y 0 0 0 0 0 1 1
8
10 15
Figure 7 Akima
data Modification
A
28
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osculatory methods , now in the interval bletween x
8 and 10, are more
pronounced in Figure 7 than in Figure 6. The third and fourth curves
resulting from
the
original A method and
interim
method) look good. The
second curve from
the
bottom
resulting
from
the
improved A method with n
=
3
exhi
bi
ts
a
small
undulation
in
the
interval
between x
8
and 10,
but
the
bottom curve
resulting
from
the
improved A method with n
6) does
not. In
the bottom
two curves resulting
from the improved A methods), the negative
slope of the
curve
at
x may
l.oo.k
a
l i t t le
strange, but the second curve
from
the
bottom looks good as a whole i f monotonic
ty of the
curve is not
required. The bottom curve changes i ts direction so fast around x
= 13 that
i t
looks as
i f i t
were deflected
at this
point. Although the bottom curve behaves
better than the
second curve, from the, bottom in
the
interval between x
= 8
and
10, the lat ter behaves better than the former around x =
13 .
The data
point
set fo r Figure 8
is
Modification B. I t consists of the
data
points
r Figure 7 Modification A
and an additional
point at
x
=
10.5,
i e at the center
of the line
segment that has
the
steepest slope. With this
additional
data
point,
the
top two curves resulting from
the
two
osculatory
methods)
are
almost
unaffected
and remain
unacceptable. The third
and
fourth
curves
resulting
from the
original
A method and interim method) are totally
unacceptable; both the original A method and interim method join the two
osculatory methods and produce
large
undulat ions in
the interval between x = 8
and x
=
10.
The
undulation
in the
interval
between x
=
8
and
10
in the
second
curve from
the
bottom
resulting
from
the
improved A method with n = 3
is
a
l i t t le more pronounced
in Figure 8 than
in Figure
7.
Even in
the
bottom curve
resulting from
the
improved A method with n
=
6 , a
small
undulation .emerges
in
the
same interval. The slope of the curve at x
=
in
the fourth curve is
smaller
than
the
same curve in
Figure
7. The behaviors of the bottom two
curves
around x
13
remain almost unchanged from Figure 7.
The data
point
set
fo r
Figure 9
is
Modification C. I t consists of the
data
points
for Figure
8 Modification
B
and an additional point at x
9.
With
this
additional data point,
the
top
four
curves
resulting
from
th e two
osculatory methods, original A method, and interim method) are not improved to
an
acceptable
level, while th e bottom two curves resulting from th e improved A
method)
are
improved
considerably.
Undulations that existed
in the interval
between x = 8 and 10 in
the
bottom two curves in Figure 8 are nonexistent any
more in Figure
9. Improvement
of the
behavior
of the
curve by
insertion
of an
29
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AKIMA
MODIFICATION
8
22
........ r w r r ....... ... yo .. .po ... . . . ....... ...... . ....... ....... ...........
2
9
8
7
6
5
4
3
. - - - . . ~ ~ - - - - . - - - - - - ' -
9
8 t J l ~ p ~
7
6 . . . . . . . . - I ~ ~ - - - - - - -
5
4
. . . . - . . t ~ -- M---
3
2
- - - - i . . . . . - - - - - M - - - - : l ~ . . .
1
o
4 ~ ~ ~ ~ w j ~
2
~ a A o ~
o
2 3 4 5 6 7 8 9 2 3 4 5
X
x = 1 2 4 6 5 8 1 10 5
11 13 14
Y = 1 1 4 5
8
1 15
Figure
8 Akima
data
Modification
B
3
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AKIMA
MODIFICATION
C
22 . . . . . . . . . . . . . . . . . . . . . . . ~ . r . . . . . . . . ~ ~ ~ ~ . . . . . . . . . . . . .
2
2
9
8
7
6
5
4
3
2
> 10
~ ~ ~ ~ ~ W I f C ~ . . . . . . .
9
~ ~ ~ ~ ~ ~ ~ ~
7
6 ~ ~ f J l l l p ~ ~ ~
5
~ ~ f ~ ~ M ~ ~
3
2
~ ~ ~ ~ r t w j ~ ~
1
O ~ ~ f i ~ ~ ~ ~
1
2
a... a. ~ ~ a Ioo I o a. . . . . . I .
o
1 2 3 4 5 6 7 8 9
1112
3
4
5
X
x
=
1 2 4 6 5 8 9 10 10 5 13 14
Y 0 0 0 0
0 1
0 2
1
4 5
8
10 15
Figure 9 kima data Modification C
3
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additional data point in the troubled area is a desirable
characteristic
of an
interpolation
method and i t seems that the improved A method has that
character is t ic . As is expected, the slope of all curves at x = 13
are
unaffected
by
the
additional point
at x = 9.
The data point set fo r
Figure
10
is
Modification D.
t consists of the
data points for
Figure
9
Modification
C and an additional point at x = 12.
With
this
additional data point,
interesting
results are observed. The f irs t
curve result ing from the original osculatory method is
degraded;
an
inflection
point emerges in each side of the newly added data point. The
second curve
resulting
from the modified osculatory method is improved;
i t
does not have inflection
points
near the newly added data
points. The
third
and fourth curves
resulting
from the original A method and
interim
method
are
degraded; a
straight line
segment
is
embedded
in
a
g n ~ r l l y
curved
line.
The
bottom
two
curves
resulting
from the improved A method are improved; the
slopes of the curves
at
x
13 look
more
natural
than
the
same
slopes in
Figure
9.
Again improvement of
the
behavior of the
curve
by
insert ion
of an
additional
data
point is a desirable
characteristic of
an
interpolation
method
while degrading
of
the behavior
by
insertion
of
an additional
data point
is an
undesirable character is tic of an interpolation method.
In Figure
10
the
bottom
curve resulting
from the improved A method with n = 6 is
better
than
the second curve from
the
bottom with n = 3 ; a higher-degree polynomial works
well without
adverse
side effect
in this
example.
The data point set
for
Figure 11
is
Modification E which
is
another
modification of Modification C but
not
a direct modification of D. t consists
of the data points fo r
Figure 9 Modification
C and an addi tional
point at
x = 13.5. With this additional
data
point, the i r s t and third curves
resulting from the original
osculatory
method and
original
A method
are
almost unchanged from
Figure
9, while all other curves are changed to some
extent. Although i t
is hard
to
say
that the second and
fourth
curves
resulting
from
the
modified
osculatory
method and
interim
method
in Figure
are
better than
the
same curves in Figure
9 we can say
unequivocally that the
bottom two curves
resulting
from the improved A method in Figure
are
better than the same curves in
Figure
9; the slopes
of
these curves at x
13
look more
natural
than
the same slopes in Figure 9. Figure is another
example in which a higher-degree polynomial works
effectively.
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AKIMA
MODIFICATION
22
r r r r . . . . . . . . . ~ . . . r ~ ~ .................... .......... ........... ......... .... . ... . ..............
2
8
7
6
5
4
3
2
1 ~ ~ ~ ~ ~ - - - w - - f t - - ~
9
8 ~ ~ l l f - - - - - ; ~ - - - M - - - ~ ~
7
6
~ ~ ~ - - - - . ; N - - - - - o - ~ ~
5
4
~ ~ ~ ~ t f . - - - - . . . . - ~ ~
3
2
1 - - 4 ~ ~ ~ - - - ~ - - - - j ~ ~
1
o ~ ~ - - - . ; ~ _ ~ _ ~ 7
1
2 ..... ..... ... . . . . .a ............ ...... ........ ...........I o.oL r......a. ...t
o
1 2 3 4 5 6 7 8 9 1112 3 4 5
X
x = 1 2
4 6.5 8 9
1
10.5
11 12 13 14
Y = 0 0 0 0 0.1 0.2 1
4.5
8 9
10 15
Figure
10. Akima data Modification D
33
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AKIMA
MODIFICATION
E
22 ....... . .... ..... .. ....... . .r . .... .. ....... ..... .......................................... ...... .. ......... ......... ..
. . . . . .
2
2
9
8
7
6
5
4
3
2
> 1 ~ ~ ~ ; . . . . . . w _ ~ ~
9
8 ~ ~ t ~ ~ w ~ ~
7
6 ~ ~ ~ ; ~ w ~ ~
5
4 ~ ~ ~ - - - ; I l f - - - - - w - - ~ -
3
2
1 4 ~ ~ ~ ~ w J ~
1
O ~ ~ P ~ ~ w ~ ~
1
2 . . . . . . . . . . . --Io-II lIo-o.I.-
. a . . . . . a . . . . a . . . . a . ~ . . . . . . ...................................
o 1 2 3 4 5 6 7 8 9 1112 3
4 5
X
x
=
1 2 4 6.5 8 9 1
10.5
11 13 13.5 14
Y
=
0 0 0 0 0.1
0.2
1
4.5
8 10 12.5 15
Figure 11. Akima
data
Modification
E
34
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7 CONCLUSIONS
e have
identified
properties
desired or
required for an interpolation
method,
di scussed the i r mut
ual compatl bil
i
ty, established
our goals, and
deri
ved
th e
guide line s for
developing
an
i m p r ~ o v e
method
that
will
produce
a
good-looking curve when th e
method
is used for
smooth curve
f i t t ing e
have
r ea li zed that one
of
ur goals
is
to
develop
l
method that has the accuracy of
a third-degree
cubic)
polynomial, i e a method that accurately
inter polates
the
gi
ven set
of
data
points
when th e
data points
l ie on a
cubic curve.
e
have
also
realized that
th e
uni
var ia te interpo la tion
method
based
on local
procedures originally developed by
Akima
1970
and called the
origina l
A
method has some
of the
desired properties. e have improved the original A
method
in
such
a
way
that
th e
improved method
called
the
improved A method
has
the accuracy of a third-degree polynomial while
retaining the
desired
properties of th e original A method As
demonstrated
in
the
examples, th e
improved A method
generally
yields a curve that looks much more
natural
than
what will result from th e original A method
The improvement has been made in the procedure
of estimating
the f i rs t
derivative of
the
interpolating
function
r
the slope of
th e curve) at each
gi ven
data
point. The improved A
method f i r s t
calculates
four primary
estimates
for the f i r s t
derivative,
each as the f i r s t
derivative of a
third
degree
pol
ynomi
al
f 1t
ted to
a
se t
of
foul
consecuti
ve
data
points.
I t
calculates th e
final
estimate of th e f i r s t
derivative
as th e
weighted mean
of
the
four
primary estimates. The weight for each
primary
estimate
is
the
reciprocal of the
product
of th e
measure
of
volat i l i ty in
the ordinate
and
th e
measure
of dispersion in the abscissa of the set of four data points.
The
sum
of
squares
of th e
deviations
of the ordinate values of the four
data
points
from
the
s tr ai gh t l in e of least-square f i t is
used
as the
measure
of
volat i l i ty
of the set of data points.
The
sum of squares of the distances
from
the data
point in question of the
remaining
three
data
points
in the
set
is
used
as the
measure
of
dispersion
of the
set of data points.
Like the original
A method
the i m p r o v ~
A method
uses
a
third-degree
polynomial
in an
interval
between
each pair of data points as
a
defaul t In
addition,
we
have
also
implemented possible
use of
a
higher-degree polynomial
for
an
interval
between a
pair of data
points
as
an
option.
Al
though
undulations a re general ly reduced by
the use
of a higher-degree
polynomial
as
35
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demonstrated
in some
examples, our
other
examples
indicate that the
use of a
higher-degree polynomial
sometimes is tor ts curves that
would
look
good
otherwise.
A higher-degree polynomial
option
should
therefore
be exercised
prUdently and sparingly when the method is used for smooth curve f i t t ing and
naturalness of the
resultant
curve
is
of
primary importance. Note
that
the
use
of
a
higher-degree
polynomial will
inevi
tably void
the
accuracy
of
a third
degree polynomial even though estimation of the
f irs t deri vati
ve at a data
point is based on a
third-degree
polynomial.
Like the original A method, the improved A method does not always
preserve
monotonicity
or
convexity; we have not intended to
preserve
i t
in
developing
the improved A method.
propose the improved A method as a
replacement fo r
the
original A method when natural appearance of the
resul
tant curve is
important;
we do
not propose
i t as
a replacement for the F-C-B method
Fritsch
and
Carlson
1980; Fritsch 1982;
Fritsch
and
Butland
1984 or
th e
M R method
Roulier
1980;
McAllister
and Roulier 1981a, 1981b
when monotonicity or
convexity
must be preserved.
The improved A method can easily be implemented in a computer program. A
Fortran subrout ine sUbprogram that implements the
improved
A method is
described
in
Appendix A with
i ts l ist ing.
Since the original A method has been improved without changing
i ts
basic
concept
most remarks given
to
the original A method
apply to the
improved A
method
as
well.
Some
remarks
pertinent
to
proper
application
of the
improved A
method follow.
1 The method does not smooth the
data.
In other words, the resultant
curve passes through a ll
the
gi ven data points
i f
the method
is
applied to smooth curve f i t t ing.
Therefore the
method is applicable
only when
the
precise
y
values are g
ven
or
where the
errors are
negligible.
2 As
i s true fo r
any method of
interpolation the accuracy of the
improved A
method cannot
be guaranteed unless
i t is
known that
the
given data
points
l ie on
a curve of a
third-degree polynomial.
Unless
the
option
fo r a
higher-degree
polynomial
is
exercised
the
method has the
accuracy
of a
third-degree polynomial i e. ,
the
method gi ves
exact resul ts when
y is a
third-degree
polynomial in x
even when the y
values
of the data points are
gi
ven
at
unequal
intervals.
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