akima 1986 -Accuracy of Third-Degree Polynomial.pdf

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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


2/76


3/76

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

5


4/76

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

2


7/76

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.

3


8/76

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,

4


9/76

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

5


10/76

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

6


11/76

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


12/76

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


13/76

(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)


14/76

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


15/76

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


16/76

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


17/76

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


18/76

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


19/76

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.


20/76

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


21/76

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)


22/76

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


23/76

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.


24/76

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


25/76

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


26/76

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


27/76

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

3


28/76

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


29/76

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


30/76

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


31/76

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


32/76

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


33/76

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


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


=

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


34/76

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


35/76

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


36/76

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.

32


37/76

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


38/76

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


39/76

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


40/76

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.

8/11/2019 akima 1986 -Accuracy of Third-

akima 1986 -Accuracy of Third-Degree Polynomial.pdf

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