A Comparison of Quadrature Rules

(1)

A COMPARISON OF QUADRATURE RULES

by

Ren-Shiaw Yang

Abstract

的分法\lIJ 之一比較

In this paper

,

we sha11 develop different numerica1 methods to eva1uate a finite integra1

and then write computer programs to compare the efficiency of these methods. By the

comparisons and ana1yses of these

resu1ts、 we

can construct a decision tree for deciding

which ru1e to use for a particu1ar integrand.

(1)

INTRODUCTION

The numerica1 eva1uation of integra1s is one of the oldest prob1ems in mathematics. In

this paper

,

we sha11 examine the Romberg ru1e

,

adaptive Simpson's ru1e

,

Gaussian

Quadra-ture ru1e

(7

-points

,

15-points)

,

Gauss-Kronrod ru1e (7-points

,

15-points

,

21-points

,

31-points) and Newton-cotes ru1e (5+2 31-points). A1so

,

we sha11 review some of the mathematica1

background of the various approximation formu1ae and their accompanying

eηor

estimates.

In each numerica1 method we are given the integrand f(x)

,

the interva1 of integration

[a ，b] 徊， b

finite)

,

abso1ute error Ea

,

re1ative rrror Er

,

and we want to find an

approxima-tion S which hopefu

l1

y satisfied the fo11owing condition:

b

IJ f(x)dx -

51 a

b

手 max(Ea ，

ErlJ f(x)dxl)

a

The numerica1 methods considered in this paper can be used to approximate integra1s

whose integrands have no antiderivative in the a1gebra or e1ementary functions. A1so

,

these

methods can be used when the integrand has discontinuous or singu1ar points in the interva1

[a

,

b].

(2)

師大學報

第卅一wl

(11)

STATEMENTS

OF

THE QUADRATURE RULE

2.1 Romberg Integration Method

Romberg integration is a method that has wide application.

It

uses the trapezoidal

rule to give preliminary approximations and then applies the Richardson extrapolation

process to improve these approximations. The algorithm is as follows:

b

Suppose we want to evaluate an integral

,

J

f (x) dx

. (a

,

b are finite). The Tra

a

pezoidal rule says that

、

••

,

HUF Fι 司₄

-n

N7

.、 J

X

F-h

t.-可-4

-vb=

m-3

勻，-+

hu

s-‘

+

a

z--h-2

冒

X

AU

X

F-hu

rJa

where

a

<口〈 b ， hzhp

and x.

==

a

+抖，

for each

_j=O

_,

₁

_,

₂

_,...,

_m

3 Letting

_hkz?丸，

b-a

consider the trapezoidal rules

k-1

b

IE

,

2 - 1 . b - a 2"

J

f(x)dx

=于 [f(a) + f(b) +

2 E

f(a+ih

k

)]- 了 hltf{υk)

,

i=1

a

where

a

<口 k

<

b

,

k =

1 ,

2 ,

3 ,...,

Define

h.

z 」 [f(a}+f{b)1=2[f(a)+f{b}}

1 ,

1

2 then

RJ屯， 1

bh

',

hu

t4

.l

+

a

'，.、 ZL 司4

-1ι

bhfuz

2.l

-gA

LA

hu

+

'，辜

,

L

RK

l-2

=

for each

_k

₌

₂

_,

₃

_{,... ,}

_n

-

642 一

(3)

積分法則之一比較

Note how the error in the extended trapezo idal rule

1 )

-K

E

--..

.

l

+

a

{

z ﹒晶

,

L

'斗， L

-vb--ki

勻，-b

h.

J

f(x)dx

=去 [f(a)

+

f(b)

+

2 a

(b-a}h4

k

,,

(4)

一守主百

f

\

~

I

(川)

+

[f'

(b) -

f'

(a)]

2hkE

υk

<

b

,

<

a

f or each k

=

1 ,

2 , . .

.‘

and some

Rk - 1

,

1 b

.r

f(x)dx

~

a

Implíes that

)

A 崎 -bkh

1 I

{

nυ

Applying Richardson extrapolation procedure

,

with the error in the above approximation

4 丸， 1

-

~-1 ， 1

3

司 4

,

-k

nuu

Define

--we obtain the approximations:

、.

-i

E

可﹒-

--

-l-R-1

-一

.••

2

--24 .14 司 •••• 司 •• 4

.l-4.

R

'.

-.、 J

Aq

=

R

i

,

j

2 ,

3 ,... ,

i

0(h:j)

The approximations are often presented in a table of the following form:

=

2.,

3 ,...,

n

--i

for each

with error

R1

,

1 R2

,

2 R2

,

1 R4

,

4 R3

,

3 R4

,

3 R3

,

2 R4

,

2 '-A

'

,

J

R

R4

,

1 R

n

,

n

-

643 一

R

n

,

3 R

n

,

2 Rn

,

l

(4)

第卅一期

師大學報

2.2 Adaptive Simpson's rule

[事， b)

G

υ

for some

From the 3-point Simpson's

ru 峙， we

have

}

HV as-、

}

aq

'

..

_、 ZL

5-o

hu-qd

}

hu

'

a

{

nhu

--X

AU

}

x

(

ZL

hu

a

pld

=

I

h

[f(a) + 4f(a+h) + f(b) )

h-3

=

S(a

,

b)

where

f(a+h)]

f

(b) ]

+

3h

4f(a+ 言.!.)

...

,

-n

需

4

+

4f(a

+

(f(a+h)

‘ ..

,

a

(

6.-[

hu-ro

-u-6

--b)

a+b

S( 一γ ，

S(a

,

Let

and

term in the error is assumed to be constant

,

then for arbitrary

toler-)

HH

(

)

A 唔， z_﹒、 SEA

If the

a+b

S

(一2'

a-恥 b 、

2 '

ance

é

the inequality

15ε

<

b)

I

S(a

,

IS(a

,

b)

(#)

a+b

S("'2~'

a+b 、

2 implies that

ε

(See reference [7]).

If

(#)

does not hold

,

apply the estimation procedure individually to the subintervals

<

b)

I

S(a

,

a+b

,

2 '

bJ

、 with a 忱mceof3

. If this tolerance is not obtained on

{a+b

-2-'

and

[a

,

each of the intervals. split each into two and repeat the above estimation procedure.

The interval splitting process can be represented by a binary tree of the form:

(5)

積分注目IJ之一比較

/

)'

'\

/\-8-'-4-/ \ b

z \ / \

「吃 γ 竿 b

] [

2.3 Gaussian

Qu

adrature rule (for Legendre)

Consider a quadrature of the form

1 n

J_f(x)dx

~

_{.E.wif(ai )}

.

ι1

i=l

We say the above quadrature ru1e has degree of precision k if the ru1e is exact for a11

po1ynomia1s of degree 1ess than or equal to

k and is not exact for a11 po1ynomials of

degree k

+

1. Since we can choose the n nodes

,

a

₁

,

a

₂

' • • •

,

a

n

,

and n weights

,

W

, ,

w....

, . . . ,

w..

,

we can hope to obtain 2n-l as the degree Of precision.

1'"2

Let the nodes a" a....

_{l ' "'2 ' • • •}

, . . . ,

_,

a_

_Qn

be chosen not equally spaced

,

but as the roots of

the Legendre po1ynomial L..(x) oforder n . Theweights W"w....

_{n''-'I -... _...-...}

_.

_..&..-

'1'0

...0.&...

"1 '"2'. . .

,...,

'"n

w_ arethen

determined by the system

0 1'

equations:

(6)

rw大學線

第卅一}的

W

1 +

W

2 +

W

3 + •

.

• +

w

n

=

2 a1w 1

+ a

2 w

2 + a

3 w

3 + •

.

• + a

n n

W

=

0

2

2 a1w 1

+ a

2 w

2 + ajw

3 + •

.

• +

a~wn

n n

=

3"

.

+

可 J

W

--n3

a

斗，

呵，

w

唔，-n2

a

+

'L

W

',-n'L

a

ququ

.，占﹒'.-nnn

ed

EVE-d

-le--。

02-n

fjtlL

--n

w

'i

nn

a

4. With the weights and nodes chosen as above

‘

the desired degree of precision

,

2n-l

,

will be obtained. For more details

,

see references [11 and [51.

2 .4

Gauss-Kronrod rule

Let L_

(x)

be the Legendre polynomial of degree n and define K

.,_ •

,

(x)

by the

2n+1

equation

K '"'_ •

_2n+1'~1

,

(x)

=

-

L__ (

_~n'~/~n+1

x)

P _ •

1 (x)

with pn+1(X)

,

a polynomial of degree n+1

,

chosen

SO

that k2rI+1(X}is orthogonal on

K

V'

n

T _ . ___

.1____._

. L . _ _ _ _ . _ _L"

K..._ •

, (x)

[-1

,

11 to all powers

x"

,

K

=

O

,

l

,...,

n . Let us denote the roots of"2n+1 '^'by

‘毛，

₁

,

x..

_x

, . . . ,

X

.,_ •

1 '

and the weights of the corresponding quadrature rule by

2 ,.. . ,

x

2n + 1

,

w 1

J

W

2'

, . .

,

w 2 n+ 1.

It

can then be shown (reference [11) that the quadrature rule

2n+1

1:

w.

_{i .- \ ""i}

f ( x

_{, )}

i=1

has degree of precision 3n+ 1 for even n

,

and 3n+2 for odd n. Note that since the n

K

(f)

=

nodes of the n point Gauss-Legendre rule are a subset of the nodes for this rule

,

we may

(without increasing the number of function evaluations) also compute the n-point Gaussian

approximation to the integral of f(x). The accuracy of the Gauss-Kronrod rule

approxi-mation can then be determined by a comparison with the Gauss-Legendre rule.

(7)

情分法則之一比較

2.5 The adaptive Newton

-c

otes Quadrature rule

Suppose that [a.b

1 is divided as follows:

a

h-2

+

nu

x

h2.4

4. nu

X\

xO+h

Xo

+τ-XO+1-b {h=-z一)

3h

__

• 7h

._

,,_

b-a

11

11 "

_"

11

11 X

_o

x

l

x

2 x

3 x

4 x

5 x

6 The 5 point Newton-cotes rule on

徊， b

1 is given by

b

S

f(x)dx~A~[7('f(xn) + f(x

_45'.'-'''0'

_-'''6''

t:))

+ 32(f(x

_--'-'--2.

.,)

+ f(x

_-'--4

_{A ))}

a

with er

r.o

r

+ 12f(x

3 )]

~

7 ~(6)

h

f'VI (;)

15120

a

<已<

b

e

=

(6)

Using f ( XO) through f

(文 6)

to approximate f \

V I

(f;),

and then subtracting the

cor-responding approximation of the error term

,

we obtain the 5+2 point rule approximation

b

lf{X} 心晶宮[圳的0) + f(x

6 )) + 2048(f(x

1 ) + f(x

5 + 2016 (f(x

2 ) + f(x

4 --}) + 4004f(x

3 )] •

(111) STATEMENTS OF THE TEST RESULTS

In this paper

,

we use nine methods which are listed as follows:

Method

(1):

Romberg Method

Method (2): Adaptive Simpson's Method

Method (3): Gauss-Legendre Method

(7

points)

(8)

向1; 大咚!悟

;n月f 一朋

Method (4): Gauss-Kronrod Method (7 points)

Method (5): Gauss-Legendre Method

(1

5 points)

Method (6): Gauss-Kronrod Method

(1

5 points)

Method (7): Gauss-Kronrod Method (21 poings)

Method (8): Gauss-Kronrod Method (31 points)

Method (9): Newton-Cotes Method (5+2 points)

Numerical experiments have been conducted on Kahaner's test problems to investigate

the performance of various numerical methods. Eåch

0 1'

the nine methods above were

tested on eighteen different

1'

unctions over three di

1'

ferent intervals and three different

tolerances. The results of these tests are listed in the following form:

Integral

1'

orm a to b

EPSABS=.

Integrand F(x) = . . .

Method

(l):

Integral Approximation = . .

Number

0 1'

Function Evaluation = . . .

Tolerance

0 1'

the integral = . . .

Error code = . . .

Method (2):

Integral Approximation = . . .

Number of Function Evaluation = . . .

Tolerance of the integral = . . .

Error code = . . .

Method (3):

a:

lower limit

b:

upper limit

EPSABS: absolute error

Tolerance

0 1'

the integral: max

(E

PSABS

,

relative error)

Error code:

(l)

0: normal return

(2) 1: over 100 subdividions

0 1'

[a

,

b]

(3) 2: round

-ó

ff error flag.

( 4) 6: 1 nput to lerance erro

r.

(9)

的分法則之一比較

The sample results of these tests are listed in Appendix 1

,

and from these results

‘

we

pick the number of function evaluation to obtain the comparison tables listed in Appendix

2. Analysis of the test result reveals the following:

(1)

The Gauss-Kronrod methods (7 points and 15 points) usually require the least number

of function evaluations. In 5 cases (out of a total of 36 cases) the 7-point

Gauss-Kronrod Method used the least number of function evaluations. In 9 cases

‘

the

15-point Gauss-Kronrod Method used the least number of functionevaluations.

(2) The adaptive Simpson's rule yields the least results for the integral f(x)

=

SQR

T(

x)

叫e叫吋吋叫

(ββ

圳

糾

3-

-1 ， 3-2

圳

3-

只刊

-2

引

2))， f(你仲

x

f叫(忱川

x叫}

=

J

↓一

(Tabl

l

x~+l

e^回 1

(Table

(1

4-1

,

14-2))

,

f(x)

=

alog

(x)

(Table

15寸，

15-2)). Each of these

inte-grands is in some way ill-behaved near the origin

,

and well-behaved away.

It

is therefore

not surprising that an adaptive quadrature rule

,

which will naturally execute more

function evaluations near a singular point

,

yields the best results.

(3) ln theory

,

one should expect good results from the 21 and 31 point Gauss-Kronrod

rules

,

as these rules have a very high degree of precision. But as indicated by our test

results

,

a very large number of function evaluations may be needed to overcome

round

-o

ff error. However

,

this may not be as serious of a problem if double precision

arithmetic is used.

(4) The Gauss-

Kr

onrod rule always used less function evaluations than the corresponding

Gauss-Legendre rule. This is due to the previously mentioned fact that once the 2n+ 1

point Gauss-Kronrod rule has been computed

,

no more function evaluations are

needed to compute the n point Gauss-Legendre rule.

(5) The Romberg Method yields the best result for the uniformly well-behaved functions;

l

f(x)

=

EXP(x)

,

(T

able (201))

,

f (x)

=-1平Ëxp 忍了，

(Table

(1

0-1))

,

f(X)

=

Sin(x)

,

(Table 02-2))

,

and f(x)

=

Cos

(l

00 Sin(x))

,

(Table

(1

7-2)).

(6) In general

,

the Newton-cotes

quadra直u're

rule (5+2 points) does not yield the least

number of function evaluations.

(7)

If the integrand is undefined at an endpoint

,

we can't use the Romberg or adaptiv

,

e

Simpson's method.

(10)

n，Ii 大 f~1f有

m Jtt一!的

(IV) CONCLUSION

Suppose that we want to evaluate the integral

b

J

f

(x) dx

,

where a and b are finite

a

By the previous discussion

,

we can construct a decision t

l"

ee as follows:

To evaluate

b

J

f (x)dx

a

Do you care about computer

time and are you willing

to do some analysis of this

problem?

no

Gauss-Kronrod (21 points)

or

Gauss-Kronrod

(31

points)

。、 d ρ-w

vd

li--jIll-ll.

,

Is the integrand uniformly

well-behaved on

峙， b)

一一一-IRomberg

ves

method

E

no

(11)

日

Is the integrand

ill-behaved in

some place and

well-behaved

away?

o

n

il--i 』

Iz--••

Are the

discontin-uities or

singularities of

the integrand

within the

interval

,

and do

we know where

theyare?

o

n

--till--li--Has the integrand

end point

singularities?

0-

1 no

Gauss-Kronrod rule

(21 or 31 points).

If this routine

returns an error

flag ，、then

we use

Gauss-Kronrod rule

(7

or 15 points).

yes

..

yes

一一-yes

_..

一-一一一-Adaptive Simpson's rule

Split Jbf into

a

c l C 2 n c 3 n b

J _

~f

+ L.-f + L_- f +...+ J

1;;

1 -';2

c n

where ci are the discontinuous or singular

points. Then on each subinterval use

Gauss-

Kr

onrod rule (7 or 15 points).

Does the integral transform

to an integral with no

sìngularities?

的分注目IJ 之一比較

Y郎，因

no

Gauss-Kronrod rule

(7

or 15

points).

-

651 一

(12)

師大學報

第卅一期

(V) PROGRAM LISTING

In this part

,

we summarize the subroutines as follows: .

Method

Subroutine

Number

Quadrature rule

Name

(1)

Romberg Method

ROMB

(2)

Adaptive Simpson's rule

ASIM

(3)

Gauss-Legendre rule

(7

points)

GS7

(4)

Gauss-Krontod rule (7 points)

GK7

(5)

Gauss-Legendre rule

(1

5 points)

GSl5

(6)

Gauss-

Kr

onrod rule

(1

5 points)

GK15

(7)

Gauss-Kronrod rule (21 points)

GK21

(8)

Gauss-Kronrod rule (31 points)

GK31

(9)

Adaptive Newton-cotes rule (5+2 points)

ANC7

We use the computer IBM 370 (Music) and coded in FORTRAN language.

(13)

的分法HlJ 之一比較

DATE • THU JUN 06

,

1985

t1

AIN

"""甜甜聶曲曲曲曲曲曲*曲曲曲曲""*"""壘*"*"甜甜食費，壘*"""曲曲曲食費*轟轟 """1、"飛禽食費 4、"*'， "1‘

食費甜食 *1、

最*晶

A

CDt1PA成 SIO"

OF

QUADRATURE

RULES

會￡舞

曲食費

YANG

,

REN-SHIAW

甜食告，、食由于世會，可貴""""負責甜甜"甜甜食*食費""""#1 11 繭 "111悟脅，甜甜甜甜甜 ft***""ftft 食費 """ftftftl悟*食，， "1悟，時 4、會會 Ir it

RELEASE 2.0

食費""""負責曲曲曲賣會 """ftft"""""""""""""""""食費甜甜甜食食費會負責舜，、由"*'，由甜食，、 j，

"*"‘快，、

會曲曲 t1

A 1 N

P R 0 G R A

t1 橫飛*

"*"

"甜食

"""11""""""" 晶晶甜食晶晶#1"甜甜食費食費最 "ft"" 甜甜食青青侖，悟甜食，‘"*"晶晶晶負 *"*"1、 4、機"*食甜食食壘

.

)

nv

nu

1 (

Hn

)

nu'

,

nu' 。

.,-'、， E

',

nu

'、

w-11

.，』 F." ，旬

,,',

',

E 、 yo

nu-nV-E

-'

,

10 ?'nu

p--,

1

•

•• 』 nu 勵自﹒

' •.•

,

.

••

、，o

nu-nvnv

',‘',,

J

'tnu--h

'，而 U'E

P3.'"n

.，'，、， 2 .LT'"VE a 肉，、'，且

',.,.

' ••• p3 、 ''"HR 自

nv

••

"

nu

‘

FPa

',

nunr

'tnu'E

',','

， 3' ，、?'

,,

-L.

,

tom

-nn'

,.,

-L'

,

“呵，

nURE--‘

.,

.•

R 's-RMH MHaH

',

haan'

,‘

MHT'? •

.

,

-nnvan

nwmu't

an

pa.p-MFLPLFEMP--' ，句 FLPLPLPUP-旬， LPL

FORTRAN

IV 區 1

0001

0002

0003

PRINT THE HEADING

.

• A 唔，'

.

• _h 『帥 "u

'

..

.

• nu

',

• .

•

“呵

.

"。

',

-MH

nu

"n

',

-L -R

.

RR

‘''

,

u'

'，』‘.'

?',

VA HHV <,

1

.， F 呵， '"r

',

VA

',

E

ee'

,

IH

E--‘

J

."va 呵呵，‘ van '、"、

•.

'

..

_‘

"r ﹒'"苟，'， FE'

,

E

.,

r-mu

nn'

,'""

',

fm."

、'，."'，、 dnn 、 J' 、 1." 。 "u

zd

••••••••••

肉， t

',

..

F3?' ，。， 1 ， b' ，、 nrM 間， E 、'，'‘、'， PE ﹒' ， Ea 肉 Ev--" ，' ?'"H?'"H 自 '"H' ， nn

nnnu"nnu

wFWF

."‘'"

FLF-wFU

3 OOO~

0005

0006

0007

CAll RO/\B SUBROUTINE AND PRINT OUT THE RESUlT

CUl RO/\B (F

, ^ ,

B

,

EP5A8S

,

EPSR

E l ,

II

t1

1 T

,

RESUl T

,

NEVAl

,

1 ER)

CAllOUTPUT(RESUlT

,

NEVAl

,

EPSABS

,

EPSREL

,

IER)

PIMP-upzu

0008

0009

CAll ASI

t1

SU8ROUTINE AND PRINT OUT THE RESUlT

CAll AS 1 " (F

，^，閉， EÞSA8S ， EPSREl ， ll 州 IT ， RESULT ， NEVAl ， IER)

曲曲曲曲，由

PR

IIIT

THE RESUL T

甜甜食會曲曲

CAllOUTPUl (RESUlT.NEVAl.EPSA8S

,

EPSR

E l ,

IER)

P-up--PL

。010

E

。 011

CAll GS

,

GK AND ANC7 SUBROUTINE AND PRINT OUT THE RESUlT

D08KEY"I

,

7 CAll AGQ(F

,

A

,

8

,

EPSA8S

,

EPSREl

,

KEY

,

LI

t1

IT

,

RESUlT

,

NEV^l

,

IER)

由晶晶曲曲"

PRINT THE RESULT

自食晶晶晶，

CALL OUTPUT(RESUlT

,

NEVAL

,

EPSA8S

,

EPSREl

,

IER)

CONT1NUE

WR

1 TE

(6

,

1 1)

WRITE(6

,

11)

F OR

/l

AT (' 1 ' )

S10P

END

FiwFLFtu

8

11 E

。012

0013

。01~

0015

0016

0017

0018

0019

0020

NOTERt1， ID.E8CDIC ， SOURCE ， HOlIST ， NOOECK ， lOAD ，"。門 AP ， NOTEST

NA/\E •

t1

AIN

• llHECNT •

56 --

653 一

對 OPTIONS

IN

EFFECT 負

(14)

第卅一月1

師大學報

OATE • THU JUN 06

,

1385

甜食 ftlti悟，甜甜疊疊，食費曲曲曲聶曲曲曲，晶晶脅，轟轟轟 ftft 費甜甜，壘，食繭 ft 甜甜 ftftftft 聶食費禽，、食*費長青

甜甜食費

甜食

SUBROUTINE ROMBERG HETHIO

食費

ft 費由疊疊 ft 轟轟轟 ftftftftftffftftft 甜甜食費由 ftft 轟轟曲甜甜 ftft 曲曲曲曲曲 ftftftftft 由晶晶晶食甜甜費 4、 ft"ftftftft"ft

MAIN

RELEASE 2.0

PIMP-旬， EMPIMP-up--PL

FORTRAN

I'

V G 1

5UBROUT HIE

RO門自 (F ， A ， B ， EPSABS ， EP5REL ， lIMIT ， RE5UlT ， HEVAl ， IER)

。 IMENSION

R(2

,

100)

IER"。

IROFF-O

NEVAl-O

RESUlT-O.O

IF (EPSABS.lT.O.O .ANO. EPSRE

L. LT.O.O)

IER胃6

IF(IER.EQ.6) GOTO 999

0001

{)002

0003

OOO~

0005

0006

0007

0008

"開 B-A

R (1，1)團"費 (F

(A)

+F

(B)) /2.0

00 515

1 個 2 ，lI MIT

NEVAl-NEVAL+l

sυ門﹒0.0

11-2 禽 ft

(1-2)

DO 525 K..I

,

ll

FX圖 F (A+ (K-0.5) 曲 H)

!:U門-SUK+ FX

R

(2, 1)

-0.5 ft (R (1

,

I)+Hft 5UM)

DQ"535 J.2

,

1 R t2

,

J) ﹒(圳青 (J-I)

ftR (2

,

J-

1)

-R (1

,

J-l)) /

(~負責 (J-

1)

-1)

RESUlT-R(2

,

1)

E

525

REO-' 唔，‘、 3. 句 PD' 旬，'， noqdnu' ，

0111111111122

nunununvnununununununununu

nunununununununununununvnu

535 TEST FOR ROUNDOFF ERROR ANO 5ET ERROR FlAG

.G

E. ABS(R(2 ， 1)-R(I ， I-I)))IRO fF "IROFF~I

CI • L

T. 2) GOTO 520

(ABS (R (2

,

1) -R

(1,

1 -1))

(IROFF .G

E. 6) IER-2

'

•.••

,

•••

111

PEMF-"

,

.•

0022

0023

002~

n3

qJ

倆 wd

)

‘

ynu

',?'

-LHU

HUFU

，且‘ F

',

L-aHU

HHMH

'，、"目

,

-nH

RE 個院 ."，巴

,"

.

• _L'

,

'，』

..

』

"n.

"、斗，‘.'， "r 、'， '，』

•..

..

•

、 J •••• "。

.

".'

，、 d' ，‘.

"rmm

',

L •••

{}}

.,.,.,

-L

Van--•

• m 鬥呵， •••••

‘'‘

1(-{

f

.

l a 叭 "n

••

‘"",

E

"，.、 nu"Hu nυ ，3.PHJ' ‘'， MH HHREq's-u 『 -L ﹒' "。."， rphJ' ，' "間， 2 、 Hn ﹒'"" "間， F 間 nu' ‘ .nu

pE

.

'"nnu"nplw

E

520 0025

0026

0027

0028

0029

0030

}

'

..

}

1 (

IER-I

tlEVAL-2 甜甜(II EVAL)

+ 1

WR ITE (6

,'

5 1)

FORMAT(/

,

15X

,

'METHOD

RETURN

END

5 "

5

515 E

999

51 0031

0032

0033

003"

0035

0036

ftOPTIOHS IN

EFFECT 曲

NOTERM ，

1 D

,

EBCD 1 C

,

50URCE

,

NOL 1 5T

,

NODE CK

,

LOAD

,

m"AP. HOTE5T

費 OPTION5

IN

EFFECT 晶

NAME

- ROIIB

,

LI

NE CIIT ..

56 費 STATISTICSft

SOURCE STATEIIEtlTS •

36 ,

PROGRAH SIZE •

0008F 。

由 5T

ATI STI

C5 曲

NO

DIAGH05TICS GENERATED

(15)

積分 i去!IIJ 之一比較

1985

DATE -

THU JU" 06

,

Ir lrlrftlri悟 ft fr. ftftftfcftft 晶晶晶食費 Ir lr fr. lrflft 曲曲 Ir lrflfl 由 f.lrftflM.flftft 負責 lritftft 食，、 ft t:食費 It f， ft:':

It i、台才有

Ir

lr

SUBROUTI"E ADAPTIVE SIMPSON RULE

吭吭

甜食 'Id、

最甜食 ftft* fr. i悟fr. ft fr.fr.fr. ft fr. ftft fr.fr. ftft 會食費 4、 4時，堅青青fr. ft 禽 ft fr.j、 fr.fr. i， fti、 {.flM、 4、fr.fr.會突如飛快 d堅決 ;d、，'，

SUBROUTINE ASIH(F

,

A,

B.EPSABS.EPSREL

,

LIMIT

,

RE5ULT

,

"EVAl

,

IER)

DIHE"510N

FAlIST(IO叫.

FBlI ST

(1

00)

,

FCl I 5T

(1

00) • TOL (100) •

費

A lI ST(IOO)

.Rl I ST(IOO)

,

H(IOO)

IER興。

NEVH~。

RE5ULT"'0.0

IROFF~。

IF (EP5^B5.LT.0.0 .^ND. EP5RE

l .

L

T. 0.0)

IER~6

IF (IER.EQ.6) GOTO 999

1-1

TOL

(1)-1 。由^，，^ X

1 (EPSM5. EP5RELlrABS (RE5UL

T))

H(I)..(B-A) 晶。 .5

ALI ST (I)..A

FAlI ST

(1)

-F

(川

rcllST (I)-F

(A+刊 (1

))

FBlI ST (I)-r (B)

RlI ST

(1)

-H

(1)費 (FAlIST(I)+4 叫 ClIST(I)+FBlIST(I))/3.0

t1AIN

RHEASE 2.0

FEMF-MPLFLPL

FOnTR^N IV GI

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

0011

0012

0013

0014

0015

0016

'‘'‘fr.j、食

nvnv

..

，旬， b

,

r

,

r

}}

))

11 {{

?','

，、"，、呵，

-L-L

PLHM

‘,',',FAR

O-?+ 膏、'，-hUPE 費

‘.'‘Jhvr'?an

.，'，'‘ m 聶壘， 1 、 1.q ﹒肉

qdH1++

肉 "JAR'，、、E'、.， qd'pun--a 悶 T﹒魚， 1 ， 1' ， 1nunu司，TTA

HHT'++RVP3"u

.，nu﹒'‘.'、'，.，.， ELFU4，.'，，EL--h't、'，、'，、'，

••

L'‘、'，‘aRFLHV、'，

••••

，.『、自' ，'、'，."?'?'，r，r-n﹒'，‘、'，、'，、'， -nuMVP3，耳，‘、，tp3t、?'，fT'、 1 ，t u叭-PE.，.'‘“‘"，'，2p3，3.'?1 nu、“H-L-LE'、'，='，.，.，、，'，tpa

nUPEH.R.H.,.,.',L-LgL.,-L.,

nu--L'‘‘'，、'，、'，、-n-La"，LRU'‘‘nu-L﹒' ‘ 4. ，.叭，F'，HHHnaH﹒叭，r，VPFHn'，RH ﹒ 'twv""，胃。"四"間，闕，， nu''，EHU'E.，司 4. “.，啥，‘'、 Je ﹒旬，hd'。可'， nu.，M"，FPF 電 dpaauvMVHVMVMVMVMV'，

t1AIN DO-lOOP

u 青 ft fr.

c

E

c

100 C

。 017

0018

0019

0020

0021

0022

0023

0024

0025

0026

00 2]

0028

0029

0030

0031

TE5T nOUNDOFF ERROR AND SET ERROR FlAG

IF(I.L

E.I)

GOTO 110

1 F (^BS (5 I+S2-V71 .GE. ABS (S

I+

S2-Rll 5T (1-

1)))

IF (InOFF .G

E. 6)IER-2

PLFLFL

。 032

0033

0034

IRorr..IROFF+1

1 F (^BS (5 1+52-V71 • L T • V6) GOTO 150

1-1+1

A

l I ST(I)..V

I+

V5

FAL 1 ST (1) "v3

FCLl 5T(I)-FE

FBlI 5T

(1)

..v4

-

655 一

E

110 。 035

0036

0037

0038

0039

00 1

,

0

(16)

師大學報

第卅一期

FORTRAN IV.Gl RElEASE 2.0

ASIH

DATE • THU JUN 06

,

1985

。0"1

00"2

00"3

H(I)-V5/2.0

TOl

(1)

-v6/2

.0

R

lI

ST

(I

)-S2

E

)

,

}

唔，‘

{

‘ ..

,

唔，‘"u '、 enu

+"

.'?'

，、凶， t

‘',,

1 ‘3MH

1++'

﹒?'嚕，‘.

,amu-‘J.'

.•

L-Rwan

--HV,rHVE',

1. ,HU--‘‘FPE

M--圓圈，'，-L'、d'、d."﹒'，' ，‘'，‘'，‘F.O-EHVP虫，

‘

y

.',',,.','‘',nunnnu'E.','

，'， E 、，tf.， 1

••

'。'。“"，旬，t 'E 、T'?'，'""、.'，.‘﹒'，，.，.，.、'，“" .'，'，、dpap3

••

',',,.,-L'E-RRR

&.,3.,.,

••

‘'',

E

‘panUHURU-R

••

",'"HHU

-1.

,-L-L-L.,-L.'?',3?'"RHV.'"n?'HU

E-L-RPL"白，tnu-Lnu'Enu'E'E"nnuv'EMH .，."，r，『'『HnT'"nnu 侃"刊 "u心，MHHW'，RH'E

nununy

EJoqJ1

.，司，‘n3PD t 句

EJ'。'，89012缸，iqEJ6'，由υqJO

'",.“

•.•

h 『 -h呵，-h 『 -uvphdphJphJphJphJP、 4 ，PHJP、dp、JP、d'hu

nvnununununununununununununvnununu

nununvnunununvnunvnunvnununununvnu

機 OPTIO"S

Ih ErrECTA

"OTER~ ， ID ， EBCDIC.SOURCE ， NOlIST.NODECK.lOAD.NO"AP.NOTEST

~OPTIO"S

IN EFFECTA NA"E - ASI"

• LINECNT -

56 食 SHTI S Tl CS 晶

SOURCE STATE~ENTS

-

6o.PROGRA~

SIZE

.

00112"

~STATISTICSA

NO DIAGNOSTICS"GENERATED

(17)

f1'i 分法具IJ 之一比較

DATE - THU' JUH 06

,

1985

、 Jnu

RO

E-

--.',

-LRd

aH.

,

VL

amaπARAR袋，ER **ARA"恥"，

‘n&n.‘',

aH品"?'nu 食品"-Lnu

*LHHU',

真 H&npb'，、

-RAn'tRU

**"nnnH

*禽，nu

an&H',.'

‘RAn-'.

‘na“MH‘,',b

*M間*.，nu 曹 *﹒肉食-Lnunk

‘“.'‘"

••

'，且 ‘n'、4‘nv''，、.'

.",34n't',‘',

-nHuanu叭，、dhu

-En

•

• *"u‘"-L-LhU

ARA"，巴，E﹒ 禽，且袋"n﹒'，

‘mMV‘",3‘.,.L

-n﹒'。"nrnu--n'，食，巴nu-L

‘nhr‘“

••

'

,

E

‘的﹒肉食，、4，‘‘HR &""u-n 個 MUT'，、d &En--n‘"﹒肉，、eRV *禽，3.'，E ‘四"，E*"r-L -RM間."，Emn﹒

AR.'an--munE

&RT'‘""。、'，MHRWJ'

*HU‘m.o."nE

*nu‘"-mnu.

nn

••

'。 ‘四 B&nFS 、 OT -nHU。"，‘、'，-mu ‘npaa 伯伯 u、'、dnuFU

**nu.,.

-n‘H.R-LRURV'，、'，甜食 -m

••••

‘,

b

*E 一 ooaBOO- **M 開 MH

•••

"，-nu' 這 nq ‘“‘"，，nununu、'，‘'，‘'，‘''，馳"，E ‘阿*'，.，nu--'，，.，.， -1 ‘'，-an﹒ ‘"-nHU'、dRRUTEhn'，、a'、，.、，巴、﹒'，3"間由 ARnvN0.L'.LnnT'，'?'?'，1，『PZ .“‘nn刊，EE--"，'"u'EP且，、dp3，、dnu'E.，也們 AH"。"HRHH"vp=pbp3.，.，.，.'"啊，.、'‘、 *&n*."*HU﹒'，EPEa-"，E" 也 -L-L-L-LRU'，-PV .R.“‘"‘"‘"，、dhu-'MH-LRR.R.-nnnm叭，E.，.，.，

MAIN

R

El

EASE 2.0

piwpbpzupup--FORTRAN IV Gl

0001

0002

000)

OOO~

0005

0006

0007

0008

000'l

0010

0011

0012

001)

001"

flRST APPROXIMATION TO THE INTEGRAL

CALL

GS7(F ，_， B ， RESULT ， A 自 SERR ， OEFABS ， RESABS)

CALl

GK7(F.A ， B ， RESUlT ， ABSERR ， OEfA 肘， RESABS)

CAll GSI5(F.A

,

B

,

RESUlT

,

ABSEnR

,

OEFABS.RESABS)

CAll GKI5(F

,

A

,

B

,

RESUlT

,

ABSERR

,

OEfABS

,

RESABS)

CAll GK21 (f

,

A

,

B

,

RESULT

,

ABSERR

,

OEFABS

,

RESABS)

CAll GK)1 ".A

,

B

,

RESUlT

,

ABSERR

,

OEFABS

,

RESABS)

CAll ANC7(F

,

A.B

,

RE5UlT

,

ABSERR

,

OEFABS

,

RE5ABS)

kEYF-l

KEYF..)

kEYF"KEY

1F

(kEY .l

E. O)

1F

(KEY .GT. 7)

C-kEYF

NEVAl"O

1F

(kEYF .EQ.

1)

1F

(KEYF .EQ.2)

IF (KEYF .EQ.))

IF (KEYF

.EQ.~)

IF (KEY

Fo

EQ.5)

IF (KEYF .EQ.6)

IF (kEYF .EQ.

71 lAST"1

RLI 5T

(1)

-RE5Ul T

El

1 5T (1) -AB5ERR

1 ORD

(1

)-1

FEMF-切，、

0015

0016

0017

0018

0019

0020

0021

0022

002)

002~

0025

0026

0027

0028

0029

00)0

由壘，

ERRBIIO-AMAXl

(EP5ABS ， EP5REl 貴 AB5

(RE5

I1

l T))

1 F

(ll 川 T.EQ.lllER叫

IF (IER.N

E. O .OR. (ABSERR.l

E. ERRBIIO .AIIO. ABSERR.II

E. RESABS)

A AB5ERR.EQ.0.0)GOTO 60

TE5T ON ACCURACY

11 壘，

PUP--FEW

00)1

00)2

00))

.on

1111 壘

INI Tl A

Ll

IAT10N

-

657 一

11""

ERR

/l

AX"ABSERR

MAXERR"'l

AREA"RESUlT

ERR5U

/I

"AB5ERR

NIUlAX.l

IROFF 1"0

FIWPLFL

。O)~

0035

00)6

0037

00)8

00)9

(18)

師大學報第卅一期

FORTR^N IV Gl

RElEA5E 2.0

AGQ

OATE _ THU JUN 0&. 1985

OO~O

_IROFF2-0

E

c

1<費由

_們AIN

_{00 lOOP}

最曲曲

E

00~1

_。o

_{10 lA5T-Z}

_,

_lIHIT

c

E

費曲曲

BI5EtT THE 5UBINTERVAl WITH THE lARGE5T ERROR ESTIHATl

E

00'.2

Al 個All

5T

(tI

^XE

Il

R)

00~3

þl"O.5 1< (A 1I 5T(們AXERR)

+B

lI

5T

(們AXERR)

)

OO~I.

_A2-Bl

。M5

自 2-B lI

5T

(I\

^XERR)

001.6 IF(KEYF.EQ.

l)

tAll G5

7(

F

,

Al

,

Bl

,

AREA1

,

ERROlll.RESABS.OHABll

00~7

IF(KEYF.EQ.2)tAll

GK7(F ， Al ，因 I ， AREA1 ， ERnORI ， RESABS ， OEFAB1)

oM8

I

F(

KEYF .EQ.3

)t

All G5 15 (F

,

A

1 ，因 I ， AREA1 ， ERROR1 ， RESABS.DEFAB1)

001'9

IF(KEYF.EQ.~)tAll

GKI5(F

,

Al

,

Bl

,

AREA1

,

ERnOR1

,

RESABS.DEFAB1)

。050

IF(KEYF.EQ.5)tAll GK21(F

,

Al

,

Bl

,

AREA1

,

ERRORI

,

RESABS.DHflBl)

0051

IF (KEYF.EQ.6)tAll GK31 (F

,

Al

,

Bl

,

AREA1

,

ERROR1

,

RESABS.DEFAB

o'

0052

IF (K

E-Y

F .EQ.

J)

tAll AN

C]

(F

,

A I

,

Bl

,

AREA1

,

ERROR1

,

RESABS

,

DEF AB

1)

c

0053

IF (KEYt .EQ.

l)

tAll G57 (F

,

A2

,

B2

,

AREA2

,

ERROR2

,

RESABS.DEFAB2)

005~

lF(KEYF.EQ.2)tAll GK7(F

,

A2

,

B2

,

AREA2

,

ERRO

Il

2 ,

RES^BS

,

DEFAB21

0055

IF(KEYF.EQ.3)tAll G515(F

,

A2

,

B2

,

AREA2

,

ERRO

Il

2 ,

RESABS.DEFAB2)

0056

IF(KEYF.EQ.~)tAll

GKI5(F

,

A2

,

B2

,

ARJA2

,

ERROR2

,

RESABS.DEFAB2)

0057

rr (KEYF .EQ.5) tAll GK21 (F

,

A2.B2.A

Il

EA2.ERROR2

,

RESABS DEFAB2)

0058

IF (KEYF .EQ.6) C^ll GK31 (F

,

A2.B2.AREA2.ERROR2.RESABS.DEFM21

。059

I F

(~EYF

• EQ.

J)

CAll ^N

C]

(F

,

A2

,

82 ,

AREA2. ERROR2. RESABS. DEF AB 2)

E

IHPROVE PREVIOU5 APPROX. TO INTEGRAl AND ERROR

E

0060

0061

0062

0063

006~

0065

0066

NEVAl-NEV^l+1

A

Il

EAI2-^REAI+ARE^2

ERROI2-ERRORI+ERROR2

EnRSUH-ERRSUI件 ERROI2-E Il R /I^X

AREA-^REA+AREAI2-RlI5Tþ1AXERR)

IF (OEFAB1.EQ.ERRORl .OR. OEFAB2.EQ.ERROR2) GOTO 5

1 F (AB5 (Rll 5T

(/l

AXERR) -AREA 12) .lE. 1 .OE -5"AB5 (^RE^ 12)

1<

.AHO.

ERIl 012.GE.9.9E ﹒ 1 晶 E IlIl M^X)

1 ROF F

1 鹽 IRO rF l+1

IF (lA5T .G

T. IO :ANO. ERROI2.GT .ERRHAXI IROFF2-IROFF

2+

1

5 R

lI

5T

(例AXERR)"AREAl

RlI5T(lA5T)-^REA2

ER

Il

BNO"A

/I

^X 1 (EP5AB5

,

EP5

1lEl

"'^,

'5 (AREA) )

1 F (ERR5U

I\

.l

E. ERR8tto) GOTO 8

。067

0068

0069

0070

0071

PIMP-up--TE5T FOR ROUHOOFF ERROR AHO 5ET ERROR Fl^G

。072

00 ]3

IF(IROFF1.G

E. 6 .OR. I

Il

OFF

2. G

E. 20) IERs2

IF (lA5

T. EQ.

lI

HIT) IER"I

007 "

0075

0076

F-up--F •• "HU

IF(ERROR2.GT.ERRORI) GOTO 30

A

1I

5T (lAST) "^2

BlI5T(MAXE

Il

R)"Bl

曲曲曲

APPEND THE NEWlY-tREATED INTERVAlS TO THE lA5T

晶聶聶

-

658 一

(19)

的分法則之一比較

FORTHAN IV GI

RElEASE 2.0

AGQ

OATE - THU JUN 06

,

1985

。077

0078

0079

0080

0081

0082

0083

008 ,.

0085

0086

0087

0088

0089

0090

0091

0092

0093

009"

0095

0096

0097

0098

0099

0100

0101

0102

0103

010"

0105

0106

0101

911 ST (lAST) "92

ElIST(MAXERR)-ERRORl

ElIST(lAST)-ERROR2

GOTO 20

30 AlIST(

I\

AXERR)"A2

E

All ST

(lAST) 吋 1

9l

1 ST (lAS

T)

"'9 1

RlIST(

I\

AXERR)-AREA2

RlIST(lAST)~AREAI

El

IST(IIAXERR)"ERROR2

ElIST(lAST)-ERRORI

C

CAll SORT TO

"AINTAINτUE

OESCEIIO I IIG OROER IIIG 111 111E

1I

51 C

OF ERROR ESTIMATES ANO SELECT THE 5UBINTERVAl WITH 1HE

C

lARGEST ERROR ESTIMATE

c

20 CAll SORT

(lI

M

IT

,

LAST

,

"AXERR

,

ERRI'AX

,

El

I ST • I ORO

,

IIRI'AX)

IF(IER.IIE.O .OR. ERRSUM.lE.ERRBNO)GOTO "0

10 CON

Tl

NUE

c

C

COllrUTEτHE

RESUlT

c

"0

RESUl T..O". 0

00 50

K 自 I ， lAST

RESUlT"RESUlT+RlIST(K)

50 CONT IIIUE

ABSEnR~ERRSU們

60 IF(KEYF.EQ.I)

IIEVAl-18 脅 IIEVAl+9

IF(KEYF.EQ.2)

NEVAl-l"~NEVAl+1

IF (KEYF .EQ.3)

NEVAl 四 "2 叫nVAl+21

IF (KEYF.EQ.")

NEVAl"'30~NEVAl+I5

IF(KEYF.EQ.5)

NEVAl-42 甜 NEVAl+21

lF(KEYF.EQ.6)

NEVAl"62 曲 NEVAl+31

IF(KEYF.E~.1) NEVAl-l" 曲 NEVAL+7

I

COurlT ~KEY F+ 2

WRITE(6.11' ICOUNT

11 FORMAT{/

,

I5X

,

'METHOO

('.11.') :')

999 RETURN

END

壘。 P T1 0llS

It

l

E Fr ECT 費

NOTERM.10 ， E9CDIC ， SOURCE ， NOL1ST ， NOOECK ， lOAD.II0 I\ AP ， II0TEST

壘。P T1 ÒIIS'

IN

E Fr ECT費

NAI\ E

• AGQ

,

LIIIECIIT •

56 會 STATISTICSA

SOURCE STATEMENTS -

l01.PROGRAM SIZE • 001658

禽 STATI

STI

CS 曲

NO

DIAGNOSTICS GENERATED

(20)

師大學報

第卅一期

FORTRA" IV GI

RElEASE 2.0

I1

AIN

DATE .. THU JUN 06

,

1985

E

甜食""""""""甜甜食*"*"""""****""*曲曲""""**甜甜甜甜食"甜食費曲曲貴晶晶晶晶晶晶食費

C

"*

"i悟

c

壘，

_{SUBROUTINE SORT(ORDERING ROUTINE)}

費 4悟

E

t."

f.

t.

E

"食費 1‘甜甜食""""會聶""食費曲曲曲，甜甜1r壘，脅""""甜食，貴 ""ft"""" 甜甜 4、食聶 4時會飛 fd聖

。 001

SUBROUTINE

50RT(ll l1 IT ， l^ST ，們AXERR ， ERI1AX ， ElIST ， IORO ， NRMAX)

0002

。 I I1ENSION

ElIST(IOO)

,

IORD(IOO)

c

---

FIRST EXECUTABlE STATE

I1

ENT

c

0003

If.(lAST.GT.2)GOTO 10

OOO~

10RD (J)吋

0005

1 ORO (2)-2

。006

GOTO 90

。007

10 ERR

I1

AX-

El

I 5T

(,

'AXERR)

0008

If(NR

t1

^X.EQ.I)GOTO 30

。 009

100-NR們AX-I

。010

。o

20 1-1

,

100 。011

ISUCC叫。RO (IIR州AX-I)

E

。012

IF(ERRM^X.lE.ElIST(ISUCC)~GOTO

30 0013

10RO(NRM^X)-ISUCC

001~

_t

lR/1

AX-NR

/l

AX-1

。015

20 CONTINUE

c

。016

30 JUP8N四 lAST

。017

IFtlAST.GT. (ll

t1

IT/2+2))

JUP8N-ll 州1T+3-lAST

。018

ERR

t1

IN-ELIST(LAST)

E

。019

J8NO-JUPB

f/

-1

0020

• 18EG':NR

t1^X+

1 。021

If(IBEG.GT.J8ND)GOTO

SO

。022

。o

"0 I-IBEG

,

JBNO

00 2)

ISUCC 叫 ORO

(1)

002~

If(ERR

I1

AX.GE.ELIST(ISUCC))GOTO 60

。025

10RO(I-

I)

-ISUCC

0026

~O

CONTINUE

0027

50 IORO(J8NO)"

/l

AXERR

。028

10RO(JUPBN)-LAST

0029

GOTO 90

c

。030

60 1 ORO

(1-1)

""^XERR

0031

R ﹒ JBN。

0032

。o

70 J.I

,

JBtl。

0033

1 SUCC"I ORO (K)

003~

IF(ERR

t1

IN.LT.ELIST(ISUCC))GOTO 80

。035

10RO

(K+ I)叫 SUCC

0036

K-K-l

。037

70 CON

Tl

NUE

0038

10RO(

I)

"lAST

0039

GOTO 90

OO~O

80 10RO(K+I)"LAST

-

660 一

(21)

積分法則之一比較

DATE .. TIIU JUU oG

,

1985

SORT

RELEASE 2.0

FORT

1t

AN IV GI

SET MAXERR AND ERMAX

---們AXERR~IORD(NRMAX ，

ER"AXftElIST(MAXERR

,

RETURN

END

nu

ptupbpbny

O

Olt

l

00"2

0 0lt

3 00""

脅。 PTlOtlS

1"

EFFE r:T曲

NOTERM ， ID ， EBCPIC ， SOURCE ， NOlIST ， NODECK ， lOAD ， NOHAP ， HOTEST

會 OPTIOH5

IN EFFECT*

NAME - 50RT

,

ll"ECNT -

56 *STA

Tl

S

Tl

CS*

SOURCE

$T

ATE

IlE

NTS ..

"",

PROGRAM SIH - 00

0lt

E2

STATISTICS

NO DIAGNOSTICS GENERATED

DATE - THU JUN 06

,

1985

費 4、費由*************食費曲曲，甜食費 **ft*** It*1t甜甜食It ltlt 疊1t*1t曲 ftftft 曲 4、飛食，、寅貴

聶甜食 4、

It

ft

SUBROUTINE ADAPTIVE NEWTON-COTES

(7 個 PTS)

州

甜甜狗食

晶晶費由甜食It i悟，食費壘 ftft 費負甜甜It ltlt 賣會 Mtftftltftlt 飛飛禽It ft 費食費It lti、會食食，悟 4、最*，:fd悟 fd.fti、

SUBROUTINE AHC7(F

,

A

,

B

,

RESUlT

,

ABSERR

,

RESABS

,

RESASC)

DIMENSION FV(])

MAI"

RELEASE 2.0

PLPLPLFzubL

FOI月 TR ^，'

IV G 1

1I

"0.5

1t

_(B-A'

FV(I'''F( 的

FV (2' -F

(A+

II/". 0'

FV

(3,

-F

(A+1I

/2 .0'

FV ('" -F (A+II'

FV (5' -F

(A+3.0州 1 2. 0'

FV (G

,

-F

(A+'7.0 州 1".0)

FV(])-F (B)

E

0001

0002

0003

000"

0005

0006

0007

0008

0009

0010

COMPUTE THE INTEGRAl

n"

,

-p3.+

.. _ι 呵‘圖，

,

1 ••

，由 ‘ 1 也可 tnu 、 ynv ， 1

• .

"."v ﹒旬

,

1 ,

b'

,

nu

MV.',1nv

prnv,

3."

攪勻，‘""壘， E

nv41."

‘

1 .)()

、 4 •• ， &R 、.'

,',bnUFHJ

ap{-(

‘1'"vnouv

、， Ezt 刊 r

RJ+O{

， E. 、

••

_'‘

dpa

v2+B

Prf、。‘F-n

+V-m'+

‘.'，pphJ'、.'、'，

‘‘JSE‘

•••

唔，'‘.' ，.、‘"，h"，E、'‘ d

Mvnv'OM--'t

F.,fFV

'.、aMU‘'， a ﹒‘，r 會."、''， 3 ， t nvnv﹒肉"間，= ﹒呵，‘'，‘-R" 回 2+V+ ﹒闕，‘J'‘.'，r、''，1

+‘',AR‘',

、 1 可 Inv.' 。 0

){-{

..

'，HV.""，'，唔，hJ

,1,

••

nu',.,.,

MVAvnu,E‘nu',

'，、'，."， 3 旬，‘'。

+1.TB+,f

‘'',.‘

••

,-nn‘.'‘',

﹒'"，‘F，1、'，‘F 'tprpp。"、'，、 J MV'‘、'，、nu'。﹒肉

FARMV.'

••

,1

,10Fnpvuv

*.

+hFF

num苟，、'，F 『 J'，、，E、 •• HTZJ ， 1 ，扭， 2 ，'，phJ'，、‘nRE'" '，、，巴、HVHH.R-R -n 膏，'，

Hnun,ts

••• 圓 "UURn-a 間，、"，、 4. ，、J' ，且， EE

RHH"n"n

。_"-m ‘"

PLFLPL

0011

0012

0013

RESKH-RESK/(B- 的

RESASC"II 會 (5

_"

9. 。由 (ABS

(FV

(1)

-RESKII

,

+ABS (FV (])

-RESK川 '+20

1 _{.8.0 份}

青

(ABS

(FV

(2)

-RESK川 +ABS(FV(6)-RESKII"+2016.0ft(ABS(FV(3)-RESKII)~

It

ABS(FV(5)-RESKH))+"00".0州 BS(FV(")-RESKH')/6615.0

RESUlT"RESK

ABSERR'"ABS(RESK-RESG)

IF (RESASC.tl

E. O.O .AIIO.

^目 SERR.N E. O.O，

* ABSERR-RESASC

It

A"INI

(I

.O

E+

OO

,

(2.0

E+

02 It

_{ABSERR/RESASC}

,

_{UI .5}

_E+

₀₀₎

E

。01"

0015

。016

0017

0018

E

REτURN

END

機 OrTIO"S

IN

EfFECT 曲

"O'ER" ， ID ， EBCDlt ， S~URCE.NOlIST ， NOOECK ， lO^O.NO"^P.NOTES ，

ftor'IONS IN

EFFECT 晶

NA"E

- ANC7

,

llNECNT ..

56 食 ST I\Tl STICSft

SOURCE STATE

I1Et

ITS -

20.PROGRA

I1

SI

lE ..

00053且

ftS'A'ISTICS*

HO DIAG"OSTICS GE"ERATED

- 661

0019

0020

(22)

第卅一期

師大學報

DATE • THU JUII 06

,

1985

曲曲曲晶***曲曲曲曲曲疊疊，脅，有曲曲曲*會曲曲，曲曲曲曲*禽，曲曲曲 ftft 甜食費**'悟聶聶食費*聶晶晶聶 4、 4時

ftft

**

膏 ft

SUBROUTINE GAUSSIAN

APPROXIMATION(l~PTS)

由*

ft 音機會

ftft 晶晶晶食費 4、食 ft*ftftftftft 甜甜食費""甜甜食費 4、 ftft 由#費 """frfrfrft 食費由 frf.f.frf. 舜，﹒\j、繭 *:'1悟 fr

SUBnOUTINE.GS7(F

,

A

,

B

,

RESULT

,

ABSEnn

,

RESABS

,

RESASC)

DIMEIISION FVI

0)

,

FV2

0)

,

WG (2)

,

WGK (")

,

XG (2)

,

XGK (")

們AIN

RELEASE 2.0

pbpivPLFEWFLFL

FonTRAN IV GI

0001

0002

DATA

.',,.',,.',,

nv.

,.,

nu.

,.,

nvnunvnunvnv

4 ， --4

冒--'

..

_‘'

...

,

•.

_h'

•.•

，.』'.』 nvqd-qnu 可 1." nURn-" 。'。"耳。 "Enyo-' ，， b ono-30 嚕，‘." 。"。 qdnuqJ-3 nURNHVR 司， nu 仇"》可'， nu"Huwphdnu ﹒旬， b

nunBRJO--3

。"。 nonuqdmo 。"。 10 ‘ 3.' 。"間，'。， PR3 。"。 -3OORJ nvnOPDnv 可 tqd

nuao-fnuny'

,

nv" 。 ."nu 唔，.，

..

,

f

••

,

r

.•

RVRU ‘，'，'， nu 、.'‘ 4. 甸 ﹒旬，"

({

',,

URHURH

‘,

EFU

2 ••

x

••

w

••

(11.11.--Funu 自υ ‘ Fnvnv ‘ Fmvnv

w

••

3 ••

3--'， E ，巴， -PEPE ， teEPE E' 。， bwnpp2w 叭， IR3 .'，‘ dp 旬，， FU""" ，'， nunEno

{8EJX59w61

6. 旬， 2' ，'， 3.mo- uw' ， PD ‘'，呵，‘嚕，'、 1.EPD ，亂"，口呵，&亂呵，'，'.， hvnv ‘'，呵，‘ RJt 、電 dzdf 、.，'"，唔，‘ mwd ，崎 Juan-4." ，包叭， hunu ， 1' 。 phd"u ﹒'，旬， '"u' 。 PD

RU'ORd-Any-wnyo

van' ， EJ' ，'，虫， ."nu -REP33' 。﹒旬‘''，旬， 3 、'， PPPD.' ，，自 8. ，."，自 .， -u 守，『 J' ‘‘ HWdp 崎J' ，‘肉 wd.-