5,· Calif

. .

. ,RECOMP II USER'S PROGRAM NO. 1119

'~OGRAM TITLE: CURVE GENERATION PROGRAM CLASSIFICATION

*

General

. AUTHatS:··

." ,DAm,.

Robert Quin1chett, Louis C ^e Fargel North American Aviation

Oolumbus, Ohio

To r~d the linear function of one curve vhichbest approximatesa· second curve b,y

the least mean squares method. The per- cent error of approximation is uso determined. . .

8

December

1961

.PubliShed "by

~ECOMPyserl $ Library"

at·

. . : AUTONETICS' IN'DUsmIAL PRODUCTS. . ' . A DIVISION OF· MORTa· AJmR,:tCAlrAVIATltJN, INC • ."

:"'3400 E' •. 7Oth.··Street, "u>ngBeach

5,· Calif

^{e '}

o · '.' ' . . • • . ,

. . ' . ' , . DISCLAIMER . . .

' . • . . . . . i . ' .

AlthougH ttlS.assumed that atlthe precautions have been '. taken 'to check out tflis .program thQroiJghly, no responsibility .

. is takenbv the .originittor.of ·this prog'rllfll, for any erroneous ,tesuftS; miscOf\te(:Jtions, or rf*'rli'ptesel1tations that may appear . infhl$ proarem. furthermore, n~respOrisibility., is taken by.

. ',A\Jtol1etr~, 'lncJusttJ'h Products for t~e correcl reoroductions of .,,-: this ... pf.Qgt~m:~' ',N~' ,warrarityi'expre$Sor ~mpliec;l, .. is .extended

'by the:.t.is~·bt·Ip.,PHcatlon 9f the pt~gram.··:· .

CURVE GENERATION

^{Page One}

1. 'INTRODUCTION

It is orten·desirous to have a medium

or

correspondence between two sets or data, empirical, or otherwise" when there exists a degree of linear! ty between the functional rept'esentations of the data. This program cal- culates the linear function of one, curve which best approXimates a secon,d

ourve

by the least mean squares method. One curve X can be used to generate curves Ii by fitting X over' each Ii in 'the best linear fashion~

The program was principally designed to economize on time and non-linear eqUipment in analog COIDpllier empirical curve simulation.

2. METHOD,

2~1

Let

curve X and curve I be fUnctions, empirical

or

otherwise, of the same variable Z. Choose' H points from curve ,y (the curve to be generated) at

Zi

and N corresponding points from curve X at the. same

Zi-

Setting Ia

=

^A

⁺

BX, the problem now resolves in finding the values of A and B

wh,~chmake the mean square value of Y - Ya a minimum. The' mean square of Y '';;''Ia

=

This ~educes to

+ t l

(y~)2

N

'Equation I: ~ {v.\~ ~ . N

.~

- 2A 't=l Ii - 2B

~l

^Yilt

N N H

, ,

, ' N H ' .

+ ~ +~

^}(=lXi

⁺

^B2

~=l

^{(Xi) 2}

=

^{F (A,B)}

N· H '

. F(A,B) has a mini.Inup1 wl~h ^respect to, A at

d

^{(mean square} _{(Y-Y a}

» = o.

" ~A

. Hence ,.A

=

^{Y i- BX}_i ^{• ~bsti} tuting for A in eqUation I and minimizing we

, have .. ' ' , ,

(~)2

H The percentage error;, at each point is computed as

'.(P.E.)i = lOO'CYi - Ysi)

-

^' O~ i ~ N

TITLE: CURVE GENERATION

Page Two 3. RESTRICTIONS :

3.1 Components R~quired .

The photoreader and typewriter are required.

3.2

Subroutines utilized.

3.2.1

3 ..

2~2

.).2.3

.3~3

4.

4.1 4.;1.1 '4.1.2 ,

4.2 4.2.1

Floating Point Input AN-007.1

'Floating point to ,fixed point output AN-Q15.l Decimal output AN-Q16.0

2 ~ N~ 10 where N is 'the ll\JlD.ber of sets of points from both curves. It isobviou,s that N varies with the complexity of the curve being generated.

Thus, in generating a straight line from a'straight line, only tvo sets of points need to be chosen along the cammon variable

Z. A

more complex curve. would require N to be larger £or increased accuracy. (Note:

Judicious selection of Zils will improve approximation; i.e., points 'should be chosen at slope intersections on curve

Y). '

Nop,oint may be chosen which has a value of zero.

If curve X is a straight line with slope zero, 3.3 becomes 2C::: N~ 10.

'Failure to. meet thi,s condition results in a possible overfioW'.

USAGE'

Computer set-up.

Set sense-switch C down

it

heading is desired.

All other switches are in normal position.

Se~ence of manualoperation~.

. Load and verifY program' tape.' The program W'ill print out heading, if sense , switch C is on, and halt waiting for input data. If the tape does not have

,L 0651.0 $ at ,the ~nd~ press start 1 to achieve the same results.

Set sense switch~ down, and press start 1. The typewriter will carriage . return and print "H:

'I.

Enter, on the typewriter, the Dumber of pairs of . : points chos~n from curves X andY by typing an integer H, 2 ~ H ~ 10.

Begin entering'datao 'Enter N values, 2 ~ N ~ 10, f'ro. the Y curve .nr~,

followedby' the cOlTespondingN values trom the X curve. The numbers i"1lBY

be int,egers, tractions, or mixed numbers.

TITLE: CURVE GENERATION

Page' Thre'e' ComPUtation '

, ,

When all the dat'a bas been enter~d, depresss "the' start button to inItiate cOmpUtation .. ' The values or A

and

B of the eqllation Y =A+ BX vi11 ,be' typed out. The program \Iil1 also type the difference be~veen Y and the actual curve generated

at

each 'discrete po~nt, along with the percent

er~ at, th~t, point. The &v,erage percent error, is also determined.

Error codes and restart :procedures.

, ,

, If 'a correction is to be made in the data, put sense switch D down and press start' 2. ,Enter first, on the typewriter, the number of correct pieces of data preceding the incorrect entl7. Then re-enter the correct ,piece(s) of'q,ata. ,Attar corrections have been made, put sense switch

n

up, press error reset and start 2 to begin calculations.

, ,

Computation Errors

No ~r~ors are anticipated.

,,4.5' " Extent or Storll.ge, R~quirements.

, 4.5

4.,5.2, , , ',4,.5~3,

4~6" ':, 4.6.~'

,

'" Program Looations

SUbroutine Locations

~ata Lo,cations OltpU~ Fo~t. , '

0470S

to

~2008

12508

:to ^{14478 ,}

'~lS'aDd 00028

~108 to' O~78' 12008 to 12478 '

The

,outPut format is the Same as,that shown in the example prob1em,llhere:

. ^{. .} ." . ' . . .

' , " ."

':(1)' : mi'

^{(1 ~ , i}

⁼⁼

10) 1s' the' difference at point ibetveen the , ~~tual~e generated and the

t"curve. '

,

(~l,:

P:.E'.

is1ihe"p~cent'

error' at the' conesponding

point~

. " . ' . ' . .

'(3)',

,.'::1 "(Ave~ag~) 1S

^{the average, percent}

err~~.

^'

. ^~.

TITLE I

rage'

^Four,

S.o :

EXAMPLE' "

I , ^I

l

I _I.

, y

I 'I

i, I

' I

I

,

^"

,I 1 \'

i' ^I

f

:r

r

I

I

. , . '

I ^"

l ^'/ ,I

; I

I _"

, ,1,

~i---·-·-~·---

I I I I

~-~

Z,

:...;...,----t---r---

z.~ Z!I ^{Z '} _~ ,Zs ' z •. ,

,,'

'~=A+ex

CURVE-GENERATION

THE

pURPOSE OF'

^THIS

PROGRAM

^IS, TO,GIVE

THE

BEST

APPROXIMATION

^{OF ONE}

CURVE 'As A

L I NEAR FUNCT-l0N OF ANOTHER \-/HEN' BOTH CURVES AR:: FUNCTJ ONS ':OF THESM-1E VARIABLt.THE PERCENT ERROR OF APPROXIMATION IS ALSO DETER-

·MINEDo' , , , " , '

N:6, "

. 2,

~:6, "

2~6

' 2·9 ' : 2.:9,

.:0001 '

• l' , .2

.2 .3 '

.'3

, 'A:,

"1" '.,

99980?- 4

1

~34 " ,

B:'

3 ..

oo08dt94~

17 '

DY1: -, ' ,,~ooOl02

DY2': .000117

g~~. .. :ggggu

'OY'l: ;"'" , . . oooQIi.ij.

, D.Y6: -,' " • 000041~

"E(~VERAGE)

::-, .

,

^P.E~­

P.E.

P.E.

P.E. " ,

P.E.:"

P.Eo~

,.0000 '

',' ~OO5125

, .005

⁰

91

,.001408

.001408

. , .OOl5'14

• 001 511~,

TITLE:

6.0

6.1 6.1.1

6.1.2

6.2 . 6.2.1

CURVE GENERATION .

CODING INFORMATION

Constants and their locations (Ootal).

. Fixed Point Constant 1060 at BlB Floating Point Constant

100.0 .2.0 0.0 1.0

Timing Estimate.

In

LOoation 0473

Looation' 0470 0476 1260

1262

Page Five

'(16

+.

ION) seoonds, when N is the number of points chosen from the common variable of the. X and Y curves.