[:ional Bureau of Standarclg

Library, K.W. Bldg

MAY 2 519^a

PB 161626

^ecUnlcaL ^^ote 125

OMNIFORM I:

A GENERAL PURPOSE MACHINE PROGRAM FOR THE

CALCULATION OF TABLES OF FUNCTIONS

GIVEN EXPLICITLY IN TERMS OF ONE VARIABLE

JOSEPH HILSENRATH AND GERALD M. GALLER

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March 3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and maintenance of the national standards of measurement and the provision of means and methods for making measurements consistent

with these standards; the determination of physical constants and properties of materials; the development of methods and instrimients for testing materials,- devices, and structures; advisory services to government agencies on scientific and technical problems; invention and development of devices to serve special needs of the Government; and the development of standard practices, codes, and specifications. The work includes basic

and applied research, development, engineering, instrumentation, testing, evaluation, calibration services,

and various consultation and information services. Research projects are also performed for other government

agencies when the work relates to and supplements the basic program of the Bureau or when the Bureau's

unique competence is required. The scope of activities is suggested by the listing of divisions and sections on the inside of the back cover.

Publications

The results of the Bureau's research are published either in the Bureau's own series of publications or

in the journals of professional and scientific societies. The Bureau itself publishes three periodicals avail- able from the Government Printing Office: The Journal of Research, published- in four separate sections,

presents complete scientific and technical papers; the Technical News Bulletin presents summary and pre- liminary reports on work in progress; and Basic Radio Propagation Predictions provides data for determining

the best frequencies to use for radio communications throughout the world. There are also five series of nonperiodical publications: Monographs, Applied Mathematics Series, Handbooks, Miscellaneous Publications,

and Technical Notes. A complete listing of the Bureau's publications can be found in National Bureau of Standards Circular

460, Publications of the National Bureau of Standards, 1901 to June 1947 ($1.25), and the Supplement to National Bureau of Standards Circular 460, July 1947 to June 1957 ($1.50), and Miscellaneous Publication 240,

July 1957 to June 1960 (Includes Titles of Papers Published in Outside Journals 1950 to 1959) ($2.25); avail- able from the Superintendent of Documents, Government Printing Office, Washington 25, D. C.
NATIONAL BUREAU OF STANDARDS

technical ^^ote

125

MAY 1962

OMNIFORM I: A GENERAL PURPOSE MACHINE PROGRAM FOR THE

CALCULATION OF TABLES OF FUNCTIONS

GIVEN EXPLICITLY IN TERMS OF ONE VARIABLE

Joseph Hilsenrath and Gerald M. Gailer

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature. They are for sale by the Office of Technical Services, U. S. Department of Commerce, Washington 25, D. C.

DISTRIBUTED BY

UNITED STATES DEPARTMENT OF COMMERCE

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WASHINGTON 25, D. C.

Price "^^
Table of Con-Gents

Abstract 1

1. Introduction . 1

2. Omniform I 3

2.1 General Characteristics 3

2.2 Mathematical Foundation 5 2.3 Input 10

a. Preliminary Input 10

b. Control Matrices 11

c. Output Instructions 12

2.4. Print Format 17

3. Properties of the- Function Generator 20

3.1 Elementary Functions 20

3.2 Special Functions 22

3.3 Column Sums 24. ^. Acknowledgments 24.

Fig. 1 OMNIFORM I Input Form (front) 26

Fig. 2 OMNIFORM I Input Form (back) 27

Table I Function Generator 8

Table 2 Typical OMNIFORM Printout in Fixed Decimal 13

Table 3 Typical OMNIFORM Printout in Floating Point 15

Table ^ Sample Input for OMNIFORM I 19

Table 5 Initial Values and Recursion Formulas for

Certain Special Functions 23

Sample Problems 28

11
OMNIFORM I : A General Purpose Machine Program for the Calciilation

of Tables of Functions Given Explicitly in Tenns of One Variable.

Joseph Hilsenrath and Gerald M. Galler*

A general purpose computer program, designed for use by

non-programmers requiring computation of tables of functions

written explicitly in terms of one variable, is described.

The program features include: 22 types of elementary and

special functions; provision for taking of functions of functions, table read-in, flexible print-out; and standardized

input to a variety of problems. The use of the program is

illustrated with a number of sample problems.

1. INTRODUCTION

In spite of the successful application of digital computers to

important problems in virtually every subdivision of modern science, they

are not yet being employed widely and effectively to assist in the routine day-to-day computations of a busy laboratory. They have not yet

replaced the multicolumned work sheet which is such an indispensable part

of both experimental and theoretical research.

A brief catalogue of these routine numerical calculations would

include: interpolation, differencing, smoothing, curve fitting, differentiation, integration, and table look-up for elementary and special

functions. These mathematical operations obviously have general applicability. Their day-to-day use in any individual situation is, however,

quite specialized - too specialized, it would seem, to justify the preparation of a multitude of computer programs tailor-made for each

'"" Present address. Systems Engineer, General Electric Computer

Department, Phoenix, Arizona.

1
application. Access to digital computers has been made easier for both

the experienced programmer and the novice by the development recently of

a variety of systems, compilers and languages. These aids have encouraged many scientists to learn to program their own problems. In spite of

the expanding corps of "scientist-programmers," there is now and will

continue to be a need for "painless" access to digital computers by

scientists who are not programmers.

A solution to the mechanization of a sizable number of ad hoc calculations now being performed by hand or being avoided altogether, seems -to

lie in the development of general purpose programs for the various digital computers. Such programs could involve either a number of interpretive general purpose codes in which a large number of different computations are included, or an operating system containing a large number of

individual programs with highly simplified input. In any case, such

programs should be complete packages permanently assembled with input and

output, rather than subroutines. In this way, they would be useroriented rather than programmer-oriented.

OMNIFORM I is one of a series of such programs under development at

the National Bureau of Standards. It was initially intended for thermodynamic calculations. As the planning of the program progressed, indications of its wider applicability became apparent and it was expanded

accordingly.

The motivation of the program and its user-oriented specifications

arose from the need of a general purpose program by one of us (a non- programmer) . The final OMNIFORM I Program, as described herein, was
made possible largely through a successful communication of these specifications to the second author (an experienced programmer)

.

2. OMNIFORM I

2.1 General Characteristics

OMNIFORM I is a completely assembled interpretive program for the

IBM 704. - 7090 which permits direct use of the machine by scientists or

engineers without knowledge of programming. Instructions, given in decimal form, control the flow of calculations in a manner highly analogous

to the logic, which prevails in carrying out computations on a desk

calculator. More precisely, OMNIFORM I simulates desk computing in that

it replaces the desk calculator, mathematical tables, and the miilticolumn work sheet but not the logic which the scientist employs in such

operations.

The name of the program has the following significance. It is one

in a series of omnibus programs, hence the OMNI. It is specifically

restricted to substituting constants and a single variable into an algebraic formula given explicitly in terms of one variable, hence FORM and

I. It should be noted that the formula or function must be written

explicitly in terms of the independent variable. It may, however, be

given in terms of more than one variable if the other variables can be

treated parametrically. The function may involve any of the elementary

functions and many of the frequently used special functions. The function must however be explicit.

The function types built into the program are identified and called

for by numbers. For example the substitution of a set of values in a power series is accomplished by placing a number 1. in the
appropriate column of the instruction sheet. This provides a power

n b.

series of the form f, = 2 a. X . A number n then tells the machine

how many terms to generate, and n numbers pairs (a. and b.) provide the

J J required coefficients and exponents.

The versatility of the program, over and above the wide range of

functions available in it, results from the fact that in any of the

functions f (X) (defined in Table l) , the variable X may be replaced

easily by any function of X either previously computed or read into the

memory as a table of functional values. Thus the program can readily

compute functions of functions, such as: loglogx, e , [f (x) ] ,

si a. + Jh + N^c , continued fractions. In (e + a) , In sin (l + bx)

to name a few. Since the order of operations is completely open, any

explicit formula involving sums of products of sums of functions can be

handled - - a feature which is the basis of OMNIFORM I ' s extensive

applicability

.

The program was assembled using the FORTRAN programming system with

subprograms from the FORTRAN Library of routines. In addition, specialized subprograms were developed to provide a complete program giving

flexibility in input, computing, and output capabilities. These programs

are a permanent part of OMNIFORM and hence provide a self-contained com- puting deck suitable for immediate use. The deck, comprising about 500

binary punched cards, can be read by the computer in less than 2 minutes,

after which the computer is ready to interpret the cards containing input data which direct it to the individual problem,

OMNIFORM I is primarily a table generator. It can produce a table

^ a., b. are real numbers but not necessarily integers.
of up to 50 columns (50 different functions) for each of 200 arguments.

Among the tables which have been computed on OMNlFORM I thus far are:

ideal gas thermodynamic functions for neon and for electrons, anharmonic

corrections for diatomic molecules, vapor pressure of solid nitrogen,

virial coefficients for nitrogen gas, negative exponentials corresponding

to certain atomic energy levels, polynomial substitution, table conversion to other units; table editing (column switching etc.) , and various

ad hoc mathematical tables for selected arguments.

2.2 Mathematical Foundation

a. The Operator Equation

The generality of OMNlFORM I rests upon two factors: a Function

Generator which provides 22 functions f (X) as defined in Table 1 (the

symbol f indicates a function of type a having n terms) and an Operator

Equation

R (Xj = G, (Xj + f"^ OjxJ Q H (X,) ^ k i k i a L t i _ c 1

which causes these functions to operate on the values H (X.) stored in

in col"umn c; and to store that result by addition to the values G, (X.)

stored in column k.

The integers k, a, n, t, Q, c constitute the first six numbers in

each line of instruction in the Operator Equation. (See Table 3.) These

numbers serve to select the function type and the number of terms in each

function, to indicate certain operations upon functions, and to specify

^ Q is a symbol for the operations +, -, x, ^ , and the Xj_ are the arguments for which the calculations are performed. Whenever instructions

are given for a line of calculations, these are performed for the entire

set of arguments (a column of X. or some function of them)

.
storage locations. For each line of Instruction, the six numbers

(defined below) are followed by a variable number of constants - -

a., b.; or a., b., c, d., - - as required by the particular function

type selected and the number of terms In It.

' b» Definitions of the Subscripts

k assigns the result of any line of computation to the k

column In the memory by addition , 1 < k < 50.

a selects the function type, 1 < a < 22.

T Vl n Is the number of terms In the a function, 1 < n < 200.

t Is a transformation switch which Is Interpreted as follows

on the first pass through the function generator. If

t = 0, 0^(X.) - X^;

t = 1., 2., . . ., 19., O-^CX^) = ^^\^ + h^.

On a subsequent pass through the function generator,*

0, (X.) = F (X.), a function previously computed and stored In

column t.

Q Is an operator Indicated by an Integer. Thus:

Q = 1. adds f^ ro,(X.)1 to H (X, ) ; ' a L t 1 J c 1

' ^ = 2, subtracts f" [^.(X.)] from H (X.);

fi = 3, multiplies f^ ^t^^l^] ^^ ^c^^l^

S = ^, divides H (X.) by f^ [^.(X.)]. ' c 1 -^ a L t 1 J

c Is the Index of the functional value upon which f 0, (X.)

^ A pass through the Generator consists of a series of computations

Initiated by a BCD IGENOaa Instruction. The repetition of the

BCD IGENOaa instruction initiates the second pass through the function

generator.
operates, 1 < c < 50.

If Q = the function computed does not operate on any other

function. A value must nevertheless be supplied for c, since

the machine expects six integers.

0. Arguments, Transformations , and Arithmetic Characterisitcs

Instructions for the arguments (the independent variable X) can be

supplied in either of two forms: as discrete numbers X, , Xp, X„, . . .,

Xp„p^, or in the incremented mode X, , A^ , X^, A-, . . . , X„. In the

latter mode the A's represent uniform intervals of tabulation between the

point X-, , Xp, etc. This is analagous to the notation employed in mathematical tables where 1. (l.) 100, (10.) 1000. indicates that values are

tabulated in unit steps from 1 to 100 and in steps of 10 between 100 and

1000.

Provision has been made to read in as many as 19 triplets of

constants p, , y, , b. which provide a like number of transformations

4t(X) = p/* + 6^

prior to substitution in the Function Generator. Thus the variable X

may be replaced at will by (l + l/x) , (4.. 7 X

-^'^^ + 17.8) , (X^ - l) , etc.

This provision simplifies the instructions and the flow of the calculations

in complicated problems.

The above remarks concerning the role of the integer t apply when

the Function Generator is entered for the first time - - when either

T+

X. or (B, X. + 6, ) is used as the variable in the function. On subse1 ^t 1 t

quent passes through the Function .Generator, the integer t no longer

denotes a transformation but instead denotes the column number of the

7
Table 1. Function Generator

\ - S ^/^ f" = S a

J [cosh xfi

n t,

fo = Zl a X hogX 13 jz'i

J ^

j=l J

^

n

f = Yl a.Tf/a + b. arc sin X ^ b. j=l

^ f = 2Z a X JlnX

j = l

J

n n b

f = 5^ a^TC/a + b, arc tan X

15 trf J J J=l

^4 n

= S ajX ^exp [cjX ij

^

j=l f = 52 a T (X) (Chebyshev Polynomial)

f^ = C (a/bpx^J Hsin x] J ^ j=l -^

n " bi nH n ** f = ^

ajX - [^sln CjX] J f = S ajU^.CX) (Chebyshev Polynomial)

f-j = 22 (a./b.)X ^ [cos X^ i f'^g = S ajPni(X) (Legendre Polynomial)

fg = iil a-X ^ [cos Cj x] ^ f

19 = ^ a L^ (X) (Laguerre Polynomial)

fg = II (a./b JX'^J [tan xf-3 f2Q = ^11 ^jH^^ (X) (Hermite Polynomial)

^ ^ ^^ r -\d- n ^ _ _d.

flO = H a.X J [tan c X^J £ = ^ a. r(b,)CP(c.X)J J j=l J j=l -^ -^ -^

f 11 = IZ a. [slnh X]^i f^2 = E] aj erf(X)
functional values which replace X, in the Operator Equation. Here t can

range from 1 to 50 since space is provided for as many as 50 columns of

functions in the memory.

Each function which is computed is forced, by Q, to operate on

another function before it is stored in a designated column. In fact,

the only way in which two columns of numbers can be added or multiplied

is if one of them either originates in or is made to pass through the

Operator Equation. Thus, if we desire the product A-B of two functions

previously computed, it is necessary to pass one of them through the

1 1

Operator Equation via an identity transformation f, =^ 1*A .

It should be noted that OMNIFORM I is programmed in such a way that

when the result of a line of computation is assigned to a column, k, it

adds the resiilt to the values already stored. For example: if A, B, and

C denote functions stored in the first three columns of the memory, and

we compute GsinB, assigning the result to column 1 yields A + C sin B.

If, however, only GsinB is desired, it must be stored in a blank

column. At the beginning of each problem all columns are blank.

By adding rather than replacing in the storage operation, it is

possible to achieve an additional operation. Thus, in one line it is

possible to compute one of the following:

A + f (B) + C or A - f (B) + C

A ± C-f(B) or A ± C/f(B)

Provision has been made in OMNIFORM I to read in tables of functional

values 9, (X. ) . The niimber of functions thus read in may be as large as

50 provided enough room is left for further storage for necessary

functions generated. An important requirement is that the array must be

9
rectangular (the same number of entries for each function) and it must

be in one - to - one correspondence with the specified arguments. Once

read in, these functions may be operated on in the same manner as if

they were computed in the generator,

2.3 Input

The input to OMNIFORM I consists of three major parts : a preliminary input, one or two control matrices, and print out instructions.

The input is via cards which, except for a few mnemonic instructions

contain data in decimal form.

a. Preliminary Input

The preliminary input consists of five distinct parts, three of

which are mandatory and two optional. The program first reads a title

card and prints the title. It then reads a second card containing three

numbers N , M , S . These are written as integers (without decimal

points) and are separated by commas. The numbers N , M , S provide instructions for the preparation of the arguments and the table read-in as

indicated below.

The third card (or cards) contains N numbers which are either a

table of arguments, or parameters from which the arguments may be generated in accordance with the instructions given on the previous card.

These are interpreted as follows:

If S = 1 then N values of X will be read in. i.e. X-, , . . . X„

If S = then N parameters of the form:

X, , A , Xp, A , X„ , . . .

, X „ are read in and the corresponding

arguments are generated.

10
As indicated in Section 2,Z OMNIFORM allows a number of transformations of the variable (X) to be made prior to entering the function

generator. If the transformation instruction t = 1., the first set p..,

Y-, 5 5-| 5 is used; for t = 2. , the set p^? To? ^2' ^^ "i^sed; for t = 3.,

the set p„, Y„, 6„, etc. The transformation constants, if any, constitute the fourth set of preliminary input, and are omitted if not

required. A TRA 2,4 instruction must follow the above input.

Columns of functional values G, (X.) , k = 1, . . ., M, may be read iu

K. 1

at this level of the program. These constitute the fifth set of input

data. The control for this read-in is vested in the integer M of the

second card. It is interpreted as follows: if M = then G input is bypassed, if M > then M columns of G ' s are read in row by row as follows:

^11' ^2 1' ^3^'^i^ ' * * •' ®M 1

^12' ^2 2' ^3^-^2 ' * * *' ^M 2

e^(x^:

These data are stored in the first M columns in the machine and contain

a number of rows corresponding to the arguments X. - a maximum of 200

values. The format for the input is illustrated through typical examples. These are given in Tables 4 et. seq.

b. Control Matrices

After the preliminary input, a single mnemonic instruction 1 GENOaa,

instructs the machine to generate a a functions , in accordance with the

a a lines of instruction immediately following. Here a a denotes a two

digit integer. The lines of instructions may be of variable length.

11
Each line starts out with the six integers k, a, n, t, Q, c which were

defined earlier. These integers are followed by either 2n, or 4j1) nujnbers depending on the functions selected. The control matrix specifies

"^t the calculations in terms of the variable X. or its transforms p.X. + & ,

.

It is followed by a TRA 2,4..

In order to compute functions of functions, it is necessary to enter

the Function Generator a second time. This is accomplished by another

BCD IGENOaa instruction followed by appropriate lines of calculations.

In the second pass through the generator, the transformation integer t

denotes which column replaces the variable in the Operator Equation. It

should be noted that in the trigonometric functions f^ through f, -^ the

transformation applies only to the X in sin X, cos X, or tan X, and not

to the factor in front of the trigonometric function. Thus in one line

of instruction, it is possible to compute either

Jl c . r- Yh -- " c . _ -,d

,

Y^ (a^./b^.)X^^'[sin (p^X^^ + 6^)] or £

{a./h.)xjs±n [f(X. J

c . Output Instructions

The remaining two branches are the temporary and final output

branches. These may be entered after computations have been completed

in a pass through the Generator branch. The temporary output branch

must be entered at least once during the entire computation. It is

entered via the instruction BCD l^TyOn^. The word BCD IptJT selects

^ In preparing instructions for the key punch operators the symbol ^ is

used for the letter "0" to distinguish it from the number zero, which is

written simply 0.

12
Table 2. Typical OMNIFORM Printout in Fixed Decimal

OMNIFORM I TE ST .P_F__.eL£^ME_NTARy „F U.N£:jJ_ON^_. _ _.MAXJL5_6_1_

X LOGX LNX SINHX COSHX TANH

.1000

.2000

-1.000000

-0.698970

-2

-1 ,302585

,609438

,100167

,201336

,005004

,020067

,099668

,197375

• 3000

,4000

.5000

.6000

-0.522879

-0.397940

-0.301030

-0.221849

-1-0

-0

-0

-0-0

,203973

.916291

,693147

.510826

.356675

,223144

,304520

.410752

.521095

.636654

.758584 ' " .888106

,045338

.081072

.127626

,185465

d,

0,

.291313

.379949

.462117

.537050

,7000

.8000

-0,154902

-0.096910

,255169

,337435

,604368

,664037

1.,9000

.0000

-0.045758

-0,000000

-0

-o<.105361.000000

11..026517

.175201

,433086

,543081

0,0, 716298

,761594

.1000 0,041393 0..095310 1<.335647 ,668518 0, 800499

,2000

.3000

0.079181

0,113943

0.0..182322

.262364

1.

1 .509461

.698382

,810655

,970914

0.

0,,833655

,861723

1 .4000

.5000

0,146128

0.176091

0.

0. 336472

.405465

1<2.,904301

,129279

2,

2, 150898

352409

0,

0.,885352

905148

1 .6000

.7000

0,204120

0,230449

0..470004

.530628

2.

2 ,375568

.645632

2,

2,,577464

,828315

0,

0,,921669

,935409

11 .8000

.9000

0,255272

0,278754 0..587787

.641854

2<

3,.942174

,268162

3,

3,,107473

,417731 0,,946806

,956237

2 .0000 0.301030 0..693147 3 ,626860 3, 762195 0,,964028

2.

2i,1000

2000

0,322219

0,342423

0,

0, 741937

788457

4,

4<,021856

,457105

4,

4, 144312

567908

0,

0, 970452

,975743

2

2 .3000

.4000

0,361728

0,380211

0,

Oi 832909

,875469

4i

5. 936961466228

5,5,,037220

556946

0,

0,,980096

,983675

2

2..5000

.6000

0,397940

0,414973

0.0, 916291

955511

6,

6, 050203694731

6,

6, 132288

769004

0,0, 986614

989027

2

2,,7000.8000

0.431364

0,447158

0,

1

1

,993252

029619

7,

8, 406262,191917

7,

8, 473467

252726

0,

0, 991007992632

2

3,.9000

.0000

0,462398

0.477121

1

1

1

1

064711

098612

9, 10. 059559

017873

9, 10,114583

067659

0.

0.,993963,995055

3,.1000 0.491362 1 i,131402 11.,076448 11, 121498 0, 995949

3

3,.2000.3000

0,505150

0,518514

1 I 1 ^,163151

,193922

12, 13.,245881

537874

12,

13,

286643

,574757

0,

0, 996682

997283

3,

3..4000

.5000

0,531479

0,544068 1 4 223775

252763

14,

16. 965359,542623

14,

16. 998733572820

0,

0, 997775

998178

3,

3, ^000

'^000

0.!'t>63o2 0,5682o2 1

1

280934

3o8333

18. 20. 285451

211285

18, 20< 312774

236008

0<

0" 998508

998778

33,>8000

•9000

0,579784

0.591o65

1 i 1 1 335ool

36o976

22.

24,

339401

691096

22,

24,

361771

711338

0«

0< 999000

999181

<fi 0000 0.6o2o6o 1

1

386294 27. 28W09 27. 308224 0« 999329

4,

4,.1000

2000

0.612784

0,623249

I i I I 410987

435084

30

33,

,161847

335655

30,

33,

178420

350651

0.

0, 999451

999550

4.

4,.3000

.4000

0,633468

0,643453

I 4 I

I

458615

,481604

36,

40,

843099

,719278

36,

40,

856667

731554

0,

0, 999632

999699

4

4 .5000

.6 000

0.653212

0,662758

1 t 504077

526056

45,

49,

,002992

737105

45,

49,

014098

747157

0,0, 999753

999798

4

4 .7000

.8000

0,672098

0.681241

1 i 1 4 547562

568616

54, 60.,969005

,751055

54,

60,

978100

759287

0,

0. 999835

999865

4

5,.9000,0000

0.690196

0,698970

1 4 1 4 589235

609438

67,

74,

141121

203163

67,

74,

148569

209900

0,

0, 999889999909

Note Above X = 2. the values for sinh X and cosh X are not correct to

six decimals though they are correct to within one unit in thesixth significant digit.

13
the temporary output branch and is used to save all or part of the

results computed up to the time this branch is entered. The y stands

for either an F for floating point or a D for fixed point and n refers

to the number .of columns of printout desired per "page" (l < n < 7) .

Table 2 shows a typical fixed point printout and Table 3 shows a floating

point printout format. If the total number of designated columns is not

an even multiple of n, the last page of the printout for that group will

contain the remaining n' columns (n' < n)

.

Following the "J^fUT" ijistruction is the column selector card. This

is a data card containing integers, separated by commas, representing the

indices of and order in which the columns are to be printed. This enables

the user not only to rearrange the computed functions, but to print only

selected ones and to repeat columns if desired.

If all of the computed functions are desired in the printout, in

the order in which they are computed, the column selector card merely

contains the single integer, zero (written as DEC O)

.

For example, suppose a user computes 14 functions (k = 1, 2, 3,

. . ., 14,) but only wants eleven of them printed. Furthermore, he would

like to rearrange the columns in a more suitable order for a two page

printout in the floating point mode, with six columns on one page and

the remaining five on the second. The instructions. would be as follows:

BCD 10UTFO6

DEC 5, 4, 1, 6, 7, 8, 13, 10, 11, 2, U

Note that columais 3, 9, and 12 are not requested. Furthermore,

column 5 will be printed first, column 4- second, column 1 third, etc.

14
Table 3. Typical OMNIFOM I Printout in the Floating Point

TEMP. H c s F 740.0000 »405223E 04^ 0.]L08942E 02 0.,801557E 01 0.251957E Ol

750.0000 .416139E 04 O.JL09386E 02 0..816209E 01 ,259351E 01

760.0000 .427099E 0^ 0..L09818E 02 0.830726E 01 0<,266742E Ol

770.0000 .438101E 04 0.:L10241E 02 0.845108E 01 0<,274129E Ol

780.0000 ,449146E 04 0..L10653E 02 .859359E 01 281510E Ol

790.0000 ,460231E 04 O.JL11056E 02 0.,873481E 01 0.288884E Ol

800.0000 .471356E 04 0.]LH450E 02 0.887474E 01 0.296249E Ol

810.0000 ,482520E 04 O.JL11835E 02 0-,901343E 01 .303605E Ol

820.0000 ,493722E 04 0.:L12212E 02 0.,915088E 01 0.,310950E 01

830.0000 0<•504961E 04 0.]L12581E 02 0<.928711E 01 0.318283E Ol

840.0000 0<,516237E 04 0.]L12942E 02 0,942215E 01 0.,325602E ol

850»0000 527548E 04 0.:L13296E 02 0-955602E 01 0<»332908E ol

860.0000 ,538895E 04 O.JL13643E 02 ,968873E 01 0<.340199E ol

870.0000 550276E 04 O.JL13983E 02 0<982030E 01 347474E ol

880.0000 0(,56169lE 04 0.3L14317E 02 0.995076E 01 0.,354733E ol

890.0000 0<573138E 04 0.]L14645E 02 0<.lOOBolE 02 0.,361974E ol

900.0000 0.,584619E 04 0.^L14966E 02 0,,102084E 02 0.,369198E ol

910.0000 0.596131E 04 0.]L15282E 02 0.,103356E 02 0-,376404E 01

920.0000 0.,607674E 04 0.]L15593E 02 0.,104617E 02 0.,383591E ol

930.0000 0*.619248E 04 0.]L15898E 02 0<>105869E 02 0.,390758E ol

940.0000 0«»630852E 04 0.]L16198E 02 0*107110E 02 0*397906E ol

950.0000 0«>642487E 04 0.]L16493E 02 0.,108341E 02 0*.405033E ol

960.0000 Oi,654150E 04 0.]L16784E 02 0,,109562e 02 0.,412139e ol

970.0000 0,,665843E 04 0.]L17070E 02 0,110774E 02 0«,419225E ol

980.0000 0,,677563E 04 0.]L17351E 02 0.,111976E 02 0.,426289E ol

990.0000 0..689312E 04 0.]L17629E 02 0.,113169E 02 0«,433331E 01

1000.0000 0.,701088E 04 0.]L17902E 02 0,,114352E 02 0.,44o351E 01

1010.0000 0,,712891E 04 0.]l18172E 02 0.115527E 02 0.,447349E 01

1020.0000 0,.724721E 04 0.]L18437E 02 0,.116692E 02 0.,454324E 01

1030.0000 0*736578E 04 0.]L18699E 02 0,,117849E 02 0.461277E ol

1040.0000 0.,748460E 04 0.]L18958E 02 0, 118997E 02 0«,468207E ol

1050*0000 0<.760369E 04 0.]L19213E 02 c..120137E 02 0.,475114E ol

1060.0000 0,772302E 04 0.].19465E 02 0,,121268E 02 0.,481998E ol

1070.0000 Oi784261E 04 0.]L19714E 02 0,,122391E 02 0.»488859E 01

1080.0000 0,,796244E 04 0.]L19960E 02 0.,123605E 02 0.495695E ol

1090.0000 0,,808251E 04 0.]L20202E 02 0,,124612E 02 0.502509E ol

llOO.OOOO 0«.820283E 04 0.]L20442E 02 0«,125711E 02 0.509299E ci

1110.0000 0.,832339E 04 0.]L20679E 02 0*>1268o2E 02 0.,516065E ol

1120.0000 0.,844418E 04 0.]L20914E 02 0,,127885E 02 0.,522807E ol

1130.0000 04856521E 04 0.]L21146E 02 Oi128961E 02 0.529526E ol

1140.0000 Oi868647E 04 0.]L21375E 02 0.a30029E 02 Oi,536221E ol

1150.0000 0,.880796E 04 0.]L21602E 02 0,.131091E 02 Oi542892E 01

1160,0000 0<,892966E 04 0.]L21827E 02 0,,132144E 02 0.,549539E ol

1170.0000 0< 905160E 04 0.]L22049E 02 0.,133191E 02 Oi,556162E ol

1180.0000 0.917375E 04 0.]L22270E 02 0*,134231E 02 Oi,562762E ol

1190.0000 0<,929612E 04 0.]L22488E 02 0,,135263E 02 0,,569338E ol

1200.0000 0.,941872E 04 0.3L22704E 02 0,»136289E 02 Oi»575889E ol

0.405223E 04 0.405223*10^ = 4052.23

15
Columns 5? Iv, 1) 6, 7, and 8 would be printed on the first page with

columns 13, 10, 11, 2, and I4. on the second page.

If the user had decided to print all I4. columns in the order in

which they were computed, he would merely follow the "^UT" instruction by

a DEC 0.

It should be noted at this point that "page" as referred to above

means a physical page only when there are 50 or less arguments. Since

OMNIFORM I has provisions for 200 arguments, it will take up to four

pages of 704.-7090 printout to complete a "page".

The final output branch can be entered only once during any one

complete run. That is, the final output branch indicates to the machine

that the user is finished with his computation and is now ready to have

the results processed and printed. This branch is entered via a

^CD lENDOOO instruction. Its primary function is to process all information which was placed in temporary storage on the magnetic tape by one or

more "0'UT" statements.

Following the "END" instruction are headings for the columns of

information to be printed. There must be one heading instruction for

every "page" of printout. Headings begin at the extreme left side of

the form and may be from one to six characters in length. Each column

heading is separated by a comma. The only two characters restricted from

use in headings are the comma and blank. A heading must also be provided

for the argument.

To continue with the example previously used, the "END" instruction

would be followed by X, COL$, COL4, COLl, C0L6, C0L7, COLS, C0L13, COLIO,

16

1
COLll, C0L2, COLI4. or other headings suitable to the particular problem.

Since the program expects a heading instruction for each "page", a

blank card will give a "page" without headings. It is clear that the

order and distribution of headings must coincide with the instructions

in the several ptJT statements . OMNIFORM is so written that when all of

the results on tape have been printed, the machine is ready for another

OMNIFORM run.

2.4. Print Format

AS indicated above, the printout of OMNIFORM can be arranged for a

variable number (1-7) of columns of entries plus one column of arguments.

The present version provides six decimals in the fixed decimal mode and

six significant figures in the floating point mode. In either case,

however, the results are good to at most seven significant figures. The

results are printed 50 rows to the page in 5 blocks of 10 numbers each

with a single space separation between blocks (See Tables 2 and 3)

Since the printout is selective and can be rearranged at will, it

is possible to repeat columns on succeeding pages for ease of interpolation or for other reasons. The program automatically prints the argument

as the first column and repeats it on succeeding "pages". It is also

possible to repeat the argument on the right side of the page. In order

to accomplish this, it is necessary to generate the arguments in the

Function Generator by asking for an identity calculation (l«X ) and

storing it in a blank column.

It will be observed that the printout in Table 2 is suitable for

direct publication. Had some of the functions been of such size as to

17
require a floating point mode, they could be arranged to appear on a

second "page" by repeating the 0UT instruction with appropriate information. Thus, if in addition to the functions shown in Table 2, there

were two other functions in the memory which required floating point, the

print instructions would be as follows

BCD l$2fUTD04.

DEC

TRA 2,4.

BCD l{2fUTF02

DEC 5,^

TRA 2,^

Figures 1 and 2 show the front and back respectively of a form

which has been found useful in preparing the input instructions. The

form is especially useful for beginners to insure proper order and form

of the instructions. Later one may be able to dispense with its use and

prepare data on a form as is shown in Table 4-? o^ indeed, in a blank

piece of paper.

18
Table 4, Sample Input for OMNIFORM I

-ROBLt>/l

'.OC OP

I

'.DDWFSS, TAG, DEC JrVit-N!"

C.^MJ^lF^^/m^T^^£^/iAlA^^A.l^

I M/c "FfZ^^cr^C^s^i^pj^^. MH I

'T^MF.^

V^C

F

DacL

i)gC

z2eA

3cj>

VEC

JllE^

^j£^

TjEA.

{fhCO

T)EC

TjEA.

I^CJ>

_3,_o^o

.^^^^<^^S1^STfH).

j--^^^^

lCr£f\/00'iLs-f^.j /y ^^^>- Aj-^i-^yO

^

i-,/f^ /y <^/^,yr^ -AZX^^-y ^,

v-^

l purl>o^

^-V^^

^A^

lLBfJl>06c>

-E/)^7^ /^-^//fr^ s/^^ e//p

1 GMNrrOfWI'lDEAL GAS'TFIFRMDD?MARIC FUNCTION FOR H + DEC 3«0»0

DEC 50.»50.,5000.

TRA 2»4

BCD 1GEN004 "

'

' ' DEC l.»3. *1. »0»»0« t l.»2. 5»0«

DEC 2. »1.»1. jO. »1. »1. -3.65354»0.

DEC 3. »1. !. 0.»0. »_3_. »_2_._5»_0. _ _j

" D EC^ 4 . n r» r."» ov» I . » i •» - 1 . 1

5"3 5 4 , o .

" ___jrRA 7^1^ .__ "BCD' 10UTD04 " i DEC 2 , 3*4^3

n^AT^T? ~

BCD IFNDOOO^

^T EM'P'/.T'- E7"RTrH-E7 RT,S/R»C/R

19
3. PROPERTIES OF THE FUNCTION GENERATOR

3»1 Elementary Functions

The properties and limitations, if any, of the functions available

in OMNIFORM I arfe discussed briefly in this section. Limitations result

either from the mathematical properties of the functions or from the

particular sub-routines which are used for their evaluation. Space does

not permit a full discussion of the sub-routines used in OMNIFORM I,

However, a brief review of the computational logic and the capabilities

or limitations of the major sub-routines is important to insure proper

ase of the program and to establish confidence in the results.

OMNIFORM I is designed to substitute real numbers in a real function

given explicitly in terms of a real independent variable. Thus operations

leading to imaginary or complex numbers are ruled out as is also any iterative operation.

In functions of type 1 and indeed in many of the others, a provision

is made for terms in arbitrary powers of the variable. No problem exists

when the exponents b . are integers , since the functions are then defined

for all real values of X. Since OMNIFORM I performs calculations on real

b.

numbers only, care must be taken not to require X "^ for negative values

of the variable when b. is nonintegral.

The program is written to examine the argument for its sign. When

the argument is positive, X is evaluated through a logarithmic routine

by evaluating the antilog of blogX. When the argument is found to be

negative, only integral exponents are allowed, and results are obtained

by successive multiplications. To avoid a program stop when a negative

20
number is raised to a nonintegral exponent inadvertently, the program

truncates the exponent for negative arguments, thereby ignoring anything

to the right of the decimal point. Thus (^2) is +4. and (-2) *^ is

taken as +4.. While the latter answer is obviously wrong, the user

should be aware of such occurrences, and should arrange for a listing out

of intermediate results if there is a chance that certain arguments or

functions might give complex or imaginary values.

The logarithmic functions are obviously defined for positive nonzero numbers. When a logarithm of a negative number is called for, the

result is automatically set equal to zero. Here again the user must

exercise caution.

In the trigonometric functions, the angle X is considered to be

given in radians. If evaluation is desired for angles in degrees, it is

possible to use the transformation feature in functions 5, V, and 9, or

by setting the coefficient c. = 57.29577 in functions 6, 8 and 10. In

J all of the trigonometric functions, the transformation provision applies

c. b.

only to the variable inside the bracket and not to the factor X "^ or X

'^

in front. The program gives the proper sign to the functions. Obviously

the tangent function is undefined for angles of -tt/2 and +tt/2.

The inverse trigonometric functions provide the principal values - -

in the case of the arc sin from to ±rT/2, and from -Tr/2 to +tt/2 for the

arc tangent. It will be observed that by putting a. = in functions I4.

and 15 gives arc sin and arc tan, respectively. If, however, a. = 1 and

b. = -1, these functions yield arc cos and arc cot, respectively, since

arc cos = tt/2 - arc cos 9 and arc cot G = Tr/2 - arc tan 9.

21
3*2 Special Functions

Among the special functions incorporated in GMNIFORM I are

the Chebyshev polynomia^^

T (x) = cos (n cos x) " (for -1 < X < 1 and n = 0, 1, 2, 3, ... .) ^J^) = [T'^-L(x)]/n+ 1,

2

the Legendre polynomials

^n^""^ " 2^nrr ~d^ ^"^ ~ ^^ ' 3 the Laguerre polynomials

\i^) ^ ®^

dx^ t^^ ® ^] ?

the Hermite polynomials

Hjx) = (-l)^e^ f^ (e-^ ).

The above listed special functions are computed from initial values and

the recursion formulas given in Table 5. The subroutines for the calculation of these functions were written especially for OMNIFORM I and

have been described in the Communications of the Association for Computing

5 Machinery.

1. Anonymous. "Tables of Chebyshev Polynomials Sn(x) and Cn(x)",

National Bureau of Standards, Applied Mathematics Series 9. Government

Printing Office, Washington, D. C. (1952).

2. Kopal, Z. "Numerical Analysis", p. 368, John Wiley and Sons, New

York (1955)

.

3. ibid, p. 370.

4-. ibid, p. 371.

5. Gerald M. Galler, Communications ACM, ^, 353 (i960).

22
Table 5. Initial Values and Recursion Formulas

for Certain Special Functions

Symbol n = n = 1 n = 2 n = 3 Recursion Formula

T^(x) 1 X 2x^-1 4x^-3x Vlf'^^ = 2-Tjx)-T^_^(x)

U^(x) 1 2x 4x -1 8x^-4x Vl(^) = 2-Un(^)-Vl'=^)

P,(x) 1 X 2^ 2 2^ 2^

H„(x) 1 2x 4x^-2 8x^-12x Hj^j^(x) = 2xH^(x)-2iiH^_^{x)

L„(x) 1 -x+1 x^--4x:+2 -x^+9x^-l8x+6 L^j^(x) = (l+2ii-x)L^(x)

-n\_j^(x)

23
other special functions include

the error function

erf(x) =^

,x ^2

e at,

and the gamma function

r(n) = x"" e ^dx (for n > 0).

^0

It should be observed that for positive integers

r(n) = (n - l)f

and that this function can^therefore^be used to generate factorials of

positive integers. It should be noted that the gamma function is

infinite for negative integers.

3.3 Column Sums

The program automatically sums each column in the memory and prints

this sum automatically as an additional line of output. The column sums

are not shown in the sample outputs since they appeared on the following

page when more than 50 lines are printed.

4.. ACKNOWLEDGMENTS

The authors wish to acknowledge the help received from a number of

their colleagues. We are indebted to Dr. Harold W. Woolley, Mr. Max

Klein, and Mr. Joseph H. Wegstein for various helpful suggestions and

stimulating general discussions; to Mr. William T. Chen, Mrs. Esther C.

Cassidy, and Miss Carla G. Harms for their patience in learning and

relearning to use the program during the period when it was under

continual change ; to a larger group of colleagues for their patience

during the birth pangs of OMNIFORM I; and to Mrs. Eleanor L. Rozsics for

24.
the preparation of the typescript. We are indebted also to Dr. Max

Goldstein of the Institute of Mathematical Sciences, New York University,

for making available to us an improved version of his Bessel Function

Subroutine, and to Mr, John W. Cooper for suggestions resulting in an

improved notation.

I

25
Problem OMNIFORM I Sponsor Date

1

1 6 7 8 10 U

1 72

'

i^0miF9!^h'' i. DEC

JDEC :

1 ,DEC

1 JDEC

i JDEC

JTRA ; 2, A

1

'

i

1 Insert data from pagfi

1

1 TRA i 2, A

1

1 BCD IGENO. .

! k,a qD^tq^^qC ,

1

1 DEC 1 DEC ! DEC

DEC DEC

DEC

DEC '?*!•?

DEC DEC * ) 5 5 ) J

j DEC

i

DEC

1 ? ' ' ^ ? • DEC DEC

DEC

DEC

DEC DEC

TRA 2. I

- BCD_ DEC

10tlT .

TRA 2, A

BCD lENDOOO

Figiire 1. OMNIFORM I Input Form (Front)

26
OMNIFORM I

"~I—

i

6 7

J BCD I

I DEC

DEC

72 IGMCL

DEC

DEC

DEC DEC

DEC

iDEC

I DEC

iDEC

OMNIFORM I : FUNCTION GENERATOR

^1 = n b. Sa,X^ f" =

11 2 a ("sinh x] J j=l > "- -^ ^2i=2-j^(^-)[^(=/).f^ -Gjomia

^3 =

n b. S a.X 'logX

j=l J

b 2 a.X •'lux J=l •

^12 =

^13 =

n _ 2 a. cosh

j=l •> ^

n2 a. tanh

j=l ' ^

n fpp = S a erf (X) j=l •' Error

OMNIFORM I

^1

=

n b. d S a X -"exp o.X >

_j=l J L J J

n . d

.

2(a_j/b.)X J [slnX] •" ^15 =

n2 a .Ti/2 + b . arc sin X

j=l •' ^ n2 a Ti/2 + b. arc tan X

j=l ^ ^

Operator Equation

F^(X,) = Gj^(X.) + f^ [^,(X.); Q H^(X^)

^6 =

n b d. S a.X J sin c.X ^ ^16 S a T (X) Chebyshev

'l-

" d. 2 (a /b )X J cos X ^ j=l J J L J f" = f_,^W^ Chebyshev

^8 = 2 a .X J foos c . x] ^ ^18

- £VV^) Legendre

^9

=

n c H. S (a./b.)X •' tan X ^ j=l J •) -

-J f" = ^9 l^^n/^)

Laguerre

^10=

n b d, 2 a.X J tan c.X f" = "^20 £Vn/^) Hennite

USCOMM-NBS-DC

Figure 2 OMNIFORM I Input Form (Back)

27
Problem 1. Prepare a table of values V = 1226. 877/0. 001316PF for values of F supplied in a one column table corresponding to 33 values of pressure P (given in millimeters of mercury) , which

are taken as the arguments.

M

1 SPECIFIC

33 .LI

50.75»66.5

109.8.114.

153.3.162.

230.4.244.

350.4.399.

2.4 1.008

1.011 1.012

1.013

1.015

1.017

1.017

1.019

1.02

1.02 1.022

1,023

1.027

1.028

1.032 1.034

1.037

1.04

1.043

1.049

VOL. SB 26 .74.4.79.2.91.8.98.2.101.8

55.117.9.125.05.132.75,144.2 __ ._ 7. 172. 2. 175. 8. 185. .190. 7. 207.

05. 257. 1.2 75. 3. 305. 15. 33_Q. 5, 25.421.35.455.1.474.05^498.55.522.3

DEC 1.064

DEC 1.^4,

DEC 1.086

DEC 1.104 . DEC 1.146 v.— c (c^,)

DEC 1,18_ DEC i.223

DEC 1.356

DEC 1.431 DEC 1.54

DEC 1.619

DEC 1.718

DEC 1.832

BCD lGENq02

d->DEC 2.. 1..1..0..0. .2. •.001316.1.

e-»DEC 3..1..1.t0.i0<>*3.tl226.877.0*

TRA 2.4

-f-»BCO 1GEN002

?-^DEC 4..i..l.*l.>3..2..1..1.

-iDEC 5..1..1.»'*.»^_»»3..1..1. TRA 2.4

i —>.BCO 1OUTD03

j -^ DEC 1.2.5

TRA 2.4 k -^ BCD lENOOOO

P»MM».F.P»ATM*»V

Remarks

Card a.

Cards b. Cards c.

Card d. Card e. Card f.

Card g. Card h. Card i. Card j. Card k.

Card 1.

indicates that there are 33 numbers in the argument instructions following, that a one column table will be read in, and that the 33 numbers are to be taken as written for the arguments

.

list the arguments for which the calculations are to be made. comprise the table of F values read in corresponding to the 33 pressures P which comprise

the arguments.

computes 0.001316P which converts the pressures to atmospheres,

simply sets up a constant, 1226.877 in column 3. instructs the machine to reenter the function generator. Now the transformation para- meter (the fourth number in each line of instruction) denotes the column which replaces

the variable. multiplies column 1 by column 2 and stores in column 4. (PF) . divides column 4. into column 3 and stores in column 5 (1226.877/PF) . asks for a dumps out of 3 columns in fixed decimal form, asks for coltimns 1, 2, and 5 to be printed. indicates the end of the computation at which point the machine is ready for the headings

and the final printout.

provides the column headings for the results.

Note that four headings are supplied, the fir-^t being for the argument column.

28
Problem 1 (Re suits)

OMNIF-ORM 1 "SPECIFIC VOL. S 3 26

p *Mm¥" F p*ATM* V

50<.7500 1.008000 0.066787

0. 08 7 5 14

18224.204102

66..5000 1.011000 ly666. 676392

'

74,.4000 1.012000 0.097910 12382.025879

79..2000 1.013000 0.104227 11626". 1 18T64 "

91..8000 1.015000 0.120809 10005.446289

98,.2000 1.017000 ~0Vl'29r2"3X~" "9 334.-^65 942

101..8000 1.017000 0.133969 9004.849121

109..8000 1.019000 ' 0."1444 9

7

8 332.3'72681

114..5500 1.020000 0.150748 7979.027344

117.,9000 1.020000 0.155156 -' 7752.312012-"

125,,0500'" ir,oz2~oo'^o'""' orre'i^^e'S' ~ ^ "'"T294."75~28U8""

132,.7500 1.023000 0.174699 6564.912537

144,.2000

~ i.-TyzTmyo--- O.TBS-fST-' "6-29^5 .1^^914 - 153,.3000 1.028000 0.201743 5915.751282

162..7000^ i.o32odo '" o^mu^ "55"52T36"4075

'" 172..2000 1.034000 0.226615 5235.902100

175,.8 00"^0 "T. 37000"" """

b'."2~3T353 5ir3"".~B"449T.0"~"

185.»0000 1.040000 0.243460 4845.517029

190..7000 1.043000 " 0."25 09 61 4 687 .16 4-5 5T - 2o7< 0000

.4000

1.049000 0.272412 4293.380127

230. 1.064000 0.303206 3802.954102

244,,05 00 1.074000 "tr. 321170 " -355IJ-.8ZITI6

- 257..1000 1 .086000 0.338344 3338.975555

275..3000 r.T.04000 -Q-,3B-2295--- - -3ij6-r.-3'96-o"5-r -

- 305..1500 1.146000 0.401577 2665.920471

330..5000 "1 ."lt3tX000"

" ^ - 0.43493"8" 2390. 516327

350..4000 1.223000 0.461126 2175.477570

399,.25 00 1.356000 ~ D. 5-25-41 3 -lT22-.-D29^4-r--

421 .3500 1.431000 0.554497 1546.188858

455,.1000

.0500

1.540000 ~

" "1.6190D~0" "

_.._.. -Q,i,9y9j-i --- "ir3TJ'."2"0Tr94---

"474, "" "0.S23850 - 1214. 714447

498 .5500 1.718000 0.656092 1088.462128

522..3000 - " 1.^32000 -- 1T.6-ST347 - - - - -9-74-.-Ti5-r2-"r -

29
Problem 2. Prepare a table of the form

1 1 .67^3 .67A5 .8^53 .84-33

-Tn N/n(n-l) N/Tn-l) N/n(n-l) n -Vn-l '/n(n-l)

for integral values of X = l(l)lOO.

Omniform Instructions

1 OMNIFORM I FACTORS FOR PROBABLE ERROR CALC,

a. DEC 3»0,0

b. DEC l.»l.»100.

c. DEC l.»l. »"1.

TRA 2»4

BCD 1GEN007

(^DEC 1. »1. »1. >0**0. )1. »1. t-0.5

) DEC 2»»1*»1«)1*»0* »2. »0.67^5»-0.5

d./ DEC 3«»1«»1«»1«»3» )1. »1. .-O.S

/ DEC 4.»1.»1. .0.»3. 2. »1. -0.5

/ DEC 3*>1*)1*>1«)0« »5# >0«8453»-0.5

I DEC 6«»1*»1«)0*(3* »5. »1. »-l.

V DEC 7«»1«»1«»0«»3# »5, »1. »-0.5

TRA 2»^

BCD 10UTD06

DEC 1»3»2»4,6»7

TRA 2»4

BCD lENDOOO

X,l»2»3»^»5»6

Remarks

a. This line indicates that there are 3 numbers on the following card

defining the arguments 5 that zero columns are being read in; and

that the arguments are given in the incremented mode.

b. This line sets up the arguments 1 to 100 in steps of unity.

G, This line sets up the transformation (n-l).

Line 1 computes l/>/n.

Line 2 computes .674.5/7 (n-l)

,

Line 3 computes 1/7 (n-l) and multiplies it by l/'^n

yielding >- ,—

. "vn N/n-l

Line 4. computes l/s/n and multiplies It by .674.5/\/Tn^

Line 5 computes .84.53/^/(n-l)

Line 6 computes l/n and multiplies It by .8453/7 (n-l)

Line 7 computes I/n/t and multiplies It by .84.53/s/(n-l)

30
Problem 2 (Results)

OMNIFORM 1 FACTORS FOR PROBABLE ERROR CALC.

X

12

34

56

7

8

9 lO

1112

13

14

15

16

17181920

21

2223

24

25

26

27

2829

30

3132

3334

3536

37

38

3940

4142

4344

45

4647

484950

0000

0000 0000

0000 0000

0000

0000 0000

0000

0000 0000

0000

0000 0000

0000 0000

0000

0000 0000

0000

0000 0000

0000

ooon 0000

0000 0000

0000

0000

0000

0000

0000 0000

ooon

0000

0000

0000

0000

0000 0000

0000

0000

0000 0000

0000

ooon 0000

0000

0000

0000

1 1.000 0.707 0.577

0.500

0.447

0.408

0.377

0.353 0.333

0.316

00034217727

107 350 n 000

214 248

964

553

333 228 0.301511

0.288675 0.277350

0.267261 0.258199

0.250000

0.242536

0.235702

0.229416

0.223607

0.218218 0.213201

0.208514 0.204124

0.200000 0.196116

0,192450

0.188982

0.185695

0.182574 0.179605

0.176777

0.174078

0.171499

0.169031

0.166667

0.164399

0.162221 0.160128

0.158114

0.156174, 0.154303 0.152499

0.150756

0.149071

0.147442 0.145865

0.144338 0.142857

0.141421

00000090529856

707107 408248

288675

223607

182574

154303

133631

117851 lo54o9 095346

087039

080064

074125 069007

064550 060634

057166

054074

051299

048 79_5. 046524

044455 042563

040825

039223

037743^ 036370

035093

033903

032791 031750

030773 029854

028989

028172

027400

0266^9 025976

025 3L8_ 024693

024098

023531 022990 022473

021979

021507 021054

020620

020203

00000090529856

674500

476944

389423 337250

301646 275363

254937

238472 224833

213296

203369

194711 187073

180268 174155

168625

163590

158981 154741

J50823

147188 143804

140643

137682 134900

13228_0 1298C8

127469

125252

123146 121144

119236

117415 115676

114011 112417

110887 109418

108006

106648 105339

104078 102860

101685 100549 099450

098386

097356 096357

.ill

a

476944

275363

194711

150823

123146

104078 090134

079491 071099

064311

058708

054003

049997

046545 043539

040898

038559 036473

034601

03.2512. 031381

029985 028709

027536

026456

025457

024531

023670

022868 022118

021415 020756

020]37 019553

019002

018481

0.17 988. 017521

017077

016656 016254

015872

015507

015158

014825

014506

014201 013908

013627

5 6 34=;4243. 00 00013454244.000000

422650 .^97717

199239 345092

0. 122009 244017

084530 189015 063005 154330

049299 130433 039937 112958 033207 099620

0. 028177 089102 02430Y" 0^ 080596

021239 073574

018771 067678

016746 062658

015061 058331

013641 U54564

012431 U51254

011390 .048323 01U486 U45709

009696 . 43 363

_

^ ao9i_oi_^_ 0083850..W41746 039327

007836 J3757a_

0. 007344 035978

006902

006502

. _Q .0 34 5_05_ 033155

0. 006J40 ^^ U31904

005810 030743

0055U9 29664

005232 -^28658 004978 027718

004744

004528

>J2i>a3.fi_ 026012

004328 025236 . 004142 024504

003969 ? ? 8 1 4__ 003808 023161

u 3_6 57 0. 022543

003516 .021958 003384 .021402

u 003260 .020873 003143 .020370 003033 002930

^19891 019433

002832 U18997

002739 .018579

.002652 .018180

.002569 .U17797

.002490 0 017430

002415 .01707fe

This line is the result of trying to divide by zero.

31
Problem 3. Prepare a 'table of the form

X X^ -/x VlOX X^ ^n/x

for integral values of X = l( 1)200.

Omnlform Instractlons

1 OMNIFORM I SQUARES»CUBES»AND ROOTS

DEC 3»0»0

a. DEC l.»l.»200*

b. DEC 10. !. 0. 100. !. »0,

TRA 2»^

BCD 16EN008

lOX

DEC 1. »1. I. jO. tO. «1.

DEC 2. »1. >1. »0.»3. »1.

DEC 3. !. >!• »0.»3« 2.

DEC ^. !. »1. »0. jO. »4.

DEC 5. I. »1. »1. »0. ,5.

DEC 6. »1. >1. *0, >0. »6«

DEC 7. >1. >1. !« *0» »7.

DEC 8. !. »1. 2.»0. jB.

TRA 2.4

d. BCD 10UTF07

DEC 2»4»5»3»6»7»8

TRA 2»4

BCD lENDOOO

N»2»l/2»l/2»3»l/3>l/3»l/3

VlOOX

• >1*

.*!.

• » • 5

• »0.5

. 0.3333333

. 0.3333333

. 0.3333333

Remarks

a. This line seta up the arguments 1. to 200. in unit steps.

b. This line provides for two transformations lOX and lOOX.

Line 1 simply computes f(X) = X and stores it in column 1.

Line 2 also computes f (X) = X but multiplies it by the value

in column 1 (yielding X^) and stores it in col-umn 2.

Line 3 similarly computes X-^ and stores it in column 3»

Line 4 computes vX and stores it in column 4-»

Line 5 computes vlOX and store.s it in column 5«

Line 6 computes vX and stores it in column 6.

3 I Line 7 computes -VIOX and stores it in columji 7,

.Line 8 computes vlOOX and stores it in column 8.

d. This line asks for a floating point print out, .seven columns to

the page.

32
Problem 3 (Results)

OMNIFORM I SQUARES. CUBES. AND ROOTS

N

123

5

.0000

.0000 .0000

.0000 0000

0000

.0000 0000

0000

.0000

0000

0000

0000 0000

.0000 0000

0000

0000

0000

.0000

2 .lOOOOOE 400000E

900000E

.160000E 250000E

0101

01

02

n?

1/2

0. lOOOOOE 0.141421E

0.173205E 0.200000E

Q_,223607E

01

01

01

01m n

1/2 316228E

447214E

547723E

632456E

7071n7E

01

01

01

01

01 n

3 .lOOOOOE 800000E

270000E

.640000E

il25llDa£

01

0102

02Qi

n

n

n

n

1/3 .lOOODOE 125992E

.144225E

.158740E

01

0101

01

01

1/3 215443E_ 01 271442E 01 310723E 01 341995E 01 36R403F 01

6

7

89

1

.360000E 490000E

.640000E 810000E OOOOOOE

121000E

I'^'^OOOE Ife'OOOE 196000E

.225000E 256000E

02

0202

02

02

0^ ^

0^

03

03

0303

244949E

264575E 282843E

300000E

316228E

ol01

01

01

01 1

774597E

836660E 894427E 948683E

OOOOOOE

01

01

0101

01 1

2I6000E

343000E 512000E

.729000E

.OOOOOOE

03

0303

03

03

n

n

181712E

191293E 200000E

208008E

215443E

01

0101

01

01

391487E

4 12128E 430887E 448140E

01

01

01

01 10 454159E 01

11

12

1314

li

16

331662E

34641oE

36o555E

374166E

387298E 400000E

ol

olol

Ql

Ol

01

l04881E

l0954bE

114018E 118322E

122474E 126491E

02

02

02

02

02

0?

133100E

172800E 219700E

.2744ooE 337500E

409600E

04

O'* 0404

04

04

n

nn

222398E

228943E

235133E 241014E

.246621E 251984E

0101

0101

01

01 n

479142E 01 493242E ol 506580E 01 519249E ol 531329E 01 5428B3E 01 17

18

1920

.289000E 324000E

361000E 400000E

03

03

0303

412311E 424254E 435890E

447214E

01

01

01

01 _0

130384E

134164E

137840E 141421E

02

02

0202

.491300E 583200E

685900E

800000E

04

04

0404

n

n

n

257128E

262074E 266840E

271442E

01

01

01

01

553966E

564621E 574890E 584803E

01

01

01

01

21 0000

0000

0000

0000 0000

0000 0000

44I0OOE 03 458258E 01 144914E 02 925100E 04 275892E 01 594392E 01 22

2324

2i26

27

484000E

529000E

576000E

625000E 676000E

729000E

03

03

0303

03

03

469042E 479583E

489898E

500000E 509902E

519615E

01

0101

01

0101

148324E

151657E

154919E 156114E

161245E

164317E

02

0202

0202

02

106480E

121670E

138240E 156250E

175760E

196830E

05

05

05

05

05

05

n

n

n

280204E 284387E

288450E

292402E

296250E 3O0U00E

Ul

0101

01

01

Ul

u

U

603681E

612692E 621446E

629960E 638250E

646330E

0101

0101

01

01 28

2930

0000

0000

0000

784000E 841000E

900000E

03

0303

529150E 538516E

547723E

0101

01

167332E

170294E

173205E

02

02

02

219520E

243890E

270000E

05

05

05

nn

n

303659E

307232E

310723E

01

0101

654213E 6619)0E

669433E

0101

01

3132 0000

0000 961000E

102400E

0304 556776E

56p6o5E

01

01 176068E

17fa885E

02

02 297910E327680E

05

05 n 314138E

317480E

01

01 676790E 68399nE

01

ni 3334

3536

3738

0000

0000

0000

0000

.0000 0000

0000

.0000

108900E

115600E

.122500E 129600E

.136900E 144400E

0404

04

04

0404

574456E

583095E

591608E 6OOOOOE

608276E 616441E

0101

01

0101

01

181659E

184391E

187083E

189737E 192354E

194936E

02

02

02

0202

02

359370E

393040E 428750E

466560E

.506530E 548720E

05

05

05

05

0505

n

n

n

320753E

.323961E 327107E

330193E

.333222E 336197E

01

^101

01

^1

01

691042E

697953E 704730E 711378E 7179-5E

724315E

01

01

01

0101

01 3940 152100E 160000E

04

04 624500E632456E

0101 197484E 200000E

02

02 593190E

640000E

05

05

n 339121E

341995E

0101 730614E

7368U6E

01

01 4142

43

0000

0000

0000

I68I00E

176400E

184900E

04

04

04

0. 640312E

648074E

655744E

01

ol01

202485E

204939E

207364E

02

0202

689210E

740880E

795070E

05

05

05

n

'n 344822E

347603E 360340E

01

0101 U

742a96E

748887E

754784E

0101

01

4^4" 4b46

4748

49

0000

0000 0000

0000

0000 0000

.0000

-D

iy3600E

.202500E 2II600E

220900E

230400E

240100E

0404

04

04

04

04

663325E Qi 67o82oE 01 678233E Ol 685566E 01 692820E 01 70000nE 01

209762E 212132E

214476E

216795E

219089E 221359E

02

02

02

02

0202

851640E 911250E

973360E

103823E

110592E 117649E

05

05

0506

06

06

n

n

n

353035E

355689E

358305E

360883E

363424E

365931E

ul

01

01

Ul

01

01

760590E

766309E 771944E

777498E

782973E

788373E

01

0101

01

01

ni 50 250000E 04 707107E 01 223607E 02 125000E 06 368403E 01 7937J0E 01

_Q 0^1.6ii4E_Q2_

1/3 464159E 01 584803E 01 669-433£:-ai_ 736806E 01 7937nnF 01 843432E 01 8879 04E 01 928317E 01 965489E 01 OOOOOOE 01 103228E 02 ln6266E 02 109139E 02 114471E 02

1169ftlF 07 119348E 02 123856E 02

L25_aa_2£:_ji2_

12.8058F n? 130059E 02

132QD1_E_ 0_2_ 133887E 02 13572 1E 0? 137507E 02 139248E 0? 14U946E 02 JA2M4j;.,D2_ 144225E 02 145810E 02 147361E 07 148880E 02

1 50349 E_02_ 151829E 02 153262E OZ

154668E 02 156049E 0? 157405; 02 15874 0E 02 160052E 02 161343E 02 162613E 02 163864E 02 155096E 0_2 166310E 02

167507E___0_2 168686E 02 169850E 0? 0.170998E 02

33-
Problem K ' Compute the function

N + (.OOpk

1.98726

for N = 0.01(. 01)1.0; k = 0, 1, 2, . . .9, and print results in cols 1 through 10.

Omniform Instructions

OMNIFORM

DEC

Q.' DEC

"DEC

.DEC

TRA

BCD

Idec

^•{[

I CONVERSION TABLE I^OR N/ 1.98726

' 3»0>0

C. 01, 0.01, 1.0

1. , 1. . 001, l.»l., .002,1.,!. ,_._C0_3_» 1_. ' 1 . . 004 »i_._»i_»_»jf_0_0^

1. ,1. ,.006,1. ,1.

,

,007,1. ,1. ,.008,1. ,1. ,.009

2,4 ^ IGENOIO

l.,l.,l.»0.,0.,l., 0.50 320542,1.

0.

0.

0.

0.

0.

0.

0.

0.

',0

1. ,0.50320542,1.

1. ,0.50 320_54_2_,_1._

1. ,0.50320542,1'.

1. ,0.50320542,1. _

1. ,0". 5 320 542ri.'

l._,0. 5 320542 »1._

Y. ,0V5 32 542.1.

1. ,0.50320542.1.^

Vl~. ,0.503 2 54"2 , 1 ,

TRA

BCD

2,4

lENDOOO

JN,0,. 001,. 002,. 003, .004

*'tN,.005,.006».007,.0 08,.C009

Remarks

a. This line sets up the argument from .01 in intervals of .01 to 1.0

b. These lines set up the transformations N + .001, N + .002, . . N + .009.

c. Lines 1 through 10 compute N/1. 98726 = .50320542N for the values of N

indicated in line a, and for each of the 9 transformations specified

in b.

d. This line asks for a printout of the columns in the order in which they

were computed.

e. These are the column headings for the 2 pages of printout.

It will be noted on the reverse that the conversion table has been 'computed

for 1,000 values of N and arranged in a rectangular array as is usual in

logarithmic tables.

3^
Problem 4 (Results)

OMNIFORM I CONVERSION jABLE FOR N/1.98726

N .001

0.0100

0.0200

0.0300

0.0400

0.0500

0.0600

0.005032

0.010064

0.015096

0.020128

0.025160

0.030192

0.005535

0.010567

0.015599

0.020631

0.025663

0.030696

0.0700

0.0800

0.^0900' 0.1000

a. 035224

0.040256

-a. 045288

0.050321

0.035728

0.040760

0.045792

0.050824

0.1100 0.055353

~ ^ Cr.0603B5

0.065417

0^.070449^"

0.075481 . G,-aso5i3

0.085545

0.055856

0.1200

0.1300

O."I40O 0.1500

a;^r600~" 0.1700

0.060888

0.065920

0.070952"

0.075984

^ "0.081016

0.086048

0.1800

0«1900

o.-zooo"~"

0«2200

0.0905 7 7 0.095609

G»IG56TS--'

0.1l0705

If. 0910^0

0.096112

•"1011^4

"0".~I0-6rrT6 0.111208

0.2300

0.2400

GiZ'^GG--

0»26oo

is*;rai'2"To-cr~~

0.115737

0.120769

o"rr?5-S0r--- 0.130833

0.116240

0.121273

0.131337

.002

0.006038

0.011071

0.016103

0.021135

0.026167

0.031199

0.036231

0.041263

0.046295

0.051327

0.056359

0.061391

0.066423

0.071455

0.076487

0.081519

0.086551

0.091583

0.096615

0.101647

0«106680

0.111712

0.116744

0.121776

0.126^08

0.131840

-0.136872

.003

"0.006542

0.011574

0.016606

0.021638

0.026670

0.031702

0.036734

0.041766

0.046798

0.051830

0.056862

0.061894

0.066926

0.071958

0.076990

0.082022

0.087055

0.092087

0.097119

0.102151

0.107183

0.112215

0.117247

0.122279

0.127311

0.132343

0.137375^

.004

007045

012077

017109

022141

027173

032205

037237

042269

047301

052333

057365

062397

067430

072462

077494

082526

087558

092590

097622

102654

107686

112718

117750

122782

127814

132846

137f

N .005

0.0100 0.007548

0.02TJO ' " " "D.D12580

0.0300 0.017612

0.0400 0.022644

0.0500 0.027676

0.0600 0.032708

0.0700 0.037740 " O.DS'OO a.042772

0.0900 0.047805

0.1000 0.052837

r ' 0. rioo 0.057869

0.1200 0.062901

0.1300 U.D67933

0.1400 0.072965

o.r5oo 0"."OT7997 0.1600 0.083o29

O.TTOO 0."0ffff06I 0.1800 0.093o93

0.1900 0.09Sli!5

_ q._2ooo 0.103157

0.2100

QiZZGo"

.006

0.008051

a. 0T3OB3

0.018115

0.023147

0.028179

0.033212

0.038244

0.043276

0.048308

0.053340

a.a5ff3T2

0.063404

0.068436

0.073468

. 0T8 500

0.083532

0.088 564

0.093596

0.098628

0.103660

0.108692

lX3JZti

.007

0.008554

0.013587

0.018619

0.023651

0.028683

0.033715

0.038747

0.043779

0.048811

0.053843

0.058875

0.063907

0.068939

0.073971

0.079003

0.084035

0.089067

0.094099

0.099131

0.104164

.008

0.009058

0.014090

0.019122

0.024154

0.029186

0.034218

0.039250

0.044282

0.049314

0.054346

0.059378

0.064410

0.069442

0.074474

0.079506

0.084539

0.089571

0.094603

0.099635

0.104667

.009

0.009561

0.014593

0.019625

0.024657

0.029689

0.034721

0.039753

0.044785

0.049817

0.054849

0.059881

0.064913

0.069946

0.074978

0.080010

0.085042

0.090074

0.095106

0.100138

0.105170

3g

U. S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical work.

In general, each section carries out specialized research, development, and engineering in the field indicated by

its title. A brief description of the activities, and of the resultant publications, appears on the inside of the

front cover. WASHINGTON, D. C.

Electricity. Resistance and Reactance. Electrochemistry. Electrical Instruments. Magnetic Measurements

Dielectrics. High Voltage.

Metrology. Photometry and Colorimetry. Refractometry. Photographic Research. Length. Engineering Metrology.

Mass and Scale. Volumetry and Densimetry.

Heat. Temperature Physics. Heat Measurements. Cryogenic Physics. Equation of State. Statistical Physics.

Radiation Physics. X-ray. Radioactivity. Radiation Theory. High Energy Radiation. Radiological Equipment.

Nucleonic Instrumentation. Neutron Physics.

Analytical and Inorganic Chemistry. Piu'e Substances. Spectrochemistry. Solution Chemistry. Standard Refer- ence Materials. Applied Analytical Research. Crystal Chemistry.

Mechanics. Sound. Pressure and Vacuum. Fluid Mechanics. Engineering Mechanics. Rheology. Combustion

Controls. Polymers. Macromolecules: Synthesis and Structure. Polymer Chemistry. Polymer Physics. Polymer Characterization. Polymer Evaluation and Testing. Applied Polymer Standards and Research.' Dental Research.

Metallurgy. Engineering Metallurgy. Microscopy and Diffraction. Metal Reactions. Metal Physics. Electrolysis and Metal Deposition.

Inorganic Solids. Engineering Ceramics. Glass. Solid State Chemistry. Crystal Growth. Physical Properties.

Crystallography.

Building Research. Structural Engineering. Fire Research. Mechanical Systems. Organic Building Materials. Codes and Safety Standards. Heat Transfer. Inorganic Building Materials. Metallic Building Materials. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics. Operations Research. Data Processing Systems. Components and Techniques. Computer Technology. Measurements Automation.

Engineering Applications. Systems Analysis.

Atomic Physics. Spectroscopy. Infrared Spectroscopy. Solid State Physics. Electron Physics. Atomic Physics.

Instrumentation. Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical Instru- ments. Basic Instrumentation. Physical Chemistry. Thermochemistry. Surface Chemistry. Organic Chemistry. Molecular ^ectroscopy. Molecular Kinetics. Mass Spectrometry. Office of Weights and Measures. BOULDER, COLO.

Cryogenic Engineering Laboratory. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Cryogenic Technical Services,

CENTRAL RADIO PROPAGATION LABORATORY

Ionosphere Research and Propagation. Low Frequency and Very Low Frequency Research. Ionosphere Research. Prediction Services. Sun-Earth Relationships. Field Engineering. Fiadio Warning Services. Vertical Soundings Research.

Radio Propagation Engineering. Data Reduction Instrumentation. Radio Noise. Tropo spheric Measurements.

Tropospheric Analysis. Propagation-Terrain Effects. Radio-Meteorology. Lower Atmosphere Physics.

Radio Systems. Applied Electromagnetic Theory. High Frequency and Very High Frequency Research. Modulalation Research. Antenna Research. Navigation Systems.

Upper Atmosphere and Space Physics. Upper Atmosphere and Plasma Physics. Ionosphere and Exosphere Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.

RADIO STANDARDS LABORATORY

Radio Physics. Radio Broadcast Service. Radio and Microwave Materials. Atomic Frequency and Time-Interval

Standards. Millimeter-Wave Research.

Circuit Standards. High Frequency Electrical Standards. Microwave Circuit Standards. Electronic Calibration

Center.
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