71900-10 December 1956

1103 CENTRAL EXCHANGE NEWSLETTER NUMBER 10

December 1956

PX 71900-10

DIVISION Of SPERRY RAND CORPORATION

1 gO 2 WE S T MIN N E H A H A A V E. ST. PAUL W 4. MIN N E SOT A

o o v

0"- N

CONTENTS

4

0F PUBLICATION 'Page 1 of 2

Titl~ =' llO') CENTRAL EXCHJ.NCE NEWSLETTER l~u"~BER 10, . December 1956

C las.sifica tion _U _ _ -,-- Publi<;ation Date_.,....,...---. _ _ ...- Charge_93~~~ _ _ ' TeXt pa:ges ___ :~ _ _ jO--,' _ _ ~ Photogl'aph$ . ..---___ --...---.,....,...,.' Drawings -, ._._. _. -_: -

Page No.

(of ) i

thru

i i i

10-1 thru 10-5

Contents ^{. Print} Number

itlePage, 71900-10

ont Hatter (Newsletter) ,71900-10

nere1 Card ~ed-In

Rouline

Drawing N-umher

10-6 thru

10-15 Eigenvalues of a

Tr1-Die-

gor.al

Me

trix .

71900-10-16 (RW~169) 10-l6thru

10-17

Floo 1;.ing Point Cere Dump 719~lo.1 (l1.W~l70)·,

10-018

thru 10-24 llD3 to l103A Conversion ,

Routine 71900-10-1

~0-25 thru 10-48·· Floating

';Vector

Arithmetio , .,

Pa'okage' 71900-10-1'

10-49

thru

10-73 Sierdficance

Preservirg F100ting

Binary Point

Arithmetic

for Digitol

Computers

10-74 thru 10-135 A

Method

for

Gene~~tin,

Random Numbers on

the

EPJ. 1103';',

"1

71900-10..l 10-136 thru 10-1 Evaluation'

or IA- ~

to.'t' "

': Matrix A (Complex ~ingle

10-142 thru

1()..14

Floating Point Linea~ .

t . -

(M-17l) (~172)

.(ilR-1731

. ',' j'

Precisio~noc.ting Po~t) 71~1~l, ' . . l-1atrix F.qpe tior" rol "lor

. . (AX=B) .', 7l9~lo..l .(R)1 ..

176)

io-150 thru 10-15 ,Complex ~ingle

Precision

, 'Floating PQint Linear Matrix F£luatiorJ Sol vel'

(AX ::: B)' . '71900-10-.1 ' :(RW .. l77) , 1~158

thru

10-16 Complex Oil1 Method

Rout1)')

.719QO..lo..l .

(RW-17S)

10-170 thru'l~l. ~lgebre.ic Eq,uation 'Solver 71~10-1

(RtV.l7<?)

10-181 . thru 10-1 Unpacked 'FloGting Point

Card Output 71900-.10-1' ,(CY ... 180) 10-,191 thru 10-21 Continuous Mtltrix 14ulti-

pIler

Using

.PLIP III

719~lo.1 (CV..-1Sl), ontlr;uous

Matrlx

Multi-

plier Using Sivgle or Multi-Precision Arith- Metic

10-241 thru 10-3 PUR - Single Precieion

Unpopked Rounded Floating Point

Package

for ERA-

1103 719QO.IO-l

(CV' .. 182)

(CV-lS),

CONTENTS OF PUBUCATION Page

2

of 2

' . : " 1··.:tl",i~ ... :~~"',"I·'~"'" , ^'~ .' ; . '

.,. Classification . .. ________ . ___ ' U lca Ion U . ' PhI' t· ^{D '} a e ____ t .____ Ch arge ... 9.,388_ . . -... 8-4._._ .

,'. '~' :Text p~ g e s . J'.:...L _. ____

o ...

~...l,

.

Pl" ,loto g r a ph s '._ _ __ . _. __ ~ ____ ._2__ _ __ . _.' , D raw in g s

J_ r.,,~age No.

. . .. J9t .. ", .. J

';"'l"¥f3 ". ·,~~:,.'l ,";

10-33.1,;,:th;r;u:l0-J3 Canputing Cer:t. r OrC8T.iZe-

~ ,.~.:. 'tioh ~'. The Rarr;o .... Wdoldrid[

Print Number'

Corpor~tior. " ·'·71900.:10t 10-335 th~ 10-33 themt. t.lcvl S~rvicee

Bra.nch

l'0-J~€,.trn:u. 1~3

Unpacked

Flofltlr;e, Poir:t ea rd Reao'<'!'

1~3J7thru) lQ..;.)5 Utility' Routine Librery '-'

Draw,ing Number

(10:27)

. (C'V-l'54'

'pm ).

,(R\<J-? 1 ~EV

10-J53"thrtl'lO:j6

.LheRtSmo-Wooldrldpe' Corpo .

·One-Pass AsseinblY Routine

71900-10-

(:P.\J":'7:2 R'E\,~

lQ-J6A>'thro 10-3 TieClnite Ir.te£r81"' Eve:lui;-. ' .

~ ,_~,.;" ". "tio,~

Rcrutire. . . . .

~" 719O('~10-~(Rl~-B;9 REV)

· , inc Poitlt r8cl:arEl ·'7190C):.10-1'···

'(pt:roe

KrVX 10-372 ~~hru 10-38 ~NAP - Interpretl va floflt-" ~' ,

I

1Q-.382 thru 10-J8~Sr;AP Sempler Tr8c~' '. 71906-10-1 (m,]-140 R"f.\1)

1~3~! tr .lo-.~~S,~JI.~'

^-

~pterpri.tive

^Floc·t-

1

1

, .

I .

^I

. . . ': ... , -jlong

^P01:I;t PfAc}.:Bce-C"rrplex

:71~lo-14p.. (R~:~l4l~)

10-390 ,thr.,U,

,:~,

0-39,,)'1', l,ie, ",F.,e, rrar.ti

.~r~l:t ',J~~ti'fel

^71900-.. ' 10.:..6,

~;'

^{, ",} <ffi.!.;..(,3

?..F.V~'

10-)94 !-t.ru lC-J91Arcsine-Aroosl.lle R011tlT a, . , '. "

: """":, ''';,\: ,."t'

·~:tf't~ POiTJ~, . 719?O-lO- " (m~:-148 BE

.>

10-398 thru 10-40~F108.t1ne Point r.rcslr~e- . , , ' I ' . '

· i Arcosine Routine' 71900-10-1

r

(P'·';-149

PJ'1)

· . Routlr;e ' . . , 71900""110-7 '(f'Jt-74 P:F.V

10-4., .02:ttla:-U'

l~~"'n08t~rie

^pOirt,

Arcta,r.g~r,t, ... , ," , " '.' "",'"

19-406

tr.ru ~O-4 Ch8nged ,W6rd post-Mortem': ~ ^.1 '.' ' "Routi'1le 71(01)...;10-1. (Rw-102 HEV.) 10-408 tl.oru 10-41 th Root, Routir.A.'. . 7190()....lo.ii ,- '.

(F.1..\r~116 nB-a,),

lq-4l2 thru 10-42 i l l lltethod ~(;brou~ine,,' 71~0f\-10-9' (p^ll-91 REV~

10-'~~ ",,~hru

lO-J" .

F;L~ tir.r. ^j 'Poir:t elll Method! ?1900-10-U' (U\>!-.!43 P.E\l) 10-1.30 ~~."1O-4}· +~h'~ing'.Poh:t Sjre-... Cof:ir ',71900-10-. .. (RV-:l!+L.) (~V) lq-/+J5, . th,~:l0-:44! ~~lE 71900-10 (~}.-86 ~f,V~.,

, • I ~ ,'" _. '-" '

I I !,' 1\,1 t

i"

^{l\ (I I}

<.

j

I

I

Number

Correction:

Correction:

Newslet t\;r Number 10 Deoember 1956

EDITOR'S PAGE

On page

9-392

of

Newsletter 9, line 2, paragraph 4, of RR-162 should read " • • • will be p-l = ^{2 35 _32=34,}

359,738,336 • • • ".

'The ootal equivalent given for the constant A4 in the descriptions for the arsin-areos floating point (RR-75) and the arosin-areos stated point

(RR-25)

routines is in error and should read

37

50417

41233

instead of the listed value of

37 04174 41233.

This oorreotion should be entered in Newsletter

3 (pg.

3-108) for RR-25 and in Newsletter 6 (pg. 6-73) for RR-75.

Editor,

Central Exchange

-

_... ^... ...

6 ~

~

t:

NEV1SLETTER

10 DECEJviBER 1956

ENCLOSURES

RW-168 RW-169 RW-170 RR-171

.~'\~

~172

~

( 173

\., . ..-./

W'8-174 RV-l75 RW-176 RW-177 RW-l78

RW~l79 .CV:~l80

OV-181

CV-182

CV-183

10.2:)

10:28 REVISIONS CV-154 RW-71

General Card

Read-In

Routine

Eigenvalues of a Tri-Diagonal Matrix Floating Point Card

Dump

l103 to l103A Conversion Routine Floating

Vector Atithmetic Package

Significance Preserving Floating Binary Point Arithmetic for Digital Computers

A Method for

Generati~g

Random Numbers on the ERA 1103 Evaluation of IA-iU I for Matrix A(Complex Single

Precision Floating Point) .

Floating Point Linear

Matrix

Equation Solver

(AX=B)

Complex Single Precision Floating Point Linear Matrix Equation Solver

(~)

.

Complex Gill

Method Routine

AlgebraicEquat~on

Solver

Unpacked Floating Point Card' Olltput

Continuous Matrix

MUltiplier

Using

FLIP

In

Continuous Matrix .Multiplier Using Single or Multi- Precision Arithmetic

SPUR - Single Precision Unpacked Rounded Floating Point

Package r~

ERA-l103 Computers

Computing Center Organization -.The Ramo-Wooldridge CorPoration Mathematical

Service

Branch - Eglin Field

Unpacked Floating Point

Card Read

Utility Routine Library

1. Table of Contents

RW-72 RW-89 RW-108 RW-140 RW-l4l RW:-63 RW':148 RW-149

RR-74

RW-I02 RW-116 RW-9l RW-143 ,... RW-144

oM ori

oM

RR-86

...,

0 I

'1 8

'"

f!

><

p..

2. Conventions 3. Reminders 4. Tape Bootstrap 5. Pool of Flex Codes 6. Cumulative Errata

7. Utility Routine Transfer Drum to Magnetic Tape 8. Utility Routine Transfer - Magnetic Tape to

Drum

The Ramo-Wooldridge One-Pass Assembly Routine Definite

I~tegral

Evaluation Routine

SNAP - Interpretive Floating Package SNAP Sampler Trace

SNIP - Interpretive Floating Point Package, Complex The Ferranti Input Routine

Arcsine-Arcosine Routine, Stated Point Arcsine-Arcosine Routine, Floating Point Arctangent Routine, Floating Point

Changed Word Post-Mortem Routine

~

Root Routine

Gill Method Subroutine

Floating Point Gill Method Subroutine Floating Point Sine-Cosine Routine

FLEXIE - Flex Code Paper Tape Input Routine

-

CO' -0

...

'-'

,

o

.... ,

o o

0"-

....

...

><

a..

Identification Tag:

Type:

Special Storage:

Program Entrance:

Program Exit:

Alarm Exit:

Coded by:

Approved by:

THE RAlv.lO -\vGOLDRffiGE CORPORATION Los ili1geles 45J California General Card Read-In Routine

Specifications

CRI-3

CRI-3 Pg. 1 of

5

11-5-56

Service Routine. (with subroutine entrance) The constant pool and temporary pool are not

used by' this routine 400l7b

40020b

The alarm routine is used by this routine

M. Perry November,

1956

w.

F. Bauer November,

1956

10-1

"

Description

Rtv-168 CRI-3

Pg. 2 of 5

11-5-56

This routine reads cards produced by

MDP-l

(binary cards), CPO-O (fixed point output), CPO-l (floating pOint output) and cards~key-punched on the 4 field format (described below) for input in tloating point, double precision floating point, fixed point, or octal. These input forms may be intermixed on a card, and the cards may be in any sequence desired. The routine automatically

differentiates the 2 card forms, and for

4

field cards, recognizes the type of input in each fieldo All input is rounded properly. This routine reads

cards at :full card reader speed and loads the memory as directed by the address or addresses appearing on the cards. Once activated, it continues to read cards until it has read and stored a card containing a stop code as described below. The input need !not be normalized to retain full significance. The routine stores high speed memory on the drum, operates in high speed roomory, and restores high speed memory from the drum prior to leaving the routine.

Qperating Instructions

1.

When routine is used as a service routine, set PAK to 40011b and start.

Routine will read ~ards until a stop code is recognized, at which time the ma:chine will stop (MSO ) with P AK set to 40011b.

2. When routine is used as a subroutine, enter the routine with

37

40020

4OOl7b.. Routine vill read cards until a stop code ·is recognized, at which time control is transferred to cell 40020b and hence to the cell following the return jump mentioned above •

To restore high speed memory at any time, start at 40040b. A andQ will not be restored.

Card Positioning

Card positioning is required before the initial read only. Card reading

automatically positions the next card to be read,and a card will be positioned for punching before leaving the routine. If the routine is, used as a service routine, the initial'positioning must be manual. If the routine is used as a' subr'outine, initial card positioning should be programmed. This can only be done

by.

one instruct ion, ~ 00000

VVVlf'iI

where vvvvv contains

00 00000 001l4b.

S

Stop Codes

.. ..

I ) of

1. Read stoE, 12 col

80.

When this code is detected on e~ther a binary card ora four field input card, the' routine will stop reading, pos'ition a card 'on the punch side of the bull, and then exit properly (see operating instructions).

2. Machine stop, 12 col 790 When this code i6 detected on a

4

field input card, the routinewi1l stop reading, position a card on the punch side of the bull, and stop MBO at 7243lb. If the machine is started, the exit from the routine will proceed as described in operating instructions.

-

co ...0

....

-

o

.... •

o I 0' o

....

t- :><

a..

RW ... 168

CRI-3

Pg.

3

of

5

11-5-56

Either of the above codes may be entered on a blank card and the routine will sense them. The prograIDI!ler is cautioned not to place the machine stop code on a binary card as this will result in improper loading (see MOP -1 write -up) •

Alarm Conditions

1. Binary Cards - As a binary card is read, the words are summed and the result is compared with a sum punched in the card (see MOP -1 write -up) • If the sums do not agree, the flexowriter will print

"ALARM 00211 000000000000 OOOOOOOvvvvv QQQQQQQQQQQQ"

a.nd the ^ma~hine will stop. vvvvv is the storage address appearing on the card. The contents of Q are not important. Starting the computer will bypass the alarm, the words will be stored as read, and in the absence of a stop code, reading will proceed.

2. 4 Field Cards - An alarm on a four field card indicates' that a number was too large to be entered appropriately into the computer. For floating point numbers (single and double precision) this is equivalent to exponent over- flow. For fixed point numbers ^J this indicates that the input properly scaled and rounded was too large to be. entered into a single cell. If an alarm'condition is detected, the flexowriterwill print,

"ALARM 00164 000000000000 OOOOOOOvvvvv 000000000000"

and the machine will stop. vvvvv is the storage address of the right-most incorrect number. ^Any or all of the other numbers may have been incorrect.

vvvvv equal to 20,000 indicates, that the indicated address field was blank.

Starting the computer will cause the correct numbers to be stored, the incorrect numbers to be ignored, and in the absence of a stop code', reading will proceed.

4

Field Card Format

The format described below is one of the formats used'by this routine and is the for.mat used by CPO-O, CPO-I, CRI-2, and SNAP Read.

The card columns are deSignated as follows:

Col 1 ... 4 Identification This field is ignored by the reading routines

Col 5-23 Field 1 Col 24-42 Field 2 Col 43-61 Field

3

Col 62-80 Field 4

10-3

,...,

co ...0

...

...,

•

0 ...

8

I 0-...

t-

>It c..

RW-168

CRI-3 . Pg.

4

of

5 11-5-56

Each field (except identification) is divided as ·follows.

digit

1-5

digit

6-15

digit

16-17

digit

18-19

Location or address Value 'or Mantissa Decimal exponent Binary exponent

(x digit 1 for octal location) (x digit

15

for negative value) (x digit

17

for negative exponent) (x digit

19

for negative exponent) Addressing Options

The following Addressing options are allowable on the

4

Field input card. Option I is used by CPO-O, CPO-I, CRI-2, and SNAP Input.

1. Decimal - Stra.ight conversion to the octal equivalent of the decimal address in the card. No indication is necessary.

2. RAWOOP Decimal.;. Straight conversion to the octal equivalent of the decimal address except that 1+0,000 decimal is designated as the first drum address

(40

,OOOb). No indication is necessary.

3.

Octal -The octal. number appearing. on the card is the actual.. address.. This mus-r-be indicated by an x(ll punch) ,over the first digit of the address.

4.

Blank'-' ,':A completely' blank a.ddress field indicates that the number in that field is not to be stored ~ · This is differentiated from an address containing 1 or more zeros which will load cell zero.

Input Numbers

The following varieties of input may be punched on'the

4

field input card. They may be in any combination on a card with the exception that for

a

double precision floating point number. the two fields 'muSt be co:gsecutive, and on the same card.

For all input, thedecima.l point 1s presl.l1Iled to be at the extreme left end of the value "field. It is 'recoIIl.Il.kended that the codes listed below be used for ea.ch field.

However, cards from

CPO-o

and CPO-l are differentiated by the fact that their

"B" (Binary Rxponent) fields are non-blank and blank respectively.

1.

2.

Floating Point .., RS. An input n:umber is designated as floating po:Lnt by the code RS (Read Snap) in the ''B" field. There is no restriction on the tiD"

field. The resulting floating binary number is rounded and is' in the form used by SNAP and by the internal floating' painton the l103A computer. A floating point number consisisof

3

parts; a sign bit., follo'W'ed by an

8

binary.bitcharacteristic biased by 200b, and'a 27 bit normalized mantissa • To negate a floating point number, the complete cell 'is complemented •

Double Precision Floating Point - RT. An input· number is designated as double preciaion floating point by the code RT (Read Two) in the ''B''field. The

"value~1 portions of 2consecutlve fields are joined to allow 20 decimal digits of in~ut. Both fields must contain addresses but the ,control information

(RT code, algebraic sign, and decimal exponent) is taken from the first field only and ignored on the second field. The resulting floating binary number is rounded and is in the form used by double precision SNAP. The upper cell

-

co ..0

...

--

o

... ^,

o

6

a-

....

t-

><

0..

RW-168

CRI-3 Pg. 5 of 5

11--5

-56

consists of a sign bit and

35

binary digits which are an extension of the

27

binary digit mantissa in the upper cell.

The programmer is again cautioned that the two fields concerned must be consecutive and on the same card.

3o.

Fixed Point - No code. An input number is' designated as fixed point by the fact that the r~rr field is not blank and does not contain an R. Since the

tlB" field is used to express the binary scale factor desired, the only caution' is that it must not be left blank.

00

(zero zero) must be punched if a scale factor of zero is desired.. The binary scale factor is allowed to be ne gat i ve and the only restriction is that the combination,of the B and

D

:fields result in a number which is not too large for a single cell. Normalization is, not important since the conversion is done in double precision. The resulting binary ntimber is rounded p r o p e r l y . "

.,

4.

Octal- RU. An input number is designated as being octal by the code RU (Read Unc.onverted) in the liB" field. Since 12 octal digit's are d.esirable, the "value" and ''D" fields are joined giving 12' digits. Each digit is loaded modulo

8,

such that an

8

becomes a zero, and a

9

becomes a 1 /'

Examples. The following ,list of, in,pu~

..

^w~d be. p~<?hed on two cards. The. second card would have a trStop ReSod." code in' col 80 because of the "+tt signeD the iast line. The octal address and translation are listed in the comments.

QUANTITY LOCATION VALUE ^l:d:z ,D d::- B

:::b

^COMMENTS

4" 0.1 2 2 1 2

O,·it 1 I j I

1

o 1:

R S:

40122 20 14631 46315

j

.. ,

1 6.1 8.4 4 5 a

0

a

b,b,0~0,4t

2

1

5:

RIU' ^I

40000 45 00000 00425

I

o 1:

^I

4·0 0

6.

4 3 1 4 1

5 ~

9,2 ,6,5 ^3:-

R,TJ

40100 57 51557·00452

I I ,

4,0 06 5 5 8 9 79 3' 23,8 5:

^I

,

40101 75· 67513 47562

I

, ^,

o~o

1 2 3 001 5

^{r , I} I I ^{I I}

0:

,

o 4' 0 OQ123 , 00 00000 . 0001 7

,

I I ,

99

¹

5 ,

o 2: 1 5: 00143 00 0001 7 00000

, ,

^{, I ,}

,

P,l:

2: 00144

1 0,0 1

^,

, ,-

^, _I

77 77777 67777

o ^0,1.2,2

¹

5

I ^I ,-,

P Ii

R s:+ ^I

00122 57 61777'77777

Identification Tag:

Type:

Storage:

Regional ^Addresse~ Used:

Entrance and Exit:

Machine Time:

Mbdeor Operation:

Coded by:

Approved by:

mv-169 EGN-l

Pg. 1 of 10

10/1/56

THE RAlviO-}lOOLDRIDGE CORPORATION

J~s Angeles

45,

California Eigenvalues of a Tri-Diagonal ~~trix

Specifications EGN-l

Subroutine available on cards for assembly.

145

words of storage needed to assemble this routine.

18 + 2n cells of temporary storage inunediately following the temporary pool used, but not stored with subroutine. (n =' order of matrix).

The constant pool and temporary storage pools are used by this rout ine •

OOR, OlM, 02M, OlR J OOK, OCT J FOO, COO RJ OOROI OOR02 No Punching }

. See Instructions RJ OOROl OOR03 Cards Punched

See table in text.

Floating Complex Arithmetic requiring SNIP be activated.

w.

Frank September

1956

W. Bauer October

1956

.-..

0"- ...0

--

r-I 0 ^I

"""'

^I

0 0 0"- t-r-!

:><

c..

Description

HW-161.J EGN-l

Pg. 2 of 10 10/1/56

This subroutine Co.il~utes the n eigenvalues of any real or complex tri- diagonal matrix D, having the form:

a1 b

l 0 0

1 a2 b

2 ⁰

0 1 a

3

^b

3

⁰

D =

U

I a n-l b n-1 J ^{I 1}

I

1

anJ

Since complex arithmetic is employed, the elements of the matrix must be presented according to the specification for number representation tor use wi th SNIP. The n elements of the main diagonal, ai' must be followed by the n-1 elements of the upper adjacent diagonal, hi' in a region whose initial address is specified by a parameter word.

The more general tri -diagonal matrix J, (also called a "Jacobi Matrix") has all its non-zero elements on the main diagonal and on either of the two immediately adjacent diagonals:

al b

l ⁰

o

cl a

2 b

2 ⁰

o

0 c

2 a

3

^b

3 o

J

=

0

10-;

RW-lb'i

EGN-l

Pg.

3

of 10 10/1/56

This routine can also treat this case if in place of the n-l elements of the upper adjacent diagonal, the n-1 products, c

i b

i, are supplied.

The subroutine occupies 145 cells and uses the constant and temporary storage pools. In addition 18 + 2n cells of temporary storage must be provided immediately following the Ramo~ooldridge Temporary Pool. Practical limitations are imposed on n by the available 1024 words of ES storage and. the use of SNIP, hence n must

be

less than

157.

In all cases the eigenvalues can be found in the last 2n cells of the (18+2n) cells of temporary storage. In addition the eigen- values tti and associa.ted residues in the characteristic polynomial p(;:\.. i) can be punched on cards.

pC (\

i) appears in fields one and two while (\. i is in fields ti ve and six. The eigenvalues are also identified serially in the identification field. At the option of the progra.xrmer thesl,1ccessive approximations to the

{\. i obtained during the iterative process can also be punched with their associated residues in the reduced polynomial. In the event that no punching is desired then a third entrance is available. _or ~

PrOgrammdng

Instructions

1. Complex mode of SNIP must be activated.

2. Entrance to the subroutine is made as follows:

a. ,RJ OOROI OOR03

xx

^{,OOAOO vvvvv}

where

OOROO is the location of the first word of the subroutine

OOAOO is the location of the real part of the first element of the diagonal

vvvvv is the order n of the tri~iagonal matrix

,...

0"- ...0

~

'-"

0 I r-t 0 I 0 0' r-t

t-

><

~

XX gives the option selected

RW-169

EGN- 1

Pg.

4

of

10 10/1/56

xx =

20, only the eigenvalues

2l

i and their respective

residues in the characteristic equation are punched.

xx =

00, in addition to the above, the successive approximation to the eigenvalues are punched.

In either case the eigenvalues tp.emselves are stored in the machine starting at the location

51 =

63b.

b. Should no punching of cards be desired then one must use the entrance:

RJ OOROI OOR02

20

oOAdo

vvvvv

3. Control is returned to the cell following the parameter word.

Machine. Time

The time taken to find the n eigenvalues can be estimateqfrom the following table of empirical times in se.conds.

I

No Punching

Punclling

Order of -Punching Eigenvalues and

Matrix Eigenvalues Iterations

3

5

Sec.

8

Sec. 13 Sec.

5

17

"

^{22 f! 36 n}

8

50

" ⁵⁸

^{fI 92}

^"

10

81 " 91 "

₁

4

₃ ^fI

27 776

"

⁸⁰³

"

39 2743

"

2782

•

"

"

:>

.. ..

I )

..

I ) )

"

..

Given the tri~iagonal matrix J al b

l 0 cl a

2 ^b2

0 c

2 ^a3 J =

o

b

3

0

a b n-l n-l

RW-169

EGN-l

,Pg. 5 of 10 10/1/56

The following recursion formula evaluates the characteristic polynomial p (it) of J for given

A:

n

where

P =0

-1 P = I o

i = 1, 2, •.• , n

'l!he problem loses no generality by assuming all c i = 1 so that only the a

i and b i need be given. Alternatively the general problem can be solved by this routine if one supplies Cib

i in place of b i ,

Using P (-1), P (1) and P (0), the program enters a modified version of the

n n n

Algebraic' Equation Solver (POL-O) and finds, by iteration, the first root 1\1 of Pn(a).

, at

Having found r roots the (r+l), root is fotind by considering the polynomial

RW-169

EGN-l

•

Pg.

6

of 10 10/1/56

where~l' ~2'

···,21

_r are the eigenvalues already found. This code prevents the re-computation of a multiple eigenvalue by not allowing JLr+l to approach the' value of any of the r roots already found. This, however, dId not prohibit the determination of multiple roots in any of the matrices tested, for the reason statec below.

Convergence

A

convergence criterion

~ i+l-

.Ai L

^10-k

:it i+l

is applied to determine the end of the iteration. In this code ^k

= 6.

This gives an accuracy of

6

to

7

places in many low order cases. For large order

(n~20)'accuracy is reduced since not enough figures are carried in the

27

bit word of

SNIP

to accurately define the roots after a large number of arithmetic operations have been performed. In the case eigenvalues are repeated the accur- acy deteriorates.. Furthermore, the residues of the characteristic polynomials in 'the neighborhood of such a root are exceedingly small. Rence, a second convergence criterion was introduced in order to avoid exponent overflow. If the residue

P

(Ll)i

2 -100 n-r

then Ai is accepted as a root.. If Ap has multiplicity p then the code will find p estimates of

A

such that

p

a) no two estimates are equal

b) ail estimates have residues

L

2 -100

If Ak is an eigenvalue then det(D -AkI) should be zero. Inspecting successive iterations and associated residues can therefore give some, indication of the convergence of the procedure.. The programmer is, however, cautioned in regards

10-11

...

,

I'

_..

'"

I

..

) I ) ) ...

of

..

RtV-169 EGN-l

Pg.

7 of

10 10/1/56

to using this quantity as a measure of accuracy of the root. It is possible, for example, to have a root accurate to 6 places and yet obtain a residue of high order .

0

OOROO

00100

0

OIMOO

00118

0 02MOO 00151

D

OIROO

00204

D

OOKOO

00239

D OOTOO 00033

0

FOOOO

00002

D

coooo

00003

OOROO 00 00000 01R16 OOROI

MJ

00000 00000 OOR02

MJ

00000 02M37 OOR03

TP

01R34 02M50 OOR04 SP

OOROI

00015 OOR05

TU

ADOOO OOR07 OOR06 TU AOOQO 02M32 OOR07

TP

00000 AOOOO

OORoa TP

00013 00T15 OOR09

TP

00013 00T14

OORIO TV

AOOQO 00T15

OORll

TU AOOOO OOROO OOR1Z TP OOKOS QOOOO OOR13 R5 00T15 00016 OOR14

LA AOOOO

00016

DORIS

QS AOOOO 01R06 OOR16

TP

00115 00T16 OOR17

TN

00016 00T17

OIMOO RA OOROI

00016 OIMOI RP 10005 OlM03 01M02 TP 00013 00T09 01M03 TP OOK01 00T12 01M04 TN

OOKOO

00T08 OlMOS TP

aOKOO

00T10 OlMU6 TN 00K01 00T06 01M07 TP 00013 00T07

OlM08 RJ

OlR30

OIRoa

OlM09 TP 00029 OOTOO

OlHIO TP

00030 OOTOI

":'"

OlM!l TP

OOKOl 00T06

0' -..0 OlM12

RJ OlR30

OlROC

r-f 01M13 TP 00029 00T02

"-'

I 01M14 TP 00030 00T03

0

r-f I 01M15 TP 00013 00T06

0 01M16

RJ

OlR30 01ROO

0

0' 01M17

TP

00029 00T04

...;

t- OlM18 TP 00030 00T05

>< _OlM19

MJ

00000 02M31

P..

OlM20

LDMP

OOTOO 00T08 OlM21 ADST 00T04 00023 01M22

MPST

00T08 00025 01M23

LDMP

OOT02

OOTlO

OlM24 STSU 00027 00023 ⁵

00144 00166 00227 00314 00357 00041 00002 00003 00144

EXIT 00145

ENTRANCE 1 00146 ENTRANcE 2 00147 00150

S 00151

E 00152

T 00153

00154

A 00155

0 00156

0 00157

R 00160

N-1 ^[ 00161

2N-2" $ 00162

5 00163

E 00164

S 00165 00166

SET

00167

00170

UP

00171

00172

STARTING

00173

00174

VALUES

00175

00116 00177 00200 00201 00202

;)0203 J0204 00205 )0206

i 0207 ( 0210

r~ 0211 ::>212 :J213 : ')214 0)215

(I 216

l l \ _ l ' l

00 00 00 00 00 00 00 00 00 45 45 11 31 15 15 11 11 11 16 15 11 23 54 53 11 13 21 75 11 11

13

11 13 11

37

11 11 11 37 11 11 11

37

11 11 45 14 14 14 14 14

RW-169

EGN .. 1

Pg. 8

of

10 10/1/56

00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00334 00000 00000 00000 00274 00356 00311 00145

OOOl7

20000 00153 20000 00267 00000 20000 00015 00060 00015 00057 20000 00060 20000 00144 00364 10000 000)60 00020 20000 00020 20000 00322 000'60 00061 00020 00062 00145 00020 10005 00171 00015 00052 00360 00055 00357. ,00051 00357 00053 00360 00047 00015 00050 00352 00314 00035 00041 00036·00042 00360- 00047 00352 00314 00035 00043 00036 00044 - 00015 00047 0035200314 00035 00045 0003;6 000 46 0000000266 30041 14051 04045 34027 14051 34031 30043 14053 34033 11027

01M25 lDMP 00T04 00T10 01M26 ADNO FOOOO 00000 01M27 TN FOOOO 00029 01M28 TN COOOO 00030 01M29 ADMP FOOOO 00T08 OlM30 MPLD 00023 00104 5 01M31 SUMP 00027 00TI0 01M32 ADMP 00025 FOOOO S 02MOO ADRT 00023 00023 S 02MOl TN COOOo

COOOO

02M02 LDMP FOOOO 00025 02M03 SJ 02M04 02M06 02M04

TN

00023 00023 02MG5 TN 00024 00024

02M06 LOAD 00025 00023 5 02M07

PMNO

00000 00000

02M08

ZJ

02MIO 02M09 02M09 1P

OOKOI

00023

02MIO

LODV 00029 00023

02Mll STMP 00T08 00T12 S 02M12 ADNO 00T06 00000 S 02Ml3 RJ 01R30 OIROO 02M14 OVPM 00104 00000 02MlS 1J 00K02 02M20 02Ml6

·TN OOKOO

FOOOO 02Ml7 TP 00013

cocoa

02M18 LDMP Fooeo 00T08 02Ml9 MJ 00000 02Mll 02M20 TP OOKOI

FOOOO

02M21 ADNO 00T08 00000 02M22 TP Fooao 00TI0 02M23 TP 00T09 00T11 02M24 RP 30004 02M26 02M25 TP 00T02 OOTOO 02M26 TP 00029 00T04 02M21 TP 00030 00T05 02M28 LDDV 00T12 OOT06 02M29 PMNO 00000 00000 02M30 T J OOK03 02M40 02M31 TP 02M29 Aoaoo 02M32

TJ

00000 0,lM20 02M33 RP 30004 02M35 02M34 TP 00T04 00006 02M35 PDNO 00010 00000 02M36 MJ 00000 01M20 02M37 TP 02~39 02M50 02M38 MJ 00000 OOR04 02M39 MJ 00000 02MS1 02M40 Tp 00114 00004 02M41 LOST 00T06 00008

02M42 5TST 00025 OOT18 B

00217 00220 00221 00222 00223 00224 00225 00226 00227 00230 00231 00232

F 00233

I 00234

N 00235

0 00236

00237 00240 ITERANT 00241 00242

AND 00243

00244

FUNCTIONAL

00245

,j 00246

VALUE 00247

00250·

00251 00252 00253 00254 00255 00256

SET UP 00251

00260 . FORNEXT 00261

00262 ITERATION 00263 00264 CONVERGED 00265 00266 00267 00270 00271 00272 00273 00274 00275 00216 00211 00300

14 14 13 13 14 14 14 14 14 13 14 46 13 13 14 14 47 11

r~W-169

EGN-l

Pg.

9

of 10

10/1/56

30045 14053 04002 00000 00002 00035 00003 00036 04002 14051 15021 30045 10033 14053 05031 14002 04021 51027 00003 00003 30002 14031 00233 00235 00027 00027 00030 00030 30031 05027 24000 00000 00241 00240 00360 00027 14 30035 20027 14 34051 15055 14 05041 00000 37 00352 00314 14 20045 24000 42 00361 00253 13 00357 00002 11 00015 00003 14 30002 14051 45 00000 00242 11 00360 00002 14 04051 .00000 11 00002 00053 11 00052 OOO~4

75 30004 00261 11 0·0043 00041 11 00035 00045 11 ·00036 00046 14 30055 20047 14 24000' 00000 ft2 00362 00277 '11 00264 20000 42 00000 00212 15 3.0004 00272 11 00045 00006 14 74012 00000 45 00000 00212 11 00276 00311 45 00000 00150 45 00000 00312 11 00057 00004 14 30047 34010 14 34031 36063

02M43 RA 00T14 00K04 02M44

RJ

01R1S OlROl 02M45 TP FOOOO 00006 02M46 TP COOoo 00007 02M47

RA

00T17 00016 02M48 SA 00016 00015 02M49 TU AOaeO 00005 02M50 PDPD 00010 00010 02M51 IJ 00T16 01MOI 02M52

MJ

00000 OOR01 OlROO TV OOROO 01R15 01R01 RP 10003 01R03 01R02 TP 00013 00030 01R03 TP OOK01 00029 01R04

TU

OOROO 00004 01ROS

TP

00T15 00023

01R06 LDMP 00000 00031 B S OlR07 ^LDSU 00000 00r06 B

OlRoa

MPSU 00029 00031 01R09

TP

00029 00031 OlRI0 TP 00030 00032 01Rll

TP Fooeo

^0.0029

OlR12 TP

coooo

00030 01Rl)

RA

00004 QOK04 01R14

IJ

00023 OlR06

CJ1Ri5 MJ

00000 00000

OlR16 TP

00T17

Aoaoo

01R17 SJ a 1Ri8 01R18 01Rla TP 00T11 00023 01R19

TP

OOK01 00031 01R20 TP 00013 00032 01R21

TP

00013 00004

01R22

LDSU

oor06 OOT18 ^'8 01R23

MPPM

00031 00000 S 01R24 ^Z,J 01R25 01R31 OlR25

RA

00004 OOK04 01R26

IJ

00023 01R22 01R27

LDDV

00029 00031

;:;-. 01R28 STPM 00029 00000

-D OlR29

ZJ OlR30

02M40

...

'-" _I 01R30

MJ

00000 00000

0 01R31 TP

aOKOO

FOOOO

...

I 01R32

ADNO

00T06 00000 S

0 .:::> 01R33

MJ

00000 OIROl

J"

,-i 01R34

PDPD

000.10 00010

t- OOKOO 05 0.0000 00000 -01 F x ~ OOK01 01 00000 00000 F OOK02 01 00000 00000 1 F OOK03 01 00000 00000 -06 F OOK04 00 00002 00000

OOKOS 00 00777 00000 B START

00302 00303 00304 00305 00306 00301

PUNCH 00310

ITERANT 00311 00312 00313 00314 00315

E 00316

V 00317

00320

L 00321

U 00322

00323

T 00324

E

00325

S 00326

C

⁰⁰³²⁷

H 00330

A 00331

R 00332

A

00333

c

⁰⁰³³⁴

T 00335

E 00336

R 00337

ISTIC 00340

00341 00342 00343 EQUATION 00344 00345 00346 00347 00350

CONVERGED

00351

00352 00353 00354 00355 00356

C 00351

0 00360

N 00361

S 00362

T

00363

ANTS

00364

00000

RW-169

EGN-l

Pg. 10 of 10

10/1/56

21 00057 00363 37 00333 00315 11 00002 00006 11 00003 00007 21 00062 00020 32 00020 00017 15 20000 00005 14 74012 74012 41 00061 00167 45 00000 00145 16 00144 00333 75 10003 00317 11 00015 00036 11 00360 00035 15 00144 00004 11 00060 00027 14 32000 15037 14 32000 10047 14 14035 10037 11 00035· 00037 11 00036 00040 11 00002

00035

11 00003 00036

21

00004 00363 41 00027 00322 45 00000 00000 11 00062 20000 46 00350 00336 11 00062 00027 11 00360 00037 11 00015 00040 11 00015 00004 14 30047 12063 14 15037 24000 47 00345 00-353 21 00004 00363 41 00027 00342 14 30035· 20037 14 34035 24000 47 00352 00277 45 00000 00000 11 00351 00002 14 05047 00000 45 00000 00315 14 74012 74012 20 04000 00000 20 14000 00000 20 45000 00000 15 54143 36750 00 00002 00000 00 00777 00000·

45 00000 00000

Identification Tag:

Type:

Service Entrance:

THE RAMO -HOOLDRIDGE COHPORATION Los Angeles ~'5, California P10ating Point Card Dump

Specifications

HDP-5

mV-170 MDt.'

-5

Pg. 1 of ).

10/15/56

Service Routine (with subroutine entrance) Address 40024 b

Program Entrance and Exit: 37 40020 ~,o024 b Other Routines ,Used:

Coded and Checked by:

Approved:

This routine uses MDP-4 and SNAP Output

R.

Beach

w.

^{F. Bauer}

October,

1956

October,

1956

RW-170

MDP-5

Pg. 2 of 2...

10/15/56

Description:

This routine operates exactly the same as MDP-4 (Octal Card Dump) with the following exceptions:

1. Entrance Address is. 40024 b

2. Output is floating decimal on SNAP output cards. Addresses are five digit octal numbers with leading zeros suppressed.

3 • A parameter word of zero will dump cells 00000 - 00777 b.

The routine ,treats each word to be dumped as a floating point (SNAP form) number and converts it to a,floating decimal number. Non SNAP numbers (i.e. instru.ctiOns and fixed point numbers) may be included among the words to be dumped but their converted values will be meaningless.

The listing will be double-spaced,; however, if a card was omitted because it would have contained all zeros, no additional spacing is provided on the listing.

The routine is essentially a driver for the SNAP output routine and MDP ..

4,

modifying 'each so that MDP-4 uses the SNAP output routine instead of its octal output section.

10-17

RR-171

REMINGTON RAND UNIVAC

ST. PAUL DEPARTMENT-INFORMATION SCIENCE

IS

December

1956 110) TO llO)A

CONVERSION ROUTINE

I. TYPEs Service routine or subroutine.

II. STATUS. Gode cheeked and machine checked by Bill Wallace.

III. PURPOSEs This routine changes A and Q machine addresses from 20000 and 10000 to 32000 and 31000 respectively, and detects magnetic tape and external function instructions. Various options are provided for print out of those addresses where an

A

or Q reference i8 modified, (indicating also u or v portion) and punching the converted program in bioctal or flex code.

IV. USAGE:

A.

STORAGE

REQUIRED:

The program 1s coded in RECO form ^and it is therefore possible

to

operate the program from ⁸ location providing 320 ootal drum address and 2000 additional octal drum addresses for a

nss

image regIon. Such a location of the program Rnd image region is done b.y Rssigning the desired starting addresses

to

regions BB end' IR respectively, of the reco tape (see coding)

'811

other regions being in

HSS,

and hence remaining the same.

The regional assignment can be ana separate

tape

from the main program reeo tape, but this tape should have END. c .r. at the end.

(See RECO write-up.)

In addition to the REeO tapes, 8 bioetal tape of the program is availBble where the program ie stored at 66000-66320, with the image region 76OCJO-7?7n.

B. INP11f-OUTPUT: Output is a punched tape in bioctal or nex code of the changed program if desired. Also the following is printed out as the conversion routine is operating: (This is also optional.) u a88Ba or VSS8SS, where 8B6a8 is the sddress where an A or Q

reference has been modified ^8nd u or

v

shows whether the u or v.address of the instruction hes been modified. Also,

TAPE 1s

printed out when an 1103 magnetic tape instruction is enco~tered,

snd

EF

and address when an externel function eommand occurSo

c.

OPERATING

INSTRUCTIONS:

(1)

Used ^8S e. service routine proceed as follows: (the term

"program" here refers to a program to be converted.) 8) Master cleer, MD stsrt

b) Set PAK to 66000,

(or bb)

c) Insert in ~ the first addreB~ at the prosra- d) Insert In ~ the last address ot" the program e) Insert in v address of

Aa

the address

ot

the last

instruction ot the program, or the la8t address ot

BWapeo

RR-171

- 2 ...

r) Insert in v address

ot AL

the following codes tor thevarlous options:

00000 00001 00002 00003

00004 00005

bioctal punoh of converted program and print out of addresses where modification occurs.

same as above but no print out.

flex code punch of converted program end print out.

same as above, but no print out.

print out, but no punch of converted. progreme no print out and no puncb of converted progrmn.

A 56 0 66010 (bb10) stop occurs it a gross error is made in the set-up, e.g. transposition

ot

^~ and ~.

(2) Used as a subroutine, proceed as followsl

(3)

a) Program the transfer of par8llleter as listed above to the

A

and

Q

registers.

b) Execute the instruction}

RJ bb2 bb

c) The options are selected in the same manner a8 previously shown.

a) The use of this conversion routine assumes that tbe program to be converted is stored either all in core storage or all in drum storage.

b) The conversion routine is coded for operation on either an 110)!, or on the 110,3 (Serial 9) at RRU, St. Paul.

10-19

RR-111

v. COPINQ

A. Hegions

re bb66000 r. un124

re

lr76000 re val37

re rf30000

re up155

re orO re prI62

re ob33 _{re tp173}

re

od54 re ef200

re oMI re os205

re kk75 re dd212

re mm1l2 re tt.3l0 B.

Program

bbO 45

^{0 bblO}

^Entrance

1

56

^{0 bblO}

Errer step

2

45

^{0 (rt)}

SubrCiJUtlne exit

.3

^{0 0 0}

storage first address

4

^{0 0}

⁰ storage last address

S

0 0

0

storage initial A

6

0 0 0

stQrage initial Qr

7 4~ 0

tt Constant

10

tp

Q

bb6

III

It

10000

bb5

12

It

00000 A

13 ^tp

0

ir

14 tp

bb 0

IS ^rp

31m

bbl7

16

tp 1

irl Store HSS

17

rp 30400 cb

To start ot core program

20

tp

bbl

orl Conclusion ot ^program

21

rJ) )1777

bb23

22 tp irl

1

Restore HSS

23 tp ir

0

24 tp bb6

Q

Restore

Q

tor dump

25 tp

bb7 A

-

^,...

^{26 eJ bb2 bb31} Test, subr. or

BerY?

t- 27

rj 700)6 tTOOO6) No, 8'Qbr.

,....

...

)0

45

^{0 bb2}

^{To exit}

•

0

31 rj 70036 f10006)

,...

I 32

56 ⁰⁰⁰⁰⁰

^bb

0 0 ::t-,....

33 ^{obO ej dd43}

^{od No punch}

to-

=-<

34

1

e.1

^{dd64 od2}

Punch fiex

~

35

2

e.1 ^{dd6S odS tJo} print, no punoh

36 .3 e.1

dd66

cd13 No print, punoh flex

37 4 e.1 ^{dd60 od6} No print, punch biootal

40 5 ^{tp or;}

^A

Laet address

RR-171

- 2 -

41

6 iJ ^{dd ot}

^HSS?

42 7

^.qt

ddl or4 Store first address

43 ^{10 lq}

^q

²⁵

J.4

11 qt ddl or3 Store last address 45 12. tv or3

^DUB

Set up transfer

46 ¹³

^la

^{or3 20017} of Modified Contents

47

14 tu

^A kk

Set up first addres8

S<>

15 ra or4 dd60 to be modified

51 16 st or3 tt No. of words

52 17 ij tt kk

53 20 45 0 bbl Error

54 ^{odO ,tv dd67} mm4

55 1 45 0 cb5

^No

punch

56

²

^{ra bb27}

^dd60

57 3 rs bbJl dd60

60 4 ^{45 0 cb5} Flex punch

61 5 rj cd1 cd 62 6 tv up4 'ualO 63

7

tv

up4

ua12 64 10 tv va15 va7

65 11 tp va15 vall No print

66

12 45

⁰

cb5

67 1J rj cd4 cd2 70 14 45

0

cd6

71 efO ra or5 dd3 Add 76000 to V

72 1 ra

Q

dd2 Add 76000 to U and V

73

²

tp cs pr Arrange to print core address

74 3 45 0 cb7

75 kkO tp

(ff) Q

76 1 tp

Q

ttl

77 2 qt dd5 tt3

^Mask

operation code 100 ₃ tp tt3 A

101 4

^ej

ddlO ef External function

102 5 ej ddll

mm

Final stop

103

6

ej ddl2 mm Interpret

104 7

^rp

20014 kkll Commands where

105 10 ej dd13 tp V only to

be

modified and

tape

commsnda

⁽ⁱ

--- ^{106 11} rp 20004 kk13 Split instruction,

~

1(17 12 ej dd2'1

mmlO Modify

U

^on~

t-P""4

110 13 rj ua6

ua

Modify U

'-"

I III 14

rj va5

va

Modify V

0

P""4

112

^JIIIIlO

tp ttl (ttl) Transfer modified

I

113

1

tp

kk A

Content

0 0

114

2 It

25

^A

Obtain current

0-- P""4

115 3

^{dd6 A AddreS8}

l"-

rs

><

116 4

^ej

or5

^blQl

Test, end ot

p."

117 5

ra

^kk

dd7 MOdifIable address

10-21

RR-171

... ;3 -

120 6 ra

JmJl

dd60 Add 1

121

7

45

0 obI?

122 10

rj

ua6 ua

Modify U

only

123 11 45 0

^IIDJl

124 uaO tp

tU Q

125 1

lq ^Q

25 . st

l26

2

qt dd33 tt4 Mask 1 octal digit .127 J tp tt4

1

130 4

^ej dd34 \18?

Q?

131 5

^ej

dd35 uall A1

132

6

45

0 if

133 7 ra; ttl ddJ6 Add 21000

134 10 45 0 up To print

135

^II

ra ttl dd3'Z Add 12000 136 12 45 0 up

137 vaO tp

ttl Q

140 1 qt ddJ3 tt4

141

²

^{tp tt4 A}

142 J

^ej

dd34 va6 Q?

143 4

^ej

^{dd)5 va 10}

^A?

144 5 45 0

^ff

1.45

⁶

ra

ttl

dd40 Add 21000

l46 ^{7 45}

⁰

^{vall To Print}

147 10 ra ttl dd41

Add 12000

150

11

pr 0

dd42

Carriage return

151 12 pr 0 dd43

Space

152 13

^pr

0 dd44 "V-

153 14

rj

prl0

pr

154 15 45

0

va5

155

upO pr 0

dd42 Carriage return

156 1

pr 0

dd45 "U.

157

2

pr 0 dd43

Space

160 J rJ prl0

^pr

161

₄

45

⁰

ua6

162

prO

tp kk

Q

163 1 1q

_Q 6

164 2 tp dd43 tt2

Index

165 .3 1q

Q

:3 166

4

qt dd46

^A

167 5 at

dd47

pr6 Print

digit

170 6 (pr

0 rt)

171 7

i~ ^{tt2 ~3}

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Remi ;igton Rnnc! Un i vac

Floating Vector Arithmetic Package

GENEHAL DESCHIPTION

This package contains four subroutines: vectbr roll-off, vector unpack, scalar product of two vectors, and vector sume Each of the subrou- tines is self-contained and can be used independently of the others. These operations are performed on arbitrarily located vectors.of not more than 108 elements. The arithmetic is floating point with one biased characteristic serving for all of the elements of a vector. The bias of the characteristic

PACKEO FOI!)1 OF A VECTOR

Associated with every vector. X

=

(xl,.o~tXn)t is a set of numbers, (bl, •• o,hn), each element of which is either 0 or 1. This set of numbers is defined in the following t'my: if xi=O, then bi=O; if xi10, then bi=l. The

three binary numbers,

blx235+b2x234+ ••• +b36.

35 34

b37x2 +b30x2 + ••• +b72 , 35 34

b73x2 +b

74^x2 + ••• +b I08t

where bn+l=bn+2= ••• =bl08=O if n~I08. are called the Q-words of the vector

X.

It is clear that u vector is well-defined if the Q-words, the number

of

ele- ments, and an ordered list of the non-zero elements are given.

The operand vectors of the floating vector subroutines must be packed (or stored) in the manner which we now describe. The first three addresses of a vector storage location are occupied by the three Q-words.

...

N r-

r-t ' - '

o I r-t

o I

o :J' ...

r-

~ c...t

-2-

RR-172

The mantissae of the non-zero elements of the vector are stored sequentially in the addresses immediately following the address of the last Q-word. These mantissae are sealed so that th-e numerically largest has 32 binary' digi ts.

There are no blank addresses between successive vectors.

Each vector hus a keyword. The v-address of the keyword contains the biased characteristic of the vector; the u-address contains the starting address (address of the first Q-word) of the vector. The address of the key- word of a vector is called the directory address of that vector; the aggregate of all the directory addresses of a system of vectors is called the directory of that system. The ke~vords are$ored in the same order as the corresponding vectors, and there are no hlank addresses between successive keywords. Fol- lowing the last keYWQrd in the directory is a pseudo keyword. If the last non-zero mantissa of the last stored vector is in location y. then y+l is entered in the v-address of the pseudo-keyword.

NOTATION

1. Throughout the subroutines three blocks of addresses are utilized for vector work spaces and temporary storane. We shall refer to these blocks as Ri" S. and T. By Ri we will mean the ith address in the block R. We denote by m the number of elements in the operand vectors •

PROGRAM PAHAMETERS

1. Locations 00005 through 00017 are reserved for constants and program parameters.