Q Curves:- Five-function Curve-fitting Computer Program
MEMORANDUMRM-5 762-PRDECEMBER 1968QCURVES:- A FIVE-FUNCTIONCURVE-FITTING COMPUTER PROGRAMH. E. Boren, Jr.PREPAWD FOR:UN I7D STATES AIR FORCE PROJECT RANDSANTAMONWCA*CALIFOIRNIA-
MEMORANDUMRM-5762-PRDECEMBER 1968CURVES: A FIVE-FUNCTIONCURVE-FITTING COMPUTER PROGRAM11. E. Boren, Jr-?-clldtItI ,lpll t mro1til -\ii 14r w id-i]IR\ 1',N01I lrh'tlt,f4ip jI::11,1lltfljiclfltni
1h-oudI- lilicatl wT114, R- i rCoypild,If IlitTaIIIofl%-ll lt'rilicl tliI ofhePublished by The RAND) C7roratiotijIts% trtiiit c rcitircof pith.,%t
IllPRE'SAGEThe computer program (CURVES) described in this Memorandum was&developed in support of estimating-relationship research e.orts beingconducted in the RAND Cost Analysis Department. This programreprese'ntsa compilation of various parts of existing programs writtenby theauthor, with modifications being included where necessary.The authormakes no claim to originality or to efficiency of operationwith regard to the program.The main purpose of writing such a program wasto have available for cost analysts an easily workable, user-oriented,curve-fitting computer program--especially adapted to handlethiathematical functions most commionly used in the development ofestima tingrelationships,I
-V-SUMMARYThis Memorandum describes a k'RTRAN-IV curve-fitting CO(mputprograment.that has been developed within the i2ND Cost Analysis Departlhe p-ogram makes least-squaresdeterminationsof the param-eters of any of five mathematical functions selected by the user,given a set of observations on the dependent and independent variablesof interest. The functions available in the program are the line,parabola, power, asymptotic-power, and exponential. Up to three independent variables may be used for the line and power functions.Also, the Y-intercept may be specified for the line, parabola, orasymptotic-powerfunction.A discussion of the characteristics of the functions is presentedin Section I, including an examination of those nonlinear functionsthat require special methods for solution.Also included is a briefdiscussion of the statistics used in the pregram.Specific detailson the operation of the program are presented in Soction II.section also treats the options available to the user.puts are discussed inSection III.sample outputs from twoMathematicalThisProgram out-For the benefit of the reader,.,ns are shown.considerationsrelating to nonlinear-least-squaressolutions are treated in Appendiccs A and B.A listing of the FORTRAN-IV computer program is presented in Appendix C.
-vii-CON4TENTSPREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . .SU11MARY . . . . . . . . . . . . . . . . . . . . . . . . . . .vLIST OF FIGURES.ixLIST OF TABLES.xiSectionI. 1,1TROEDiLflON.Program Description.IFunctionType's-.2Function Caracteristics. .3Nonlinear-Least-Squares SoIluions.6Statistical Considerations.7II. INPUT PROCEDURES.8Title Card.8Order Card. . 13format Card.17Scale Card. I.I.19Data Cards.21Blank Card.21End Card (Optional).23Sumary.23III.PROGRAM .AJTPUTS.25App enidixA. NONLINEAR-LEAST-SQUARES CONSIDERATIONS.35B. LEAST-SQUARJES SCLUTIOh FOR ASW(TOTIC- POWERFUN4CTION4.39C. LISTING OF CURVES FORTRAN-IV C((PUTER PROGRAM .47
-ix-FIGUXES1.urves used in Program for a OneExamples of.Independent-Variable Case .49.92.Flow of Operations3.Example of Title Card . .164.Example of Order Card .185.Example of Format Card .186.Example of Scale card .7Example of Data Card .8.Arrangement of Two Data Cdrd Packs forTh.re e Run s .9.First Page of Oitput,f O t put.(Linear Case) .(Linear Case) . .Second P ,ts-11.First Page of LW tput12.Second Page of Output13.Listing of In.puts for Two Runs .whose Outputs.are Shown in Figs. 9-12 . .(Power Case) .222426. 2710.(Power Case) .20.28.2930
xi -TABLESI.Surmmary of Information on 'Atle Card.142.Statistical Equations Used in Program. 32
I.INTRODUCTIONPROGRAM DESCRIPTIONA FORTRAN-IV curve-fitting computer program (CURVES) has beenwritten by the author that makes least-squares determinations of theparameters of any of five types of functions, given a set of observations on the dependent and independent variables of interest.Thesefunctions are commonly used in the derivation of cost analysis estimating relationships, and are:(a) line, (b) parabola, (c) power,(d) asymptotic-power, and (e) exponential.They are described in de-tail in subsequent parts of this section. Standard statistics relating to "goodness-of-fit" measures are also calculated in the program.No predictive statistics are included, however, because of thedifficulty of obtaining such statistics for the nonlinear functions-the latter three above.Consequently, the prog-am is intended essen-tially for curve-fitting.The CURVES program can handle up to 200 data points for each regression and is so structured that if a set of data cards containsdata for several separate regressions, that set needs to be enteredonly once.This obviates the need for duplicating such input datadecks for each regression run.KA variable-formatprocedure isprovidedthe user so that data may be entered in any order on the input cards.Also, an option is provided to allow the user to specify the Y-interceptvalue (regression constant) for the line, parabola, and asymptotic-IJpower functions.The program is written completely in FORTRAN-IV, using A4 formatsfor all alphanumeric information.programs are used.No matrix-inversion or other sub-All solutions are made through either standard,algebraic methods for the linear and parabolic cases or through iterative methods f-or the other cases.Consequent'y, the program shouldbe readily adaptable to other computer systems.*In this Memorandum, a linear function is defined as oue whichis linear with respect to all of its parameters.Under this definition,the parabola is consi4ered to be a linear function.
-2-FM!CTION TYPESThe functions availablein this program wurc.hos.:ncn the basis of their application to tLL derivationLne (containiniDtothreeindepndentvwriabcs".(XI),A BY)f cost analysisThey are:estimating relationships.I.principallyY; A(Xi)C (X,,Y-interceptA BY(Xl)(X2) C, (X3(A) may bespec if ied.2.Parabola(XI)Y A B3.4.Power (containing up to three independent variables)BYA(XI)YA(Xl)BYA(Xl)B.(x2 C)(X2) C (X3) D ,Asytotiz-PowerY5. C - (Xl) 2 . [A (X)B CY-intercept (C)may be specified.Exponential [A 4 B - (Xi)]As in FORTRAN notation in the program, the independent variablesWhen only on,. indart, itpresented by Xl, X2, and X3, respectivelypendent var lab.IIs considered, XI is us odAlso, a.; t xplainedonly pusitive values aie considered for all variables.later,
-3-whereY dependent variao-le,Xi, X2, X3 - independent variables,A, B, C, D - parameters to be determined by least-squares methods,e constant.2.71828.FUNCTION CHARACTERIS TICSExamples of some of the types of curves that can be obtained fromthe five functions are shown in Fig. I for a one-independent-variablecase.LineThe linear form is the most simple of the forms treated her-.Itscharacteristics are well knov,n and, in the opinion of the author, needno further elaboration.The user has the option of using up to threeindependent variables and also the option of specifying the Y-intercept(A).ParabolaSometimes the parabolic function is used to represent points thatlie along a curve having a Y-intercept(including zero).However, onemust be aware that since this function is actually a polynomial of degree 2, its curve always has a maximum or minimum point(vertex).Th1ismeans that the effect of the independent variable (XI) on the dependentvariable (Y) is reversed once tnis point is traversed.Again, the userhas the option of specifying the Y-intercept (A).PowerThe power function is one of the more common functions used incost analysis work.linear form,cost curve."isA plot of its logarithmic counterpart, the log-known as the "learning curve" or "improvementHowever, for reasons discussed later, the power, ratherchan the logarithmic, form is used in this program.For this function,
LYParabola-X 2AB XI-Y1VVertexYAsymptotic -powerY-,AX',PorA*X18B 1B QasymptoteGoes through oriain for B 0OrExIlpential-ribeCsYC
-5-the user has the option of using up to three independent variables.For positive exponent B, thi,, curve always passes through the or-igin,as shown in the figure.Thereforc,itshould never be used where apositive Y-interc.ept is desired or logically required.For negativeB, the curve is undefined at XI 0 and is a declining curve, approaching asymptotically the Xl-axis as Xl becomes large.AsyMptotic-PowerAn examination of the fourth function, the asymptotic-power, showsthat the curve has a horizontal asymptote of Y That is, as X1 becomes large, the first term (AC for negative B.Xi B ) approaches zero,and hence the value of Y approaches that of the constant term C.Con-sequently, there is a level-off effect associated with this curve fornegative B.This function may thus be used to represent points thatlie along a curve either increasing or decreasing to a horizontaiasymptote.Like the power curve, this curve is undefined at X1 0for neg -ive B.For positive B, there is a Y-intercept equal to C.As XI becomes large,the first X1 B ) ultimately becomes largeterm (Acompared with C, and therefore Y approaches a pure power function(AX1B ) in this region of Xl.A plot of the asymptotic-power function on log-log paper producesa curved line at low values of XI that approaches either a horizontal(B 0) or inclined (B 0) asymptote at high values of XI.Where apositive Y-intercept is desired, the user may specify a value for theconstant term C.Exponent ia 1The last form, the exponential, is used to represent points thatlie along a curve having a positive Y-interceptbe either a rising (B 0) or falling curveapproaching asymptotically the Xl-axis.of the exponentialfunction isstraight line on semilog paper.(eA).(B O)--the falling curveThe logarithmic counterpartthe somilog function,Thatis,The curve mayIn Y iswich produces aa linear function
-6-As was the case with the power function, for reasons discussedof XI.later, the exponential, rather than the semilogarithmic, form is retained for this program.NONLINEAR-LEAST-SqUARES SOLUTIONSIt can be shown mathematically that the least-squares solutionsof the parameters of any function are always exact and unique providedthat the function is lineat with respect to all of its parameters.Therefore, for this program, the line and parabola produce exact andunique solutions.(The term exact is used to refer to a solution thatcan be obtained algebraically.)However, the latter three functionsare not all linear in terms of their parameters.Thus, their solutionsare not exact and, as shown later, may not represent absolute minimumsof the sum of squares of the Y residuals.They must be obtained insome other way--usually through some type of iterative procedure.(Thegeneral principles of such procedures and other mathematical considerations relating to the solutions of nonlinear-least-squares equationsare presented in AppenJix A.)Foi the power and exponential functions, a modified Gauss-Newtonmethod is used, in which initial estimates are obtained from the logarithmic solutions (which are exact) and then correctiou., guaranteedto produce convergence to a solution, are applied to those initial estimates.fliis procedue is repeated until the absolute change in thevalue of each parameter becomes equal to, or less than, some predetermined value (10-8in the program).-The solution of the asymptotic-power function is based on anotherbecause there appears to be no easy way totype of iterative procedu*iAll logarithms discussed hereit are natural logarithms (base e)and are represented by In.This procedure is described in detail in RM-4879-PR by C. A.Graver and H. E. Boren, Jr., Multivariate Logarithmic t,,d ExponentialIt may be notedRegression Models, The RAND Corporation, July 1967.that conceptually, the solutions for the power and exponential functions are each different than for their logarithmic counterparts (seehe above referenced RMAlso, the term "exponent ial" inAppendix A).is equivalent to ti, term. power" in this Memorandum.kII
-7-obtain the initial guesses that are requtred for the modified GaussNewton method.,iisprocedure is treated in Appendix B.STATISTICAL CONS IDERATIONSBecause of the difficulty of calculating and applying predictivetype statisticsitfor the rnonlinear functions,"goodness of fit" statistics in the program.was decided to use onlyConsequently, this pro-gram should be regarded as essentially a curve-fitting program withonly those statistics being used that relate to how well the curve fitsthat particular set of data.Also, it '-hould be noted that the statis-tics may not have exactly the same meaning for the power, asymptoticpower, and exponential functions as for the line and parabola becauseof the nonlinear chazacteristics of the former three.In general,statistics for nonlinear functions should be used with care.For ex-ample, unless there is proof to the contrary, the F statistic for anonlinear function probably should not be compared with the F table.Such statistics should generally be used only qualitatively--notquantitatively--until a thorough investigation is made into the application of such statistics to nonlinear functions.The principal reason for omitting the logarithmic formsin thisprogram is that it is very difficult to compare their fits statistically with those of the nonlogarithmic forms (see Appendix A).res lt,no logarithmic cirv#-.nre .sodianthe statisticalAs aresultsrelating to the five functions used in this program can be comparedmore directly.However, since the iterative solutions for the powerand exponential functions require that their logarithmic solutionsbe determined for the initial estimates, these solutions are alsoprinted in the output (without any related statistics) for the benefitof the user.A sumary of the statistical au-atioos iPL4 LLedin Se:tionIII, following the discussion on program outputs,The use of predictive statistics for the powerfunction is treatHowever, that program requiresed in RM-4879 (see previous footnote).many additional subroutines, which in the opinion of the author, wouldmake the CURVES program prohibitively large, slow in operation, andrestricted to 50 or less data points.
-8-II.INPUT PROCEDURESThe flow of operations within the program is depicted in Fig. 2.The program is so structured that many sets of data may be entered, inwhich each set (200 data points) constitutes a run.As soon -, eachset is read in. the program operates on that set before proceeding tothe next set of input data.deck of cards.Each data set may be entered on a separateOn the other hand, several or all of the data sets,if space on the cards permits, may be entered on one deck of cards,thus effecting considerable savings in the use of cards and in theeffort of duplicating a deck of cards containing data for severalruns.A variable format procedure is used, allowing much flexibilityin the format of the input data.Listed below are tho types of cards that must be entered for thefirst run.I. Title card2.Order card3.Format card4.Scale card5.Data cards6.Blank card7.End cardNeed to be entered only once if input datafor all runs are to be entered in sameformatUsed only if data are to be scaled or ifY-intercept is to be specified.Optionp!TITLE CARDThe title card must b- entered for each run.In addition to thetitle (alphanumeric), this card also contains other information aboutthe run.If the two cards relating to the variable-format procedures(order and format cards) are to be read, a "I"the title card.quentFor th, first run, a "I"must bu entered.runs involving different input formats,However,ifis entered in Col. I ofizFor subse-still must be entered.data for all subsequent runs are to be entered on separate
-9START[SET PAGE NUMBER TO 1.IWPAGE 1[SET RUN COUNTER TO 1.NRUNSUBROUTINE INPUTREADS INPUT DATA FROM CARDSOR UTILITY DISK AND, IF SODESIGNATED, WRITES DATA ONTOUTILITY DISK FOR REREADINGDURING THE NEXT RUN.1ISUBROUTINE CHECKPRINT MAIN HEADINGCHECKS INPUT DATA FOR ERRORSAND PAGE NUMBER O NFIRST PAGE.AND, IF SO DESIGNATED,SCALES DATA.STEP PAGE NUMBER BY I[PAGE WIAGE ISUBROUTINE SUMSORDERS THE DATA FROM LOW TOHIGH VALUES OF Y IF INDICATED;OBTAINS VARIOUS SUMS ANDMEANS OF THE INPUT DATA; ANDOBTAINS THE STANDARDDEVIATIONS OF THE INPUT DATA.CLEAR COMMON.A(SET VALUES OF MOSTGWVARIABLES IN COMMONTO ZERO)SET SUBROUTINE.55CHKVAUINDICATOR TO I.IND 1SUBROUTINEREADSUBROUTINE PkiNTPRINTS APPROPRIATESUBHEADING FOR3EQUATION BEINGUSED .READS TITLE CARD A D,IF SO DESIGNATED, ORD[RCARD, VARIABLE 4FCNAATCARD, AND SCALE CARD .I.3 (Power)E u t o s u.d ir g oLINE)intercept A S (XI)C.KYVARIABLE-FORMATDEINAO INq A (XI C (X2) 4AFTB"l)*C(2)D(3CALX)PARABOLAY A B (XI) C (X I )ICARD2. 9DFEIGATIORNEINTRAsymtotic-power3.POWERY A (X1)'(X2 )c (X3)D4.4.ASYMPTOTIC -POWER IC5.EXPONENTIALy e(A-0I ( X )]Fig. 2--Flow of OperationsIniterceptmay beA may becified
-10-5)fmpfoti c -powerY LineSUBROUTINE LINESUBROUTINE i SYMSELECTS APPROPRIATE SET OFNORMAL EQUATIONS FORLEAST-SQUARES SOLUTIONOF EQUATION OF A LINECONTAINING UP TO THREEINDEPENDENT VARIABLES.(INTERCEPT A MAY BE SPECIFIED.)DETERMINES L AST-SQUARESSOLUTION OF ASYMPTOTICPOWER FUNCTION.o-2ParabolaSUBROUTINE PARASUBROUTINE SOLVESELECTS APPRC('RIATE SET OFNORMAL EQUATIONS FORLEAST- SQUARES SOLUTIONOF EQUATION OF A PARABOLA.SOLVES SIMUtTANEOUSLYLfP TO FOUR LINEAREQUATIONS.--(INTFRCEPT A MAY BE SPECIFIED.)ISUBROUTINE EXPOFIRST. DETERMINES LEASTSQ0ARES SOLUTION OF SEMILOGARITHMIC EQ UATION.THEN USLS THAT SOLUTION A'STARTING POINT FOR ItERAI'SOLUTION OF EXPONENTIALFUNC TION.SUBRO.TINE POWRFIRST,SELFCTS APPROPRIATE SUT.OF NORMAL EQUATIONS FCR4LEAST-SQUARES SOtUTION OFLOGARITHMIC-LINEAR EQUATION.THEN USES THAT "OtUTION ASSTARTING POINT FOR ITERATIVESOLUTION OF POAER FUNC'ION."-----"AIBROUTINE ITERDETERMINES LEAST SUARESSOLUTION OF POWEL ANDEXPONENTIAL I.)'LATIONSSOLUTION 1S gAkED ONMOO IF IT)6 ALUSS- NEWTONME THO).Fig.2--Flov of Operat tons(Cont.)
-II1--SUBROUTINE5 IAT,ALCULATES STANDARDSTATISTICS RELATING TOGOODNESS OF FIT."SUBROUTINE OUT]PRINTS SOLUTION ANDSTATISTICS RELATING TOGO000NESS OF FIT."SJRW.UiINE QUT2PRINTS LISTING Of INPUTDATA, CALCULATED YVALUES, Y RESIDUALS, ANDPtRCENT Y DEVIATIONS.rSTEPRUN COUNTER}NRUNNRUN - II!AIT PAGE NLUMM.RTO 1,PAGIFig.2--Flow ofOperations(Gont.)
-12-decks of card&: usingiLhe same format as the firstonly needs to be entered for the first run.run,then the "1"Whenever the 1 is notentered, then the order and format cards are not entered.If input data representing several runs are entered on one deckof cards so that for subsequent runs those same cards will, in effect,be reused, and hence re-read (in different fields), then a "1" is entered in Col. 2 of the title card for the first set of such data to beread in.This causes the machine not only to read the first set ofdata but to write all of the input data onto a utility disk for rereading during subsequent runs.Unless the user selects another util-ity disk or tape, the program automatically uses utility disk S.SU04(FORTRAN logical unit 4) for this operation.For the remaining runs that use the input data from the same deckof cards, a "2" is entered in Col.2 of each title card for those runs.This caaxses the machine to read the input data from the utility disk(instead of from new cards) in accordance with the format instructionsso entered.Column 3 is used for the function designator.An integer from"1" through "5" is entered to designate which function is being considered for that run.The integer designators are as totic-powerExponentialThere must be an integer of one of the above values entered in Col.3 for the first run.If Col. 3 is left blank after the first run,then the value for the previous run is used.Thus, if the same typeof function is being examined for a series of runs, its designatorneeds to be entered only for the first run.Column 4 is used to designate whether the input data are to beordered from low to high values of Y.the data are to be ordered.A value of " 'signifies thatFor the first run a blank (zero) signifiesthat the data are not to be ordered.However, for subsequent runs a
-13-blank (zero) signifies to the machine that the value of the orderdesignator for the preceding run is to be used.Again, this is doneso that if all runs in a series are to be either ordered or unordered,the order designator will only have to be entered for the first run.In the case where a zero is desired fur the designator after a "1"has been entered previously, a "2" must be entered.This in effectsets the value of the designator to zero.The scale designator is entered in Col. 5.If any of the dataare to be scaled (using the Scale Card described later), a "1" isentered in this column.Otherwise it is left blank (zero).Column 6 is used to designate whether the Y-intercept term forthe line, parabola, or asymptotic-power function is to be specified.This is done by entering a "1."Otherwise Col. 1 is left blank (zero).If a "1" is entered in either Cols. 5 or 6, or both, then the scalecard is entered in the order shown previously.Columns 9 through 72 are reserved for the title.The title mayconsist of any alphanumeric symbols.A sunmmary of the information on the title card is given in Table1. An example of a title card is shown in Fig. 3, in which a linearregression is to be made on the input data ("I" in Col. 3).The data;ire to be ordered with respect to Y ("1" in Col. 4) and are to bescaled ("l" in Col. 5).The "1" in Col. 1 indicates that the orderand format cards are to be read next.ORDER CARDThe next card (when used as specified by a "I" in Col. 1 of thetitle card) indicates the order (from left to right) in which the dataare located on the data cards.The order (designated by alphanumericsymbols) is entered in Cols. 1-2, 4-5, 7-8, 10-11, and 13-14.Depend-ing on the number of independent variables being used and on whetheridentifiers are being used, Cols. 7-14 may not be required.bols used to show the order are as follows:The sym-
-14-Table 1SUIOIRY OF INFOATION ON TIT1E CARDColumnsRemarksA "I" indicates that the order card and formatcard are to be read, respectively, following thetitle cardIf blank, no such cards are to beread. A "I" must be entered for the firstrun.2A "I" indicates that th 0 input data cards containdata for several runs and are to be written onto autility J"k(Sysr :9 Unit S.SU04, FORTRAN Logicallit 4), A "2" indicates that the input data areto be read from the disk.If blank, the inputdata cards for this run are to be read unly once.3An integer from "1" to "5 is used to designatewhich function is being used. This is done c-powerExponentialIf blank after the first run, the value for thepreceding run is used.4A "1" indicates that the data are to be orderedfrom low to high values of Y. For the first runa blank indicates that the data are not to beordered. If blank for subsequpnt runs, the valuefor the preceding run is used. A "2" is used forsubsequent runs to restore the designator to zerowhen desired.5A "1" indicaces that the data are to be scaled.Otherwise, it is left blank.6A "" inuicates that a Y-inte, pt is to bespecified for either the linear, parabolic, orasymptotic-power case. Otherwise, it is leftblank.
-15-Tabl1c 1 (Cont.)SUI4ARY OF INFORMATION ON TITLE CARDRemarksCclu'trn,-7-8Not used.9-72Title for run.symbols.May consist of any alphanumeric
IRNIL1II- EiR00000 0 q 00 00 0 0 0 00 000 0 0 00a4444444444 4 4 A4 4 4 4 4 400 000 00DO004 44 444 44 4 44 4 44 44 4 44 444 4 44441'1 1R7I1 iSl 1781181791'41711 1185130 c o04 44 4 44 4 4444 49488178877 177 7 1718884488i9 9 9 993 9 3939 9499399 9 9j9 9 599 9 9 9 99 3 9 99 939 9 39 99 9 999 39 9 9F ig.3- -Examp I p of Ti t ICa rdGJ000 , J99 99
-17-SymbolType .of DataIDYlXIX2X3Identifier (alphanumeric) (uptional)Dependent variable (requireJ)First independent variablt- (r-quired)Second independent -vdriable (optional)Third independent v riable (optional).MAy be in anyorder from leftto right./Suppose that apzf data is to be entered in which values forthe three independent variables and the dependent variableare locatedin Cols. 1-12, 13-24, 25-36, and 37-48, respectively. Supposealsothat an identifier (a six--digit integer) is in Cols. 55-60.Then,"XI,""X2,"! "X3,f1,"Yand "ID"would be entered respectively,inCols. 1-2, 4-5, 7-8, 10-11, and 13-14 to show the aboveorder acrossthe card.All alphanumeric information is treatedin A4 formats in thisprogram in order to be adaptable to IBM-360 systems.In addition, theidentifiers may be entered in either A4 or 2A4 formats.This is indicated in Col. 16 on the order card by either a "I" (A4)or a "2"(2A4). If Col. 16 is left blank, after the first run, thenthe valuefor the preceding run is used.The order card is shown in Fig. 4 for the above example,in whiicha 2A4 format is to be read for the six-digit identifier.Since thecomotnas separating the symbols are in columns that are notread by themachine, they may be used for the purpose of clarification.FORMAr CARDThe format card indicates where the data are located on thedatacards. Again, this card is used only if a "I" is enteredin Col. Iof the title card. This card must begin with a left-handparenthesisand end with a right-hand parenthesis beforemation within the parenthesesformats.Exceptand the infor-must conform to the rules for FORTRANfor the alphanumeric identifiers, all input datamuit be in real-number (floating point) formats.shown in Fig.,The format card5 would be used for the previous example.
-18-1002 2 2 2 22 22 22 200023d2,3O* 3 3 3 33 l 33*32 22?33 3325 5552225 5 55 5 5 S5 s000000000100A2222222272222 2122222?2?2?2222?2?222?22?4 1j4 A44 4 4 4 44 44 44 44444 4 41 4 Aq 44 4415 5955555 5 5 559595S95 526650000j'1i 401l322:3 3 133 3 33 33 3303 33 33 3133 3 333 3 33 3 33 3 333 33 33 3 3 31303 33 1334 44 4 4 4 4 4 4 44 44 4 4 4445650?100000000AO5 555 5 595 5 55 5 s6E 656555&666555S6 6 6 56 30 4 44 44 44 44 44 455S96C6 6666666I50111010000010111I1011100011771 711710771 Fig. 4--Example of order Card(4F12.0 6X# 2A4I4?42S2S4I2f22 21 11 12'2 2'I 111 181110111111SII2 ? ?,2 2 ? .47242'222222 2)2U2 C2111110 1 111011 1 1111 ;I I IlII1101011110001b1111q181111359213SbI ;11 11 11111 I III101II1IIII 111101111a0e10a0000001111e480"18 11q3331930q81q3qIq 949090923I 885391 13118111 249Fig. 5--Example of Format CardI'4''''101 j049
-19-SCALE CARDAs designated by a "1" in either Cols. 5 or 6 (or both) of thetitle 'ard, the scale card is used either if the data are to be scaled,or if the Y-intercept is to be specified for the linear, parabolic,or asymptotic-power function.Data ScalingThe first four sets of two columns each on the scale card areused for scaling the input dependent and independent variable datawhen required.These scale indicators (integers) are located as-ol-lows.Column Location ofScale DesignatorVariable ScaledYXlX2X3(dependent variable)(first independect variable)(second independent variable)(third independent variable)1-23-45-67-8An integer (fixed-point) number is used to indicate the number ofplaces that the decimal is to be moved.A positive number indicatesthat the decimal is to be moved to the right as many places as is thevalue of the number.A negative number indicates that the decimal isto be moved to the left as many places as is the absolute value of thenumber.For example, suppose f 4ar a "3" is entered in Col. 2 (must beright-justified).Then each input Y value will have its decimal pointmoved three places tohe right, e.g., 50.123---o-50123.,0.6127----*612.7, etc.Care must be taken to enter all positive integers in the rightitand column of the two-column set.If, for example, a "3" were enter-ed in Col. 1 Lnstead of Col. 2, the machine would read the number as"30"instead of "3."Any scalefactor entered applies,of course,the entire set of the corresponding variable for that run only.toAnexample of this card is shown in Fig. 6, in which the first-independentvariable data (Xl) ai.o be scaled down by a factor of 10.Data may also be scaled by using P-formats.
-20-A -I33 I I ;171I I3 P311808 313*'.I I33313131,,'.lii5555555555555555555.jt1II IIII I Ii I i4iOI.44l 4III33333333331311333314 . 44]4444444. 414.11444454-II1.644A 64 4141I I44,;fl.',1I iiI113333383333] 3333 Fig. 6--Example of Scale Card4 4'
CURVE-FITTING COMPUTER PROGRAM H. E. Boren, Jr. PREPAWD FOR: UN I7D STATES AIR FORCE PROJECT RAND SANTA MONWCA *CALIFOIRNIA-MEMORANDUM RM-5762-PR DECEMBER 1968 CURVES: A FIVE-FUNCTION CURVE-FITTING COMPUTER PROGRAM . lhe p-ogram makes least-squares determinations of the param- .
behringer ultra-curve pro dsp 24 a/d- d/a dsp ultra-curve pro ultra- curve pro 1.1 behringer ultra-curve pro 24 ad/da 24 dsp ultra-curve pro dsp8024 smd (surface mounted device) iso9000 ultra-curve pro 1.2 ultra-curve pro ultra-curve pro 19 2u 10 ultra-curve pro ultra-curve pro iec . 7 ultra-curve pro dsp8024 .
Creating a new design curve from brushing over a French curve is simply a matter of generating a sub-curve [10] from the formula-tion of the digital French curve. Editing design curves using French curves is based on the premise that the section of the design curve being edited has the same general shape and position in space as the
The following graph shows the kinetics curves for the reaction of oxygen with hydrogen to form water: O 2(g) 2H 2(g) 2H 2O(g). Which curve is hydrogen? a. the dashed curve b. the gray curve c. the black curve d. either the gray or the black curve e. Any of these curves could be hydrogen. 3
Figure 5 (below) shows a B Trip Curve overlaid onto the chart. The three major components of the Trip Curve are: 1. Thermal Trip Curve. This is the trip curve for the bi-metallic strip, which is designed for slower overcurrents to allow for in rush/startup, as described above. 2. Magnetic Trip Curve. This is the trip curve for the coil or solenoid.
SOLIDWORKS creates surface bodies by forming a mesh of curves called U-V curves. The U curves run along a 4-sided surface and are shown in the magenta color. The curves that run perpendicular to the U curves are called the V Curves, and they are shown in the green color. Certain commands may offer the preview mesh where these
order curves. Most of these curves have had elds of size around 2256, and thus security estimates of around 128 bits. Recently there has been inter-est in a stronger curve, prompting designs such as Curve41417 and Microsoft’s pseudo-Mersenne-prime curves. Here I report on the design of another strong curve, called Ed448-Goldilocks.
At each interior joint j, we have: 1. Last curve ends at j 2. Next curve begins at j 3. Tangents of two curves at j are equal 4. Curvature of two curves at j are equal The m segments give: _ interior joints _ conditions The 2 end joints give 2 further contraints: 1. First curve begins at firs
501 Concrete 501.1 Description (1) This section describes proportioning, mixing, placing, and protecting concrete mixtures. 501.2 Materials 501.2.1 Portland Cement (1) Use cement conforming to ASTM specifications as follows: - Type I portland cement; ASTM C150. - Type II portland cement; ASTM C150. - Type III portland cement; ASTM C150, for high early strength. - Type IP portland-pozzolan .
