AHIBtHunliSlllv inMlnli b iirr I mTr ■HI i IB HHBII Hi Hi HUE MHBiBB flf Inni mumiD flnBIHi wmSBBm msRSN &m HBffiiH iE fflH i W HBHMlHMra^^ MUI»1iMWOT llMmlS«Mgl¥liti BHHUIillUHiyffifflutfllUfii ilHH H HaHHttil nniKwa BntfinunflH Mali ill LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 510.S4 I-fitor -no. 629-534 co/o- 2. CENTRAL CIRCULATION AND BOOKSTACKS The person borrowing this material is re- sponsible for its renewal or return before the Latest Date stamped below. You may be charged a minimum fee of $75.00 for each non-returned or lost item. Theft, mutilation, or defacement of library materials can be causes for student disciplinary action. All materials owned by the University of Illinois Library are the property of the State of Illinois and are protected by Article 16B of lllino'n Criminal law and Procedure. TO RENEW, CALL (217) 333-8400. University of Illinois Library at Urbana-Champaign ' 2001 5 2001 aVr MAY 2 6 21 When renewing by phone, write new due date kalnnt nrOi71AI1C /IIIO /llto I 1 A9 below previous due date L162 UbAs r + uiucDcs-R-72-530 vr/^ z^ IMPLEMENTATION OF BASIC SOFTWARE FOR SIGNIFICANT DIGIT ARITHMETIC by Steven See Sun Lai June, 1972 IMPLEMENTATION OF BASIC SOFTWARE FOR SIGNIFICANT DIGIT ARITHMETIC BY STEVEN SEE SUN LAI 1972 Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 61801 This work was supported in part by the National Science Foundation Grant No. US NSF-GJ-328. Digitized by the Internet Archive in 2013 http://archive.org/details/implementationof530lais Ill ACKNOWLEDGMENT The author wishes to thank his advisor, Professor J. R. Phillips, for his suggestion of this thesis topic, his subsequent guidance, and his patience throughout its preparation. Special thanks go to Bob Bloemer of the OL/2 implementation group for his many suggestions and stimulating discussions which greatly- contributed to this effort. Furthermore, the author is indebted to the Office of Uni- versity Administrative Data Processing for their financial sup- port and to the Department of Computer Science for the computing facilities necessary for this work to be completed. IV TABLE OF CONTENTS CHAPTER Page 1 INTRODUCTION 1 2 PRECISE/IMPRECISE ARITHMETIC 5 2.1 Unnormalized Floating Point Numbers 5 2.2 True Zero and Relative Zero 5 2.3 Internal Representation of Numbers 6 2.4 Base Operations 7 3 ADJUSTMENT RULES FOR FUNCTIONS 15 3.1 Function Error Propagation 15 3.2 Propagation of Relative Error: Richtmyer Adjustment Rule 16 3-3 Application of Richtmyer' s Adjustment Rule 18 4 INPUT AND OUTPUT 23 4.1 Objectives of Significant Digit I/O 23 4.2 Decimal to Binary Conversion 24 4.3 Binary to Decimal Conversion 25 4.4 The I/O Routines 27 5 DISCUSSION 29 5.1 Matrix 29 5.2 Complex Numbers ■ 29 5.3 Some Problems with Significant Digit Arithmetic 30 REFERENCES 32 APPENDIX A. EXAMPLES OF SIGNIFICANT DIGIT ARITHMETIC 34 B. EXAMPLES OF DOCUMENTATION OF THE BASIC ROUTINES 62 CHAPTER 1 INTRODUCTION Automatic error control has been studied and implemented since the late fifties (15) • Over the past decade refinements have been made and further analytic results have come to light (^ s 5 » 8, 9, 16). It appears that the state of the art suggests that designers and implementers should try to embed these rules into their software systems. This provided the motivation for this thesis, which was to implement the rules of "unnormalized significant digit arithmetic" of Metropolis (15> 9) and Ashen- hurst (2, 4) and embed them in the array language OL/2 (20). However, these software packages are also adaptable to other com- patible software systems, since they are written in assembly language for the IBM 360/370 machines. Numerical errors in digital computation arise from three principle sources: "inherent" errors which result from inaccur- ate data, "analytic" errors which result from replacing an in- finite process with a finite process, and "generated" errors which result from using finite-precision arithmetic (1). Often, computed results are reported without specifying their accuracy because inherent and generated errors have been treated rather lightly. Presently, most floating-point arithmetic is performed in normalized form which leads to the retention of nonsignificant digits in final computed results. There is no control of significant digits when experimental data is used as input. Even with precise input, ill-conditioning or instability in the programmed algorithm may cause erosion of precision, thus a specification of error or significance should accompany out- put numbers. As a consequence of its practical importance, analytical investigations into, as well as implementation of, significance control has been made by several earlier investigators on sev- eral different machines. Notable among these are Gardiner and Metropolis (10), Goldstein (11), Ashenhurst ( 3 ), and Bright' ( 7 ) . Each agree to the two rules of significant digit arith- metic: (1) for addition and subtraction, the resulting exponent is equal to the larger of the input exponents, and (2) for mul- tiplication and division, the number of significant digits in the result is equal to that of the less significant operand. They differ, however, in their implementations of the internal representation of significant digit numbers. Because each mach- ine employed a different exponent base, corresponding significant adjustment rules for library functions also reflect this differ- ence . All numerical quantities may be regarded as either exact or approximate. However, the finite presentation of number in computer requires a further characterization. If an exact number can be represented in some preassigned number of digits (the computer word length), then it is said to be precise . Other exact numbers belong to the approximate class are called 3 imprecise . Using these two categories and base of two, Gardi- ner and Metropolis ( 9 ) introduced the ABC (analyzed binary computation) form of arithmetic. The goals of ABC arithmetic are: (1) to determine if the computed result is precise or un- precise, and (2) to specify the number of significant digits in the imprecise result. Thus, ABC arithmetic uses the rules of significant digit arithmetic and defines an additional set of rules for combining both precise and imprecise numbers. When- ever we use the term "significant digit arithmetic" we mean to include the new rules of ABC arithmetic as well. The present work follows the approach of Gardiner and Metropolis ( 9 ) > ex- cept that the base two rules must be replaced by base sixteen rules, since the implementation is for the IBM 360/370 machines. Overview of the Thesis The next chapter defines the basic rules for combining precise and imprecise numbers in the context of ABC arithmetic. Two kinds of zeroes, true and relative, are also discussed in this context, along with the internal format for imprecise and precise numbers. Chapter 3 discusses the significant digit adjustment rules for elementary functions. In particular, Richtmyer's adjustment rule for functions is derived. The discussion follows the rules that were prescribed by Ashenhurst (4), with suitable changes for base sixteen rather than base two numbers. The fourth chapter explains the format of input/output for precise and imprecise numbers. The significance number base conversion algorithms of Kanner (13) will be presented. Chapter 5 briefly explores some extensions to the present work, and also some of the problems which the uninitiated user must be aware of when using significance arithmetic . CHAPTER 2 PRECISE/IMPRECISE ARITHMETIC 2.1 Unnormalized Floating Point Numbers A floating point number may be represented by a pair (e, f), where e is the integral exponent and f the real valued fraction part. Most of today's computers use normalized arith- metic in which the floating point number has its non-zero frac- tional part satisfying r ^_ f < r , where r is the number base. This format alone cannot specify the number of significant digits in the fraction. By unnormalizing the number, it is possible to convey the significant digits in the fraction, since (e + m, r f) represents the same number as (e, f) for any integer m. The integer m describes the number of leading zeroes in the fraction. By definition, therefore, imprecise numbers are re- presented internally as unnormalized numbers with at least one leading zero. Precise numbers, on the other hand, are repre- sented internally as normalized numbers. This distinction be- tween precise and imprecise numbers forms the basis for the rules of ABC arithmetic (9) which are discussed subsequently. 2.2 True Zero and Relative Zero In significant digit arithmetic there are two kinds of zeroes: relative zero, which has no significant digits and true zero, which has an infinite number. Both zeroes have zero frac- tional parts, and obviously, there is no sign attached to the 6 fractions. True zero is designated by using the smallest ma- chine representable exponent, in general, this is just the usual machine representation of zero. Relative zero, on the other hand, may have any exponent value. Clearly, the rules for operat- ing with these two zeroes is quite different. True zero is a precise number in the sense that it acts like zero. In contrast, relative zero is a degenerate imprecise number with no signifi- cant digits but with an exponent field that indicates an "order of magnitude." Since relative zero contains some information about a number besides complete loss of significance, it seems appropriate to consider separate rules for operating with it . It should be clear that a relative zero may occur during a computational process as the result of very basic operations. For instance, a relative zero may occur when two nearly equal imprecise numbers are subtracted. If the original operands had a large exponent, then the relative zero has a large ex- ponent. This indicates that a relative zero is not necessarily close to zero, but is merely a bound on the result. 2.3 Internal Representation of Numbers The present work was implemented on a IBM 360/75 computer using long floating point numbers. These numbers use 64 bits: 7 bits for the base 16 exponent and 57 bits for the fraction. (See Figure 1.) The zero-th bit is used for the sign of the fraction. The exponent is represented in excess-64 code with limits of -63 and + 64. True zero has an exponent of -64 and has zeroes in all 64 bits. (This Is the standard floating point 7 zero.) Relative zero is represented by a zero fraction, a zero in the sign position, and arbitrary exponent e in the range -63 <_ e £ 63. The sign of relative zero is neither positive or negative . A precise number is represented as normalized floating point number with 14 hexadecimal digits in the fractional part. Only precise numbers are normalized. The range of IBM's long 7 7 -t7 C\ floating point number is from 10 to 10 . Since an imprecise number is an approximate quantity, it is represented with fewer fractional digits. The first 4-bit fractional byte is always zero and the last hexadecimal digit is always defined to be a nonsignificant guard byte . Thus, the imprecise fractional field uses the middle 12 hexadecimal digits. Because imprecise numbers are unnormalized, they can only repre- -77 +49 sent numbers between 10 and 10 . The guard byte is used in input/output conversion and also serves as a guard against round-off errors. Kanner (13) points out that if input conver- sion followed by output reconversion is to be an identity opera- tion at least one guard bit must be kept. Because the exponent base is sixteen, it is necessary in this implementation to use an entire byte of 4 bits. 2.4 Base Operations In this section the operations of addition (including sub- traction), multiplication, and division in ABC arithmetic are described. If one of the operands is imprecise, then the result is imprecise or a relative zero. The significant digit rules are I fraction field (8:63) exponent field (1:7) fraction sign bit (0) a) IBM 360/370 long floating point format 1 1 guard byte (60:63) .leading zero byte (8:11) b) Imprecise number Figure 1 . Internal number format 9 used for imprecise numbers. Let X, = (e,, f -, ) and Xp = (e^, fp) be the operands, and X^ = (e~, f-O the results, then the rules of significant digit arithmetic are defined as follows: 1. For addition or subtraction e_ = max {e, , ep}, 2. For multiplication or division m.3 = max {iru , nip} where m is the number of leading zeroes in the fraction. If a zero fraction results, the exponent of the relative zero is fixed according to the rules of regular arithmetic. That is, in addition e~ is the maximum exponent of operands, in multipli- cation e, is the algebraic sum of the exponents, and in division e~ is the difference between the exponent of the numerator and the exponent of the denominator. There are some obvious prob- lems with using the 360/370 floating point arithmetic. If the unnormalization of imprecise numbers Is not possible, then the ABC arithmetic routines signal exponent overflow. This is why +4 9 imprecise numbers have an upper limit of 10 instead of the +75 10 of the precise number. A necessary but not sufficient condition for a precise re- sult is that both operands are precise. The result would be imprecise if it cannot be represented exactly in the long float- ing point word. For multiplication and division, the low order part of the immediate product is generated and examined for any non-zero bits. (The 360/370 machines generate only the higher order part.) Since the product is almost twice the length of input operand, a quick test for a precise multiply is to check if both precise inputs are contained in the short floating point 10 number format which has 32 bits and 6 hexadecimal fractional digits. For addition, the precise check is to subtract the result by the operand with the larger exponent. This subtrac- tion would have no shifting of fraction to align the exponents The result is then compared with the other input operand. If they are not equal, fractional digits were lost in the addition; so the result is imprecise. The special case of a true zero resulting from the subtraction of the precise number by itself is checked by comparing the input fractional digits for equality. As noted earlier, true zero is the zero element in ABC arithmetic. In addition, it is the identity element. A true zero product results if one of the operands is a true zero. Division by true zero is signaled as a divide exception and a true zero dividend results in a true zero quotient. On the other hand, the rules for operating with relative zero are very different. If both operands are relative zeroes, then the exponent of the resulting relative zero exponent must be adjusted. In extreme cases, an exponent overflow in a rela- tive zero raises an error condition while an exponent underflow is replaced with the smallest relative zero. In multiplication and division, the result is generally a relative zero if one of the operands is a relative zero. It is interesting to note that the quotient of a non-zero divided by a relative zero is defined as a relative zero. In this case, multiplication of quotient and divisor does not give the dividend. 11 Table 1 ABC Multiplication °T °R X I x p °T °T °T °T °T °R °T °R °r °R X I °T °R °R :X I °R :X I x p °T °R °R :X I Xj:X p 0™ = true zero X T = imprecise non-zero R = relative zero X p = precise non-zero Table 2 ABC Division °T °R x i Xp °T 00 00 oo oo °R o T °r °r °r X I o T °r °r :X i °R :X I X P o T °r °R :X I X x :X p Orp = true zero X T = imprecise non-zero R = relative zero X p = precise non-zero 12 The addition of a relative zero and non-zero may have either an imprecise result or a relative zero. If the exponent of the non-zero operand is greater than or equal to the exponent of the relative zero, then the sum is imprecise. In particular, if the non-zero operand is precise, then the sum Is the rounded imprecise representation of the precise non-zero operand; if the non-zero operand is Imprecise, then the sum is the imprecise non-zero operand adjusted (signifying loss of significant digits) by the difference of exponents of the non-zero operand and rela- tive zero. Clearly, if the exponent of the relative zero is greater than the exponent of the non-zero operand, then the sum is just the relative zero. Table 3 ABC Addition and Subtraction °T °R X I X P °T °T °R X I X P °R °R °R °R :X I °R :X I X I X I °R :X I °R :X I °R :X I x p x p °R :X I °R :X I Om : X-j- : Xp m = true zero X T = imprecise non-zer< R = relative zero X p = precise non-zero 13 The rules just described for significant digit are arith- metic implemented in assembly language as subroutines. The entry point names are @ 0L2ADD, @ 0L2SUB, PIMPY, and PIDIV. (See Appendix B for the program listings.) Each routine employs a round on the digit to shift off due to unnormalization of the result to the correct number of significant digits. There are some problems with significant digit arithmetic using unnormalized numbers. One aspect will be discussed here and others will be treated in Chapter 5. The significance rule for multiplication and division calls for retention of the same number of significant digits as in the operand with the fewest number of significant digits. This rule can result in the loss of significant digits. For instance, consider the following multiplication: 01 x 09 = 09 and 09 * 09 = 81. In both cases, the rule specifies one significant digit. For the first case, it is correct; however, in the second case the second digit "1" would be discarded. Thus, a corrective shift or an auxiliary rule is necessary for multiplication and divi- sion. The auxiliary rule for multiplication is: if e~ = e, + e ? , then itu = m~ - 1, where e.'s are the exponents of the normalized operands. This corrective right shift for mutliplication re- sults in gaining one significant digit. Since division is the inverse operation of multiplication, the division auxiliary rule is: if e. = e, - e-, then m~ = m_ + 1 . Here the corrective left shift in division results in losing a significant digit. in Routines @ 0L2MPY and @ 0L2DIV are Identical to PIMPY and PIDIV except for the incorporation of the above auxiliary rules. The next chapter will examine the rules of significant digit arithmetic for elementary functions using base sixteen. 15 CHAPTER 3 ADJUSTMENT RULES FOR FUNCTIONS 3.1 Function Error Propagation Let F be a function of one argument and let (e-^, f , ) be the argument value at which F is to be evaluated. Clearly, this requires the specification of two functions, g and h: f 2 = g( ei , f 1 ) e. 2 = h(e 1 , f 1 ) e_ e^ such that r ^f ? = F(r ,f-,). This relation, however, does not imply anything concerning the number of significant digits In the result (e ? , fp) since (e ? + m, r fp) also satisfies it. The adjustment criteria must be based on further considerations. Given a number (e-, , f -, ) , let f-. be regarded as an approxi- mation to some true value f-i^. The error in the function of the T argument is defined by 6, = f, - f, , while the error in frac- tion of the function is defined by &~ = g(e, , f -, ) - g(e-. , f-. ) . If we use the generalized mean value theorem, then we have the estimate |6p| = a|6, |. The error propagation associated with a given g is thus expressible in the terms of the "amplification factor" a. This is the procedure which is described by Ashen- hurst ( 4 ) . Adjustment of the function value (ep, fp) is made so that a is within a factor of 16 (since we are using base sixteen) of 16 of any predetermined value. If a Is approximately one, then an adjustment is said to be in accordance with a "significant digit" criterion. Notice that this does not imply that all the computed digits of f„ are actually significant. Rather, it means that the error in f« due to the error in f-, is of the same order of magnitude in terms of the adjustment used. Both f.. and fp contain roughly the same number of "non-significant" digits . 3.2 Propagation of Relative Error: Richtmyer Adjustment Rule T Suppose x is a number approximating a true value x not equal to zero. The relative error between the approximate and T T true values may be defined as r(x) = (x - x )/x . Similarly, the relative error in the function F may be defined as r„(x) = (F(x) - (F(x T ))/F(x T ) . If F(x) is dif f erentiable at x, then T T r„(x) = (F'(x) (x - x ))/F(x ). Using the definition of r(x), T T T we have r p (x) = (x F ' (x)r (x) )/F(x ). If (x - x ) is small, we T replace x with x to get: , X x P' (x) , s r-o( x ) = — -p / \ r(x). ■pv / F(x) By definition l6" m-1 < |f| < l6" m , thus it follows that l6 m |6| < \^\ < l6 m |6|. Since this inequality may be applied to both the argument and function values, using respectively f ., , m, , . , and fp, nip, ~ , we can write l6 m 2 -m 1 -l 6. 6 2 f l < i6 m 2~ m l +1 | 6 2 6 l f 2 17 Thus , 6 2 f l 6 l f 2 I6 e2 fi. I6 ei f. I6 e2 f F(x) - F(x ) I6 e l6. T FTry T x - x r p (x) provided that 6, and 6 ? are sufficiently small. An approximate inequality which holds to a first order approximation may be formed : x F' (x) j| < l6 m l- m 2 which is equivalent to TJJJ < 16a l6 m 1 -m 2 -l x F' (x) x < a < I6 m l- m 2 -mo+1 x F'(x) The above inequality bounds a as a function of the argument x and the adjustment (mp-m-,). Taking the base 16 logarithm of all members of the inequality (this is permissible since log, r is a monotonic increasing function) gives log 16 x F' (x) TXT) - Am-l < loi a < log- x F T ( x ) TZx) - Am+1 . '16 " " ^ 6 16 With a "significant digit" criterion a=l the above inequality can be written as -1 < lo g l6 x F T (x) - Am < 1 18 Thus, an approximate adjustment criterion Is x F» (x) Am * l°g l6 TT7T R. D. Rlchtmyer had suggested the above adjustment rule for unnormalized arithmetic as well as for functions. The above deviation is identical to the one of Ashenhurst ( l] ) except base 2 was replaced by base 16. Notice that in actuality Am is between -1 and +1 (base 16), a defect which Ashenhurst points out can be troublesome since it implies that 16 < a < 16. He shows, however, that if it is assumed that 16 If, is uniformly distributed in the interval (16 , 16) the expected value of a is unity. Therefore, on the average, Richtmyer's Rule provides the optimal adjustment. 3.3 Application of Richtmyer's Adjustment Rule This section investigates the adjustment rules for various functions in base 16 . The difference between corresponding function adjustment rules for base 16 and base 2 will be explained. Raising to a power, F(x) = x , where n is any number gives: Am = lo s l6 / n-lv x(nx ) n = log l6 |n Since for the most practical purposes [n| « 16, an approximate adjustment rule is Am = 0. For n=l and n=2, the adjustment rule is m =m, , which means that the number of leading zeroes in the normalized fraction remains unchanged. For the square root function where n=l/2, the adjustment rule must be also mp=m, to conform to that of square function. However, in base 2, 19 Am = logp |n| dictates m 2 =m, -1 for n=l/2 and nip=in,+l for n=2. Therefore, the adjustment in base 16 is relatively coarse com- pared to base 2. That is, in base 2 we would actually adjust m for small values of n, whereas in base 16 the adjustment would be unnecessary. For exponentials and logarithms, F(x) = a =e , Richtmyer's Rule specifies Am = log l6 x x(a In a) = log l6 |x|+ log l6 | In a| Recalling 16" ■*-" <_ | f -, | <16~ ■*■ , the adjustment Am may be ex- pressed as either Am = (e-.-m-.-l) + log-./- | In a | or as Am = (e-,-m-,) + log-,/- | In a|. Once again, if a uniform distribu- tion of 16 if., is assumed, these two equations may be averaged to give Am = e,-m..-l/2 + log-,/- | In a|. Thus, the adjustment rule for an arbitrary a is m ? = e, - 1/2 + log-,/- | In a| . Clearly, the right hand side of the equation must be rounded to an in- teger. In a similar fashion, the logarithm function F(x) = log (x) where x>0 should be adjusted by the rule: a m 2 = e l + "'"/^ " l°Si 6 I ^ n a l* Notice that the only difference of adjustment rules for exponentials and logarithms is the factor of one-half. More complicated functions such as trigonometric functions, hyperbolic functions, and the error function may be expressed as a power series in x . As examples, consider the following 20 expansions with <_ x <_ tt/^I : 2 i\ x x cos x = l-py+jp-+. . . 3 5 X X sin x = x - =-r + -p-p + . . . j ! b I If we assume as Ashenhurst does that the error in the function can be approximated by the error in the first term of the series, the sine can be thought of as a power series in x and the cosine 2 can be thought of as a power series in x . Using the previous derived rule for P(x) = x , namely, rru = iru + log-,/- |n| we have for sine (n=l) nip=m, and for cosine (n=2) nip = m, + log-,/- 2. The last adjustment rule can be approximated by mp=m-, . Ashenhurst derived the cosine adjustment in base 2 as nip = 2e-. - m, . Again, this difference is due to the different bases used. Using the first term for an approximation, the ad- justment rule mp=m-, also holds for hyperbolic functions, expon- tial functions, and the error functions. Significant digit function routines have been implemented as subroutines which call the corresponding IBM PL/1 library functions. Ashenhurst in ( 4 ) suggested this strategy of carrying out the function computation by normalized and then adjusting the result at the end. The adjustment rules for imprecise arguments for functions in base 16 falls into two equations — either mp=m-, or m ? = |e-, | . Table 4 ABC Function Rules 21 Function Base 2 Adjustment Base 16 Adjustment Sin Cos Tan Square Root Exponent : e x Log Log, 10 '2 Sinh Cosh Tanh Arctan Error Function Complement Error Function mp=m, m„= I 2e-, -in-, nip=m, mp=m, -1 m 2 = I e l -1 1 m 2 -|e 1 | m ? = | e-, +1 m p I e i mp=m, m ? = | 2e-, -m-, | mp=m-| m 2 =m l m 2 =m l m„=in. mp=m, mp=m. mp=m.| m =m. m 2 =|e 1 m p = I e -i I m 2 =|e 1 | nip=m. mp=m-| mp=m-. mp=iru mp=m-. mp=m. The previous discussion assumed the arguments were im- precise or true zero. The subroutines (except for the square root) always return an imprecise number even when the argument is precise. The justification for this action is that values of the function can only rarely be represented as a precise, 22 number, for Instance, sin 30° = .5 and cos 0° = 1 . For a rela- tive zero argument, the argument exponent is examined. If the exponent is positive, the same relative zero is returned for the function value. Relative zeroes with non-positive exponents are treated as if the argument is near zero. In particular, an imprecise one with one significant digit is returned for the exponential, cos, cosh, and error functions, while a relative zero with exponent of +14 is returned for the sine, tan, sinh, tanh, arctan, and complement error functions. An error condi- tion is signaled when relative zeroes with non-positive exponents are arguments to the log functions. For the square root func- tion, the same relative zero is returned as the functional value. This implementation of relative zero as an argument for the above functions does not account for repeated functional evaluations with relative zero. Thus, the treatment of relative zero should be studied further and refined. This completes the discussion of implemented significant digit adjustment rules for functions. The next chapter is con- cerned with the input and output convention for precise and imprecise numbers. 23 CHAPTER 4 INPUT AND OUTPUT This chapter will deal with the problems of input and output. We shall also discuss algorithms for number base con- version as described by Kanner (13) for significant digit arithmetic . 4.1 Objectives of Significant Digit I/O The main objective of I/O for significant digit arithme- tic is to obtain a faithful representation of significance under the operation of base conversion from binary to decimal, as well as the inverse of this operation. For instance, the number 2.00 on input should be immediately converted on output (without any intervening computation) as 2.00, and not, say, as 1.99* Kanner points out this can be achieved only if at least one guard bit is maintained in the floating point frac- tion. In this implementation, as already pointed out, a guard byte (4 bits) is always available. This multipurpose guard byte, whose bits are considered to be nonsignificant, also serves to separate accumulated rounding error from the error inherent to the computation. Since I/O is part of the human-computer interfacing problem, there should be ample diagnostic aids to help the user interpret the computer outputs. The I/O routines of the present work were written with the above objectives in mind. 24 4.2 Decimal to Binary Conversion The following algorithms of Kanner (13) are based upon the strategy that correct significance can be retained in the conversion of integers. Briefly, the procedure involves multi- plying the number by that power of ten which is necessary to express the significant portion as an integer, then this inte- ger is converted to its intermediate internal representation in the desired number system, and finally, this result is divided by the same power of ten. The following discussions have been adapted to base 16 instead of base 2 as discussed in Kanner (13). The binary representation of N can be obtained from the decimal integer representation w,Wp...w ,w by considering the following scheme: a =0, a.=10a. -, + w. , N=a . The long floating point o 5 i l-l in to to r number (64 bits) implementation dictates, for efficiency, the use of floating point hardware instead of the general registers (32 bits). Each w. can be represented as a floating point num- ber with leading zeroes up to the right most hexadecimal digit and with an exponent of C, where C is the total number of frac- tional digits. (For the 360/370 long floating point number, C is 14.) This scheme, in effect, allows fixed point (integer) arithmetric to be performed with floating point hardware. We now consider the algorithm to convert any decimal num- ber to binary while retaining significant digits. Let E N = ± x., Xp. . .x.. y.,yp. . .y, * 10 . Clearly, (j+k) must be less or equal to D, the maximum number of decimal digits permitted. 25 (D is 15 for 370/360). The signs of N and E are saved. Next, the integers |E|, k, and x, x ? . . .x ,y, Vp . . . y R are converted to binary using the previous algorithm. If it is possible to intro- duce a guard digit, then I, the binary representation of x-. Xp . . .x . y, y ? . . .y R is then adjusted. F— k The next step is to multiply I by the quantity 10 ~ using significant digit (SD) arithmetic. In this way, the final number will have the same significance as the original number. This step can be separated into three cases, depending upon the sign and magnitude of E. The cases are: (1) E > 0. Let ES = I-k. For ES < 0, divide I by 10 ' ES ' - For £ ES < M, where M is the maximum representable power of ten, I FS I multiply I by 10' 1. (Because of unnormalization, M is 49 for an imprecise number while M is 63 for a precise number.) I F I (2) -M £ E < 0. Divide I by 10 ' ' and then divide the k result by 10 . / v M I E I -M (3) E < -M. Divide I successively by 10 , 10 ■ ' , and 10 . In this case, E has the lower bound of smallest represent- able exponent. (For the present work, E must be equal or greater than -63. ) All the operations in the above three cases are performed in significant digit arithmetic. The last step is to attach the proper algebraic sign to the number. 4.3 Binary to Decimal Conversion Let g be the number of guard digits in the fractional part of the floating point number. (In the present work, imprecise numbers have one guard digit and precise numbers have none.) 26 The first step is to convert the original number to a signifi- cant integer multiplied by a power of ten. There are three cases depending on x, the exponent of the number to be conver- ted . (1) x > (C-g). Let ES = x - (C-g) and E(n) denote the exponent of the normalized binary representation of 10 . The original number is divided by 10 , where n is chosen such that E(n) - 1 < ES < E(n+1). If the exponent of the quotient is less than (C-g), then the number is divided instead by 10 After conversion of the significant integer to decimal, the result is the decimal integer times 10 or 10 . (2) (C-g) - G < x < (C-g), where G is the maximum repre- sentable exponent in the floating point number. The original number is multiplied by 10 , where n is chosen such that E(n-l) < ES < E(n) . If the exponent of the product is less than (C-g) , the number is mutliplied instead by 10 . The result is the converted integer times 10~ or n0~ ~ . M (3) x < (C-g) - G. The number is multiplied by 10 , where M is the maximum representable power of ten. Then the resulting product is treated as in case (2) above. Here the result is the converted integer times 10 ^ , p is the power of ten used in the second multiplication. Finally, if a decimal point is to be inserted in any de- sired position, the power of ten is modified accordingly. 27 4.4 The I/O Routines The input and output of imprecise numbers conform to the conventions that are used for PL/1 E and F formats. The special symbol P is used to distinguish precise numbers from imprecise numbers, e.g., LOOP and -1.23P+01. Like PL/1, the maximum number of digits per item allowed is 15, due to the computer word length. The I/O routines consist of three modules: 0L2PILK, which accepts the input numbers as character strings in the PL/1 GET LIST manner, 0L2PICP, which converts character strings into long floating number, and 0L2PIPD, which converts the internal numbers into character strings for output. Both 0L2PICF and 0L2PIFD perform an operation with a ten of power, as described by Kanner , in two successive steps in the ten's place and then in the one's place to preserve precise numbers. It was found that a precise number multiplied by 10 became 20 imprecise, but the same precise number multiplied by 10 and 10 remained precise. This is a result of the computer word length used. Besides returning the correct significance for a number on output, 0L2PIFD also supplies other information to interpret the number. There is significance pointer to indicate the last significant digit of the output number. This is useful, for example, when a user does not want to output all significant digits of a number. A common user error is to specify an in- sufficient field width for output. In this case an error flag is returned along with as much of the number as possible. For 28 output of arrays it is also useful to have indicators for a maximum significant digit count and a maximum field width. This allows the user to analyze his array output with respect to maximum significance and field width. Each of these consid- erations are valuable and contribute to a reliable and useful significant digit I/O package. 29 CHAPTER 5 DISCUSSION Precise and imprecise arithmetic, as described up to this point, can be used to carry out more complicated mathematical operations. However, separate routines to handle operations with complex numbers and matrices may be advisable. Further in- vestigation is needed to study the effects of correlated errors with imprecise numbers. More significance rules may be required or perhaps, the present significant rules must be modified. 5.1 Matrix Most matrix operations consist of a series of inner prod- ucts of vectors. The inner product subroutine 0L2PIVP was written to extend ABC arithmetic to matrix operations. Follow- ing the same approach of the significant functions, the addition and multiplication are done normalized. At the end, the result is unnormalized to the correct number of significant digits by keeping a count of the maximum number of leading zeroes encount- ered. In this way, the resulting inner product is more accurate because of less round-off errors with normalized numbers. 5.2 Complex Numbers A complex number can be represented as a number with real and imaginary parts. Routines using the approach of the inner product can extent ABC arithmetic to include complex numbers. Some definitions are necessary: a complex number is precise if 30 and only if both real and imaginary parts are precise; the sig- nificance of a complex number is that of the less significant part; a complex true zero consists of two real true zeroes. The possibility of a relative zero part and non-zero part must be studied and appropriate definitions implemented. 5.3 Some Problems with Significant Digit Arithmetic The adjustment rules for imprecise numbers are based on the assumption that their associated errors are statistically independent. If their errors are correlated, the resulting imprecise number may have a representation which is not a re- liable measure of significance. An example of correlated error is found in the multiplication of an imprecise number by itself. The resulting product should be more significant since the approximation error was squared. The auxiliary rule for multi- plication attempts to correct this situation by increasing the number of significant digits by one. Similarly, repeated sum- mation of an imprecise number by itself, under the present ad- justment rule, would allow the number of significant digits to increase. A combination of these correlated errors can be found in the vector norm operation. Thus, correlated errors in signi- ficant digit arithmetic should be studied. To complete the implementation of significant digit arith- metic, routines to compare two operands are needed. Detection of a relative and true zero is done by check for zero fraction. If the exponent is also all zero, then the number is a true zero. To compare two relative zeroes, the exponents are examined. 31 Since relative zeroes have no significant digits, only the ex- ponents of a non-zero and a relative zero can be compared. True zero would always be "lower" than any number which is not a true zero. Care must be taken to avoid using the guard bits of an imprecise number in a compare. After removing the guard bits and normalizing, comparison of imprecise numbers can be made. Finally, precise numbers are compared with any special treatment. As noted in Chapter 3, the present implementation of func- tional values for relative zero arguments did not account for the possibility of repeated function evaluations using relative zeroes. Each function should be studied separately to provide a more comprehensive treatment of relative zero arguments and correlated errors in functional evaluations in significant digit arithmetic . There are other methods of error monitoring. The interval arithmetic of Moore (19) also provides error bounds on computed results. Metropolis (17) gives techniques of analyzing algorithms in unnormalized arithmetic. Computational errors are also dis- cussed in (21). Metropolis ( 9 ) states "... different algor- ithms for the same problem give different representations is connected with the observation that mathematically equivalent approaches are not computationally equivalent." Significant digit arithmetic is one way to aid the error analysis of a com- putational algorithm. It is hoped that present implementation will aid further study and experimentation with imprecise numbers and significant digit arithmetic. 32 REFERENCES (1) Ashenhurst, R. L., and N. Metropolis. "Error Estimation in Computer Calculation," The Am. Math. Monthly , Vol. 72, No. 2, Part II (1959), 47-58 . (2) Ashenhurst, R. L. , and N. Metropolis. "Unnormalized Float- ing Point Arithmetic." J. ACM , 6 (1959), 415-428. (3) Ashenhurst, R. L. "The MANIAC III Arithmetic System." Proc . Joint Computer Conf ., 21 (1962), 195-202. (4) Ashenhurst, R. L. "Function Evaluation in Unnormalized Arithmetic." J. ACM , 11 (1964), 168-187 . (5) Ashenhurst, R. L. "Number Representation and Significance Monitoring." Mathematical Software . Edited by J. R. Rice. New York: Academic Press, 1971, 67-86. (6) Blandford, R. C, and N. Metropolis. "The Simulation of Two Arithmetic Structures." LA-3979, Los Alamos Scien- tific Laboratory (1968). (7) Bright, H. S., B. A. Colhoun, and F. B. Mallory. "A Soft- ware System for Tracing Numerical Significance during Computer Program Execution." SJCC, (1971), 387-392. (8) Fraser, M. , and N. Metropolis. "Algorithms in Unnormalized Arithmetic III. Matrix Inversion." Num. Math . , 12 (1968), 416-428. (9) Gardiner, V., and N. Metropolis. "A Comprehensive Approach to Computer Arithmetic." LA-4531, Los Alamos Scientific Laboratory (1970). (10) Gardiner, V., and N. Metropolis. "Significant Digit Arith- metic on a CDC 6600." LA-4470, Los Alamos Scientific Laboratory (1970). (11) Goldstein, M., and Hoffberg, S. "The Estimation of Signif- icance." Mathematical Software . Edited by J. R. Rice. New York: Academic Press, 1971, 93-104. (12) Gray, H. L., and C. Harrison, Jr. "Normalized Floating- Point Arithmetic with an Index of Significance." PJCC Vol. 16 (1959), 244-248. 33 (13) Kanner , H. "Number Base Conversion in a Significant Digit Arithmetic." J. ACM , 12 (1965), 242-246. (14) Menzel, M. , and N. Metropolis. "Algorithms in Unnormal- ized Arithmetic, II. Unrestricted Polynomial Evalua- tion." Num. Math . , 10 (1967), 451-462. (15) Metropolis, N. , and R. L. Ashenhurst . "Significant Digit Computer Arithmetic." IRE Trans. Electron. Computers , EC-7,265 (1958). (16) Metropolis, N. "Algorithms in Unnormalized Arithmetic, I. Recurrence Relations." Num. Math . , 7 (1965), 104- 112. (17) Metropolis, N. "Algorithms in Unnormalized Arithmetic." Proc . Colloq. Inst. Centre. Nat. Rech. Sci., Besaucon, France (1967). (18) Miller, R. H. "An Example in Significant-Digit Arithme- tic." Com. ACM , Vol. 7 (1964), 21. (19) Moore, R. E. Interval Analysis . Englewood Cliffs, N. J. Prentice Hall, 1966. (20) Phillips, J. R. "The Structure and Design Philosophy of OL/2 - An Array Language." Part I. (To appear.) (21) Rail, L. B. Errors in Digital Computation . New York: John Wiley, 1965. (22) Richtmyer, R. D. "The Estimation of Significance." AEC Research and Development Report, NYO-9083 (i960). (23) Yasui, Toshio. "Significant Digit Arithmetic on ILLIAC IV." ILLIAC IV Report 211 (1969) . 34 APPENDIX A EXAMPLES OF SIGNIFICANT DIGIT ARITHMETIC 35 APPENDIX A This appendix contains several examples of significant digit arithmetic . Each program illustrates how a software sys- tem may generate the necessary linkage or CALLS to the basic significant digit routines. The unnormalized significance out- put was provided by 0L2PIPD. The significance pointer is the position of the last significant digit, counting from the left- most position of the field. This significance pointer can also return negative numbers: -3 for true and relative zeroes, -2 for insufficient field width (severe error), and -1 to indicate that not all of the significant digits in the fraction were dis- played for the given field width (warning error) . 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O -l in ?■ ^ o^ QN UJ TO z < O O 1*1 Q < 33 _> o O **> < r*l N- JJ — . O *t O < >»■ -^ a. o o r- o — n M ON ON UJ CO z O O U. o < 'X '•> O u. — ' u. X — < ~ — o o o o o l/> x -* r ■* -r -r -» n H ii ii h II X X X X X X 11,1 ill ! 1 1 til 1 1 ; LU I I X I X X Z Z z z z z z o» -• o o o* c o o — - o & co — o o o o <^ O 00 -^ o o CO -* O"* o O 0"» 0^ O CO *-« o o* CO •■ * s o o o* O^ O CO ^n o 0^ CO •■* o> o O O 0* O CO -* o CO CO -* o* o • o^ 0* O CO — • -* • co o o> o o 0> O CO -D • • a* o -• o • • o o o o o <7> o o o o 0> o o o 0> o o a o <3> o o o o> o o o o «T> o o o 0> o a o o o 3 o o O o o o 0> o o o o 0» o o o 0> o o o o CO 0> o o N" N- o o o <7> 0> o o o- 0> o • • • • o 0» • t o o n o • o a o o o o o o> • o o o o • o o o o o o o I I I I I I a. a a. a a. a. IX o TO m — . •o u. < z < o CO lt\ a. a. X o in * < (T> LL o LU — u- »* l/> ~r a. o I 1- CD 33 u. ■— a. (^ < u-» LU u. ^* z ai < •o O 0» u. o at lo -O o < -0 o u. LU LU Q o o o o O N- ac u. o o o o a Z Q. — < o u. o #-J ■3 — LU o o o 3 o o or »M f\J —i «* m ~r * * •r * -»■ •r 30<<0-r -< * -< -< oo-«oooooo II II II II II II O O O O O O O O O ♦ ♦ I I + ♦ UOUOUO * I I I I |. + |LULULUL1JLULU LULULUaJLULUajLUlij— lO-TO^O o^oco-^crN-^-^N-coooomo co ^u>r>r~cNLrtir\o>N-oo>.r'Ni-ftir\(\iooi* , c > oo 2eoeoo>-^*f\j^^rvicNO^LrioO OCcocooLriLri^-^m^o-iajoo 0^3300>^cocooOCOO»000>00 LU^0>OC0O0 > 0>OO0 N C7 > OO0 > OO »J3>OOCNU*000»00 — l-U>Tje0OC7'0 , >OO0 > L7 > OOC7 > OO -IZCO — 0>-«(7>0>00 0>0 o»ooo>oo xloco-*o>— 'O^o^ooocNoocfOo HNl^ D QCLU LP O CO O U> C7* O O C7* cu O O O^ N- O Oct « - 0> — 0>CPtJOC , 0>00>N-0 > ZO-O^OCOOO^COOOCO • • • • « • lu • • • • • • • • . -. o a* a> -> o at:.*. - o x .*o s o>^-*o > II II II II II II II II II II II II II II II — — — -. — — -..«.•-..» — _ _. _ o-»rgm*m •^CMI»>*l/>-ON-CO0»^^^-<-<^ h ii ii ii n n — — _ — _-.-, — __«._«. — _ 60 LU U Z of — o Z a O Z o o o o o o o o 0> o a 0> 0« o o ee r- o -. 0> — * 0> -u o 0> o o 0» 0* o 0> o> o o >- ^ o o- o o • • • • • o 0* • • o < CO o CO -. o o — * ■rf o _4 o • o — o 1— o> o 0> o o o o o z • • CO o o o o UJ ■4 o 0> o o o o O to • • o o o UJ UJ o o o o o *u ct a o o — a. o o o -J UJ o o ■n o .r -J- j- as o < 0£ o o o o < < < J- UJ r o o o o o o o -^ — < cc UJ • • • o h. r- r~ LU J"\ a u — < o o o Q Q o < co Z z o o o u. a. a. u. u. z < o o o o -4 o O -* 3 O o n o o o o o o — « o o o o o o o O u. o o -3 o o -3 o o M o o o o o o o o z o o o o o a -J o o o o o o M — ■ o o o o o (/I -^ CO as CO CO co •*■ •T J- -r -f -J- H II H II II II X X X X X X UJ UJ UJ UJ UJ UJ I I I X I I I I I I I I z z z z z z 3 ce o o < m u. o a o o a a o r- sa co o z « -0 o o> -^ u. o o u. LL o a u. u. o x 3 u) o < -o o O — * o O a. o o LU — a o o o O o o a o O O o o o o X ►- u. o o o o o o o o o o o o o o < -- o u. o o o o o o o 3 o o o o -I ►- o o o o o o o o o o n o o n o < z o o o o o o o o o o o o o o o Z UJ o o o o o o -5 o o o o o 3 o o aC to o a o o o o o o o o o o o o J UJ UJ o o o o o o r> a o o n o o o o t- oc o a o o u o u o o o o o o o o Z a n o n 1 o o ^ o r> n -5 1 *^ -1 -1 — UJ o o o a o o o o o o o o o o o at 0> o-> CO CO Q u u o u u o CO o u u * •«• * # * ■»• >»• •* •* >»• -»■ * •J/ > CD — O Z — O a a to to 3 . o • . o o o o o a a o o o o o o o o o o o> o o <*■ • 0> 0> 0^ CO 7* O. 0> • 3 s s • •*••••••••#• _._..,_ .................. o o ---• o o ...... o~f-^>ro'-oo-O>fOOO ~g. JOi' r iirioo — i— o^r-^o">ooJA-o>coa , o^oo^'MOO x o^o Zcococoo>ruocooococooo^4'coo ^ruro^u^ug ocjj tr o co O O -^ocooococroxcDo^o ZQ.<^OCOOOCOOOCO • • t • • • in . • • • t • • • • -i o o> o> -i o ae^-«a> — ^o>^- H M H II II II II II II H II II II II || — -. — — — — «..•-. — — -. — .» — 0—«lN< ,, '^U> ^(M(*»*l«u3^coo»^-^-«^^-H II II It H II l| __w V — — w___ — __ — — 61 UJ U Z or < UJ O t- — z u. — — o Z a o fM m rg ft o o o o o 0> 0> o o- o CO o h- o — 0> o 0> o o o 0- o» o 0» o .0 o c* o H- • o 0> o o o 9 • o • o 0* o t • < — o • o o o — « *-. o —4 o • o .-* o r- o o o o o o o — o o z 3 a o o o o o 3 UJ o o 3 o o o o o o o o o o o o o o jj m o o o o o o o o — 0. -i UJ o o o o 3 o o o o o o o a o o o o o o o o o 3 ^ O t*~ UJ CO < oe o o o o o o o O Q a. a < u. z o o o o o o o OIL- u. a. — « Of UJ • t • • • • • o o o 3 o o o o o o o o o 3 o o 3 3 O o o z z o © o o 3 o o 333 a a o z < o o o o o o o o o o o 3 o 3 O o o o o o o o O O -3 o o 3 — o o o a o o o 333 o o 3 u. o o o o o o o O O -5 o "3 "3 MM o o 3 o o o o o o o o O o z 3 3 3 o ■3 o o o o 3 o 3 o M --3 3 o 3 o l/l — 2 'J m •D o •» * -r •T -1- J- « H II II II II X X X X X X III III 111 III 111 III I X I I X I I I I I I I x z z z z z I I I I "~l I a a a i. a a Q o *~ o o o a o o o o UJ o r~ -4 z U. o Q 3 o o u. u. o u. o r- o u. o X 3 o LL •a 3 © — * o "J u. 3 o UJ — o o O 3 o o o o o o o -- o o o X t- 3 o 3 o o o a o o o -3 o o 3 o < 3 o o o o o o o o o o o o 3 o -J >- a o o o o o o o o o o 3 o 3 o < z 3 o a o o o o o o o a o o 3 o Z UJ o o o o o o o o o o o o o 3 o ec wi o o o o a a o o o o o o o 3 a UJ UJ o o o -3 o © o o o o -5 o o 3 3 »- ec u o o o o o o o o o o o u a o Z a. o 3 3 3 3 o »5 3 •D 1 -1 o o 3 n _ UJ o o o o O o o o o o o o o 3 o ec o o go S3 — o o o u o o CO o u o * <»■ *»• * * * f •»■ ■*■ <*• ->■ Z X a o < ^ • -« x I 3 •- < -J uj o z 1/1 */» X 3 •- < — 3- > S3 — O l- J 3 Z o "> <* 3 • 0> » • r" • ^^ ^^ ^ CO (^ a. a- i c 3> • •• •• ...... .. _,_,.._ .................. o s 3 3 3 n> ..-•»- 03-<333333333S33 HUH II II II 333303333 + ♦ ♦ ♦ * I <*) U <-) <-} *• * | + + *-*.* + ujujujjjujuj ujujujujujujujujujcj>3a300 333333003333333 OOUO 030000 00000 000030330333000 Z00000303330IT>000 a33ir\33300003l~-300 O— OOf~003000000D030 Uj*-00r0000000^0^0iri3 ixno'M3r-3^-3 — ►-^O^OOO'NJfNO^O-^T'^O JZ-^0,-\JOOO^-<3C03-03-rif> oO x oo , *o > zo-oOcossoe^o^s • • • • t • UJ • • • • • ♦ t • . — * o — < O ^ rn at-^oo^ooo-^-js II II II II II II II II II II II II II II H — — -» — -» — — . -» — - ._ — — .__o- — -j-<^-<— 4 (1 II H II II I! _~.__ — ww — -. — — ww^.. 62 APPENDIX B EXAMPLES OF DOCUMENTATION OF THE BASIC ROUTINES 63 APPENDIX B This appendix contains the assembly listings of several significant digit routines: @0L2ADD, @0L2MPY, and 0L2PIVP. One of the assembly language macros used to generate the sig- nigicant digit function is also included. It should be noted that in its present form, 0L2PIVP (inner product) assumes the elements in the input vector occupying consecutive locations. The indexing scheme can be modified to handle more complicated storage schemes as found in the array structures of OL/2 lan- guage (20) . 64 Cg OOOC-OOCOCOOOC.or'OOOC. 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Report No. UIUCDCS-R-72-530 IOGRAPHIC DATA h le and Subtitle IMPLEMENTATION OF BASIC SOFTWARE FOR SIGNIFICANT DIGIT ARITHMETIC ' JistU N SEE SUN LAI (rforming Otganization Name and Address ]3pt . of Computer Science Jiiversity of Illinois at Urbana-Champaign "°bana, Illinois 61801 jonsoring Organization Name and Address litional Science Foundation ishington, D.C. 20550 3. Recipient's Accession No. 5. Report Date June 1972 6. 8. Performing Organization Rept. No. 10. Project/Task/Work Unit No. 11. Contract/Grant No. US NSF-GJ-328 13. Type of Report & Period Covered 14. upplementary Notes bstracts This report is co dgit arithmetic usi ad Ashenhurst . One isic software modul bwever, these modul ;)re are adaptable t uchines. A discuss Dntrivial problem o povided in Appendix ncerned with the implementation of significant ng the unnormalized arithmetic of Metropolis of the primary objectives was to design the es for inclusion into the OL/2 array language; es are written in assembly language and there- o other software systems for the IBM 360/370 ion of significance arithmetic, including the f input/output, is presented. Examples are A. ley Words and Document Analysis. 17a. Descriptors Significance arithmetic, unnormalized significant digit arithmetic. Hi Identifiers/Open-Ended Terms 7COSATI Field/Group B-vailability Statement ^limited 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages 77 22. Price I NTIS-35 ( 10-70) USCOMM-DC 40329-P7 1