AD-AI20 107 ARMY ELECTRONICS RESEARCH AND DEVELOPMENT COMMAND WS--ETC FIG 12/1

ASI MPLIFI C ATI ON OF LENTZ S ALGOR!THM.( G82wjLNZ

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TR-01 18[A Reports Control SymbolQSD 1366

lye A SIMPLIFICATION OF LENTZ'S ALGORITH14

AUJGUST 1982

By

Wd. J. Lentz

Approved for public release; distribution unlimited.

9 US Army Electronics Research and Development Command

Atmospheric Sciences LaborstoryWhite Sands Missile Range, NM 880-12

82 10 12 018

NOT IC ES

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The findings in this report are not to be construed as anofficial Department of the Army position, unless so desig-nated by other authorized documents.

The citation of trade names and names of manufacturers inthis report is not to be construed as official Governmentindorsement or approval of commercial products or swvicyerefereoced herein.

Disposition

Destroy this report when it is no longer needed. Do notreturn it to the originator.

ERRATA

ASL-TR-0118

A SIMPLIFICATION OF LENTZ'S ALGORITHM

Please change subject document as follows.

Page 7, sixth line from bottom of page, change and the

denominator [-1,1]" to and the denominator [1,-i]"

NI

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_______________________________________READ INSTRUCTIONSL~PNP~(A REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM

1. REPORT NUMBER 127 GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

ASL-TR-01184 t4)~~/~r4. TITLE (anid Subtitle) S. TYPE OF REPORT & PERIOD COVERED

A SIMPLIFICATION OF LENTZ'S ALGORITHM Final Report6 PERFORMING ORG. REPORT NUMBER

7. AUTHOR~'.) S. CONTRACT OR GRANT NUMBER(*)

W. J. Lentz

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASKAREA & WORK U NIT NUMBERS

US Army Atmospheric Sciences LaboratoryWhite Sands Missile Range, NM, 88002 DA Task lL162111AH71-DO

it. CONTROLLING O FFICE NAME AND ADDRESS 12, REPORT DATE

Us Army El1ectronics Research and August 1982Development Commnand13NUBROPAE

Adeiphi, MD 20783 114. &ONITORIN4G AGENCY NAME & ADDRESSII dillfernt from Controflinif Office) IS. SECURITY CLASS. (of this report)

IS. DISTRIBUTION STATEMENT (of this Report) SHDL

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the. abstract entered In Block 20. It differeni from Report) j

It. SUPPLEMENTARY NOTES

IS. KEY WORDS (Continue on reverse side If necesry and Identify by block num~ber)

Continued Fractions AlgorithmComplex Bessel Functions Spherical Bessel FunctionsM4ie Scattering

AUSTrAcr F(cautame sr- New .*i nggeway mad idevify by block number)

The Lentz algorithm is a method of computing the continued fractionrepresentation of functions. It was originally developed to calculatespherical Bessel functions in Mie scattering calculations. A simplificationof the algorithm improvement is presented for the case in which a given termis exactly zero. A FORTRAN IV program is presented which calculates threetest cases correctly when the numerator or denominator terms are exactly zero.

DD 1473 Eve-noes or i Nov 6s is oUSOLETE __________________

SECUITY CLASSIFICATION OF THIS PAGE (When Date Entered)

CONTENTS

INTRODUCTION.................................. ..................... 5

BACKGROUND......................................................... 5

Original Algorithm Improvement ....................................... 6

Numerical Examples ................................................. 7

DISCUSSION OF THE SIMPLIFICATION ..................................... 8

Simplified Error Removal............................................ 8

SUMMARY............................................................ 9

APPENDIX A FORTRAN IV PROGRAM SIMPLE.FR ............................. 10

COpW-Np~e

2P '

,I.

INTRODUCTION

The ratios of spherical Bessel functions of complex argument are needed forMie calculations of scattering and absorption. Lentz has developed a newcontinued fraction technique' for evaluating these ratios, with an improvementto eliminate errors when certain terms approach or become zero. The authorsof a recent note,' however, had difficulty in applying the algorithmimprovement when either the numerator or denominator was exactly zero. Thiscase is special, and a simplification is possible. The purpose of this paperis to present an algorithm for this special case, together with someclarifying remarks concerning its development.

BACKGROUND

The Lentz algorithm is simply a way of calculating continued fractions; thealgorithm begins at the beginning of the continued fraction rather than atsome indeterminate final coefficient. An improvement formula allows one tobypass any step in which a given numerator or denominator might be zero (tothe accuracy of the computer). In addition, roundoff errors may be reduced byusing the algorithm improvement. When applied to the continued fractionrepresentation of ratios of Bessel functions, the method is easy to use andfree of most of the weaknesses of the other algorithms used to generate ratiosof Bessel functions. 2

Let us review the method of computing a continued fraction by defining thesimple continued fraction. The simple continued fraction is defined forarbitrary an not equal to zero as

F = a1 + 1.0a2 + 1.0

a3 + 1.0a4 + ... (1)

which may be written as

I ~F a l+Li_

F a + ... , (2)

and the n'th partial convergent of the continued fracticn is

'Lentz, W. J., "Generating Bessel Functions in Mie Scattering CalculationsUsing Continued Fractions," Applied Optics, 15:668, 1976.

ljaaskelainen, T., and J. Ruuskanen, "Note on Lentz's Algorithm," AppliedOptics, 20:19, 1981.

. .. p~jj4 Pk5 5ha .lmo1 11PA

n a 2 + a3 + a4 + + an (3)

The Lentz algorithm is simply a way of computing the Fn without sta-tir.y at

some indeterminate last an. A further simplification of notation is needed:

Fn = [a,, a2 ., an (4)

Then the Lentz algorithm becomes

[aI [a2, a1) [a3, a2, 1] . .. [ar, an-I' . a1]

n - a2J La3, a2] ... Lan, an_ 1 ' .... a2 " (5)

Although the formula looks complicated, the successive numerators anddenominators are generated by taking the reciprocal of the previous numeratoror denominator and adding it to the next am.

When a given numerator is equal to its corresponding denominator (to thenumber of digits accuracy that is desired), the continued fraction is saie. tohave converged.

Original Algorithm Improvement

The original algorithm paper included an improvement technique to avoidinaccuracy when any term in the numerator or denominator approached or becamezero. The problem step is bypassed without loss of accuracy even if the termis exactly zero. Consider the following case in which I/a is equal to but ofopposite sign to am . When am is added to I/a, every digit that is the same in

the two terms will cancel and cause a loss of accuracy in the final number ifno steps are taken to bypass the problem. For example,

Let [a,-I, ..., al] =

If [a., ... , al] o - am + 1/a - 0

and [am+,1 ... , a = Y = am+ I + 1/B.

so [a2, ...1"2 a1 ] Z S am+2, + 1/y

Then C am+ 1 8 a, 0 + 1 (6)

-oj

where € is not necessarily 1.0.

The product of the two terms 8y E can be correctly calculated by equation(6). This avoids the problem of multiplying zero times infinity in equation(5).

& is the product of the next two terms. z, the next numerator or denominator,is then

am+2 y + 1 am+2 (am+i 8 + 1) + 8Y ~ am+1 8 + 1

am+2 + a + B/ (7

= m+2

The algorithm improvement may be applied to the denominator by replacing at by

a2 in the above equations. The algorithm improvement may be applied

independently to either the numerator or denominator or to both.

Numerical Examples

Consider the following three cases, which illustrate the implementation of thenormal Lentz algorithm and the Lentz algorithm with improvement.

Case A- Fna = -7 [1,2,2,2,2,...]

= 1* 3* 2.333*2.428571429* 2.411764706* 2.414634146

2 * 2.5 * 2.4 * 2.4166666666 * Z.41379313

Case B: Fnb = [1,1,-1,1,2] 0.5

Case C: Fnc [I,-1,1,-1,2] = 0.5

In case B the denominator [-1,1] is exactly zero, regardless of the computerword length. In case C both the numerator [-1,1] and the denominator [-1,1]are zero, causing both the numerator product and the denominator product to bezero. For case B one has:

_m [I1:] [1,1] [-1,1,1) [1,-I,1,1] [2,I,-1,1,1]nb UJLi L-I,i Ll,-l,1J LZ,1,-1,lI

Applying the algorithm improvement one has

FI ( 1 I( , 1 ] [ - ,I-1 , 1,, ,1 [2 , 1 , - 1 g ,1 ,1nb . 1 (1*0+1) {Z+Q/1}

7

where (1*0+1) corresponds to equation (6) and f2+0/1} corresponds to equation(7). The continued fraction is then calculated with ease:

F 1 * 2 * -0.5 * -1 * 1 = 0.5nb 1*1*I

In like manner, the improvement may be applied to more complicated cases, suchas case C, where corrections in both the numerator and denominator arerequi red:

Fnc L 'lIJ L-l,1,-I] J -,,1

In this case we have

F 1* (1*0+1) (-1+0/1) (2+1/-1) = 0.5

nc (-1) (-1*0+1) (2+0/1)

Notice that the term cannot be zero unless am+3 = 0, which is not allowed by

definition. If any am were zero in the derivation of a representation of a

function, the fraction could easily be transformed to the form of equation (2)by formula (5) of section 3.10.1 of Abramowitz.'

Up to this point we have not considered the Bessel functions for which theLentz algorithm was derived. Like Florida orange juice for breakfast, thealgorithm is not just for Bessel functions anymore. Rather, it is a generalway of computing the continued fraction representation of any function. Agood source of continued fraction representations of functions is found inAbramowitz together with the general properties of the functions.

DISCUSSION OF THE SIMPLIFICATION

Simplified Error Removal

In some cases it may not be desirable to include an error correction when oapproaches zero but is not identically zero. When either the numerator ordenominator is zero to the accuracy of the computer, a significantsimplification in programming is possible. This simplification reduces the

'Abramowitz, M. and I. A. Stegun, ed., Handbook of Mathematical Functions, USDepartment of Comm.erce, National Bureau of Standards, Washington, DC, 20402.

8

complexity of the algorithm and will often speed up its execution with littleloss of accuracy.

Consider equations (6) and (7) when B is exactly zero:

[am , ....a 1I] = B = 0

[am+i,...,aI] = y am+, + 1.0/8

[am+2 .... ,aI] Z = am+2 + B/ = am+ 2 (8)

And the product of the two terms By isZ = By = am+Is + 1 = 1

To include the simplified algorithm improvement, simply do not take thereciprocal of the previous numerator and denominator before adding the nextam+2. The running product is multiplied by one, so that it does not change if

the error bypass is implemented. This simplification was first published inthe Hewlett Packard 67 User's Library under Spherical Bessel functions., TheUser's Library program is no longer available, but the algorithm in appendix Adetails the methods used in that program.

SUMMARY

The original algorithm improvement in the computation of continued fractionsmay be applied to both the numerator and denominator of equation (5). Asimplified version of the improvement may be implemented in the case of anumerator or denominator being exactly zero. A FORTRAN IV programimplementing these ideas is listed for ease in use.

"Lentz, W. J., Hewlett Packard 67/97 User's Library, No 00642.

9

APPENDIX AFORTRAN IV PROGRAM SIMPLE.FR

Algorithm Example

The implementation of the simplified form of the algorithm improvement givenin equations (6) and (7) can be elusive. The best method to avoid pitfalls isto check the algorithm against well-defined cases such as Case B and Case C.The FORTRAN IV program Simple.FR, listed in appendix A, correctly calculatescases A, B, and C. The program was implemented on a Data General Nova 3minicomputer using FORTRAN V revision 6.11. The nonstandard accept and typestatements allow input and output in arbitrary format and may be replaced byread and write statements in standard FORTRAN IV. The variables have thefollowing functions:

AN the input terms of the continued fraction from equation (4)

NUM = the current partial numerator in equation (5)DEN = the current partial denominator in equation (5)

PDT the n'th convergent of the continued fraction formed the n

numerators and n-1 denominators in equation (5)

NFLAG = switch to avoid taking the reciprocal of zero in thk rator

DFLAG = switch to avoid taking the reciprocal of zero in the denominator

ICOUNT = counter to keep track of the current AN

10

S I IPLE.FR

1: PROGR~ri SIM"PLE IS DEI.TCHED TO PAVE SEPPATE LODCPS '70C NUM OND DE2: A P SiMPLIFIED Thl !- -4L- FOP-- LWi~JRkD TDErOM'lt4PTOP EO_ 4 rr P

3: C PROCPAN POISSES THE TE.T F THE HJLLJ D EPIES4: [.-12=U.53 PT '6E P T ?-Ot0,EPLENE

1,LL.L1,2JO0.5 AT T11< LPST1 fOFVJ-GEY CEr- C THE Tp 12S rEST'S THE [U O ITCP L&F (PhD THE SECOND TUE NULMFnPTJC:

EYPPS.

5: PEOL POT. NU 1.DEN, P]A

11: 1 TiPE ' ICLUHT"U, iflPUT ti'n~12: PCCEPT AH13: PDT=A4

15: DEH=0.O,

13: DFLPCG.1?-' !t*~*v' [n!)lt LOOP POINT AFTER III LiZTO*.*****w***v**wt20 10[ -if FUU. FD. DEN. rHtD.NFLPC. NE.1I.PND.OFLPC..NE. 1)TYPE "FRACTION HAS CONVERGED"'1 I COLIHI> ICOUIT-1

Y' 1 Pt 'PTT ", PDT, [COUNITT)FE "INPUT RNI"

-, ' ACCEPT il1IPkHN.E0.U) GOTO I

1: c STOP IF PN IS ZERO WJHICH Is HOT ALLOWED IN THE ALGORITHM: C****lTEST FOP fPEVIOUS NUMERATOR OR DENOMINATOR ZERO). IF SO SKIP STEP********'

-3 : [Ft [[EURO.! E. I) NUr'l=N'Ul-PN29: IF (DFLPiC. I4IE. 1 ) DEN -DEN-rAN30: HEFLAG =HFLPnG+ 1Z I: DFLPG =DFLR:C 1-l,2: P**C**t!3'% 1LO4NCI ORMAL NUMERATOR LOOP UNLESS ~ ?l*****t~*

5)3: IF(t!il.E7.fl) 5-OTO 113,1: PDT=PDTtNfUlI

35: Ull" 1 .0/"HUHob: 14FLOG=J37: C~*x~cctto1f~1-LO GIS DENOM I NATOR LO':***************38: 11 IF(DENI.EOKO) GOTO 1039: PDT=PDT/I)EIlQ J: DEN>'-- I.0O/DEN41: DFLnG=O42: COTO 1043: END

IDENTIFIER REFERENCES

AN 9 12 13 14 24 25 20 29DEN 9 15 28 29 Z P 35 4DFLAG 18 20 29 31 41ICOUNT 26 21 .2NFLAG 17 20 3U 36NU1m 9 14 20 28 3 24 35P T 9 13 22 34 29

1 11 25:0 20 :3 42

1233 38SIMPLE7COUNT=O, INPUT AN1.PDT= 3. 8fl08E I

'4PUT AN1.

PDT= 0.200000E 1 2flNPUT AM

PDT= -0.108000E1 3

INPUT AN1.PDT= 0.100000E 1 4INPUT AN2.PDT- 0.500008E 0 5INPUT AN0ICOUNT=9. INPUT ANI.PDT= 0.100000E 1 IINPUT AN-1.PDT= -0.1000E 1 2INPUT AN1.PDT- -0.10080E IINPUT AN-1.PDT 0.1080080E 1 4INPUT AN2.PDT- 0.500000E 0 5INPUT AN0ICOUNTa8, INPUT AN

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