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by: Hailey Halvorson


Hailey Halvorson
GPA 3.8


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This 125 page Class Notes was uploaded by Hailey Halvorson on Thursday October 22, 2015. The Class Notes belongs to PHYS 4 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/227134/phys-4-university-of-california-santa-barbara in Physics 2 at University of California Santa Barbara.

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Date Created: 10/22/15
Quanfum Monfe Carlo simulafions of solid 4H9 01 zero femperafure DE Galli Dipar rimen ro di Fisica gf mag l39 39 9 Email DavideGalliunimii r Coworkers L Reaffo M Rossi PhD s ruden r Universi rh degli S rudi di Milano Via Celoria 16 20133 Milano I raly n0261x10 3 no211lt10 3 n0228x10 a n0114x10 3 7 f 5 1 0 Ir r l A Summary oF my s rudies on solid 4He Full op rimiza rion oF Shadow Wave Func rion SWF Moroni Galli Fan roni Rea r ro PhysRevB 58 1998 Microscopic compu ra rion of EEG induced by a ni re concen rra rion oF vacancies in solid 4He SWF Galli Rea r ro J Low Temp Phys amp 2001 New exac r projec ror Quan rum Mon re Carlo me rhod the Shadow Pa rh In regral ground s ra re SPIGS Galli Rea r ro Mol Phys E 2003 Vacancy exci ra rion spec rrum in solid 4He SWF longi rudinal phonons SWF ex rra vacancy Forma rion energy SPIGS Galli Rea r ro Phys Rev Le r r 2003 J Low Temp Phys amp 2004 BEC in commensura re solid 4He SWF Galli Rossi Rea r ro Phys Rev B 71 2005 S rudy oF 4He con ned in a narrow pore SWF Rossi Galli Rea r ro Phys Rev B 72 2005 Transverse phonons in bcc solid 4He SWF Mazzi Galli Rea r ro proceedings LT24 Cri rical discussion on the na rure of the ground s ra re oF solid 4He commensura reincommensura re and BEC in incommensura re solid 4He SPIGS Galli Rea r ro cond ma r0602055 Is rhe ground s ra re oF bulk solid 4He commensura re or incommensurate 0 Early rheore rical works were based on rhe assump rion of zeropoin r vacancies Andreev and Lifshi rz JETP 1969 Ches rer PhysRevA g 1970 o IF ground s ra re vacancies are presen r this will have signi can r effec rs on low T behavior of solid 4He phenomenological rheory by PW Anderson WF Brinkman DA Huse Science m 2005 o Naive answer if is commensura re because compu ra rion oFlt P0 H 1P0gt in presence of a vacancy n of par ricles 100500 gives an energy which is higher of energy of perfec r solid Vacancy Forma rion energy Aev a r mel ring densi ry Fixed densi ry Me rhod la r rice 1 vacancy SPIGS hep 157108 SWF 156iO6 Aev gov 1V 8NVN 1 o This argumen r is no r conclusive rhe small size of rhe sys rem and the periodic boundary condi rions do no r allow ro explore all relevan r con gura rions allowed by PO1N in a real large sys rem Varia rional rheory oF a quan rum solid 0 In the Framework of varia rional rheory of quan rum solids The wave Func rions Fall in rwo ca regories 1 rransla rional invarian r 11 one example 7i 7jl Jas rrow 1PJ717N njf 2 1P imposes the symme rry of The la r rice localizing rhe a roms around rhe la r rice si res R q s 2 riRi N C 11117117N 11 x H e i Jas rrowNosanow N 0 Example Jastrow Function 1pJ717NHe umpQ12 Normalization constant QN Ida N e umj iltj Schematic landscape of probability distribution when density is large similarity with probability in con guration space of suitable classical particles PJ has a nite BEC Reatto PhysRev N crystal with 2 vacancies pocket Commensurate Crysml WW 1 iltj crystal pocket Vacancy POCke r Liquid pocket l ll ivyquotm itquot l M ml n Q N 1 39139 18 3 1969 and a nite concentration of vacancies For a Jastrow wF we know that overwhelming contribution to normalization 0N comes From pockets with vacancies nite concentration Hodgdon and Stillinger 1995 computed this vacancy concentration unfortunately they used an unrealistic PJ For solid 4He Standard MC computation normalize PO only in a single pocket computed energy is biased by the choice of N and cell geometry Our varia rional rool Shadow Wave Func rion Evolu rion of Vi riello Runge and Kalos Phys Rev Le r r 60 1988 WSWFltRgt MR x f d5 KltRSgt x SltSgt Direc r explici r Indirec r coupling via J as rrow correla rions subsidiary shadow variables Parlicles coordina reszR YiIquotN Classical analogy of 11152WF Shadow variables 5 S1SN N rria romic N Moms Q molecules Jas rrow rerms rR9 4555 00 gt a 3 a O SWF 39 KRS HZYJCW39E39Z Q Q Shadow variables 0 Shadow variables are s rrongly correla red Spon raneous rransla rional broken symme rry For pgtpO Crys ralline order oF 4He a roms induced by manybody correla rions in r roduced by rhe shadow variables 10 SWF simula rion oF hcp solid 4He projection of The coordina res of The real and shadow par ricles in a basal plane For 100 MC s reps 0 YA 5 6 ts 0 Shadow posi rions 4He a rom posi rions 10 1 0 10 x A SWF the solid phase PresentlI a Fully optimized SWF provides the most accurate variational description of 4He in the liquid and in the soHd phase Moroni Galli Fantoni Rea r ro PhysRev858 1998 9 A3 Accurate Freezing and melting densities Solid phase spontaneouslyI broken translational symmetry Local density hcp lattice p0029 A3 008 I I I 002 910 I 5 E K 00 I I I I I 9 Ac 0005 Equation of state Aziz potential 1979 2 39 I I I 39 393 OSWF DMC 4 x SPGS I I I 0025 0030 9 A3 0035 Local density direction L basal plane 1 I I I 39 39 I I 39 I 39 39 0 Classical in rerpre ra rion Classical analogy normaliza rion oF PSWF coincides N idiom with me con gura rional N so 0 mleCUIs par ri rion Func rion oF a classical o a o sys rem oF sui rable exible O SWF o a rria romic molecules 2 PSWF describes a quan rum solid wi rh vacancies and BEC Commensura re Rea rfo Masserini PhysRevB 38 1988 crys ral pocke r o PSWFXFQO where Fgoio only In the commensura re solid pocke r 1 90 increase We kine ric energy lt 1 WM WM 2 only a direc r calcula rion can discrimina re commensura re or incommensura re ground s ra re 0 Ground s ra re energy per parlicle of a rruly macroscopic sys rem eGEG N 0 Energy per par ricle From The simula rion of a commensura re s ra re eoEo N 0 To ral energy From the simula rion of an incommensura re s ra re wi rh one vacancy E1E0AeV eGeOXvAeV where XV is The average concen rra rion of vacancies 0 Al mel ring rhe bes r energy of a wave Func rion wilh localizing Fac rors is 0056 K per a rom above SWF gt allowing For XVAeV SWF are s rill rhe bes r For any XVltO8 Aevz7K a r xed la r rice parame rer o Calcula rion of XV For WSWF is a priori ry compu ra rion For lhe Fu rure SPIGS In principle The me rhod is exac r rwo parame rers con rrol convergence W K Evolu rion in imaginary rime E number oF projec rions P number oF monomers rime slices M2P1 otztP 9 accuracy oF rhe shor r rime propaga ror pairproduc r approxima rion Ceperley RevModPhys 67 1995 example 198 199 2o1 7 202 7 Evolu rion in imaginary rime oF rhe po ren rial energy a r p00218 A3 s rar ring From a s randard IIJSWF 3939139039 39393939quot39 3939 203 02 03 04 1 K1 01 Convergence oF rhe 493 6180K391 EN K 5 an Incommensura re crys ral 1 vacancy N107 51 M5 msx M37 M11 M15 7 x quott o 52 l l 4 o 005 01 1 015 02 T K energy per par ricle as Func rion 0F 13 in a simula rion oF solid 4He s rar ring From a Fully op rimized SWF Exponen rial convergence 2 e80K13 Commensura re crys ral N108 p003l A393 BEC in presence oF a ni re concen rra rion oF vacancies Galli Rea rfo J Low Temp Phys g 2001 0 Using a Shadow Wave Func rion For Fcc and For hcp crys ral with one or rwo vacancies a EEG was Found A r mel ring densi ry p002898 A393 condensa re per vacancy noV 022 Fcc noV 021 hcp ie For a sample with 1 vacancies rhe condensa re Frac rion per a rom is no22xlO393 39 Solid He I I I Solid He We did no r answer The ques rion p Hel Hel if vacancies are presen r in the ground s ra re Solid He BEC Specula rion on The phase diagram p T2BECz1 3 2A3v Hell Hel Hel SWF longifudinal phonons Galli Rea r ro PhysRevLe r r 90 2003 and JLow TempPhys 134 2004 We have compu red longi rudinal phonons in hcp and bcc solid 4He nding good agreemen r wi rh experimen rs Also rransverse phonon in bCC solid 4H8 Mazzi Galli Reallo Proceedings LT24 Near melting density p0029 A393 I I O FK O FM o FA SCPtheoryFA EXpFA l ExFM I 9 p I l I i I I l l g C 4O 30 20 1O 1 00 bcc i a m o 0 5 A 3 i i i t A Q j A f A A 0 AA A ngivzr F A Exp 01 O I I I 1 2 3 O 1 2 O 2 4 6 k A4 SWF resul rs ODLRO in commensura re solid 4He Galli Rossi Rea r ro PhysRev B E 2005 0 ODLRO is presen r nozSileO39 a r mel ring and For a ni re range of densi ries up To 54 bars 0 Local maxima signa rure of dis ror red la r rice 0 No ni resize eFFec rs Key process is rhe presence of VIPs 100 39 l I l 39 l 39 I Scaling analysis p0029 A3 2 2 1o Local maxuma lt p10 39 in a 1 10394 6 1 I I I I 10 o 5 1o 15 2o 7 A ODLRO microscopic origin p102 lt0 ltFgtIi1ltm0gt Snapsho r oF SWF rrimers in a basal plane o 5 e z 0 a gf o m 0689 b39quoteis 39aso ec cbw99q Q gu c 03 ycopuova s swore WWWQ New SWF ODLRO resulls p139 10 I I I 0 ODLRO in rhe less s rruc rured solid rhe 2 Email condensa re Frac rion is grea rer about 40 rimes higher 0 Wi rh rhis new SWF rhe densi ry range where ODLRO 0 is present will probably be p 0031 A393 larger 0 More damped oscilla rions 0139 1060 I 5 I 10 39A 15 I 20 7 large dISfance39 fhe Plafeau ls along neares r neighbours dis rance reached at shor rer dis rances single layer con rribu rions lo p1 along the z direc rion Agsuap 5USDaJ3UI inquot H 2 3 10quot 7 0 393 A N quotN D Lio 1 3 0 no i 10 Q 9 a or V N 2 d layer 10 60 Li 5 l 5 7 so 10 11 z z A o Resul rs 10 nonzero pla reau BEC for a wide 1 quot range of pressures 0 in lhe cen rral region of lhe pore lhere is BEC even if lhe syslem is in lhe solid phase 0 lhe oscillalions in lhe p1 lails are regislered wilh lhe crys ralline lal rice lhereiore lhe major conlribulion lo BEC comes from defec rs such as mobile vacancies p1 Agsuap 5USDaJ3UI Atmosphere and Surface Energy Balances Chapter 4 continued Lecture 10 January 28 2005 UCEUFFEHCES SD 35 Raw Score 40 45 5 Lnsolahon Re ecmn nsolanozw lnsolailon Scattering V The greenhouse effect 0 Rammed lncomin a nut to space sorlm radiation Credithttpwwwtuftsedutietciimagescli matechangegreenhouseeffectjpg components of the qreenhouse effect water vapor 60 carbon dioxide 25 ozone 8 Also nitrous oxide methane and CFC s Radiated energy relative mcreasmg a EMISSION SPECTRA OF THE SUN AND EARTH Visible Ultraviolet Infrared 5 10 20 A t E 300 K earth E E I Infrared I I III 45 Oxygen and Solar radiation 5 Ozone to Earth E 8 I Carbon dioxide 39 E A g i Watervapur L 5 I What atmosphere L frared ospace 4 5 10 20 Wavelength micrometers 1 A compare the ideal curve for earth 0 0 2 0 50 m 20 50 m above with the actual curve at left Wavelength mlcrometerst u me C Earth s temperature without the greenhouse effect Earth s average temperature without the natural enhanced greenhouse effect ie its black body radiative equilibrium temperature would be 256 K or 17 C Earth s actual average temperature is 288 K or 15 C the difference is due to the greenhouse effect The greenhouse effect on other planets atm Tblackbod T surface pressure y K Earth 1 K Mars 0007 216 240 Venus 90 227 750 what does this imply about the yes both atmospheric composition of atmospheres these planets are mostly COZ The greenhouse effect is not global warming the greenhouse effect is one of the most well validated theories in Earth science global warming is enhancement of the greenhouse effect due to gases we are adding gt coz N20 CH4 CFC s think of the natural background greennhouse effect and the anthropogenic enhanced greenhouse effect leading to global warming The greenhouse effect is not global warming Show shockwave 7 HSVIBEHOI39I to space To space a i3 9 xi 5 H in r5001 x quot 5 radiation Brut E lh 45 Incoming solar radiation reaches surface Cioudaibedo and ciouoigreenh0use forcing Shortwave re ection I 4 Cioud k Cloud greenhouss lovclng a Shortwave radiation 7 Longwave radiation Ernackcralsk 1 above IBUC39D eel Mtocumulus Smtacumrum bEIDW E feel am to lama feet 39 Ila Stratus hef w 61300 feat 51000 Almt ratu s 20 feet Cumui Inelaw 6000 feet Credithttpwwwucaredu High thin cirrus clouds Modest reflection of s or wave insulation Mostshonwave cirrus clouds have a net I a quot395quot 3quotquot warming effect greenhouse effect gt albedo effect Longwave radiation E u 2 7 m 1 a raradialed a High clouds net greenhouse towing and atmospheric warming Low thick stratus clouds ostratus clouds generally have a net cooling effect greenhouse effect lt albedo effect b Low clouds net albedo forcing and atmospheric cooling Aevoso Albadc e mm Law High Low van QWWS Amosphenc Warming Surface Coolan High Low 551722 A bedw is a furmtivaz a E 5 5 usuurugzce Farcms39 1 0227on4 crops grasslands 1 725 v Brick 551 20 40 focus on the Sahara why does it have both high reflected light but also high losses of longwave radiation now look at the Amazon Central Africa and Indonesia what causes high reflected shortwave and low longwave losses Emrw a amg m m Tmmwmrg 70 l J 39sw39rh39a g f 810qu 1 Re ecnan To 0869 Atmospheric heal llow radiation Dilluse radia on Direct Surface rad auon W heat ow 4 w Eath Energy Balance in the Troposphere Show the Shockwave Some fundamental concepts as a whole the earth is in radiative equilibrium the surface is net positive while the atmosphere is net negative the two balance one another out But there s more heat in the tropics 35D p350 Surplus Heat Energy Transferred By Atmosphere And Oceans To Higher Latitudes Net Shortwave Net Lengwave quot I W l I l l I l l I l l I D 90 YD 5D 40 30 2D l G 11 20 30 413 50 m St North Latitude r Sleuth 36 N 36 S Polar energy defliicit egy m ne y g SUWp rus Inooltmimg energy gain 1 gm Polar energ deficit Equatoniel r m i g a E n Poleward transport or energy su rplus r 7 39 r r r n If no energy transfer poles would be 25 C g equator 14 C warmer IF39aIewa transport or energy sunplus Surface enerqv budqet net radiation Qquot SWdown SWup LWdown LWup SWshortwave Atmosphere LWongwave sw w A LW Energy rate Wmz 39 I l l I l l lnsolation s75 i fuse 700 525 350 7 depends on rmed cloud cover 175 atmospheric temperature 0 i i i we 475 Re ection dependsion albedo 7350 A depends on LWl surface 525 I I Y Y Infrared I Y temperature Midnight 4 8 Noon 4 8 Midnight AM AM PM PM Time local Partitioning of surface net radiation Q H Sensible heat L Latent heat S Surface heat ux into soil or water latent heat is Solid Water Liquid Water Water Vapor I Freezing d Condensation theheat 9 lt lt7 I 80Calories d d d 500 caiories Involved In I i a I Meiting Evaporation water phase changes Credit wwphysicalgeographynet sensible heat is the heat that you measure with a thermometer what you can sense Processes of Heat Transfer Latent heat gt I Conduction Convection Latent heat flux Latent Heat Loss amp Salinity sutan now an aumce tummu moan chtrua mrmwrswrm w 2w a m o w mrworuwnonwrno39 mmmzo mww m Q can m A L wo wo39waovuowzov o39 m m m mmmammm mmmommaw own 2 x t a Latent Heat Loss amp Land Vegetation Sensible heat ux Enemy rate lWm sway me Wm 1 r m 9 m m m M 9 525 asu 1757 a 1 x x Mmgm 4 3 Mann 4 a Mmmghl m m w my me my as long as net radiation is positive nega ve air temperature will increase decrease I I 5ampth rm Solar I 53 radiati n 127mm T Tmm mart Elam Sari m n 35pm pm 2am l R a ii39acmm Sonar radiation 11 V ll l g i g tarmzatria i rariiati n r V m ii am I Will Credit httpapolloIscvsceduclassesmet130noteschapter3 Warmesttime otda Local y noon rquot I Ail 2s 2 temperature a x 5 gt r 9 39 quot quot 3 Is V 9 ur u t g I p Absorbed g E 39 t Insoiatlon 2 E i 39 g lt D ii I i er Midnight Sunrise Noon 3PM Sunset Midnight A Absorbed insolation Airtemperature for a cloudless day at midlatitudes Part IV Computer Arithmetic Division kLCICIKMiHMS AND HARDWII EF DESIGNS i NumberRepresentannn u AdditinnSubtracnnn Hi Muitipiicatinn iv Divisian v ReaiAnmmen Vi FuncnnnEvaiuatinn sethDZ Parhami vu impiememannnTnpiEs May 2007 Computer Arithmetic Division m Siidei I About This Presentation This presentation is intended to support the use of the textbook Computer Arithmetic Algorithms an ol Hardware Designs Oxford University Press 2000 ISBN 0195125835 It is updated regularlyby the author as part of his teaching of the graduate course ECEV252B Computer Arithmetic at the University of California Santa Barbara Instructorscan use these slides freely in classroom teaching and for other educational purposes Unauthorized usesare strictly prohibited Behrooz Parhami Edition Released Revised Revised Revised Revised First Jan 2000 Sep 2001 Sep 2003 Oct 2005 May 2007 Slide 2 May 2007 S B Computer Arithmetic Division IV Division Review Division schemes and various speedup methods Hardest basic operation fortunately also the rarest Division speedup methods highradix array Combined multiplicationdivision hardware Digitrecurrence vs convergence division schemes 3913 I14 15 Variationsi nDivide rs 16 Division May 2007 s B Computer Arithmetic Division Siide 3 I M WLLED AT warms1L PUT ON TELEVISION mom39s IT S NOTHING 601 magmas wRueEs MD GEA rw us D IVI39DJNG 13 Basic Division Schemes Chapter Highlights Shiftsubtract divide vs shiftadd multiply Hardware firmware software algorithms Dividing 2 scomplement numbers The special case of a constant divisor May 2007 l i39 i 3 5 Computer Arithmetic Division Slide 5 Basic Division Schemes Topics Topics in This Chapter 13 1 ShiftSubtract Division Algorithms 132 Programmed Division 133 Restoring Hardware Dividers 134 Nonrestoring and Signed Division 135 Division by Constants 136 Preview of Fast Dividers May 2007 S 8 Computer Arithmetic Division Siide 6 131 ShiftSubtract Division Algorithms Notation for our discussion of division algorithms Dividend Z ZZkriZZer 23222120 0 DIVIsor deH cl1 q QUOtient qwqu 7qu s Remainder z dxq sHsH sis0 Initially we assume unsigned operands d Divisor Quotient O O O O O O Dividend Subtracted bitmatrix O O O O s Remainder Fig 131 Division ofan 8bit number by a 4bit number in dot notation May 2007 l quot Computer Arithmetic Division Slide 7 Division versus Multiplication Division is more complex than multiplication Need for quotient digit selection or estimation Overflow possibility the highorder k bits of 2 must be strictly less than 0 this overflow check also detects the dividebyzero condition Pentium III latencies Instruction Latency CyclesIssue Load I Store 3 1 Integer Multiply 4 1 Integer Divide 36 36 DoubleSingle FP Multiply 5 2 DoubleSingle FP Add 3 1 DoubleSingle FP Divide 38 38 May 2007 S Computer Arithmetic Division Slide 8 Division Recurrence a Divisor 7 Qum39em 22 C C C gr 3 xgioe k wm mam aimsg am A githis magi Dividend Di 3 u mm Sumac 2rd q1 a 2i bitmatrix g qud 2 k bits k bits s Rem ainder Integer division is characterized by z d X 7 s Nooverflow dition for 2 2M x 2m 2 0 erac dfrac X qfrac Ziksfrac fracnons IS Divide fractions like integers adjust the remainder Zfrac lt dfrac May 2007 Computer Arithmetic Division Siide 9 Examples of Basic Division Integer division Fractional division 01110101 01110101 Flg3913392 z z quot 24d 1 0 1 0 22C 1 0 1 0 Examp399s 0f sequential 30 01110101 30 01110101 divisionwith 2slt0gt 01110101 2slt0gt 01110101 t d q324d 1 010 q31 q4d 1 01 0 q11 39f egelraln rac iona s 0100101 s 0100101 25W 010 01 01 25 010 01 01 Operands q224d 0 0 0 0 q2 0 q2d 0 0 0 0 q20 slt2gt 100101 slt2gt 100101 2slt2gt 100101 2slt2gt 100101 q124d 1010 q11 q3 1010 q41 33 10001 33 10001 253 10001 253 10001 q024d 1010 qo 1 q4d 1010 q41 slt4gt 0111 slt4gt 0111 s 0111 sfac 000000111 q 1011 qfrac 1011 Siide 10 May 2007 s B Computer Arithmetic Division 132 Programmed Division Shifted Partial Shifted Partial Next Remainder Quotient quotient igi caITy RS Rq inserted Fiagl I here Partial Remainder Partial Quotient 2k ij Bits j Bits Rd 2 d Fig 391 33 Regis terxusage for programmed division May 2007 s B Computer Arithmetic Division Siide ii I Assembly Language Program for Division Using left shifts divide unsigned 2kebit dividend H 133 zihlghizilow storing the habit quotient an remainder 9 Registers holds 0 for counter REQ39S eruisage for diVisor Rs for zihlgh amp remainder fQI39PrDQrammed Rq for zilow amp quotient dIVIsion Load operands into registers Rd Rs and Rq swam 5mm M div 1 a Rd With lelsor Emma meuzm grim load Rs With zihlgh Rs ma mgmd load Rq With zelow LEII hequot Check for exceptions m X a a n branch dibyio if Rd R0 P e i amid ow branch diovfl if Rs gt Rd N Initialize counter m mm load k into Rc Begin division loop 2 s diloop shift Rq left 1 zero to LSB MSB to carry rotate Rs left 1 carry to LSB MSB to carry s ip if carry branch noisub if Rs lt Rd sub Rd from Rs incr Rq set quotient digit to l noisub decr Rc decrement counter by l h d op 0 store the quotient and remainder quoti nt Fig 134 store Rs into remainder programmedrdms39ion quot39 using leftshifts May2007 ComputerArithmetiQDivision SiideiZ I Time Complexity of Programmed Division Assume k bit words k iterations ofthe main loo 6 or 8 instructions per iteration depending on the quotient bit Thus 6k 3 to 8k 3 machine instructions ignoring operand loads and result store k 32 implies 220 instructions on average This is too slow for many modern applications Microprogrammed division would be somewhat better May 2007 S Computer Arithmetic Division Slide is I 133 Restoring Hardware Dividers Shi Trial difference am bk Quotient q quot39 quot39 Lo fquot Partialremaindersm nitialvaluez I lt a a i Shi H 5 iQuotient MSBof E digit 250 E selector i i I i J Fig 135 Shiftsubtract sequential restoring divider May 2007 Computer Arithmetic Division Siide i4 Indirect Signed Division In division with signed operands q and s are defined by Z dxqs signs signz s lt d Examples ofdivision with signed operands z 5 d 3 3 q 1 s 2 z 5 d 3 3 q 1 s 2 not k7 2s 1 z 5 d3 3 q 1s 2 z 5 3 3 q 1 2 Magnitudes of q and s are unaffected by input signs Signs ofq and s are derivable from signs of z and 0 Will discuss direct signed division later May 2007 S B Computer Arithmetic Division Slide 15 Example of Restoring 0 01 Unsigned Division 1 1 1 0 1 1 0 1 1 1 1 1 s 0 0 1 0 0 1 0 1 Positive so set q3 1 251 0 1 0 0 1 0 1 24d 1 0 1 1 0 32 1 1 1 1 1 0 1 Negative so set q2 0 sZ251 0 1 0 0 1 0 1 and restore 252 1 0 0 1 0 1 24d 1 0 1 1 0 33 0 1 0 0 0 Positive so set q1 1 253 1 0 0 0 1 24d 1 0 1 1 0 34 0 0 1 1 1 Positive so set q0 1 s 0 1 1 1 q 1 0 1 1 FIg136 Ex am peiof testo rln g May 2007 8 Computer Arithmetic DiviSiOn Siide 16 I 134 Nonrestoring and Signed Division The cycle time in restoring division must accommodate Shifting the registers Allowing signals to propagate through the adder Determining and storing the next quotient digit Storing the trial difference if required Trial arrerence i Later events depend on earlier ones in the same cycle causing a lengthening of the clock cycle Nonrestoring division to the rescue Assume 7H 1 and subtract Store the result as the new PR the partial remainder can become incorrect hence the name nonrestoring May 2007 s B Computer Arithmetic Division I dl git seleaor Justification for Nonrestoring Division Why it is acceptable to store an incorrect value in the partialremainder register Shifted partial remainder at start of the cycle is u Suppose subtraction yields the negative result u 2kd Option 1 Restore the partial remainderto correct value u shift left and subtract to get 2U 2kd Option 2 Keep the incorrect partial remainder u 2kd shift left and add to get 2u 2M 2kd 2U 2kd May 2007 S Computer Arithmetic Division Slide 18 Example of Nonrestoring Unsigned Division NOOWWQW 0 1 113m lt 10mm Positive so subtract Positive so set q3 1 and subtract Negative so set q2 0 and add Positive so set q1 1 and subtract Positive so set q0 1 Fig 137 Exam Ie of P no nrestoring unsgned dIVIsion Siide19 May 2007 5 Computer Arithmetic Division Graphical Depiction of Nonrestoring Division Example 01110101M01010tWO 117ten 10ien Fig 138 Partial remainder variations for restoringand nonrestoring division May 2007 Pariial remainder Pariial remainder Computer Arithmetic Division b Nonresiori rig Siide 20 Nonrestoring Division with Signed Operands Restoring division 7H 0 means no subtraction or subtraction of 0 qk1 means subtraction ofol Example q 0 0 0 1 Nonrestoring division 1 1 1 1 We always subtract or add It is as if quotient digits are selected from the set 1 71 1 39 to quot quot 7 39 to addition Our goal is to end up with a remainderthat matches the sign of the dividend This idea of trying to match the sign of s with the sign 2 leads to a direct signed division algorithm ifsigns signd then 7H 1 else 7H 71 May 2007 s B Computer Arithmetic Division Slide 21 I Quotient Conversion and Final Correction Partial remaindervariation d and selected quotient digits during nonrestoring 0 division with d gt 0 z 7d Quotient with digits 1 and 1 11 1 1 17 lt11 1 1 11 Replace 1 s with Os Shift left complement M88 and set LSB to 1 to getthe 2 scomplement quotient Check 73216 a 42172576432421 1 ii 6 1 Final correction step ifsigns signz Add dto orsubtract dfrom s subtract1 from or add 1 to 7 May 2007 Computer Arithmetic Division Slide 22 Example of Nonrestoring Signed Division 0 0010 0001 A 0 0 1 0 0 0 01 signslt0gt I signd FIG 139 1 1 o o 1 so set q3 1 and add Exampie of 1 1 101 0 01 igf gggtqr39ngv 1 1 01 0 01 signslt1gt signd 69 0 01 1 1 so set q2 1 and subtract 39V39s39on 0 0 0 0 1 0 1 0 0 0 1 0 1 signs2 I signd 1 1001 sosetq11andadd 1 1 0 1 1 1 1 01 1 1 signs3 signd 0 01 1 1 0 set qo 1 and subtract 1 1 1 1 0 signs4 signz 0 0 1 1 1 so perform corrective subtraction 010 Shift compl MSB Add 1 to correct Check 3377 74 Siide 23 May 2007 5 Computer Arithmetic Division Nonrestoring Hardware Divider Com lement of Partial Remainder Sign Fig 13110 Shift u btractseqU ntiail nonresto39ring divi der May 2007 s B Computer Amhmeuc waswon Shde 24 I 135 Division by Constants Software and hardware aspects As was the case for multiplications by constants optimizing compilers may replace some divisions by shiftsaddssubs likewise in custom VLSI circuits hardware dividers may be replaced by simpler adders Method 1 Find the reciprocal ofthe constant and multiply particularly efficient if several numbers must be divided by the same divisor Method 2 Use the property that for each odd integer 0 there exists an odd integerrn such thatdx rn 2 1 hence d 2quot 1rn and Multiplication by constant Shiftadds E Z 2 72 12 quot12 2quot12 4quot 3 23974 2quot1e2quot 2quot Number of shiftadds required is proportional to log k May 2007 s B Computer Arithmetic Division Slide 25 l Example Division by a Constant Example Dividing the number 2 by 5 assuming 24 bits of precision Wehaved5m3n45x 24 1 Z 32 32 32 4 3 215 7 7i 1 2 1 2 1 2 5 2471 2417274 16 Instruction sequence for division by 5 lt z z shiftleft 1 32 computed 394adds lt q q shiftright 4 32124 computed lt q q shiftright 8 321241 2 8 computed lt q q shiftright 16 321241 2 81 245 computed lt q shiftright 4 32124 1 2812i616 computed QQQQQ May 2007 s B Computer Arithmetic DiviSiOn Siide 26 I Preview of Fast Dividers 33352 Fig 1311 a Multiplication and b division as multioperand addition problems a k x k integer multiplication b 2k k integer division Like multiplication division is multioperand addition Thus there are but two ways to speed it up a Reducing the number of operands divide in a higher radix b Adding them faster keep partial remainder in carrysave form There is one complication that makes division inherently more difficult The terms to be subtracted from added to the dividend are not known a priori but become known as quotient digits are computed quotient digits in turn depend on partial remainders Slide 27 May 2007 s B Computer Arithmetic Division II 14 HighRadix Dividers Chapter Highlights May 2007 Radix gt 2 3 quotient digit selection harder Remedy redundant quotient representation Carrysave addition reduces cycle time Implementation methods and tradeoffs Computer Arithmetic Division Slide 28 HighRadix Dividers Topics Topics in This Chapter 141 Basics of HighRadix Division 142 Radix2 SRT Division 143 Using CarrySave Adders 144 Choosing the Quotient Digits 145 Radix4 SRT Division 146 General HighRadix Dividers May 2007 S 8 Computer Arithmetic Division 141 Basics of HighRadix Division Radioes of practioal interest are powereo 2 andperhaps 10 q d I m in k digits k digits hi Division with left shifts so rswn qurkd with sioiz and shift sltki rks subtraot q Quotient d Divisor Fig 141 C O O O O O O z Dividend Radix4 q3q2twod 4 division in q1q0twod 4 dot notation s Remainder May 2007 8 Computer Arithmetic Division Siide 30 Difficulty of Quotient Digit Selection What is the first quotient digit in the following radix10 division 2043 12257968 I J 1226 39 39 39 39 122206 39 39 39 39 12252046 12257 2043 5 The omen mar tiara iamgiaxm w is maritime em mm iron am Hi macJam ame grammar arm imimvm mi mmgmwwmm ii E Bllmii f WM are to make a Suppose we used the redundant signed digit set 9 9 in radix 10 Then we could choose 6 as the next quotient digit knowing that we can recoverfrom an incorrect choice by using negative digits 5 9 6 391 May 2007 Computer Arithmetic Division Slide Si Examples of HighRadix Division fractional division Fig 142 Examples of highrradix division with integer 39and fractionalaoperands May 2007 Computer Arithmetic Division Siide 32 142 Radix2 SRT Division Before discussing highradix division we try to solve the more pressing problem of using carrysave methods to speed up each iteration gm 5 ggiwm c mm x was Q Li viii IP 17 a an M as mam iiie m mme May 2007 Computer Arithmetic Division Siide 33 Allowing O as a Quotient Digit in Nonrestoring Division This method was useful in early computers because the choice 7 0 requires shifting only which was fasterthan shiftandsubtract S May 2007 a metre e i Computer Arithmetic Division Slide 34 The Radix2 SRT Division Algorithm We use the comparison constants 42 and 12 for quotient digit selection 23 2 2 means 23 01xxxxxxxx 23 lt 42 means 23 1 OXXXXXXXX S g iaww DeCEHCof d WE 5W2 2 qj an as 12 We ampi go exit 9 113 1 E 2d d qJIj l d 2 12 Siide 35 1 May 2007 Computer Arithmetic Division Radix2 SRT Division with Variable Shifts We use the comparison constants 42 and 12 for quotient digit selection For 23 2 2 or 23 01xxxxxxxx choose 7 For 23 lt 42 or 23 1 OXXXXXXXX choose 7 1 2 srcornpl 2 srcornpl Choose 7 0 in other cases that is for 0 g 23 lt z or 23 00xxxxxxxx2lgrcomp 42 g 23 lt 0 or 23 1 1xxxxxxxx2l5700mp Observation What happens when the magnitude of 23 is fairly small 23 000001xxxx2l5700mp Choosing 71 0 would lead to the same condition in the next step generate 5 quotient digits 0 0 0 01 23 11110xxxxx2lgrcomp Generate 4 quotient digits 0 0 01 Use leading OS or leading 1s detection circuit to determine how many quotient digits can be spewed out at once Statistically the average skipping distance will be 267 bits May 2007 s B Computer Arithmetic Division Slide 36 l ADD 0 Example Unsigned Radix2 SRT Division 2 12 so set q1 1 and subtract In 42 12 so set q2 0 In 42 12 so set qa 0 lt 42 so set 4 1 Fig 146 and add Exafmple of Negative UHSiQried so add to correct radix 2 SRT diiision Uncorrected BSD quotient Convert and subtract up May 2007 Computer Arithmetic Division Siide 37 143 Using CarrySave Adders 710 of 011 derlap 5Hd Over ap Lj 712 0 Choose 71 Choose 0 Choose 1 g m mammmmaa msg zmmm fm Elna a May 2007 Computer Ammmeuc Dwon Shde 38 Quotient Digit Selection Based on Truncated PR 710 0 on Overlap 5 d Over ap 2 0 Choose 71 Choose 0 Choose 1 Sum part of 28W u LIon HAL2 0257mm Max error in 1 Carry Pan Of 230 V VrVo V71V72 392 srcornpi approxrmatron Approximation to the partial remainder lt 1A 1A 12 t UH VHN Add the 4 MSBs of Hand v Error in 0 12 May 2007 Computer Arithmetic Division Sirde 39 Divider with Partial Remainder in CarrySave Form is ulp for 2 coran Fig 14839 Blockzdiagram ofa radix2 diVider with partial remainder in storedcarr9form May 2007 s B Computer Arithmetic Division II Why We Cannot Use CarrySave PR with SRT Division S d 1 12 1 1 250 1 2d d q d 2d 1 q j 1 i d L 1 d 1 d Fig 149 39Overlapnregions in radix2 SRTLdivision Siidelii May 2007 s B Computer Arithmetic Division I 144 Choosing the Quotient Digits infeasibie region mm D cannot be 2 2o um um tmrsmaseeror am e 1 margin n umparisun nnn AUDI iEIi MEI m Choose mu 7m cnoose m eum infeasibie region p cannot be lt 72o mu JELEI Fig 1410 A p d plot for radix2 division with d e 121 pfartial remainder in d d andquotientdigits in 1 1 May 2007 l Computer Aninrneno Division lt gtlt gt 53 tn 0le m n was 4 men was I Fig 147 Siide 42 Design of the Quotient Digit Selection Logic Shifted sum uwuo uiiu 2H0mp Shifted carry v1 v0 viwvi2 2igrcomp Combinational logic NonO Sign Sign May 2007 S B Computer Arithmetic Division Siide 43 145 Radix4 SRT Division Radix4 fractional division with left shifts and 7 e 3 3 0 4sz q d with slolz and slkl4ks shift subtract s i d 7quot 77 1 l l l l 13971 74d I j 4d 45 l A d Fig 1411 New versus Shifted old partial remainder in radix4division with 7 in 3 3 Two difficulties How do you choose from among the 7 possible values for 7 lfthe choice is 3 ores how do you form 30 May 2007 s B Computer Arithmetic Division Slide 44 Building the pd Plot for Radix4 Division Infeasible region p cannot be 2 4d Choose 2 100 101 110 111 Fig 1412 A p d plot for radix4 SRT division with quotient digitsetV 3 3 May 2007 s B Computer Arithmetic Division II Siide 45 Restricting the Quotient Digit Set in Radix 4 Radix4 fractional division with left shifts and 7 e 2 2 0 4sz q d with slolz and slkl4ks shift subtract 4s01 Fig 1413 New versus shiftedold partial remainder in radixz4division with 7 in 2 2 Forthis restriction to be feasible we must have s e 7hd hd for some h lt 1 and 4hd 2d hd This yields h g 23 choose h 23 to minimize the restriction May 2007 s B Computer Arithmetic Division Slide 46 l Building the p d Plot with Restricted Radix4 Digit Set Infeasible region p cannot be 2 act3 Choose 2 m n Choose 0 V d3 m 100 101 110 111 d Fig 1414 A p d plot for radix4 SRT division with quotient digitsetV 2 2 May 2007 s B Computer Arithmetic Division II Siide 47 146 General HighRadix Dividers Process to derive the details Radix r Digit set u on for L7 Number of bits of p v and u and dto be inspected Quotient digit selection unit table or logic Iq jl d or us complemem Multiple generationselection scheme CSA tree Conversion of redundant q to 25 Fig 1415 Block diagram of radixr divider with partial remainder in storedcarry form V i quot 39 Computer Arithmetic Division Slide 48 15 Variations in Dividers Chapter Highlights Building and using p d plots in practice Prescaling simplifies qdigit selection Parallel hardware array dividers Shared hardware in multipliersdividers Squarerooting not special case of division Siide49 May 2007 Computer Arithmetic Division Variations in Dividers Topics Topics in This Chapter 151 Quotient Digit Selection Revisited 152 Using pd Plots in Practice 153 Division with Prescaling 154 Modular Dividers and Reducers 155 Array Dividers 156 Combined MultiplyDivide Units May 2007 S 8 Computer Arithmetic DiviSiOn 151 Quotient Digit Selection Revisited Radixr division with quotient digit set a a a lt r 1 Restrict the partial remainder range say to hd hd From the solid rectangle in Fig 151 we get rhd ad hd or h ar 1 To minimize the range restriction we choose h ar 1 ill r1 0 1V 0 1 o r 1 d Od d 06 Dtd rhd quotd rhd rs1 d Fig 151 The relationship between new and shifted old partial remainders in radixr division with quotient digits in a 0 May 2007 r r r Computer Arlthmetlc Dlvlslon Slide 51 Why Using Truncated p and d Values Is Acceptable Standard 39p XX XXXX 39 o d Carrysave p h 3 1d xxxxxxx XX XXXXX AH W ChooseB 3mm Noteh ocr 1 4bits ofd d min d Fig 152 Apart of p d plot showing the overlap region for choosing the quotient39digit value 3 or 31 in radixr division with quotien39t39digit set a a May 2007 S B Computer Arithmetic Division I Table Entries in the Quotient Digit Selection Logic 11 B1d lth ism 7hB1d B 7h ma Note h yrrel Origin d Fig 391 53 Apart ofp dplot showing an overlap region and itsstairoaseIikeselection boundary May 2007 s B Computer Arithmetic Division II 152 Using p d Plots in Practice I70L71 dminlr 470L dmin lgt 39b Smallest Ad occurs for the overlap region ofaanda l dmin dminAd d Ad dmin 2h71 Hg 1574 Establishing upper bounds on quot 7 n0 the dimensions df uncertainty39rectangles 4p dmmczm 1 May 2007 s B Computer Arithmetic Division I Slide 54 Example Lower Bounds on Precision p hai1d Choosea dmin 2 7 1 7ha Ad Fig 154 lt Ad gt Apdmin 2h1 l7ocr1 drnin i Choosea71 L V h0tdrniri P t For r 4 divisor range 05 1 digit set 2 2 we have a 2 dm 12 h ar 1 23 d w mm d 4371 Ad12 18 7 7232 Ap 1243 1 16 Because 18 23 and 23 g 16 lt 22 we must inspect at least 3 bits ofd 2 given its leading 1 and 3 bits ofp These are lower bounds and may prove inadequate In fact 3 bits ofp and 4 3 bits of d are required With p in carrysave form 4 bits of each component must be inspected May 2007 s B Computer Arithmetic Division Siide 55 Upper Bounds for Precision a 71 hd Choose a Choose a 7 1 My rams mm 1511 m gm m 1mm rz wink M2 4 May 2007 Computer Ammmeuc Dwon Shde 56 Some Implementation Details Choose 5 1 Chooserp Enooseigw i Fig 455 The asymmetry of quotient digit selection process May 2007 s l Computer Arithmetic Division m Fig 156 Example of p d plOt allowin large runcertainty re cta ng es if theg4 cases marked with asterisks are handled as exceptions A Complete p d Plot Radix r 4 7 in 2 2 d in 12 1 p in 83 83 Explanation ofthe Pentium division bug May 2007 8 Computer Arithmetic Division I 153 Division with Prescaling Choose 5 1 Overlap regions of a p d plot are widertoward the high end ofthe divisor range If we can restrict the magnitude ofthe divisorto an interval close torW say 1 sltdlt 1 5 when dW 1 quotient digit d selection may become simpler Thus we perform the division zmdm for a suitably chosen scale factor rn m gt Prescaling multiplying z and d by I77 Sh0Ud be done WithOUt Restricting the39divisor tothe shaded real mUltlpllcatlonS area Simplifiesquotientdigit selection May 2007 s B Computer Arithmetic DlvlSlOri l 154 Modular Dividers and Reducers Given dividend z and divisor d with d 2 0 a modular divider computes qLzldJ and szmoddltzgtd The quotient q is by definition an integer butthe inputs 2 and 0 do not have to be integers the modular remainder is always positive Example L 376123l 4 and 376123 116 The quotient and remainder of ordinary division are 73 and 7007 A modular reducer computes only the modular remainder and is in many cases simpler than a fullblown divider May 2007 S B Computer Arithmetic DlvlSlOn Slide 60 155 Array Dividers 21 11 22 12 23 13 24 Fig 157 Restoring array q divider composed of 391 controlledsubtra ctor cells V 3951 Z72 53 54 Zis 56 3 u u u u i H i i Quotlent q g1 g2 13 Remalnder s n n g4 g5 g6 May 2007 8 Computer Arithmetic Division Siide 61 I Nonrestoring Array Divider Fig 158 Nonres orihg array divider built of controlled 4 addsubtradtcells Similarity to array multiplier is deceiving May 2007 8 Computer Arithmetic Division Slide 62 I Speedup Methods for Array Dividers Idea Pass the partial remainder downward in carrysave form to speed up the operation of each row Critical path Fig 158 s 4 s 573 7 75 576 However we still need to knowthe carryborrowoutfrom each row Solution Insert a carrylookahead circuit between successive rows Not very costeffective thus not used in practice May 2007 s B Computer Arithmetic Division Slide 63 l


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