Floating-Point Arithmetic Algorithms in the

Size: px
Start display at page:

Download "Floating-Point Arithmetic Algorithms in the"

Transcription

1 IEEE TRANSACTIONS ON COMPUTERS, VOL. C-3, NO.1, JANUARY 1974 Citroe, Paris, as a Research Egieer workig with the cotrol of machie tools. I 1956 he joied the Societe Sciak, Paris, as Head of the Electroics Laborator, to desig ad develop circuits for weldig machies. He spet 1959 ad 1960 with the Compagie Geerale de Telegraphie sas Fil (CSF), as Head of the Techolog Departmet. From 1961 to 1963 he was a Teachig Assistat at Caregie-Mello Uiversit. I 1963 joied the Uiversit of Alabama, Hutsville, as a Assistat Professor of Electrical Egieerig; he became Professor i He teaches ad coducts research i the area of commuicatios ad data processig. Dr. Polge is a member of the Societe Fracaise des Electricies ad Sigma Xi. iclude optimal cotrol, digital processig of image data, ad simulatio ad aalsis of radar sstems. James M. Carswell was bor i Bishop, Calff., B.K. Bhagava was bor i Msore, Idia, o o Jauar 19, 19. Februar 15, He received the B.E. degree His major field of iterest is mechaical i electrical egieerig from the Bagalore egieerig, although he has had cosiderable experiece i electrical egieerig. For the last Uiversit, Bagalore, Idia, i 1967, the M.E. 15 ears his work has icluded trajector degree i electrical power egieerig from the Idia Istitute of Sciece, Bagalore, i 1969, aalsis; i particular, digital simulatio of sixdegree-of freedom trajectories, Mote Carlo ad the Ph.D. degree from Southem Methodist aalsis of trajector variables, ad special Uiversit, Dallas, Tex., i problems i flight mechaics. Util Jue 1973 Presetl he is a Research Associate with he was with the Uiversit of Alabama, Hutsthe Research Isitute of the Uiversit of Alabama, Hustsville. His curret iterests ville. Curretl he is with North America Aviatio, Dowe, Calif. Floatig-Poit Arithmetic Algorithms i the Smmetric Residue Number Sstem EISUKE KINOSHITA, HIDEO KOSAKO, MEMBER, IEEE, AND YOSHIAKI KOJIMA, SENIOR MEMBER, IEEE I. INTRODUCTION THE residue umber sstem is a iteger umber sstem. At preset, the techiques kow make it icoveiet to represet fractioal quatities. It is to be desired that umbers with fractioal parts ca be hadled as easil as itegers i the residue umber sstems. A few studies o the floatig-poit arithmetic i the residue Idex Terms-Cclic mixed-radix sstem, expoet part, sstem have bee published [11, [1. I these reports a power floatig-poit arithmetic algorithms, floatig-poit represetatio, matissa, ormalized form, umber of precisio, smmetric residue of or 10 is used as a expoet. This paper deals with floatig-poit arithmetic with a umber sstem. expoet which is a product of moduli i the smmetric residue umber sstem. This umber sstem has the followig Mauscript received September 10, 1971; revised August 4, ) fidig the additive iverse of a residue digit is The authors are with the Departmet of Electroics, Uiversit of advatages: ) sig detectio b mixed-radix coversio is eas, fairl Osaka Prefecture, Osaka, Japa. Abstract-The residue umber sstem is a iteger umber sstem ad is icoveiet to represet umbers with fractioal parts. I the smmetric residue sstem, a ew represetatio of floatig-poit umbers ad arithmetic algorithms for its additio, subtractio, multiplicatio, ad divisio are proposed. A floatig-poit umber is expressed as a iteger multiplied b a product of the moduli. The proposed sstem assumes existece of ecessar coversio procedures before ad after the computatio.

2 10 eas, ad 3) the result of scalig is rouded to the closest iteger. A cclic mixed-radix sstem will be itroduced first. The, based o this sstem, a ew expressio of floatig-poit umbers ad algorithms for additio, subtractio, multiplicatio, ad divisio will be described i terms of ormalized operatios. II. NUMBER SYSTEM I order to provide the appropriate foudatios ad motivatio for the ormalized floatig-poit format proposed here, a cclic mixed-radix sstem will be itroduced first. Cclic Mixed-Radix Sstem Cosider a weighted umber sstem i which a real umber is expressed i the form ± h E w.w. i=- { ci } is ad a.h is the most sigificat digit of the umber. From a practical poit of view the series (1) should be approximated b its appropriate partial sum. Here we assume that there are give radices ml, m,, m. We deote b ii the least positive (iteger) remaider of the divisio i/, ad b [i/] the largest iteger less tha or equal to i/, i is a iteger. The the approximate value of a give real umber ma be represeted i the form (1) a set of permissible digits, I wil a set of weights, I+-1 ± E iwi i=l with digits i succeedig positios, ai = ai i 1 ili +1 ) < a + i + 1 < mi i + 1 I wi= M /ml il mm () (3) (4) (lii 0) (I il =0) (5) ad et is the least sigificat digit, ad M = flh1 i. If a umber X is expressed i the form (), we call X a umber of "precisio." It should be oted that umbers i the biar or the decimal sstem are expressed b the form (). Number sstems i which the weights are ot powers of the same radix are called mixed-radix sstems. A cclic mixed-radix sstem is defied as a mixed-radix sstem cosistig of all umbers of the form (). I the followig discussio it will be assumed that the radices are odd positive umbers so chose that ml < m <.. < m ad the digits a, are restricted so that Mlil +1-M1 l l +(6) S t1il +1 6 IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 istead of (4). As a example, Table I lists the weights ad the permissible digits of the cclic mixed-radix sstem with ml = 3,m = 5,adm3 = 7. Now cosider the umbers of precisio, i a cclic mixed-radix sstem, expressed i the form Let e= [i/] I+-1 z i=l a.w.. I I S= ill (9) a.=i±+j-1i +1=ls+j-lI +±1 I = 1,,---,(). (10) The, these umbers ca be represeted i the followig form: kxme XM k is a iteger such that ki< (M-1),M= rh1i, e is a iteger, ad s s = 11J MP. Ms= 1, i= 1 Proof: From (5) ad (9), ad moreover, usig (10), w1=m=m w. I=m -.m im w I j-i e=ti-v i ma W0, v=l O the other had, b (3) ad (IO),5 Hece I11+j-1=OL 1+;-1 I + 1 l+-1 p = aaj. E Oliwi = E rl+j_ 1Wl+j_l i=lj= Here cosider the expressio (11). Let The, sice the a0 (s #0) (s = 0). (j= 15,,, ). (7) (8) ±t+ E 'a( I mi)mems. (11) /j-i k = ±a+ Ea( i0)a j= 1 v=l v are itegers such that, 3,,,, ). (1)

3 KINOSHITA et al.: FLOATING-POINT ARITHMETIC TABLE I WEIGHTS AND PERMISSIBLE DIGITS OF CYCLIC MIXED-RADIX SYSTEM WITH ml = 3, M = 5, AND m3 = 7( = 3) i [ i/l i If < < 7-x5-x3- = -5= <a <43 7- x5-1 x3- = -4= <ct <1 7-lx5-lx3- = -3= <a_ = < 7- x <_a < <cL <1 1 = 0 -= <ac < 3 0-3«a <3 5x <a <1 7 x 5 x <ca < 7 x 5 x 3 s 1-3_<a <3 7 x 5X 3 : 7x::3 we obtai kli< Therefore, m -1 a. m-i a. a. I _1 )+ E -± ) mu) lk< (m m " ma -1I). a -a1 a I view of (10) it is easil see that the sequece o1, a, is a permutatio of 1,,,. Hece, Ikl (M-1). This completes the proof. It should be oted that (7) ca geerate egative umbers as well as positive umbers or zero because of the restrictio (6). Normalized Floatig-Poit Format I floatig-poit operatio proposed here, the set of umbers cosists of 0 ad the set of all umbers of the form kxme XM k is a iteger, I(M/m8+ 1)S IkI.(M-1), (s 0) 1(MM + 1)-6 k <(M-1) (s = ) w ag (13) ormalized b the coditio that the most sigificat digit aa i (1) is ot zero, ad e is a iteger ragig betwee -E ad E, sa. We call the iteger k ad the pair (e, s) the matissa ad the expoet part, respectivel, of the floatig umber (13). Exceptioall, zero is defied as follows: O=OXM XMo. Theorem: The proposed ormalized floatig-poit format is uique. Proof: B cotradictio. It suffices to prove the theorem for positive umbers. For a positive umber A, let A = k1meims, ad A = kmems with k, # k, el 0 e, or s1 : s. Assume that the theorem is false; the it must be possible to fid a A = k Me, Ms, = kmems. with kl, el, si, k, e, ad s meetig the precedig restrictios. Without loss of geeralit we ma assume that e = el + Ae, Ae is a iteger such that Ae > 1. The it follows that which i tur implies Sice ad the Further, sice the I 11 (M/m + s i km =kmam I 1 8 k1m8 MAe =~ 1 kms S Me >M. (14) (i = 1, ) (M-1l)Ms AfAe S M + m (M-l)MS m 8 1) < ki"im.<i( 1) +1 <m M AM MAe<M But this cotradicts the assumptio (14). Hece, I S<M, si el = e. Next we assume that s1 < s without loss of geeralit. From (15), - ki M-1 k M + I m 8 11 (15)

4 1 Sice the m M ki <m k 8 O the other had, whether sl = O or #0, ki k = M1S +1 ms + ms but hece This cotradicts (16). Hece m 8al+ m~~~~~ 1 8sl+ 8s >m ki k 8 SI = S. 8~~~ (16) Cosequetl, k, =k. This completes the proof of the theorem. For a real x we deote b fl (x) either of the two umbers of the form (13) which miimize fl(x) -x I, except that if the fl(lxx)= MMM<M I x I < MM (M+ m +l1) I1a(M + I MeM+, (si -1) +IMl)Me+,M0 (s=-l) The value fl(x) will be called a ormalized floatig-poit represetatio of x. We defme the rage of floatig-poit umbers to be the iterval D= [M-1 MEM, M- MEMl ~~ 1~j Let fl(x) = kmem. be the ormalized floatig-poit represetatio of x for x e D, Ixl > ((M/m,) + IW-E. The, sice m MeM < Ixl< MAme, (SOO) s MeM8. lxi< MMl M8, (S=#O) (17) IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 it is easil see that e, s, ad k ca be determied i the followig maer: e = [logmmf Ixl -1. The iteger s is foud as follows. First compute 1x1 Me+l This quatit is used i the relatioship Gi_ M m. to obtai G,, G,, etc. This iterative procedure is cotiued util [Gi] = 0. If this occurs o the ith (i = 0, 1, ), the Fiall k is defied b s = i. k= ( \MeMS Here the R refers to correct roudig i fixed-poit arithmetic. If e is outside the iterval -E < e < E, we shall ol sa that a overflow or a uderflow has occurred ad shall ot proceed further. Coversio to the Residue Represetatio For the iteger k i (13) the least remaider i absolute value whe divided b mi ma be computed. This quatit, is referred to as the smmetric residue of k deoted b /k/mi, mod mi, ad the radices mi are called bases or modull. For a give set of moduli the residues of k ma be formed ito a -tuple Ik/ml J, klm Iklm This -tuple is called the smmetric residue represetatio of k. The iteger 1k/rm is called the ith smmetric residue digit -M-1 M-l to ma be uiquel represeted. Now cosider the coversio from a fixed-radix sstem such as decimal or biar to the residue sstem. The iteger k is specified i a fixed-radix sstem as k=d,r±+d, r-1 + +dir+do r is the radix ad 0 < di 6 r -1. The, takig this expressio modulo mi, the followig equatios are obtaied: km =/dllr'lm + d_1 Ir'-'/mi+ +d,irlmi+ do/ml (i= 1,,,).

5 KINOSHITA et al.: FLOATING-POINT ARITHMETIC Thus, if the powers of r modulo mi are directl available from the memor, lk/m, ma be computed b repetitive additio (modulo mi,) of those powers of r. Before goig ito the mai argumet, residue iteractig operatios of mixed-radix coversio, base-extesio, ad scalig will be briefl described. (For detailed iformatio o these operatios, see [3].) Mixed-Radix Coversio Mixed-radix coversio process is used to covert from the residue sstem to the mixed-radix sstem. The particular mixed-radix represetatio of iterest here is of the form k=am1m...m am1 + a1 the ai are the mixed-radix digits which are to be determied b this procedure. A iteger i the rage M-1 M-1 to ma be represeted i this form ad hece this represetatio has the same rage as a residue sstem of moduli ml, m,, m Ṫhe mixed-radix coversio is a fudametal operatio, from which other importat operatios such as base extesio, relative magitude compariso, ad sig determiatio ca be derived. I the smmetric residue sstem, the sig of a iteger is give b the sig of the most sigificat ozero digit of the mixed-radix expressio of the umber. B successivel subtractig ai ad dividig b mi i residue otatio, all the oi ca be determied, startig with al. A simple example of this procedure is give i the Appedix. Base Extesio Base extesio is used to fid the residue digits for a ew set of moduli, give the residue digits relative to aother set of moduli. I most cases, oe or more moduli are added to the origial base. The procedure is a mixed-radix coversio with a additioal fial step. A simple example of base extesio is give i the Appedix. Scalig I covetioal fixed-radix arithmetic, scalig up or dow b a power of the radix is simpl a series of right or left shifts ad is a fast ecoomical operatio. I the residue umber sstem, because multiplicatio is a simple operatio, scalig up is o problem. Scalig dow is, i geeral, a difficult operatio. However, it is eas to scale dow b a product of the mi; for example, permissible divisors are mi1 X mi X m5 or m but ot mi1 X M X m5 or M. This restricted operatio is referred to as scalig. A simple example of scalig is give i the Appedix. Ill. FLOATING-POINT ARITHMETIC Let X = kxmexmsx ad Y = kymeyms be two floatig-poit umbers i the rage D, the subscripts x ad represet X ad Y, respectivel. We defie b Z = fl(x * Y) the desired result of a floatig-poit operatio, Z is of ormalized form ad the * smbol represets additio, subtractio, multiplicatio, or divisio of two floatig-poit umbers. Adjustmet ad Overflow or Uderflow i Matissa Part I the proposed floatig-poit umber sstem, some adjustmet of a matissa part is eeded before computatio i multiplicatio or divisio as well as i additio or subtractio. Overflow or uderflow occurs a time the matissa of X * Y would fall, i absolute value, outside the iterval I1= +[f( I ),(M-1)] (s * ) I m ) '(m 1)] (s= 0). The, special methods are eeded to detect the occurrece of overflow or uderflow ad to ormalize the result. Additio or Subtractio We assume, without loss of geeralit, that e e M XM >M YM sx S (18) The expoet parts of X ad Y must be made equal before additio or subtractio. To alig the expoet parts, we rewrite Y as ex Y=KM xm S x e -e km Y xm K M The, we ca express the sum or differece ofx ad Y as e k M ZM z sz kz =kxkr± e5=e,s =s It should be oted that, i geeral, the floatig-poit umber is ot i the ormalized form at this poit. Now i order to kow how to get KR, cosider the relatioship betwee ex ad e, or sx ad s. 1) IfexZ>e +, I K I< I k IJM-Ms IMs < I k I(Mm ) < 1 /(m) Hecex Hece KR =0. ) If ex = e+ l ad sx >sy, x 13

6 14 Hece IKI= Ik 1M M /MI.I<IkI/M< 1/. SY KR =0. 3) If ex=e +ladsx <s, km1m K=k M1M8 /M8 = m S Sx m0ml.. ms m8 MIM "' M k m0m 1 s m s +1ms+ m x mo = 1. I the smmetric residue sstem the result of scalig is rouded to the closest iteger, hece KR is obtaied.... b scalig k b mom msxms+ims + m- Here, b the assumptio that all the moduli are odd, KR- K < 1/. 4) If ex = e ad sx > s, k K=k M -/M_ =Y S x msy+1 ms + msx Hece KR is obtaied b scalig k b m,+1m8y+ m.sx 5) If e = e ad sx =s, the evidetl KR =k. Note that the ol remaiig case ex= e ad sx < s ever occurs b virtue of the assumptio (18). After makig ecessar arragemet for a aligmet of expoet parts as is previousl stated, we compute k. = kx + KR. The, sice 0 < Ikz8I M - 1, a overflow or a uderflow ma have occurred, ad a test is required for overflow or uderflow detectio. Cosider the smmetric residue sstem with moduli mi, M, -- mm ad a redudat odd modulus m +, m+ is pairwise relativel prime to all the other moduli ad satisfies the coditios M-1.ia(Mm1+-1), mi <M+1' (19) Suppose we have the residue represetatios of kx ad k for all moduli, icludig ml,l If k_ is expressed i its mixed-radix form, we have kz a+1m1m m + tmim m1 +*-±+ami1 +a 1. Defie au to be the most sigificat mixed-radix digit which is ot equal to zero. The' subscript u will be equal to some iteger from + 1 through 1. The, u is equal to if ad ol if 4((M/m) + 1) < S 4(M - 1). Hece a overflow has occurred if u = + 1. If u <, the (M ) 4(S ) which implies that a uderflow has occurred. The ol remaiig case is if u =. I this case it is ot possible to IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 kow from u aloe if a uderflow has occurred, ad aother test is required. A method for the uderflow detectio requires the availabilit of the quatities '((M/m,z) + l)(sz = 1,,, - 1) modulo m1(i = 1,,, ). If s = 0, ((M/msz) + 1) < Ik I, which implies that o uderflow has occurred. The quatities :((M/m, ) + 1) are costats ad ca be permaetl stored i the residue form. If sz # 0, Ikz- 1((M/msz) + 1) is formed i its residue code ad coverted to its mixed-radix form. If the sig of the most sigificat ozero digit of this form is egative, a uderflow has occurred. After these tests, kz is replaced, if ecessar, b the iteger obtaied b scalig or multiplig kz b a factor, ad the expoet part (ez, sz) is corrected. I scalig kz, it is desirable to choose m,z+l as the factor, takig accout of the defiitio of MsZ, ad icrease sz b 1. If sz has reached, sz is set to zero ad ez is icreased b 1. It ca be show easil that kz falls ito the iterval I after beig scaled b m,z+i. I multiplig kz b a factor, if u <, sice kzmemzms be expressed as kz/ s + H mimz.=s +U+ 1 s +u z Hl M. j=1i ca we choose I=SZ+U+1 mi as the factor, decrease ez b 1 ad icrease sz b u, i is a modulo umber whe i >. If sz has reached, sz is set to zero ad ez is icreased b 1. The Sz quatities r1=.sz+u+1zi are costats determied b s ad u ad ca be permaetl stored i the residue form. If u =, we choose msz as the factor ad decrease sz b 1. The quatities msz(sz = 1,,, - 1) are costats ad ca be permaetl stored i the residue form. It ca be show that kz is ormalized after a fiite umber of this multiplicative iteratios. I the case of u = 1 or u = ol oe iteratio is required. Thus we have the desired result fl(x ± Y) = kzmezms8 which is the floatig-poit represetatio of X ± Y. Multiplicatio It will be assumed, without loss of geeralit, that sx > s We first cosider the case of s = 0 or 1. I this case, the product of X ad Y ca be expressed i the form e k M zm z sz k =k k M, e =e +e, s =s. z x s z x s z x Obviousl this is a uormalized umber. If s = 0, i m + I) Ikz8<(M- 1)

7 KINOSHITA et al.: FLOATING-POINT ARITHMETIC ad if s = 15 If we suppose that (M+l)d(M ±+1) the multiplicative overflow occurs alwas i computig kz. To cope with this difficult, cosider the smmetric residue sstem with redudat odd moduli m+ M+3, m+p i additio torl,,, m+l, m.(i = +,+3, + p) are pairwise relativel prime to all the other moduli ad satisf the coditios p k(m- l)i1.<(mm +1m+ -+ 1), m+ <m+ < < + (0) Before computatio of k5, we exted the base to iclude M+, m,3, -, m+p for the residue represetatios of kx ad k1. The multiplier M1 required for gettig k5 whe s = I is costat ad ca be permaetl stored i the residue form. Now to ormalize k5, defie ao.to be the most sigificat ozero mixed-radix digit of k_. The, if u is greater tha, overflow has occurred, ad therefore kz must be replaced b the result of scalig k_ b a factor. The scalig factor is chose as follows. For u such that u > +, we choose Il[j=ZzJ+.l as factor ad icrease sz b u -, i is a modulo umber whe i >. If s has reached, sz is set to zero ad e_ is icreased b 1. It ca be show that the value of u decreases to u = + 1 after a fiite umber of this scalig iteratios whe u > +. If u has reached + I, kz is scaled b msz+1 ad sz is icreased b 1. If sz has reached, sz is set to zero ad ez is icreased b 1. It ca be easil show that ad I m m 1<1(, 4 + I + I m < Ik M- m m 1 z 4 1, 1-1 ( M s +1 S (MRm +1 11(M 1< (>1) 1)M which implies a overflow ma have occurred. At this poit, to detect a overflow Ik_,I - (M - 1) is formed i its residue code ad coverted to its mixed-radix form. If the sig of the most sigificat ozero digit of this form is positive, a overflow has occurred, the kz is scaled b m, +I ad the expoet part is corrected. The quatit (M - 1) ca be permaetl stored i the residue form modulo mp(i 1,, ;- + 1). i Thus we have the desired result fl(x X Y) = kzmezm S5 the case of s = 0 or 1. A similar procedure is applicable to the case of s # 0 ad s # 1, usig the followig modificatios. I this case we ca express the product of X ad Y as e kzm ZM kek k = (= x1 R' e=e±+e±,is =S First of all, we exted the base to iclude m+, m+3, m,p pfor the residue represetatios of kx ad k. ad the compute kxk. Scalig kxk b ms +1msY+ - m, we have kz. It ca be easil show that i ) + 1) < ki. Hece kz must be ormalized b the aforemetioed scalig algorithm. Divisio Cosider the divisio of X b Y. If Y = 0, divisio is ot defied. Otherwise, sice kx I, k <. (M- 1), it is impossible to get the desired precisio of the quotiet kx/k b merel dividig kx b k. To icrease the precisio, we defie the followig fuctio A which is a multiplier to kx. 1) IfsX >s ad s = 0, X kxm e -e -1 MX Y M. Hece A is defied b Hece ) IfsX > s ads *O, B-= A =M. k BCM Xx XMex Y k Ms BC HI mi i=s+ 1 k kmcsx BCM e -e kmc MX k BC e -e - 1M -k MX A =BX C. s x X i=sx-s+l Note that if sx = s ad s * 0, the A = M. Mi.. (1) () is

8 16 3) If sx<s, kxms B = x x e -e Y km B s km B x s8 e -e = km MX = Hece A is defied b km B x sx e -e -1 X Mx kxms BD e -e - k x~ Mx Y M D= f[ Mi. A=M i=+s -S +1 x s x XBXD. +x - The quatit A is a fuctio of sx ad s ad is a product of moduli. It ca be easil see that A has a maximum Am = (Mm - 1 *- (+ 3)/ M(+1)/ ( is odd max (mm_ * m 1m )' ( is eve) - I (/)+1 whe SX = ad s = [1]. It follows that IkxIA < 4(M - l)amax. Therefore, the previous redudat moduli m+, m+3, +p are chose so that the further satisf the followig coditio: ~(M-l)Amm(Mx +1m+ --im+p 1). (4) Before the computatio, we exted the base to iclude Im+p for the residue represetatios of kx m+, m+39 ad k. The we form the product of kx ad A ad divide kxa b k. The value of A is a costat determied b sx ad SY ad ca be permaetl stored i the residue form. It is to be desired that the iteger quotiet kz = (kxaik)r should be foud b applig the divisio algorithm proposed b the authors [4]. The equatios required for calculatio of expoet part is give as follows. Ifsx >s, from (1) ad (), ad if sx < s from (3), ez =ex -e -1 sz =s, -s, e5 =e --e -, sz =±s -s. Thus we have a uormalized floatig umber IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 e kzmzms It ca be show that o uderflow has occurred ad a overflow ma have occurred i computig k.a/k. If a overflow has occurred, the same discussio ad procedure as described i the case of multiplicatio ma be applied to ormalizatio of kz. IV. SAMPLE RESIDUE NUMBER SYSTEM As a sample residue umber sstem, cosider the smmetric residue sstem cosistig of moduli ml = 13, m = 17,m3 = 19, m4 = 3, ad ms = 9 (M = ). The, for the matissa of the floatig-poit umber (13), the iterval of (3) defiitio is, i absolute value, [ 4889, ] [10771, ] [ 8375, ] [ 73704, ] [ 60886, ] (s = 0) (s= 1) (S= ) (s= 3) (s= 4). The redudat moduli which satisf (19), (0), ad (4) are chose asm6= 31,m7= 37,m8= 41, m9 = 43, ad mlo = 11. I floatig-poit add operatio, uderflow detectio requires a table of the quatities Lsz = T((M/is) + l)(sz = 1,, 3, 4). This ca be accomplished b storig, i a special memor, a table such as Table II. The smbol (( ))5 stads for the residue represetatio, for istace, ((L l))5 <-> I/Li/ l,, / l /1m t - Normalizatio requires tables of HlsZ m. (for u K:) ad msi (for u = ) such as Tables III ad IV. For floatig-poit multiplicatio, the quatit M1 must be = 1. This ca be provided to get kz = kxkms whe s accomplished b storig ((ml ))1 0 i a special memor. Overflow detectio requires the residue represetatio of (0' (M - )) 6. To have the desired precisio of a quotiet i floatig-poit divisio, we must provide a table of multipliers A, such as Table V. V. ALGORITHMS From Sectio III, we ma summarize the algorithms for floatig-poit residue arithmetic as follows. Suppose kx ad k to be represeted b smmetric residue digits ((kx)) + 1 ad (() Floatig-Poit Add or Subtract Operatio Assume that X = kxmexmsx ad Y = k3,meyms are the auged (miued) ad the added (subtrahed), respectivel, ad that 4eXMe > s,em 1) Compute Ae = ex-e. IfAe >, fl(x ± Y) = X. ) If Ae = 1, check whether SX > s. If this is true fl (X_ Y) = X. Otherwise, scale k b mom, *- Ms Ms+lMs+ m ad let the result to be KR.

9 KINOSHITA et al.: FLOATING-POINT ARITHMETIC 17 TABLE II TABLE COMPOSED OF LS = W/ms ) + 1) s z L S z 1 ((L1))5 ((L))5 3 ((L3))5 4 ((L4))5 TABLE III TABLE COMPOSED OF HiI=Zz++l m1 Z ((mm3m4m5 1)6 (( M34m5m1 ) 6 (m4m5m1m ))6 ((m5m1mm3 ))6 (( 1m m3m4 ))6 (( m3m4m5 ) 6 (Im4r5mr1 ) 6 ( m5m1m ) 6 r ml mr3 ) 6 ( mm3m4 ) 6 ((r3r4m5 6 (( m5r1 ) 6 r(( 1r ) 6 ( mr3 ) 6 r( 3m4 ) 6 4 (( m5 ( 6 ((m1 )6 ((r ( m 6 (( m3 ( ) 6 m((4 ( 6 4 TABLE IV TABLE COMPOSED OF MS s z i s z 1 ((ri1) F, 6 ((I ) 3 (( m3 )) 6 4 ((m4 )) 6 TABLE V TABLE COMPOSED OFA VALUES S X (( A )) ( 0 ( 0 m mm I m 1 (( r 3r4i5 ) 10 (( 10 ( r 3m4 5 ) 10 (t 3r4r5 )) 34m 10 (( 3 m m3m4m5 )) 10 ((-1mm ( 1(mlm3 4mb )) 10 1( i )) mS ) 10 (( r3r4 mii )5 10 ((m3i4ms ) 10 (( m14ff5 )) 10 r 1 r4r5 )) ) 10 4 (( m m - m m mm34 5 ) 10 (( m (( (( :4 A 10 _ (~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 3) If Ae =0, check whether sx = s. If so, set KR = k. Otherwise (if sx > s ), scale k b m +.m..+ m. ad let the result to be KR. 4) Compute kz = kx ± KR ad set ez = ex, sz = sx. 5) Fid the mixed-radix digits cal, a,.',a+, of kz which are associated with the smmetric residue sstem ml, m, m+l1 If all the digits a(i= 1,,, + 1) are zero, set ez = 0, sz = 0. 6) Otherwise, deote b au the most sigificat ozero digit. If u = + 1, scale kz b m,z+l ad icrease sz b 1. If sz has reached, set sz = 0 ad icrease ez b 1. 7) If u <, multipl k b HIzs=! fl+u+imi which ca be

10 18 read from the table b meas of u ad sz, decrease ez b 1, ad icrease sz b u. The, if sz <, retur to Step 5). If sz >, after icreasig ez b 1 ad decreasig sz b, retur to Step 5). 8) If u =, check whether sz = 0. If so, kz has bee alread ormalized. If s, * 0, form kz - ((M/ms ) + 1) ad fid the sig of the result through the mixed-radix coversio, ((M/ms3) + 1) is read from a memor. If the sig is egative, multipl kz b msz ad decrease sz b 1. Thus we ca get the desired result fl (X ± Y) = kzarzmszm. Example 1: I the same residue sstem i Sectio IV, suppose we add X-= M'M4 ad Y = M1M. Examiig these two umbers we fid ex -e = 0 ad sx > sy. The matissa of Y must be scaled b m3m4. This gives K KR=(19( X3 )R= Addig kx ad KR, we get kz = , the we set ez = 1, sz = 4. Replacig kz b (kz/m_)r ad correctig the expoet part, we get k = 48308, e =, s = 0. The desired result the is fl ( M1 M M1M) = 48308MMo0. Floatig-Poit Multipl Operatio Assume that X = kxmexmsx ad Y = k1mems are the multiplicad ad the multiplier, respectivel, ad that sx > s,. 1) Perform the base-extesio operatio o kx ad k ad fid the smmetric residue digits /kxlm i ad /k/m i (i = +,+3,, +p). ) Set ez = ex + e, sz = sx. 3) If s = 0, compute kz = kxk, ad if s = 1, compute kz = kxkml, ml is fetched out from a memor. Otherwise, icrease ez b 1, compute the product kxk ad scale this result b m3 +1msY+ -- m to get kz = (kxk/(ms +IMs8+ )R 4) Fid the mixed-radix digits - a1, a, a+p of k. If all the digits cxi are zero, set ez = 0, sz = 0. Otherwise deote b au the most sigificat ozero digit. 5) If it > +, replace kz b (kz/ily=z+lmi)r ad icrease sz b u -, i is a modulo umber whe i >. If sz =, set s to zero ad icrease ez b 1. Retur to Step 4). 6) If u = + 1, replace kz b (kic/m+1 )R ad icrease sz b 1. If sz =, set sz to zero ad icrease ez b 1. Compute Ikz Ī(M - 1) ad fid the sig of this result b meas of the mixed-radix coversio, f(m - 1) is fetched out from a memor. If the sig is positive, replace kz b (kz/msz+l )R ad icrease sz b 1. If sz =, set sz to zero ad icrease ez b 1. Thus we ca get the desired result fl (X X Y) = kimezm. IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 Example : I the precedig residue sstem, suppose we multipl X = MAM4 = fl (.9979 X 1010) b Y= M-5MO = fl (6.656 X 10-7). First, we set e =0+(-5)=- 5,s =4. Sice s =0, k = k k = z x We ca easil kow that the value of u is 10. Hece, scalig kz b M, we have kz = 16549, ez =-4, sz = 4. This matissa is i the ormalized form. Thus we have fl (310418AfM4 X M-5MO) = 16549M-4M4. Floatig-Poit Divide Operatio Assume that X = kxmexmsx ad Y= kmem3 are the divided ad divisor, respectivel. 1) Perform the base extesio o kx ad k ad fid the smmetric residue digits /kx/l ad /k,/m (i= +, + 3,, + p). ) Multipl kx b A, which ca be read from a memor b meas ofsx ad s, ad set ez = ex-e- 1, sz = S -s. 3) If sx < s, decrease ez b 1 ad icrease sz b. 4) Compute kz = (kxa/k)r (check if k, = 0; if so, divisio is ot defied). Hereafter, follow from Step 4) to Step 6) i the multipl algorithm. Example 3: Let X= 16549M-4M4 Y= 87955M-4M4 = fl ( X 10-16). The, sice sx = s= 4,A =M. ez -4-(-4)-1=-1, sz= 0. ki= ( A) = The subscript a for is 6. Hece, scalig kz b ml, we have kz = , ez =-1 sz = 1. Thus we have fl (16549M-4M4 /(87955M-4M4)) = M-'Ml. VI. CONCLUSION A ew residue floatig-poit umber expressio ad arithmetic algorithms based o it have bee proposed. The mai advatage of the residue sstem is that i additio, subtractio, ad multiplicatio a particular digit of the result is depedet ol o the correspodig operad digits. This propert elimiates carries from digit to digit for all three arithmetic operatios previousl metioed ad removes the eed for partial product formatio i multiplicatio. I cotrast to covetioal digital sstems,

11 KINOSHITA et al.: FLOATING-POINT ARITHMETIC these three operatios ca be executed i the same time as required for a additio operatio, while the followig operatios have bee criticized as awkward i residue computers: detectig sig ad the occurrece of overflow, relative-magitude compariso of two umbers, extedig the rage of the umber sstem, ad shiftig the operad. For residue arithmetic, the use of matrix uits is most attractive for implemetig additio, subtractio, ad multiplicatio modulo mi. This method of implemetatio has the advatage of the absece of carries iside as well as amog residue digits. However, a disadvatage arises whe mechaizig the matrices. I geeral, I-out-of-mi codig is used to drive the matrix; therefore, the umber of compoets required for each matrix is large. The use of the smmetric residue otatio permits foldig the arithmetic matrices, which results i a decrease i matrix compoet. A desirable form of logical implemetatio would be the use of the arithmetic matrices realized with LSI techiques. Such matrix uits would give extremel high speed to the three residue operatios. The proposed expressio is basicall differet from the other residue floatig-poit umber expressio. It represets a residue floatig-poit umber as a iteger multiplied b a product of the moduli. As a result the ewl proposed expressio has the advatage of efficiet applicatio of residue iteractig operatios to the floatig-poit arithmetics. These iteractig operatios are split ito base extesio, scalig, ad mixed-radix coversio ad are closel coected with covetioal residue arithmetic operatios. There is much possibilit of this advatage beig regarded coversel as disadvatage because these iteractig operatios are usuall take as time cosumig. The proposed algorithms have the obvious advatage of etire performace of multiplicatio ad divisio as well as additio ad subtractio, uder a certai coditio of the moduli used. The algorithms make good use of the iteractig operatios which ca cope with the awkward operatios metioed previousl. The mai disadvatage of the residue iteractig operatios is that these operatios are relativel time cosumig because of the cascaded process used. This disadvatage, however, would be little worth cosideratio whe the aforemetioed LSI logic matrices are used, because the residue iteractig operatios cosist of residue additio, subtractio, ad multiplicatio. Coversio before or after the computatio will be discussed i aother paper uder preparatio. The material of this paper forms a ivestigatio of the applicabilit of residue umber sstem to the floatig-poit arithmetics. APPENDIX A. Example ofmixed-radix Coversio For ml = 7,m = l1, m3 = 13, adm4 = 17, fid the smmetric mixed-radix digits of X < ), -5,, -1. Solutio Residue represetatio of X Subtract cea Multipl b / I1/ri Subtract ax Multipl b /Al /m. I Subtract 13 Moduli a1 = a = -l 03 = 6 At this poit, it should be apparet that the remaiig mixed-radix digit a4 must be zero. Hece the process ca be termiated. Thus we have X= 6(7 X I1) + (-1)(7) + (l) = 457. I the precedig, the quatit of the form /m is called the multiplicative iverse of a mod mi. The quatit /-/m exists if ad ol if (a, mi) = 1 ad lalm/ i 0 ad satisfies mi- 1-6I-/ a mr B. Example ofbase-extesio < ad /t/lm. X a/mi= 1. Give the residue represetatio X = 37 -, 41 for the base with moduli 7 ad 11, fid /X/1 3 ad /X/l 7. Solutio Residue represetatio of X Subtract ail Multipl b /7/rm i Subtract a The Moduli a,1= = /- X /X/l 3 + 4/13 = 0 / X/X11 7+ /17 =0. 19

12 0 IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 Hece REFERENCES I X /X/l 3/13 =-4, [11 A. Svoboda, Digitale Iformatioswadler. Brauschweig, Germa: Vieweg ad Soh, A. Sasaki, "The basis for implemetatio of additive operatios i the residue umber sstem," IEEE Tras. Comput., vol. C-17, pp , Nov [3] N.S. Szabo ad R.I. Taaka, Residue Arithmetic ad its Applicatios to Computer Techolog. New York: McGraw-Hill, [4] E. Kioshita, H. Kosako, ad Y. Kojima, "Geeral divisio i the smmetric residue umber sstem," IEEE Tras. Comput., vol. C-, pp , Feb X /X/I 7/17 =-. Cosequetl, /X/ 3 =- /X/l 7 =3. C Example of Scalig For ml = 7, m =11, m3 = 13, a I m4 = 17, scalex= {1, -5, 1, -8 b the scale fa ctor 11 X 17. Deote the result b Z. Eisuke Kioshita was bor i Osaka, Japa, i 193. He received the B.S. degree i mathematics from Hiroshima Uiversit, Hiroshima, Japa, i He taught mathematics for two ears at a high school i Osaka, Japa, ad sice 196 has 1 bee a Research Assistat i the Departmet of Electroics, Uiversit of Osaka Prefecture, Solutio Moduli Residue represetatio of X Subtract/X/I1 =-5-1 Multipl b /la /m. Subtract x/ =--/8 / Multipl b /'-m i Eter 0 ito missig colums for extesio of base Subtract Multipl b / 71/r - -3 Subtract Osaka. His curret research iterests lie i the areas of digital sstems ad computer arithmetics Mr. Kioshita is a member of the Istitute of Electrical Egieers of Japa ad the Iformatio Processig Societ of Japa. dih?f sg, l ji'z'/ ~~~~Hideo Kosako (S'58-M'6) was bor i Osaka, Japa, o November 15, He received the B.E. ad M.E. degrees i electrical egieerig ftom the Uiversit of Osaka Prefecture, Osaka, Japa, i 1953 ad 1957, respectivel, ad the Ph.D. degree i electrical commuicatio Zeigieerig from Osaka Uiversit, Osaka, i He joied the facult of Osaka Uiversit i ZZZZ g1960. Sice gsi ZgProfessor i 1961 he has bee a Associate the Departmet of Electroics, Uiversit of Osaka Prefecture. Durig he was a Visitig Professor of Electrical Egieerig at the Uiversit of Arizoa, Tucso. His teachig ad research iterests iclude hbrid computig sstems ad simulatios. Dr. Kosako is a member of the Istitute of Electroics ad Commuicatio Egieers of Japa. The / X /Z/II + 1/1 I = 0 ad /- X/Z/I + /17 = 0. Hece /Z/, = 4 ad /Z/1 7 = 3. Therefore, the residue represetatio of 6873/(11 X 17) -37 is 1,4,-,3[.Note that 686/(11 X 17) ad hece it was rouded to 37, the closest iteger;rather tha to ACKNOWLEDGMENT The authors would like to thak the referees for their ecouragemet ad ver helpful suggestios. Yoshiaki Kojima (SM'57) was bor i Osaka, Japa, o September 9, 193. He received the B.E. ad Ph.D. degrees i electrical egieerig from the Toko Istitute of Techolog, Toko, Japa, i 1946 ad 1961, respectivel. From 1946 to 1950 he was with the Sharp Corporatio, Japa. I 1950 he joied the staff of the Uiversit of Osaka Prefecture, Osaka, ad sice 1963 he has bee a Professor i the Departmet of Electroics. His preset research iterests iclude the areas of logical operatio sstems. Dr. Kojima is a member of the Istitute of Electroics ad Commuicatio Egieers of Japa, the Iformatio Processig Societ of Japa, ad the Istitute of Electrical Egieers of Japa.

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION 1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

More information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

More information

Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have

Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information

Equation of a line. Line in coordinate geometry. Slope-intercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Point-slope form ( 點 斜 式 )

Equation of a line. Line in coordinate geometry. Slope-intercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Point-slope form ( 點 斜 式 ) Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before

More information

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015 CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

Basic Elements of Arithmetic Sequences and Series

Basic Elements of Arithmetic Sequences and Series MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is 0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

More information

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern. 5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

Recursion and Recurrences

Recursion and Recurrences Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr. Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios

More information

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity

More information

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

More information

Factors of sums of powers of binomial coefficients

Factors of sums of powers of binomial coefficients ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

More information

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say

More information

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

Linear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant

Linear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is

More information

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL. Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

More information

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee

More information

Algebra Vocabulary List (Definitions for Middle School Teachers)

Algebra Vocabulary List (Definitions for Middle School Teachers) Algebra Vocabulary List (Defiitios for Middle School Teachers) A Absolute Value Fuctio The absolute value of a real umber x, x is xifx 0 x = xifx < 0 http://www.math.tamu.edu/~stecher/171/f02/absolutevaluefuctio.pdf

More information

Systems Design Project: Indoor Location of Wireless Devices

Systems Design Project: Indoor Location of Wireless Devices Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

arxiv:1012.1336v2 [cs.cc] 8 Dec 2010

arxiv:1012.1336v2 [cs.cc] 8 Dec 2010 Uary Subset-Sum is i Logspace arxiv:1012.1336v2 [cs.cc] 8 Dec 2010 1 Itroductio Daiel M. Kae December 9, 2010 I this paper we cosider the Uary Subset-Sum problem which is defied as follows: Give itegers

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

Automatic Tuning for FOREX Trading System Using Fuzzy Time Series

Automatic Tuning for FOREX Trading System Using Fuzzy Time Series utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

5 Boolean Decision Trees (February 11)

5 Boolean Decision Trees (February 11) 5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

7. Sample Covariance and Correlation

7. Sample Covariance and Correlation 1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

More information

3. Covariance and Correlation

3. Covariance and Correlation Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

More information

Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites

Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites Gregory Carey, 1998 Liear Trasformatios & Composites - 1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio

More information

Research Article Sign Data Derivative Recovery

Research Article Sign Data Derivative Recovery Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

Chapter 9: Correlation and Regression: Solutions

Chapter 9: Correlation and Regression: Solutions Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

A Guide to the Pricing Conventions of SFE Interest Rate Products

A Guide to the Pricing Conventions of SFE Interest Rate Products A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

More information

INVESTMENT PERFORMANCE COUNCIL (IPC)

INVESTMENT PERFORMANCE COUNCIL (IPC) INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks

More information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

More information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study. Normal and t Distributions. Density Plot. Normal Distributions Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

More information

INFINITE SERIES KEITH CONRAD

INFINITE SERIES KEITH CONRAD INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8 CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4

GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4 GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4 MFP Tetbook A-level Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

Domain 1: Designing a SQL Server Instance and a Database Solution

Domain 1: Designing a SQL Server Instance and a Database Solution Maual SQL Server 2008 Desig, Optimize ad Maitai (70-450) 1-800-418-6789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a

More information

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff, NEW HIGH PERFORMNCE COMPUTTIONL METHODS FOR MORTGGES ND NNUITIES Yuri Shestopaloff, Geerally, mortgage ad auity equatios do ot have aalytical solutios for ukow iterest rate, which has to be foud usig umerical

More information

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

More information

Integer Factorization Algorithms

Integer Factorization Algorithms Iteger Factorizatio Algorithms Coelly Bares Departmet of Physics, Orego State Uiversity December 7, 004 This documet has bee placed i the public domai. Cotets I. Itroductio 3 1. Termiology 3. Fudametal

More information

Institute of Actuaries of India Subject CT1 Financial Mathematics

Institute of Actuaries of India Subject CT1 Financial Mathematics Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i

More information

Chapter 7: Confidence Interval and Sample Size

Chapter 7: Confidence Interval and Sample Size Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

More information

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series 8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

More information

CHAPTER 11 Financial mathematics

CHAPTER 11 Financial mathematics CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula

More information

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

More information

Lesson 15 ANOVA (analysis of variance)

Lesson 15 ANOVA (analysis of variance) Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

More information

Lecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k.

Lecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k. 18.409 A Algorithmist s Toolkit September 17, 009 Lecture 3 Lecturer: Joatha Keler Scribe: Adre Wibisoo 1 Outlie Today s lecture covers three mai parts: Courat-Fischer formula ad Rayleigh quotiets The

More information

Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett

Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu> (March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1

More information

Trading the randomness - Designing an optimal trading strategy under a drifted random walk price model

Trading the randomness - Designing an optimal trading strategy under a drifted random walk price model Tradig the radomess - Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore

More information

9.8: THE POWER OF A TEST

9.8: THE POWER OF A TEST 9.8: The Power of a Test CD9-1 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based

More information

Time Value of Money. First some technical stuff. HP10B II users

Time Value of Money. First some technical stuff. HP10B II users Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle

More information

FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10

FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10 FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.

More information

A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design

A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 16804-0030 haupt@ieee.org Abstract:

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009) 18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

More information

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

Fast Fourier Transform

Fast Fourier Transform 18.310 lecture otes November 18, 2013 Fast Fourier Trasform Lecturer: Michel Goemas I these otes we defie the Discrete Fourier Trasform, ad give a method for computig it fast: the Fast Fourier Trasform.

More information

Subject CT5 Contingencies Core Technical Syllabus

Subject CT5 Contingencies Core Technical Syllabus Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value

More information

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

More information

Escola Federal de Engenharia de Itajubá

Escola Federal de Engenharia de Itajubá Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica Pós-Graduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José

More information

The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection

The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity

More information