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A note on beta-methods for simulating binomal distributions
207
I 0
h+
0
subjected to a rctaticn through
0
-1
1
0
0 0 0 0 0 0 0
the angle n/Z in the
(U) plane, gives
R_OOO-l
000
1000
0’ 0I
REFERENCES 1.
2. 3. 4. 5.
LOAN c., A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix. Linear Algebra and its Appl., 61, 233-251, 1984. IKRAMOV KH.D., Numerical solution of matrix equations and symplectic matrix algebra. In: Computational Processes and Systems, 5, Nauka, Moscow, 1987. IKP.AMOVKH.D., Numerical Solution of Matrix Equations. Orthogonal Methods, Nauka, Moscow, 1984. PAIGE C. and VAN LOAN C., A Schur decomposition for Hamiltonian matrices. Linear Algebra and its Appl. 41, 11-32, 1981. IKRAMOV KH.D. and SAGITOV M.S., On the factorizationof symplectic matrices as products of elementary matrices, in: Methods and Algorithms of Numerical Analysis, Izd. MGU, 111-119, Moscow, 1987. VAN
Translated by D.L.
U.S.S.R.
Comput.Maths.Math.Phys
.,Vo1.28,No.6,pp.207-208.1988
Printed in Great Britain
0041-5553/88 $lO.OO+O.OO 01990 Pergamon Press plc
A NGTE ON BETA-IETCiGljS FGk SIFIULATINGBINGKIALCISTRIBCTIGNS* B.B. POKHODZEI
Based on a theoretical treatment of the so-called beta-methods for simulating binomial distributions,hitherto investigated only by experimental tneans,theoretical estimates of their speed are obtained. The standard interpretation of a random variable (T.v.) &..P obeying a binorlal distribution with parameters 1121 and Ocp
’ *Zh.vychlsl.Mat.met.Flz.,
28,12,1902-1903,1988
208 and consequently, the problem of simulating the binomial distributionwith parameters n and and p reduces to simulating binomial distributionseither with parameters n-i (P--ivl(j-N or with parameters i=l and P/B. To simulate the latter one can apply the same device iteratively until the value of the first parameter approaches some prescribed number no (and thereafter use sequential realization of the Bernoulli scheme). One version of this compound Bernoulli scheme consists of sequential choice of an i (if n is even, one augments the value of the simulated r.v. En,p such that n=%i--i by adding ~,~,~,(a~), thus decreasing n by unity), i.e. one simulates the median of a sample of n=li-I r.v.'s uniformly distributed in (0,1) and arranged in increasing order, which has a symmetric beta-distributionwith parameter i. Iterating this procedure until the condition n/2~=no is satisfied, one sees that the required number of iterations of the algorithm is which for relatively small no is practically the same as log% k==lognnogn0, On the other hand, i can be so chosen that the expectation of an r.v. 8 having a betadistributionwith parameters i and n-i+& which is i/(n+1), approximatesp, i.e., i=[(n+l)pl. Then the speed of the resulting algorithm is estimated by the quantity
where
f.,*(z) is the density of a beta-distributionwith parameters a and b. Putting (Tn= and noting that a,f.,b(p+(m5)-r(2n)-‘” exp(-x2/2) as n-+- for any x, one readily sees that the required estimate is proportional to [np(i-p)]"*, which does not exceed n".. After k iterations of the algorithm the last estimate .becomes &, so that if the stopping condition of the algorithm is n'-"=n‘the required number of iterations is k~log(loga/logno) , and for relatively small RO this is of the same order of magnitude as loglogn. The experimental estimates presented in /l/ for the speed of the corresponding algorithms for simulating the binomial distribution are in good agreement with the above hierarchy of theoretical estimates. [P(i-P)hl’h
REFERENCES 1. POKBODZEI B.B., Beta- and gamma-methods for simulating binomial and Poisson distributions, Zh. vychisl. Mat. mat, Fiz., 24, 2, 187-193, 1984.
Translated by D.L.
.,Vo1.28,No.6,pp.208-210,1988 U.S.S.R. Comput.Maths.Math.Phys Printed in Great Britain
OC41-5553/88 $lO.CC+O.oO 01990 Pergamon Press plc
AN EXPEkIfiENTALESTIMATE OF THE EFFECTIVENESS OF l&IK UJALITY TO SOLVE LISCRETE PKOGRAM~JNG PFiOBLEKS* A.O. ALEKSEYEV, O.G. AIEKSEYEV, V.D. KISELEV and G.P. MIROVITSKII
Consider the discrete programming problem
C(X) - max
cl
(ZJ)
(1)
with the limits
aiJ(zJ)4b1,
i=i, 2,..., jn, x,=1,2, . ..( A~.
J-L
where
eJ(zJ)=-0,
and
W,@J)%
*Zh.vychisl.Mat.mat.Fiz.,28,12,1904-1905,1988
i=1,2,...,n.
(2)
I 0
h+
0
subjected to a rctaticn through
0
-1
1
0
0 0 0 0 0 0 0
the angle n/Z in the
(U) plane, gives
R_OOO-l
000
1000
0’ 0I
REFERENCES 1.
2. 3. 4. 5.
LOAN c., A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix. Linear Algebra and its Appl., 61, 233-251, 1984. IKRAMOV KH.D., Numerical solution of matrix equations and symplectic matrix algebra. In: Computational Processes and Systems, 5, Nauka, Moscow, 1987. IKP.AMOVKH.D., Numerical Solution of Matrix Equations. Orthogonal Methods, Nauka, Moscow, 1984. PAIGE C. and VAN LOAN C., A Schur decomposition for Hamiltonian matrices. Linear Algebra and its Appl. 41, 11-32, 1981. IKRAMOV KH.D. and SAGITOV M.S., On the factorizationof symplectic matrices as products of elementary matrices, in: Methods and Algorithms of Numerical Analysis, Izd. MGU, 111-119, Moscow, 1987. VAN
Translated by D.L.
U.S.S.R.
Comput.Maths.Math.Phys
.,Vo1.28,No.6,pp.207-208.1988
Printed in Great Britain
0041-5553/88 $lO.OO+O.OO 01990 Pergamon Press plc
A NGTE ON BETA-IETCiGljS FGk SIFIULATINGBINGKIALCISTRIBCTIGNS* B.B. POKHODZEI
Based on a theoretical treatment of the so-called beta-methods for simulating binomial distributions,hitherto investigated only by experimental tneans,theoretical estimates of their speed are obtained. The standard interpretation of a random variable (T.v.) &..P obeying a binorlal distribution with parameters 1121 and Ocp
’ *Zh.vychlsl.Mat.met.Flz.,
28,12,1902-1903,1988
208 and consequently, the problem of simulating the binomial distributionwith parameters n and and p reduces to simulating binomial distributionseither with parameters n-i (P--ivl(j-N or with parameters i=l and P/B. To simulate the latter one can apply the same device iteratively until the value of the first parameter approaches some prescribed number no (and thereafter use sequential realization of the Bernoulli scheme). One version of this compound Bernoulli scheme consists of sequential choice of an i (if n is even, one augments the value of the simulated r.v. En,p such that n=%i--i by adding ~,~,~,(a~), thus decreasing n by unity), i.e. one simulates the median of a sample of n=li-I r.v.'s uniformly distributed in (0,1) and arranged in increasing order, which has a symmetric beta-distributionwith parameter i. Iterating this procedure until the condition n/2~=no is satisfied, one sees that the required number of iterations of the algorithm is which for relatively small no is practically the same as log% k==lognnogn0, On the other hand, i can be so chosen that the expectation of an r.v. 8 having a betadistributionwith parameters i and n-i+& which is i/(n+1), approximatesp, i.e., i=[(n+l)pl. Then the speed of the resulting algorithm is estimated by the quantity
where
f.,*(z) is the density of a beta-distributionwith parameters a and b. Putting (Tn= and noting that a,f.,b(p+(m5)-r(2n)-‘” exp(-x2/2) as n-+- for any x, one readily sees that the required estimate is proportional to [np(i-p)]"*, which does not exceed n".. After k iterations of the algorithm the last estimate .becomes &, so that if the stopping condition of the algorithm is n'-"=n‘the required number of iterations is k~log(loga/logno) , and for relatively small RO this is of the same order of magnitude as loglogn. The experimental estimates presented in /l/ for the speed of the corresponding algorithms for simulating the binomial distribution are in good agreement with the above hierarchy of theoretical estimates. [P(i-P)hl’h
REFERENCES 1. POKBODZEI B.B., Beta- and gamma-methods for simulating binomial and Poisson distributions, Zh. vychisl. Mat. mat, Fiz., 24, 2, 187-193, 1984.
Translated by D.L.
.,Vo1.28,No.6,pp.208-210,1988 U.S.S.R. Comput.Maths.Math.Phys Printed in Great Britain
OC41-5553/88 $lO.CC+O.oO 01990 Pergamon Press plc
AN EXPEkIfiENTALESTIMATE OF THE EFFECTIVENESS OF l&IK UJALITY TO SOLVE LISCRETE PKOGRAM~JNG PFiOBLEKS* A.O. ALEKSEYEV, O.G. AIEKSEYEV, V.D. KISELEV and G.P. MIROVITSKII
Consider the discrete programming problem
C(X) - max
cl
(ZJ)
(1)
with the limits
aiJ(zJ)4b1,
i=i, 2,..., jn, x,=1,2, . ..( A~.
J-L
where
eJ(zJ)=-0,
and
W,@J)%
*Zh.vychisl.Mat.mat.Fiz.,28,12,1904-1905,1988
i=1,2,...,n.
(2)