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Probability bounds and asymptotic properties of error propagation
009~~94~~83~0~0307~04%03.00~0 c 1983 Pergamon Press Ltd
PROBABILITY BOUNDS AND ASYMPTOTIC PROPERTIES OF ERROR PROPAGATION MATYAS ARATOt Department
of Systems Science and Mathematics, (Received
Washington University, St. Louis, MO 63130, U.S.A. March
Communicated
1982)
by M. Arato
Abstract-An Ito type differential equation is given for the description of accumulated round-off error behavior in integration of systems of ordinary first order differential equations. By a convenient scale transformation sharp boundaries can be gotten for the probability of maximum error. The method used is connected with the estimates of nonexit probabilities of a Wiener process to moving boundary.
I. INTRODUCTION
In this paper we investigate the asymptotic behavior of the accumulated round-off errors in integrating a system of ordinary differential equations by one-step methods in the interval 0 I x 5 b on the basis of the related stochastic system of differential equations. Let be given the following first order vector initial value problem Y’(X)
=
(1)
f(x, y(x)), y(xo) = y(O),x0 5 x 5 6,
where y and f are column vectors and asterisk indicates the transposition A one-step method is defined by the formula
of a matrix or vector.
yn+ we, yn; h), h > 0, x, = x0 + nh, yo = y(O),
Yn+l =
where 4(x, y; h) is called the increment
function.
We assume that
4(x,,, in;h) - 4(x,, yn ; h) = Gkb,
+ hen,
where in is the numerical approximation of yn. r,(x) = r,, = 9” -yn which fulfils the following equation (see Henrici [5], Arato [ 11) I-,,+~ -r,,
(2)
= hG(x,)r,, + ~(h)pk+J
+
(3) is the propagated
~%,,k,+~, r. = 0,
error
(4)
where E, is the local error with EC,, = 0, Ee,q,* = 1. The accumulated round-off error, after n step, r, fulfils the above stochastic difference equation (4) which can be handled as the discretization of the solution of a stochastic Ito type differential equation. This stochastic differential equation we called the related stochastic equation to the system of ordinary differential equation (1). As a measure of error behavior we introduce the probability P{ max Ir,l}, which can be reduced by the well known “time” scale transformation in diffusional type processes to the calculation of probability for the absolute value of Wiener process to remain less than 1. This results are closely connected with the estimates and asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary[8,9].
tvisiting
professor from the Ehtvbs Lorand University,
Camwa Vol. 9. No. 2-E
Budapest, Hungary 307
M.ARAT~
308
2.THE PROBABILISTIC ERROR DESCRIPTION
Equation (4) indicates that the propagated error r, = r(x,) can be regarded as a function of x and the solution of the following stochastic differential equation dr, = G(x)r, dx + @p(x) dx + B;‘(x) dw(x), r. = 0,
(5)
where w(x) is the standard Wiener process with Ew(x) = 0, Ew(x)w*(x) = I.x. In this case r, is a Gaussian random process, the solution of the following linear equation rX =
[G(u)r,
+ pp(u)] du +
’ Bz’(u)dw(u), I0
r. = 0,
(51)
where the second term is a stochastic integral with respect to the Wiener process. It is possible and more adequate to regard w(x) as a wide sense Wiener process ([6], Section 15), but for simplicity we assume that w(x) is a Wiener process. LEMMA 1. Let F(x)=exp([G(u)du},
(6)
be the fundamental matrix, i.e. the solution of the differential equation
Wx) = G(x) F(x), F(0) = Ikk,
(7)
dx
Then rX = F(x) [ ~o’(F(~))e’p .p(s)ds
+Ib;(F(~))~~~~‘(~)dw(~)),ro
= 0.
(8)
LEMMA 2.
(Arato [l]) Let the elements of p(x), G(x), B,(x) be integrable functions on 0 5 x 5 b, and let rX fulfil the stochastic equation (5). Then m(x) = Er, and B(x) = Er,r,* are the solutions of the differential equations
dm(x) - dx = p(x)+ G(x)m(x),
(9)
dB(x) = G(x)B(x) + B(x) G*(x) + B,(x).
(10)
-
3.THE PROBABILITY
the natural
of the
OF THE
OF ERROR
behavior we
(11) F(x) is given by (6). In [l] we proposed this bounds in the one dimensional case, instead of the estimation of the mean m(x), as it is accepted in the literature[5]. Let us recall here that in the case p(x) = 0 and G(x) =
$$,
(12)
Probability bounds and asymptotic properties
309
with positive continuous function m(x), x 2 0, the following statement holds THEOREM1
&I
b
P {lr,ls km(x), 0 I x 5 b} exp
(13)
Proof. The following representation for the probability that a Wiener process w(t) does not exit the interval [-k, k] is known:
P(,s;~T/w(u)tak}=~~o !$$exp[-(2~+1)z~]. n
(14)
On the other side for the Gaussian random processes r,, x 2 0, defined by the relation r, = m(x) we
X m-‘(u)b$*(U)dw(u),
(15)
I0
have
=P
I
IO(x)]51,O
I0
bm-2(u)bJu)du},
(16)
where I(x) is a new Wiener process obtained by the “time” change U=
I
*
0
mm2(s) b,(s) ds
(17)
in the stochastic integral Jt mm’(u)b z’(u) d w(u) [4]. Taking one and two terms respectively in the alternating series in (14) one can get the estimates
This and (16) imply (13). In the k-dimensional case it is known that the following inequalities hold for the standard Wiener process w(t) with independent components (see Skorokhod [ 121)
P{lw(T)I2 Cl 5 Pio$y$JTlw(t)l 2 Cl 5 2P{lw(T)I 2 Cl, where p{lw(T)( 2 c} =
From (8); assuming p(x) = 0, we get as a first approximation for the ith component
(18)
310
M. ARAT~
where C(x) is a new Wiener process obtained by the “time” change[7]
(21) in the stochastic integral J/ F;‘(s)Btj2(s)dw(s). statement.
Comparing (18) and (20) we get the following
THEOREM2.
Let B = obIiF;‘(u)Bz2(u)j/2du, then I
1 - 2P{IC(B)j rC} 5 P~~y~bjF;‘(r)r,l5 Cl 5 1 - P@@)I
2 C).
Remark. From (19) it follows that
&;r&
P{ozytT jw(t)l2 c} e((“2)+c)(c2’T) = m,
&,~& P[ spT [w(t)12 c) e((1’2)-r)(c2’T) = 0.
REFERENCES 1. M. Arato, Round-off error propagation in the integration of ordinary differential equations by one-step methods. Acta Mathematics Szegediensis. (to appear (1982)). 2. M. Arato, Linear stochastic systems with constant coefficients. (manuscript) (1982). 3. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, New York (1%6). 4. I. Gihman and A. Skorokhod, Stochastic Differential Equations. Springer, Berlin (1972). 5. P. Henrici, Discrete Variable Methods in Ordinary Diferential Equations. Wiley, New York (1962). 6. R. Liptser and A. Shiryaev, Statistics of Random Processes. Nauka, Moscow (1974). 7. H. P. McKean, Stochastic Integrals. Academic Press, New York (1969). 8. A. Novikov, On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary. Math. USSR Sbomik llO(l52) No. 4 (1979). 8. A. Novikov, On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary. Math. USSR Sbomik llO(l52) No. 4 (1979). 9. A. Novikov, A martingale approach to first passage problems and a new condition for Wald’s identity. Stochastic Differential Systems (Edited by M. Arato, D. Vermes, A. V. Bafakrishnan), pp. 146-156.Visegrad (1981). 10. A. Novikov, Martingale approach to frrst passage problems for nonlinear boundaries. Proc. Steklou Inst. of Math., Vol. 158(1981). I I. H. Rademacher, On the accumulation of errors in processes of integration on high-speed calculating machines. Annals Comput. Lab. Harvard Univ. 16, 176-187(1948). 12. A. Skorokhod, Random Processes with Independent Increments. Nauka, Moscow (1964).
PROBABILITY BOUNDS AND ASYMPTOTIC PROPERTIES OF ERROR PROPAGATION MATYAS ARATOt Department
of Systems Science and Mathematics, (Received
Washington University, St. Louis, MO 63130, U.S.A. March
Communicated
1982)
by M. Arato
Abstract-An Ito type differential equation is given for the description of accumulated round-off error behavior in integration of systems of ordinary first order differential equations. By a convenient scale transformation sharp boundaries can be gotten for the probability of maximum error. The method used is connected with the estimates of nonexit probabilities of a Wiener process to moving boundary.
I. INTRODUCTION
In this paper we investigate the asymptotic behavior of the accumulated round-off errors in integrating a system of ordinary differential equations by one-step methods in the interval 0 I x 5 b on the basis of the related stochastic system of differential equations. Let be given the following first order vector initial value problem Y’(X)
=
(1)
f(x, y(x)), y(xo) = y(O),x0 5 x 5 6,
where y and f are column vectors and asterisk indicates the transposition A one-step method is defined by the formula
of a matrix or vector.
yn+ we, yn; h), h > 0, x, = x0 + nh, yo = y(O),
Yn+l =
where 4(x, y; h) is called the increment
function.
We assume that
4(x,,, in;h) - 4(x,, yn ; h) = Gkb,
+ hen,
where in is the numerical approximation of yn. r,(x) = r,, = 9” -yn which fulfils the following equation (see Henrici [5], Arato [ 11) I-,,+~ -r,,
(2)
= hG(x,)r,, + ~(h)pk+J
+
(3) is the propagated
~%,,k,+~, r. = 0,
error
(4)
where E, is the local error with EC,, = 0, Ee,q,* = 1. The accumulated round-off error, after n step, r, fulfils the above stochastic difference equation (4) which can be handled as the discretization of the solution of a stochastic Ito type differential equation. This stochastic differential equation we called the related stochastic equation to the system of ordinary differential equation (1). As a measure of error behavior we introduce the probability P{ max Ir,l}, which can be reduced by the well known “time” scale transformation in diffusional type processes to the calculation of probability for the absolute value of Wiener process to remain less than 1. This results are closely connected with the estimates and asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary[8,9].
tvisiting
professor from the Ehtvbs Lorand University,
Camwa Vol. 9. No. 2-E
Budapest, Hungary 307
M.ARAT~
308
2.THE PROBABILISTIC ERROR DESCRIPTION
Equation (4) indicates that the propagated error r, = r(x,) can be regarded as a function of x and the solution of the following stochastic differential equation dr, = G(x)r, dx + @p(x) dx + B;‘(x) dw(x), r. = 0,
(5)
where w(x) is the standard Wiener process with Ew(x) = 0, Ew(x)w*(x) = I.x. In this case r, is a Gaussian random process, the solution of the following linear equation rX =
[G(u)r,
+ pp(u)] du +
’ Bz’(u)dw(u), I0
r. = 0,
(51)
where the second term is a stochastic integral with respect to the Wiener process. It is possible and more adequate to regard w(x) as a wide sense Wiener process ([6], Section 15), but for simplicity we assume that w(x) is a Wiener process. LEMMA 1. Let F(x)=exp([G(u)du},
(6)
be the fundamental matrix, i.e. the solution of the differential equation
Wx) = G(x) F(x), F(0) = Ikk,
(7)
dx
Then rX = F(x) [ ~o’(F(~))e’p .p(s)ds
+Ib;(F(~))~~~~‘(~)dw(~)),ro
= 0.
(8)
LEMMA 2.
(Arato [l]) Let the elements of p(x), G(x), B,(x) be integrable functions on 0 5 x 5 b, and let rX fulfil the stochastic equation (5). Then m(x) = Er, and B(x) = Er,r,* are the solutions of the differential equations
dm(x) - dx = p(x)+ G(x)m(x),
(9)
dB(x) = G(x)B(x) + B(x) G*(x) + B,(x).
(10)
-
3.THE PROBABILITY
the natural
of the
OF THE
OF ERROR
behavior we
(11) F(x) is given by (6). In [l] we proposed this bounds in the one dimensional case, instead of the estimation of the mean m(x), as it is accepted in the literature[5]. Let us recall here that in the case p(x) = 0 and G(x) =
$$,
(12)
Probability bounds and asymptotic properties
309
with positive continuous function m(x), x 2 0, the following statement holds THEOREM1
&I
b
P {lr,ls km(x), 0 I x 5 b} exp
(13)
Proof. The following representation for the probability that a Wiener process w(t) does not exit the interval [-k, k] is known:
P(,s;~T/w(u)tak}=~~o !$$exp[-(2~+1)z~]. n
(14)
On the other side for the Gaussian random processes r,, x 2 0, defined by the relation r, = m(x) we
X m-‘(u)b$*(U)dw(u),
(15)
I0
have
=P
I
IO(x)]51,O
I0
bm-2(u)bJu)du},
(16)
where I(x) is a new Wiener process obtained by the “time” change U=
I
*
0
mm2(s) b,(s) ds
(17)
in the stochastic integral Jt mm’(u)b z’(u) d w(u) [4]. Taking one and two terms respectively in the alternating series in (14) one can get the estimates
This and (16) imply (13). In the k-dimensional case it is known that the following inequalities hold for the standard Wiener process w(t) with independent components (see Skorokhod [ 121)
P{lw(T)I2 Cl 5 Pio$y$JTlw(t)l 2 Cl 5 2P{lw(T)I 2 Cl, where p{lw(T)( 2 c} =
From (8); assuming p(x) = 0, we get as a first approximation for the ith component
(18)
310
M. ARAT~
where C(x) is a new Wiener process obtained by the “time” change[7]
(21) in the stochastic integral J/ F;‘(s)Btj2(s)dw(s). statement.
Comparing (18) and (20) we get the following
THEOREM2.
Let B = obIiF;‘(u)Bz2(u)j/2du, then I
1 - 2P{IC(B)j rC} 5 P~~y~bjF;‘(r)r,l5 Cl 5 1 - P@@)I
2 C).
Remark. From (19) it follows that
&;r&
P{ozytT jw(t)l2 c} e((“2)+c)(c2’T) = m,
&,~& P[ spT [w(t)12 c) e((1’2)-r)(c2’T) = 0.
REFERENCES 1. M. Arato, Round-off error propagation in the integration of ordinary differential equations by one-step methods. Acta Mathematics Szegediensis. (to appear (1982)). 2. M. Arato, Linear stochastic systems with constant coefficients. (manuscript) (1982). 3. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, New York (1%6). 4. I. Gihman and A. Skorokhod, Stochastic Differential Equations. Springer, Berlin (1972). 5. P. Henrici, Discrete Variable Methods in Ordinary Diferential Equations. Wiley, New York (1962). 6. R. Liptser and A. Shiryaev, Statistics of Random Processes. Nauka, Moscow (1974). 7. H. P. McKean, Stochastic Integrals. Academic Press, New York (1969). 8. A. Novikov, On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary. Math. USSR Sbomik llO(l52) No. 4 (1979). 8. A. Novikov, On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary. Math. USSR Sbomik llO(l52) No. 4 (1979). 9. A. Novikov, A martingale approach to first passage problems and a new condition for Wald’s identity. Stochastic Differential Systems (Edited by M. Arato, D. Vermes, A. V. Bafakrishnan), pp. 146-156.Visegrad (1981). 10. A. Novikov, Martingale approach to frrst passage problems for nonlinear boundaries. Proc. Steklou Inst. of Math., Vol. 158(1981). I I. H. Rademacher, On the accumulation of errors in processes of integration on high-speed calculating machines. Annals Comput. Lab. Harvard Univ. 16, 176-187(1948). 12. A. Skorokhod, Random Processes with Independent Increments. Nauka, Moscow (1964).