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Delta methods for numerical inversion of the laplace transform, and their convergence
A. K. Shuvaev
22 3.
SHATOV, A. K., A numerical method for solving the problem of stationary electromagnetic at a body of revolution, Zh. vj%hisI.Mat. mat. Fiz., 13, No. 1,237-241, 1973.
4.
SOBOLEV, S. I,., Some applicatio~sof fictions funktsional’nogo analiza v matematicheskoi
5.
SAVELOVA, T. I., Application of a class of regularizing algorithms to the solution of integal equations of the first kind of the convolution type in Banach space, Zh. vfchisl. Mat. mat. Fiz., 14, No. 2,479-483, 1974.
6.
MARKUS, M., and MINK, Kh., Survey of matrix theov and matrix i~e4uulities (Obzor po teorii matrits i matrichnykh neravenstv), Nauka, Moscow, 1972.
7.
SAVELOVA, T. I., The solution of an equation of the convolution type with inaccurately specified kernel by the method of regularization, Zh. vychisl. Mat. mat. Fiz., 12, No. 1, 212-218, 1972.
8.
ARSENIN, V. Ya., and IVANOV, V. V., On optimal regularization, Doki. Akad. Nauk SSSR, 182, 9-12,1968.
9.
KATO, T., Perturbation theory for linear operators, Springer, 1966.
wave diffraction
analysisto mathemati~I physics (Nekotorye fizike), Izd-vo LGU, Leningrad, 1950.
p~me~eniya
NO.
1,
10. ARNOL’D, V. I., Lecture on bifurcations and versa1 families, hp. Matem. Nauk, 27, No. 5 (1967), 119-184, 1972.
DELTA METHODS FOR NUMERICAL INVERSION OF THE LAPLACE TRANSFORM, AND THEIR CONVERGENCE* A. K. SHUVAEV Tula (Received
7 April
1971; revised 29 April 1974)
ASYMPTOTIC forms of the delta function and corresponding theorems on numerical inversion of the Laplace transform are given. An asymptotic estimate is obtained for the rate of convergence of the appro~mation to the exact solution in certain classes of originals.
Introduction Methods are discussed for the approximate inversion of a Laplace transform specified on the real axis, by means of a sequence of functions asymptotic~y convergent to the delta function 6(t - x). We shall refer to the functions of each such sequence as representations of 6(t - x), The methods represent a generalization of Post’s formula [l] , to which there corresponds the representation of &(t - x) Pn(2, t) =
(
$
>
n1 (n
I),
.
exp (-
f
t) F-1.
Post’s method has the drawbacks that the derivatives of increasing order of the Laplace transform have to be evaluated, and that the approx~ations converge rather slowly to the exact solution. The approximate methods considered in [2] have the same slow convergence. Just as in [2] , our methods do not require evaluation of the derivatives of increasing order of the Laplace transform; while our inversion by the “generalized representations” of s(t - x), introduced below, ensures a faster convergence on classes of smooth curves of the appro~mations *Zh. vychisl.Mat. mat. Fiz., 15,6,1389-1398,
1975.
Numerical inversion of the Laplace transform
23
to the exact solution than is possible with the methods given in [2] and Post’s formula. Questions of the stability of our methods mount to finding the number of the approx~ation up to which the computations need to be performed. For this, we obtain below, for the simplest case, an equation whereby the number of the approximation can be found, at which the total error, including the error of the method, the error of the computation, and the error of the transformation, is asymptotically minimal. Other cases can be treated in a similar way.
1. Approximate methods of inversion
We defme a non-negative representation of s(t - X) as the function
R%P7
b ‘) = B (n/(r _ I), n)
exp (--$$)
(1 - .~+~)~--r,
(1-l)
where B(p, 4) is the beta function, x > 0, and the parameters r and b are such that r > 1, bx = In r. We use R$ (z, t) to prove: Theorem 1 Let f(t), t > 0, be a single-valued function, integrable in any finite interval [0, 7’j , T> 0, for which there exists a real so such that 11 f (t) \ e-So’dt < 30.
(14
0
Then, at all points of continuity x > 0 of the function j(t), as n + 00:
(1.3)
Inr
ZB (n/P - V, 4 where F (s) =
1f(2) e-s’ a. 0
Proof We first obtain estimates for R$ (2, t) . Using the relationship between the beta and gamma functions, and Stirling’s asymptotic form, we arrive at
(1.4) where w(n) + 0 as n -+ m,
(1.5) In the interval 0 G t d In r/b, S(t) is increasing from zero to unity. Hence, given any E,, O
24
for some C which is independent
ofn and C.
To estimate R$) (z, t) with t>ln r/H-e*, r/&. If S(t) = G(y), then, fory 2 0,
eg>O,
we transform to the variable y = t - ln
Let
(1.7) For e2 > 0, the equation
as an equation in 6 has a non-negative solution 6 ( eo) +=c 1.Also, it is easily shown that, withy > e2, the derivative drp (6 (Q) , y) idy>O. From this, and Eqs. (1.7) and (I .8), we Frnd that
in view of which, for the same y ,
where G,(Ez)=I-F(~z))O. On now putting
and returning fromy to the variable t, we see from (1*9) that, if t>ln
exp[ -&(e2) inequality
t], where &(e2) -[ b/(r--l)]&‘(~~)
>O.
Finally,
r/b-l-ez, then S’(t) < for the same f, we obtain the
We now turn to the proof of (1.3). If the conditions of Theorem 1 are satisfied, we easily fmd, by using (1.6) and (1 JO), that, for sufiiciently large n,
25
Numerical inversion of the Laplace transform
where C, is independent of n. Recalling the equation 00 (1.11)
s R$’ (Z, t) dt = 1
0
it is obvious that, for any e, 0 < e
where I?!’ (5, t) dt,
I;., L =
J 1f(t) x+c
1R’,O’(2, t) a,
s If(t)-f(“)lR(n0)(2,t)dt.
I ‘;,c=
Z--L
Onthebasisof(1.6)and(l.lO),weget
I;,
1
<
b(E) exp -T(n1
Cn'll
i)]~lf(‘)lexp[-~~(n-i)t]dt, 0 m
J,;, L< Cn’/aexp {- 18, (E) (n from which it follows, recalling conditions (13,
I:,,-0,
1) -
sol(z i- 8)) 5 If(t) \ e-sot& x
that, as n + =,
r;J+o.
(1.13)
IfAt) is continuous at the point x, we easily fmd, using (1.1 l), that I& -+O as n + m, and from this and (1.12) and (1.13),
(1.14)
lim S j (t) R’,O)(z, t) dt = f (.x). 11-00o
Recalling that bx = In r, and the definition of F(s), (1.3) now follows from Eqs. (1 .l) and (1.14). This proves Theorem 1. As the first-order generalized representation of &(t - x) we take the function
b[n---@----)l e_~‘,’(1 _ e-bt)n-z rB(s+l,n-1) b [I - 12+ so” - ‘)I e-~!,t(1 _ e-bt)n-l
&‘@t)=
i
(1.15)
1
rB(s+
l,n)
where bx = In r, n
s=--+++[l+ r1
l/z
4rn (r -
1)2 I
The possibility of inverting the Laplace transform by means of Rc,‘) (r, t) is proved by:
(1.16)
26
A. iy. f&maw
Theorem 2
If the conditions of Theorem 1 hold for tit), then, as n + *,
rsB(s+
l,n--
1) (1.17)
The proof is similar to that of Theorem 1, using the inequalities (1.18)
(1.19) where C is independent of n and I, and CJ independent of n. The inequality (1 .18) is obtained from (l.l), (1.1’S), and (1.16). We shall speak about the proofof(l.19) in Section 2. It is easily shown that R’,O’(z, t) and lit’ (5, t) satisfy the equations DD
s
e-i&t&l)
(5,
t)
&
=
e-ib=,
i =
0,1, . * . , j,
(1.20)
0
for i = 1 and 2 respectively. In accordance with this, as the secondorder generalized representation of &(t - x) we take RF)
(g,
f)
=
2
a,_ie-sbf (1 _
e-“‘)n-l-i,
i=O
which satisfies the equations (1.20) for j = 2 under the supplementary condition n
s=-. )“-
1
The inversion of the Laplace transform by the function Rgt (5, t) is proved by: l7reorem 3 If the conditions of Theorem 1 hold for fit), then, as n +m,
2rx (s + I) B (s, n)
(1.21)
27
Numerical inversion of the Laplace transform
where s is given by JSq.(1.21). The proof is similar to the proof of Theorem 1, using appropriate inequalities of the type (1.18) and (1.19) forR@71 (5, t).
2. Errors of the approximate methods of inversion Let K, be the set of functionsfit),
t 2 0, satisfying the condition
If”‘(t~)-fY)(t;i)
Ibkltr-til
for the v-th order derivative. Further, let fnci) (x) denote the left-hand side of (1.3) (1.17), and (1.22) for i = 0, 1,2 respectively. The convergence of the approximate solution f, co) (x) to the exact solution j(x) is established by: Theorem 4 Given any x > 0, as n + 00,
Proof: If Z(t) = P(t), where S(t) is given by (1 S), and bt=ln r-i-n-“yi, expanding ln Z(t) in powers of K’/~JJ~, up to the term of order ,-1/4y15: Z(t)=exp{--ain”‘y~2[1+D(y*,
n)
then we obtain, on
I+@(% Yi)>7
(2.3)
whence we obtain, recalling (1.4), for I y1 I < 1,
R:;, (x, t) =
I” 5
G
b (
+ 0
exp {--
wW?~[~
+
D (Ye, @I
)
(2.4)
h Yl)),
where w(n, yl) + 0 as n + m uniformly with respect to ~1, 1YI 1G 1,
D (~1, n) = -
al = 2(F’ =
a 3
$
n-‘/4yl
a = (r 1)2’
2
(F -
-$
I)”
(2.5)
n-‘lay12,
1)2 + 3(r - 1) + 2, S(F-~)~
(F--)3+~(F---l)2+12(~-~)+6
24
+
’
A, K. Shuvaev
28
Relation (2.4) holds, if q,G-ctQ,
where
(1-g+,
ql=
(1 + gi)x.
q2 =
Hence it is easily seen that, as n + 00, 91
s8:’
(x, t) 1t -
x I”dt -
rlr
withy,
The function Z(t) decreases as t moves away from to =x. Hence, from Eqs. (2.3) and (2.5) =-I,andEqs.(1.4)and(lS),weget rtr s R’,O’(5, t) 1t - CE1” dt <,(‘AIn’/zexp (-
a#*
- a&i4 - as}
(2.7)
0
where A 1 is independent of n. Similarly, 02
sR’,O’
(x, t) It - z 1” dt < A&/t exp {- aln’I’ + aznli* - as}
4;
(24
where A 2 is likewise ~dependent of n. Noting that, for any k > 0, nh exp{ -hn’“fa,n”*)
+O,
as n + m, we see from (2.6), (2.7), (2.8), that, as n + m, m
s
R’,O’ (2, t) 1t -
3
I’” dt -
1.(r-l)m bmrm,2nm,2
t2n)‘,2
+f
0
e-f’/2 I t 1”’dtv
(2-9)
--m
whence we obtain, in particular, m
s
o~‘,~‘(s,t)~t-xl:jdt-
2(r-1)5 (2n;rn)‘~IIn r
0
00
s
&“(x, 4 (t - d2 dt - (rrn
0
We now obtain the asymptotic form (2.1) from the inequality ll’n0’(Z)-f(2)1~kTR~)~~,t),t-ZIdt, 0
(2.10)
I)2 x2 (In
r)a
.
(2.11)
Numerical inversion of the Laplace transform
29
which holds for any j(t) of the classKo, and from (2.10) in the light of the fact that f(t)=klt--sjEKo. Let f’(t) =g(t); then,
~~)(.z)-f(5)=~R(nO)(P,t)~[g(r)-g(5)1dTdl 0
x
+g(~)~R’,0’(2,t)(t--)dt. 0 Iff (t)=K1,
then,bytheinequality
Ig(a)--g(z)
IdkIr---sI,
(2.12)
amlusing(2.12),
(2.13)
By (2.1 l), as n + 00the first term on the right-hand side of (2.13) is equivalent to 1)zzz 2m (In r)Z *
k(r -
(2.14)
To estimate the second term in (2.13), we note that 00
s
[I -
@(f-r) ]
&‘)(x,
t) 02 = 0.
(2.15)
0
If e-b(t-xl is expanded in powers of b(r - x) with remainder term -l/ge-brP(t)--rr (t-z) 3, where O
OD
0
0
sR’,O’(s,t)(t-spt--1
R’,O’(z, t) (t-
x)2& = 0,
whence, using (2.1 l), we obtain the asymptotic form as n + w
(2.16)
The asymptotic form (2.2) now follows from (2.13), (2.14), and (2.16), plus the fact that, if g(r) = k(t - x), we have f(t) E&, this completes the proof of the theorem. The following Theorems 5 and 6 are proved in a similar way. Theorem
5
Given any x > 0, as n + =,
A. K. Shuvaev
30
where E is an arbitrary positive number, and 4-l U=
s
exp (-
t2/2) dt.
-1
Theorem 6 Givenanyx>O,asn+m,
whereY=O, 1,2,
and
where 6 is also arbitrary and positive, and
bl (r) =
7r + 1 12 ’
bs09 =
5r + 3 5-r 8 lnr+T,
b, (r> =
5r -i_ 3 8 ,
31
Numerical inversion of the Laplace transform
During the proofs of Theorems 5 and 6, we have proved the asymptotic relations
-
1 I”’ 1t + I 1dt,
from which we obtain, with m = 0, the inequality of the type (1.19) for I?:) (T, t) and R%) (s. t). Numerical realization of the above approximate inversion formulae requires that we estimate the supplementary errors due to tr~sformation errors and the computations. For simplicity, we shah confme our discussion to the inversion formula proved by Theorem 1. We introduce the notation lnr zB (n/(r - f), n) F
((A+ \ r---l
i)q)
=X,,,(i),
S, = max I N,, x (i) [,
i=O,l,...,
n-l,2
n-l,
i, 7%
i=O,l,...,
T, = max IAN&(i)], i. n
12-1,
,...,
n=1,2
,...,
no,
no,
where AN,. %(i) is the error of Nn,X(i), and no is a fairly large integer. We find by simple arguments that the error AM,, x (if is not greater than M,, =(i) /1tY-*, if L correct decimal places are used to represent the mantissa of a real number. Using the above notation for the error Arc) (z) in evaluating the left-hand side of (l-3), we obtain the inequality (2.17) We find by direct evaluation of the integrals in (2.11) and (2.16) that m
IS
RL?(TJ)(LLq&
0
I<
m
s&‘(x, 0
t) (t - x)~dt <
1p x 2rnlnr
(r -
(
1 +-
0.2r n
- lYx2 (1 +$) “,, r ln2
)
*
(2.18)
(2.19)
A. E. Umnov
32
for the limits of variation of the parameters ‘lGrG5, I
where Al =
el1”lnr 2x (2rr?-)‘/~
The number n of the best app~~mation
is given by the equation
AiAznn51+2A,Aznn”~In Az=2nAs+4A1-t-2A,Az”n”. ~~s~te~
&y D. E. Brown
REFERENCES 1.
AMERBAEV, V. M., Some numerical methods for inverting the Laplace integral transform, Candidate Dissertation, Matem. in-t Akad. Nauk SSSR, Moscow, 1963.
2.
RYABTSEV, I. I., Practical evaluation of the original, given the values of the tran~orm at ~uid~t~t of the real axis, Izv. vuzov. ~ate~tjka, No. 3,139-143,1966.
points
THE METHOD OF PENALTY FUNCTIONS IN PROBLEMS OF HIGH D~ENSIONALITY * A, E. UMNOV Moscow (Received 16 Wuury
1974)
AN ABORTS is given for solving problems of mathematics programming, based on the use of the method of penalty functions in the standard decomposition scheme. Its convergence is proved. the effectiveness of the algoritbrn, the problem of obtaining the exact solution, and the use of the algorithm for solving dynamic problems, are discussed.
1. Intmduction It is well known that practical implementation of the method of penalty functions [ 1,2] involves considerable computational difficulties; in particular, the unsatisfactory convergence of the method in a small nei~~urhood of the exact solution has often been mentioned [l] .
Experience has shown that the difficulties of the method also occur when solving problems *Zh. vjkhisl. Mat. mat. Fiz., 15,6, 1399-1411,197s.
22 3.
SHATOV, A. K., A numerical method for solving the problem of stationary electromagnetic at a body of revolution, Zh. vj%hisI.Mat. mat. Fiz., 13, No. 1,237-241, 1973.
4.
SOBOLEV, S. I,., Some applicatio~sof fictions funktsional’nogo analiza v matematicheskoi
5.
SAVELOVA, T. I., Application of a class of regularizing algorithms to the solution of integal equations of the first kind of the convolution type in Banach space, Zh. vfchisl. Mat. mat. Fiz., 14, No. 2,479-483, 1974.
6.
MARKUS, M., and MINK, Kh., Survey of matrix theov and matrix i~e4uulities (Obzor po teorii matrits i matrichnykh neravenstv), Nauka, Moscow, 1972.
7.
SAVELOVA, T. I., The solution of an equation of the convolution type with inaccurately specified kernel by the method of regularization, Zh. vychisl. Mat. mat. Fiz., 12, No. 1, 212-218, 1972.
8.
ARSENIN, V. Ya., and IVANOV, V. V., On optimal regularization, Doki. Akad. Nauk SSSR, 182, 9-12,1968.
9.
KATO, T., Perturbation theory for linear operators, Springer, 1966.
wave diffraction
analysisto mathemati~I physics (Nekotorye fizike), Izd-vo LGU, Leningrad, 1950.
p~me~eniya
NO.
1,
10. ARNOL’D, V. I., Lecture on bifurcations and versa1 families, hp. Matem. Nauk, 27, No. 5 (1967), 119-184, 1972.
DELTA METHODS FOR NUMERICAL INVERSION OF THE LAPLACE TRANSFORM, AND THEIR CONVERGENCE* A. K. SHUVAEV Tula (Received
7 April
1971; revised 29 April 1974)
ASYMPTOTIC forms of the delta function and corresponding theorems on numerical inversion of the Laplace transform are given. An asymptotic estimate is obtained for the rate of convergence of the appro~mation to the exact solution in certain classes of originals.
Introduction Methods are discussed for the approximate inversion of a Laplace transform specified on the real axis, by means of a sequence of functions asymptotic~y convergent to the delta function 6(t - x). We shall refer to the functions of each such sequence as representations of 6(t - x), The methods represent a generalization of Post’s formula [l] , to which there corresponds the representation of &(t - x) Pn(2, t) =
(
$
>
n1 (n
I),
.
exp (-
f
t) F-1.
Post’s method has the drawbacks that the derivatives of increasing order of the Laplace transform have to be evaluated, and that the approx~ations converge rather slowly to the exact solution. The approximate methods considered in [2] have the same slow convergence. Just as in [2] , our methods do not require evaluation of the derivatives of increasing order of the Laplace transform; while our inversion by the “generalized representations” of s(t - x), introduced below, ensures a faster convergence on classes of smooth curves of the appro~mations *Zh. vychisl.Mat. mat. Fiz., 15,6,1389-1398,
1975.
Numerical inversion of the Laplace transform
23
to the exact solution than is possible with the methods given in [2] and Post’s formula. Questions of the stability of our methods mount to finding the number of the approx~ation up to which the computations need to be performed. For this, we obtain below, for the simplest case, an equation whereby the number of the approximation can be found, at which the total error, including the error of the method, the error of the computation, and the error of the transformation, is asymptotically minimal. Other cases can be treated in a similar way.
1. Approximate methods of inversion
We defme a non-negative representation of s(t - X) as the function
R%P7
b ‘) = B (n/(r _ I), n)
exp (--$$)
(1 - .~+~)~--r,
(1-l)
where B(p, 4) is the beta function, x > 0, and the parameters r and b are such that r > 1, bx = In r. We use R$ (z, t) to prove: Theorem 1 Let f(t), t > 0, be a single-valued function, integrable in any finite interval [0, 7’j , T> 0, for which there exists a real so such that 11 f (t) \ e-So’dt < 30.
(14
0
Then, at all points of continuity x > 0 of the function j(t), as n + 00:
(1.3)
Inr
ZB (n/P - V, 4 where F (s) =
1f(2) e-s’ a. 0
Proof We first obtain estimates for R$ (2, t) . Using the relationship between the beta and gamma functions, and Stirling’s asymptotic form, we arrive at
(1.4) where w(n) + 0 as n -+ m,
(1.5) In the interval 0 G t d In r/b, S(t) is increasing from zero to unity. Hence, given any E,, O
24
for some C which is independent
ofn and C.
To estimate R$) (z, t) with t>ln r/H-e*, r/&. If S(t) = G(y), then, fory 2 0,
eg>O,
we transform to the variable y = t - ln
Let
(1.7) For e2 > 0, the equation
as an equation in 6 has a non-negative solution 6 ( eo) +=c 1.Also, it is easily shown that, withy > e2, the derivative drp (6 (Q) , y) idy>O. From this, and Eqs. (1.7) and (I .8), we Frnd that
in view of which, for the same y ,
where G,(Ez)=I-F(~z))O. On now putting
and returning fromy to the variable t, we see from (1*9) that, if t>ln
exp[ -&(e2) inequality
t], where &(e2) -[ b/(r--l)]&‘(~~)
>O.
Finally,
r/b-l-ez, then S’(t) < for the same f, we obtain the
We now turn to the proof of (1.3). If the conditions of Theorem 1 are satisfied, we easily fmd, by using (1.6) and (1 JO), that, for sufiiciently large n,
25
Numerical inversion of the Laplace transform
where C, is independent of n. Recalling the equation 00 (1.11)
s R$’ (Z, t) dt = 1
0
it is obvious that, for any e, 0 < e
where I?!’ (5, t) dt,
I;., L =
J 1f(t) x+c
1R’,O’(2, t) a,
s If(t)-f(“)lR(n0)(2,t)dt.
I ‘;,c=
Z--L
Onthebasisof(1.6)and(l.lO),weget
I;,
1
<
b(E) exp -T(n1
Cn'll
i)]~lf(‘)lexp[-~~(n-i)t]dt, 0 m
J,;, L< Cn’/aexp {- 18, (E) (n from which it follows, recalling conditions (13,
I:,,-0,
1) -
sol(z i- 8)) 5 If(t) \ e-sot& x
that, as n + =,
r;J+o.
(1.13)
IfAt) is continuous at the point x, we easily fmd, using (1.1 l), that I& -+O as n + m, and from this and (1.12) and (1.13),
(1.14)
lim S j (t) R’,O)(z, t) dt = f (.x). 11-00o
Recalling that bx = In r, and the definition of F(s), (1.3) now follows from Eqs. (1 .l) and (1.14). This proves Theorem 1. As the first-order generalized representation of &(t - x) we take the function
b[n---@----)l e_~‘,’(1 _ e-bt)n-z rB(s+l,n-1) b [I - 12+ so” - ‘)I e-~!,t(1 _ e-bt)n-l
&‘@t)=
i
(1.15)
1
rB(s+
l,n)
where bx = In r, n
s=--+++[l+ r1
l/z
4rn (r -
1)2 I
The possibility of inverting the Laplace transform by means of Rc,‘) (r, t) is proved by:
(1.16)
26
A. iy. f&maw
Theorem 2
If the conditions of Theorem 1 hold for tit), then, as n + *,
rsB(s+
l,n--
1) (1.17)
The proof is similar to that of Theorem 1, using the inequalities (1.18)
(1.19) where C is independent of n and I, and CJ independent of n. The inequality (1 .18) is obtained from (l.l), (1.1’S), and (1.16). We shall speak about the proofof(l.19) in Section 2. It is easily shown that R’,O’(z, t) and lit’ (5, t) satisfy the equations DD
s
e-i&t&l)
(5,
t)
&
=
e-ib=,
i =
0,1, . * . , j,
(1.20)
0
for i = 1 and 2 respectively. In accordance with this, as the secondorder generalized representation of &(t - x) we take RF)
(g,
f)
=
2
a,_ie-sbf (1 _
e-“‘)n-l-i,
i=O
which satisfies the equations (1.20) for j = 2 under the supplementary condition n
s=-. )“-
1
The inversion of the Laplace transform by the function Rgt (5, t) is proved by: l7reorem 3 If the conditions of Theorem 1 hold for fit), then, as n +m,
2rx (s + I) B (s, n)
(1.21)
27
Numerical inversion of the Laplace transform
where s is given by JSq.(1.21). The proof is similar to the proof of Theorem 1, using appropriate inequalities of the type (1.18) and (1.19) forR@71 (5, t).
2. Errors of the approximate methods of inversion Let K, be the set of functionsfit),
t 2 0, satisfying the condition
If”‘(t~)-fY)(t;i)
Ibkltr-til
for the v-th order derivative. Further, let fnci) (x) denote the left-hand side of (1.3) (1.17), and (1.22) for i = 0, 1,2 respectively. The convergence of the approximate solution f, co) (x) to the exact solution j(x) is established by: Theorem 4 Given any x > 0, as n + 00,
Proof: If Z(t) = P(t), where S(t) is given by (1 S), and bt=ln r-i-n-“yi, expanding ln Z(t) in powers of K’/~JJ~, up to the term of order ,-1/4y15: Z(t)=exp{--ain”‘y~2[1+D(y*,
n)
then we obtain, on
I+@(% Yi)>7
(2.3)
whence we obtain, recalling (1.4), for I y1 I < 1,
R:;, (x, t) =
I” 5
G
b (
+ 0
exp {--
wW?~[~
+
D (Ye, @I
)
(2.4)
h Yl)),
where w(n, yl) + 0 as n + m uniformly with respect to ~1, 1YI 1G 1,
D (~1, n) = -
al = 2(F’ =
a 3
$
n-‘/4yl
a = (r 1)2’
2
(F -
-$
I)”
(2.5)
n-‘lay12,
1)2 + 3(r - 1) + 2, S(F-~)~
(F--)3+~(F---l)2+12(~-~)+6
24
+
’
A, K. Shuvaev
28
Relation (2.4) holds, if q,G-ctQ,
where
(1-g+,
ql=
(1 + gi)x.
q2 =
Hence it is easily seen that, as n + 00, 91
s8:’
(x, t) 1t -
x I”dt -
rlr
withy,
The function Z(t) decreases as t moves away from to =x. Hence, from Eqs. (2.3) and (2.5) =-I,andEqs.(1.4)and(lS),weget rtr s R’,O’(5, t) 1t - CE1” dt <,(‘AIn’/zexp (-
a#*
- a&i4 - as}
(2.7)
0
where A 1 is independent of n. Similarly, 02
sR’,O’
(x, t) It - z 1” dt < A&/t exp {- aln’I’ + aznli* - as}
4;
(24
where A 2 is likewise ~dependent of n. Noting that, for any k > 0, nh exp{ -hn’“fa,n”*)
+O,
as n + m, we see from (2.6), (2.7), (2.8), that, as n + m, m
s
R’,O’ (2, t) 1t -
3
I’” dt -
1.(r-l)m bmrm,2nm,2
t2n)‘,2
+f
0
e-f’/2 I t 1”’dtv
(2-9)
--m
whence we obtain, in particular, m
s
o~‘,~‘(s,t)~t-xl:jdt-
2(r-1)5 (2n;rn)‘~IIn r
0
00
s
&“(x, 4 (t - d2 dt - (rrn
0
We now obtain the asymptotic form (2.1) from the inequality ll’n0’(Z)-f(2)1~kTR~)~~,t),t-ZIdt, 0
(2.10)
I)2 x2 (In
r)a
.
(2.11)
Numerical inversion of the Laplace transform
29
which holds for any j(t) of the classKo, and from (2.10) in the light of the fact that f(t)=klt--sjEKo. Let f’(t) =g(t); then,
~~)(.z)-f(5)=~R(nO)(P,t)~[g(r)-g(5)1dTdl 0
x
+g(~)~R’,0’(2,t)(t--)dt. 0 Iff (t)=K1,
then,bytheinequality
Ig(a)--g(z)
IdkIr---sI,
(2.12)
amlusing(2.12),
(2.13)
By (2.1 l), as n + 00the first term on the right-hand side of (2.13) is equivalent to 1)zzz 2m (In r)Z *
k(r -
(2.14)
To estimate the second term in (2.13), we note that 00
s
[I -
@(f-r) ]
&‘)(x,
t) 02 = 0.
(2.15)
0
If e-b(t-xl is expanded in powers of b(r - x) with remainder term -l/ge-brP(t)--rr (t-z) 3, where O
OD
0
0
sR’,O’(s,t)(t-spt--1
R’,O’(z, t) (t-
x)2& = 0,
whence, using (2.1 l), we obtain the asymptotic form as n + w
(2.16)
The asymptotic form (2.2) now follows from (2.13), (2.14), and (2.16), plus the fact that, if g(r) = k(t - x), we have f(t) E&, this completes the proof of the theorem. The following Theorems 5 and 6 are proved in a similar way. Theorem
5
Given any x > 0, as n + =,
A. K. Shuvaev
30
where E is an arbitrary positive number, and 4-l U=
s
exp (-
t2/2) dt.
-1
Theorem 6 Givenanyx>O,asn+m,
whereY=O, 1,2,
and
where 6 is also arbitrary and positive, and
bl (r) =
7r + 1 12 ’
bs09 =
5r + 3 5-r 8 lnr+T,
b, (r> =
5r -i_ 3 8 ,
31
Numerical inversion of the Laplace transform
During the proofs of Theorems 5 and 6, we have proved the asymptotic relations
-
1 I”’ 1t + I 1dt,
from which we obtain, with m = 0, the inequality of the type (1.19) for I?:) (T, t) and R%) (s. t). Numerical realization of the above approximate inversion formulae requires that we estimate the supplementary errors due to tr~sformation errors and the computations. For simplicity, we shah confme our discussion to the inversion formula proved by Theorem 1. We introduce the notation lnr zB (n/(r - f), n) F
((A+ \ r---l
i)q)
=X,,,(i),
S, = max I N,, x (i) [,
i=O,l,...,
n-l,2
n-l,
i, 7%
i=O,l,...,
T, = max IAN&(i)], i. n
12-1,
,...,
n=1,2
,...,
no,
no,
where AN,. %(i) is the error of Nn,X(i), and no is a fairly large integer. We find by simple arguments that the error AM,, x (if is not greater than M,, =(i) /1tY-*, if L correct decimal places are used to represent the mantissa of a real number. Using the above notation for the error Arc) (z) in evaluating the left-hand side of (l-3), we obtain the inequality (2.17) We find by direct evaluation of the integrals in (2.11) and (2.16) that m
IS
RL?(TJ)(LLq&
0
I<
m
s&‘(x, 0
t) (t - x)~dt <
1p x 2rnlnr
(r -
(
1 +-
0.2r n
- lYx2 (1 +$) “,, r ln2
)
*
(2.18)
(2.19)
A. E. Umnov
32
for the limits of variation of the parameters ‘lGrG5, I
where Al =
el1”lnr 2x (2rr?-)‘/~
The number n of the best app~~mation
is given by the equation
AiAznn51+2A,Aznn”~In Az=2nAs+4A1-t-2A,Az”n”. ~~s~te~
&y D. E. Brown
REFERENCES 1.
AMERBAEV, V. M., Some numerical methods for inverting the Laplace integral transform, Candidate Dissertation, Matem. in-t Akad. Nauk SSSR, Moscow, 1963.
2.
RYABTSEV, I. I., Practical evaluation of the original, given the values of the tran~orm at ~uid~t~t of the real axis, Izv. vuzov. ~ate~tjka, No. 3,139-143,1966.
points
THE METHOD OF PENALTY FUNCTIONS IN PROBLEMS OF HIGH D~ENSIONALITY * A, E. UMNOV Moscow (Received 16 Wuury
1974)
AN ABORTS is given for solving problems of mathematics programming, based on the use of the method of penalty functions in the standard decomposition scheme. Its convergence is proved. the effectiveness of the algoritbrn, the problem of obtaining the exact solution, and the use of the algorithm for solving dynamic problems, are discussed.
1. Intmduction It is well known that practical implementation of the method of penalty functions [ 1,2] involves considerable computational difficulties; in particular, the unsatisfactory convergence of the method in a small nei~~urhood of the exact solution has often been mentioned [l] .
Experience has shown that the difficulties of the method also occur when solving problems *Zh. vjkhisl. Mat. mat. Fiz., 15,6, 1399-1411,197s.