On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions

U.S.S.R. Comput.Haths.Math.Phys .,Vol.27,No.6,pp.18-2S,1987 Printed in Great Britain

0041-iS53/87 $10.00+0.00 01989 Pergamon Press plc

ON THE CALCULATION OF THE MULTIPLE COMPLEX ROOTS OF THE DERIVATIVES OF CYLINDRICAL BESSEL FUNCTIONS* M.K. KERIMOV and S.L. SKOROKBOWV

Efficient algorithms are given for computing the multiple (double) complex roots of the derivatives of cylindrical Bessel (ordinaryand modified) functions of the lst, 2nd, and 3rd kinds when the argument and index take complex values. They are based on asymptotic methods and numerical contour integration. They are realized as FORTRAN programs for the BESM-6 computer. The programs are used to compute with high accuracy many double zeros, some of which are quoted in the paper.

Introduction In some problems of mathematical physics (e.g., of diffraction or wave propagation theory etc.) it becomes necessary to study and compute the multiple as well as the simple zeros of cylindrical Bessel functions and their derivatives. We sometimes have to deal with multiple complex zeros when the argument and index of the Bessel function take complex values. There is a vast literature dealing with the simple (especiallyreal) zeros, see e.g., /l-4/. Multiple real zeros have been much less studied, as may be seen from the references quoted in our papers /S, 6/, where we computed with high accuracy the multiole (double) real zeros of the derivatives with respect to z of z,I."'(:), Y."'(z),s--l,P,3, of Bessel functions of the 1st and 2nd kinds, when the arqument r takes (zG@?)or complex (z=C) values, and the index v takes real values (v=m). As far as we know, the multiple complex zeros of the derivatives when Y takes complex values have only been studied in /6/. In the present paper we attempt to treat in detail the multiple (double)complex zeros of the derivatives of all the Bessel (both ordinary and modified) functions of the lst, 2nd, and 3rd kinds, when the argument and index take complex values. Our theoretical results enable us to offer efficient algorithms for computing these zeros with high accuracy for a wide range of variation of the argument and for the BESM-6 computer, and were index. All the algorithms are realized as FORTRAN programs used to compute the 101 first double complex zeros of the functions in question (the results are given at the end of the paper). 1. Multiple

zeros

of the derivatives

J;(Z)

and

1;(g)

of Bessel functions

of the 1st kind. We showed in /5, 6/ that Bessel functions cannot have double zeros at singular points I#_, though their derivatives with respect to z can have such zeros. It was z+‘O and of variation of, v,vE(-n-1,-n), also shown that, in the case vE(R, in every unit interval n--l.2,...,there is only one value v. at which there is a double zero of J",(z). while we have the asymptotic relation v,--n-'/,+0(1), n-m. In the case vEc,v=U+i$, thestudy of the double zeros is more difficult. By the wellknown formula of analytic continuation /4, p.361/ r.(re~m')-~'""'~,(z), ?W=Zkl, it is sufficient to confine the study of the double zeros of J,'(z)to the right half-plane of r, largz(
(1.1) J-.‘(v), largvl< It is thus sufficient to study only the simple zeros of the functions J.‘(v), n/2. In future, therefore, by the "zeros" we mean the simple zeros. Further, by the property of complex conjugation (1.2) J:.(r)-(Jy’(z))’ O&argv<;r/2. To localize it is sufficient to consider only the first quadrant of v variation, and compute the zeros of the individual functions I.'(v) , JI. (v) , we consider the cases: 1) 2) IvI-"I 3) average values of IvI. Ivl-0, In case 1) we used the well-known /4, p-360/ asymptotic relation We start with J_‘(v). v#- I, -2 ,..., IzI+O. J.(z)-(zE)‘/T(v+l), (1.3) Starting from the exact relation /4, p.361/ I,’ (z)

lZh.vychisl.Mat.mat.Fiz.,27,11,1628-1639,1987 18

(1.4)

19

and using 11.31, we obtain J.‘(v)=J*(v)-J.+,(v)

-

(v’;;;yv;v’2) =J,.’ (v)

J,(v)-(v/2)‘lr(v+1)=J.,(~).

(I&)

7

[VI--0.

(1.5b)

By using the algorithms of /7/ to compute the functions I"(Z) and j.'(Z)in ;he complex domain, we can show numerically that, when (1.5) are used with decreasing Iv],]v]
([J.(v)-J..(v)llJ..(v)},

6,(v)=max

and

l/J..‘(v)},

6,'(v)-m;x([J,‘(v)-J.,‘(v) V=/VIP,

O<+Xll,

decrease monotonically; they have order SX~O-~ for ]v]='it,3x lo-’ for Iv!-0.1, and 3x10-5 for the function J,‘(v) has no v-=0.01. It thus follows from (1.5) that, with ]v[+O,O
In case 2) we start from the well-known /4, p.368/ asymptotic relation I,’ (v) -b/v”-u/(~v”~),

where b --2’/[3’“r(‘/,)],

]argv]
(1.6)

a-2"/[3"r('/,)].

It was checked numerically that, in this case, b,'(v)decreases monotonically as Iv] increases,andhastherespectiveorders:3~10-2 for ]v]='i~, 5x10~~ for iv]==1, 10-‘ for Iv]=&,and IO-' for ]v(-10. Similarly, the error 6,(y) was computed from the asymptotic relation /4, p.368/ for J,(v) itself: -a:v'!~-0/(70v'~~), IargvlGni2, J,(v)

IvI-c=.

where

a and b are the same as in (1.6). The order of the error 61(v) then proved to be the same as for 6,'(v). We can therefore conclude that, as /VI-Lm,v+a, the function J.‘(v) has no zeros. In case 3) of average values of Iv],we used the well-known property of a function F(z) which is analytic in the domain G where it has no poles: the number N of zeros of F(z) in G, where G has the boundary r, isgiven by

(1.7) where there must be no zeros of F(Z) on the boundary r. As F(r) we took J,‘(Z), while the derivative F'(Z) was evaluated from the elementary relation of difference differentiation F'(Z)~[F(Zih)-F(z)l/h, h-IO-'. As the domain G we took the square O
*

Y

_h)dz=H &(z,+kH), H--y k-0

(W

:I

(where 2%' means that the first and last terms of the sum are taken with weight I/z),we can show that J.‘(v) has no zeros in the domain G. in summarizing what has been said about the zeros of J.‘(V), ]argv]
=cos(nv)

J”‘(v)-sin(nv)

Y.‘(v).

the Bessel function of the 2nd kind Y.(z) and its derivative asymptotic relations Y,(V)-Y.,(V)=-13a/vU-7)3 b/(70+‘), For

Y,‘(V) -Y.:(v)

-1’3 b/vVY3d(5v“*),

largvl<

(1.9) Y.‘(s)

1argv]
we know

/4, p.368/ the

(1.10)

where a and b are the same as in (1.6). It was checked numerically that the relative errors 61(v) and 6,'(v) of the functions Y.(v) and Y,'(V) behave as Iv]+= in the same way as in the case of the functions J.(v) and J.‘(v). Here we used the algorithms of /8/ to compute Y"(Z) and Y.‘(z) in the complex domain. Using asymptotic relations (1.6), (1.10) and also (1.9), we can conclude that, with v=C, v-m, the same result, see /7/, is obtained as when v=R: the function J:,(z) has a denumerable set of only real double zeros. Consider the case ]v]+O. We use the relations 14, p-358/ Y”(Z) -

J.(t) cos (nv) -J_,(z)

sin(nv)

’

(1.11)

20

y,, (z) = I.’ (J)COS(nv) -J:“(z) sin(m)

(1.12)

’

By (1.3) and (1.4) we have

J_‘.(v)=-J-.

(v) -I_.,,

(v) -

-cv’2;;;,~2)

)

lvl-0,

(1.13)

We use below the Taylor expansions v

*

(1 1--I+vln~+o(v),

r(l+v)=l+vr'(l)fo(v), [v/+0.

(1.14)

cm substituting (1.5) and (1.13) into (1.12) and using (1.14), we obtain Y,'(v)-2/(17v),lvl+O.

(1.13)

Using (1.15), it was checked numerically that, 'if Ivl-0, the error 6,'(v)decreases monotonically and amounts to 5x10-2 for lvl=O.l and 10m3 for IvI=O.OL It thus follows from (1.9), (1.5), and (1.15) that, for sufficiently small Iv/, the function r-.'(v)has no zeros. In the domain of average values of IYI , to study the zeros of J:,(v),the same quadrature formula (1.6) was.applied to integral (1.7). However, since J:,(t) has zeros on the real axis, the domain G was deformed into the domain G' with boundary r, obtained from Gbyadding the strip - i
n=l,2,..., n-00.

on the basis of the well-known /4, p.375/ relations connecting J.(z) Bessel function of the 1st kind I.(Z):

with the modified

I,(z) =e-“““*J.(ze”‘.‘*), -.zCargz
can

have double zeros only

(2.1) and third, -s
(2.2)

differentiate both sides With respect to z, put z-v, and equate the result to zero. We reduces to studying the simple then find that the problem ofthedouble zeros of Y,'(ze-"') of the function zeros v-a+@, a30, p>O, (~(v)=e”“Y,‘(v)-2icos(xv)Jv’(v). When Iv1 is small, we can show by using the asymptotic relations (1.51 and (1.15) that

m(v) has no zeros as v+O. Using asymptotic relations (1.6) and (l.lO), separating real and v-m, we imaginary parts, and finding the asymptotic expansion of the resulting System as

21 v-m, the function m(v) has a denumerable set of complex zeros find that, as where for on and &, we have the asymptotic relations

rl,=nt’/,+p/n”‘to(n-“), ii.

In 2

Y3p

2s

n-’

Ee..-_-+O(n-q’)

,

n=O, I,. . . .

v,=Z,+i& (2.3a)

n-co,

where 13 I-('('1) = -0.030012573364, ' = - 20X-6'r('.',)

(2.4a)

+ _.

(2.4b)

0.110317800076.

In other words, in every half-strip of unit width of variation of complex v,Imv>O, nGRev< nfl. n-o. 1,. , there is one value v. at which there is a double complex zero of the function Y;” (ze-"'). If initial approximations to the zeros are obtained with the aid of (2.3) (with n-0 we took v~='/~+i(ln?)/(~n)), then we refine them by Newton's iterative scheme, we can compute the first 101 zeros vn of Y;,(ze-=) to 8 decimal places (see Table 1). For average values of 1y1, we performed numerical integration over the contour r with and found only the first 20 zeros of the function cp(:), which are described by the F(r)-+) asymptotic formulae (2.3). Consider the double zeros of the functions YL(r), Y',(ze-") with O&argv
(2.5)

Thus the problem of the double zeros of Y:,(z) reduces to studying the zeros of the function (F,(')-Y,'(V)c09(~(V)+l:(v)sin(~v). Using (2.2) and (2.5), it can be shown that the problem of the double zeros of reduces to studying the zeros of the function

Y_,(X9')

When studying the zeros of m,(v) in the case lvl-0 and /VI-C=,we used the suitable asymptotic formulate for I.'(v) and Y"'(V), while for average values of Iv1 we performed numerical integration over the contour r' of the domain G', since it was remarked earlier that there is a denumerable set of zeros of Y:,(v) on the positive real semi-axis. It was found as a result that m,(v) has only a denumerable set of known real zeros. Study of the zeros of qt(v). including integration over the contour r of the domain G, showed that the function YL, (zehXi) has no double zeros in the domain O
ln3 f 0 (1) , 4.7

n-l, 2,.. . , n-m

Recalling the complex conjugation property (2.1), we can claim that, along with the double zeros obtained for Y,'(Z) with O0 it 1' has adenumerable setofcomplexdoublezerosonlyinthe lefthalf... I' 6 plane of s; for the real and imaginary parts oftheindicesofthese we have the asymptotic relations (2.3). zeros, %==ll*“Tt$,, f **. Now consider the behaviour of the double complex zeros of in the complex 2 plane. In the case of the real double Y.‘(z) -0.5 zeros the question of the formation and decay of the zeros was previously studied in /6, 81. In Fig.1 we show the trajectories I largz(l they move picture of the formation apart at right angles to their own trajectories with t
.r

-a.5

0

22 ') v(t)=n+t(v,-n), O
3. Multiple

zeros

of

derivatives

of

Bessel

functions

of

the 3rd kind.

Consider the double comulex zeros of the derivatives h','"'(:), HP' (3) of the Bessel functions of the 3rd kind H."'(:),H."'(z) (Hankel functions). As the main sheet of the Riemann surface we take --.1
H':' (z)==~-~**H:*'(:),

we can confine ourselves to the case Rev>O.

Further, from the relations

H."'(z)=lv(z)+iYv(~),

H.'"(Z)=],(~)--iY,(3).

(3.1)

and from (1.2) and (2.1), we obtain 11) (2) fi,.(t')=+=(&(z))', H,:"(z')-(H:*)(2))'. Recalling

condition

(1.11,

which

has

to

be

satisfied

in

order

for

H.'"'(s), H."" (3) to have

double zeros, we can confine ourselves to studying the zeros in the of the z plane. Also, using the relation /4, p.361/ " ”'Z,(_e-"')=e-""H * \"'(t) ?

it

cm be regarded as sufficient to study the simple zeros of

1st and 3rd quadrants

H,“’ * (v), Hi' (v), H,!" (ve-ni)

O
It was

found

that H!"'(v) Hi"' (v) have no zeros. Table 1 Complex double zeros v,of the function Y:. (xe-z')

-

Pm”

”

0.221157041 0.15624008 0.14225916 0.13507363 0.13088741 0.12810393 0.12609878 0.124.37433 U.I2337022 0.12239c63 N12157555 0.12068469 0.12029986 0.11977355 0.11931626 0.11891392 0.11855198 0.11822577 0.1 li9299i O.lli66028 O.lli41321 0.11718589 0.1169i368 O.116i8118 0.11R60008 0.11643112 0.116%06 0.11K12481 0. I tr,93.-54 0.1138541~ 0.1 l5ialj

0 1 ;

5” iI9 6

10

::

13 i4 15

16 17 is 19 20 21 23 23 24 25 -3 27 26 29 30 31 32 .33 34 35

0.11~~128~ O.ll5lJ171

0.11539621 0.11529.591 0.11.520041 0.11510936 0.11502242 0.11493932 0.114859i8 0.1147635a 0.114i1048 0.11464030 0.1 I45i2tui lJ.lli30796 0.11444551 0.11438532 0. I 1432727 0.1142il25 il.ll42li14 @.I1416484

% z 40 41 42 z 4.5 46 47 2 56

To study the zeros

-I

InlV,

-5,

51 52 53 54 5.3 56 5i 58 59 60 :: 63 64 65 FG :: 70 it i2 73 74 75 76 7i i8 i9 Pl 82 a3 84 65 E 66 z 91 :3’ 94 93 96 9i :: 106

51.33118320 52.33121035 53.33123666 54.33126217 55.33128692 56.33131094 57.33133426 58.33135693 59.33t37896 60.33140039 61.33142123 62.33144153 63.33146129 64.33148055 65.33149931 66.33151761 67.33~53546 68.33155287 69.33156987 70.33158647 71.33160268 72.3316N3 i3.33163401 74.33164915 75.33166395 76.33167644 7i.33169261 7633176548 7933172006 80.33173337 81.33174640 82.3317591i p3xt;;

0.11411426 0.11406530 0.11401789 0.11397195 0.11392741 0.11386420 EEE t; ;;;62 0:1EJ66606 0.t 1364965 o.iKi61421 0.11357970 0.1W4607 0.11351329 O.li346134 0.11345017 0.11341975 0.11339007 0.11336Vl8 0.11333276 0.11330510 0.11327806 0.11325162 0.11322576 0.11320047 0.11317572 0.113i5149 0.113127i7 0.11310454 0.11306178 0.11305948 0.11303762 O.tl301620 0.11299519 O.il29i458 0.11295437 0.11293454 O.il291507 0.11289596 “0::;3;;

65.33*:9599 86.33160779 87$;978&;6

2 89.3316-418i 9933185261 91.33166355 92.33187410 9333165447 94.33189465 9533199465 96.33191449 97.33192415 98.33193365 99.33194299 10093195218

H~J'(ve-"') we use the formula /4, H.(‘J

In view

Imv,

p.361/

0.f 1264066 0.11262291 0.11280544 0.11276627 0.11277140 0.11275481 0.11273850

of

analytic Continuation

(~e-~')-2 cos(nv) H."'(z)+~-"'~H~(*'(z).

of this, the problem of the double zeros of this function reduces to studying the simple zeros of the function

23

(:I* (II), QGrg6x/?. f(v)=2cos(nv)H."' (V)fC""'H. this, we used (in view of (3.1)) the above asymptotic relations for the functions r.'(v) and Y.'(v)as Iv(-o and [VI--~,and in the case of average Iv1 numerical integration over the contours r and I". We were thus to prove the following theorem. For

Theorem 4. Thederivative ,H!,!)' (2) OftheHankelfunction

8,("(z) has adenumerable set of

onlyinthe 3rdguadrantofthezplane. The complex double zeros when vn=r.+&, r.>O, $>O, derivative HP,'(z)oftheHankelfunction H.'*' (z)has a denumerable set of complex double zeros when the index and argument are conjugate to the index and argument for the function @"(z). For the real part 1"and imaginary part 5. of the indices corresponding to complex double zeros we have the asymptotic relations r.-n+'l,+++o(n-+), n-O,&...,

f.P-$+O(n-qS).

(3.2)

n-r-,

where p is given by (2.4). In other words, in every strip of unit width of variation of v=C, n
has a double complex zero.

It must be said that, when proving Theorem 4, numerical integration was performed over both the contour r and over r': because the zeros of f(v),see (3.2), are close to the real axis, integration over r gives a large computational error, while integration over r' does not have this drawback. Using (3.2) to find the initial approximations,and Newton'siterative scheme for refining them, we computed the first 101 complex double zeros of H(rG(se-"') to 8 decimal places (see Table 2). " We know /4, p.375/ that the functions H."'(z) If"'(s) are connected with the modified Bessel functions of the 2nd kind K,(z) by the relitions K.(z)='/,ITie""ilzH!') (c+/?), K,(z) =_‘/,~~e-“n’rH”(rr(ze-~~‘~),

-n
z
We can therefore obtain the following from Theorem 4. Corollary 2. The derivative of the modified Bessel function of the 2nd kind K.(z)has no double complex zeros on the main sheet largzln. It can easily be seen that the picture of the formation and decay of the double zeros of H!)'(Z)is similar to the picture in the case of Y.'(z). 4.

Discussion of the results. We checked our algorithm and numerical data by using the well-known Delves and Lyness algorithm, see /9/, for findingthezerosofan analytic function in a given domain of the complex plane. It is based on constructing a polynomial whose zeros are the same in this domain. For the construction we take the analytic function F(z) given in the domain G with boundary r. on which F(i) has no zeros. Then, the integrals

are well-known to have the property

where Iv==& is the number of zeros 01 of F(s) inside G. we can construct the N-th degree polynomial P.v(z):

Then, using the sequence &...,SN,

such that its zeros are the same (allowing for multiplicity) as the zeros of r(z) inside the domain. Relations connecting the coefficients b, with Sk are given in /lO-12/. The algorithm of /9/ is written in /14/ as A FORTRAN program, though the text is not given in /14/. The program was therefore written afresh in the light of the modification /13/ of the algorithm, where no computation of the derivative F’(z) is needed. In /13/, Sk was computed from the relations S,=K=+rgf(~)li. _.

24

where a

is any point of the domain c. [argF(:)], is the variation of the argument 9 of F(r),F(z)-]F(z)[e", when the contour r is circuited when cpis varied continuously. A similar requirement that the variation of Ing(z)=lnIg(z)I+iargg(z), g(z)=(z-a)-XF(:). be continuous in imposed when computing &from

(4.1). Table

2

Complex double Zeros v,,of the function Hti (ze-"') In”,,

n

0 I 2

3 4 z 7 8 9 10 I1 I? I3 t: I6

ii

IS 19 z 22

23 24 25 26 27 '23 g i g : 38 4": 41 42 43 44 45 46 47 45 49 50

0.316333132 O.l2246li32 1.31195iii0.04i'sx? 2.3liilO95 0.03154834 3.320z439 0.02451412

4.32256370 5.32339Si54 6.324&i 7.3Wi?i53 S.3i62xn79 9.32673S15

0.02o4O.T4Y O.OlXliti57 I 0.015tiS94i 0.0141S408 0.01299311 0.01202334 10.32il39240.011213il 11.32i~1%4 0.01033OSti 12.32iSLOli O.G0994Mi 13.32YIO?S? O.C0942i89 14.3w439;i O.OOS9i5~i9 15.328ZM4 0.00837392 lfi.3257X29 O.OO921415 li.3?8?27Oi O.oOiSS9i9 18.32!xeG3 O.Wi595Xi 19.32923137 0.00732727 203293Ga5 O.M)iOSl41 21.3294Si95 O.OOF.S%14 2?.B?Ri019? 0.mx41afi 23.329iOiiS O.l%45219 ?$.329?4%42 o.c062;lY4 o.oc%103.-c 23.32939860 O.OO5941:0!1 26.32998498 O.OO.ii9Wl 27.33CM~iIl OOO565951 29.33o!;449 2%?3o'lG5 O.CG552S64 3&%30%266 0.0054osI7 31.33ORlit7 0.00328817 32.33JXC.337 O.OO5l7i37 33.33O4W51 O.CO507219 o.oo497219 34.33052185 35.3305i459 O.OOdSili96 p3m'~; wc4i~~ 3.33oi:i9ot 39.330id3OS 40.33osoi33 4 1..33itx%S 42.330&85 43.33W132 44.3309:xa 45B3O!Wl3 46.xll0wi3 rt7.331O~G393 $S.,%Y IMJ Ii 19.33I Ix3.3 jO.RSII <
0.0046ltj.?i' 0.00453724 O.oo44'il23 0.03438~.12 0.oo431831 O.OO425102 0.00418639 O.cal2397 O.CQ4M39O o.oo4oo.i9s 0.00.395006 O.CWsS9iioR o.co3u.w

n

.iI

52 53 54 55 .iti 57 .5S 59 6": 62 63 Ii4 Ii5 66

tii

:: i0 71

i2 i3

74

i5

;i 79 !Y

a2 g A5 8 : :: :: 94 9.5 96 :; a9 100

IRlV,

Ilrr,

5133lliiT' 52:33120505 53.33123152 s54.33125i19 55.33129209 .56.3313C++25 57.331329il 58.331.35250 59.331374Mi 60.33139~20 613314l717 62.3314373i fi3.33145744 64.3314ifii9 65.33149X5 66.33151~01 67.33153198 68.33154948 69.3315665ti 70.33158324 71.33159953 72.33161544 73.33163099 74.33164620 75.331661Oi

i6.3316iW 77.3316898~

78.33170378 79.331ili4' 8033173078 81.33174387 S2.33li566Q S3331iti923 84.331X157 8j.33179365 86.33180549 87.33181711 88.33182P31 89.331839iO 9o.331s5lMs 91331Sd;l4i 92.33187205 93.33lSnwi 94.33180267 95.331902il 96.3319l2.iS 97.33192% 98.33193181 99.331!~4118 !00.33193040

g;;g 0.00369723 o.ocL365t.35 ylm&"," 'i O.Oo35219~ o.oo34S12~ O.OO.3441i5 0.00340334 O.OiXXWJ!l o.Qo332964 0.00329425 0003259i9 0.00322622 0.0031935o 0.003ltit59 0.oo31:@4i 0.003t0010 oO:FEE pm@mlg; o:oo295s6o 0.00293220 0.0029o63i o.oo2SS112 0.00285640 O.CO283220 O.CtWQS.il O.C027S;h1l 0.002762.?3 O.o0274OZ! O.OO27194~ o.oo2697w O.OO26ifll) 0.0026:.X! 0.002635:::~ O.OO26155Z O.QOZ.i96l'i 0 oo257R!.'I o:imm~ O.oo253!~<~ 0.002521~7 0.002504cI o.oo24S65e o.Qo246941 O.C@2452.%i O.oo24359~ o.M)24t9w

The algorithm was used to compute the zeros of the derivatives of the Bessel functions;

H,':"(Z). As C we took the domains of Sects.2 and 3 above. as F(z) we took y,'(z),H,("'(z), The value So-S was computed to 11 correct decimal places, and S,, k-l (l)iy,to 5-S places. To zeros were also further refined by Newton's scheme. The values of the complex double zeros computed by this algorithm were exactly the same as those given in Tables 1 and 2. Since our basic task was to study and compute the double complex zeros of the derivatives of the cylindrical functions, and not to examine methods of computation, we did not compare the computing times of the two algorithms. REFERENCES WATSON G.N., A treatise on the theory of Bessel functions, Cambridge Univ. Press, 1962. 2. KRATEER A. and FRANZ W., Transrendente Funktionen, Akad. Verlag, Leipzig, 1960. 3. OLVER F.W.J., Asymptotics and special functions, Acad. Press. N.Y., 1974. 4. ABRAMOWITE M. and STEGUN I., (Eds), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Nat. Bur. Standards Appl. Math. Ser., 55, US Government printing office, Washington, 1964. 5. FERIMOV M.K. and SKOROKHODOV S.L., On the multiple zeros of the derivatives of cylindrical Besselfunctions,Dokl. Akad. Nauk SSSR, 288, 2, 285-288, 1986. 1.

25

6. KERIMOV M.K. and SKOROKRODOV S.L., On computingthemultiple Zeros of the derivatives of F.(:) , WI. vych. Mat. i mat. Fiz., 25, 12, the cylindrical Bessel functions I"(r)and 1749-1760, 1985. 7. KERIMOV M.K. and SKOROKHODOV S.L., On computing the complex zeros of the Bessel functions I"(Z) and Iv(r)and their derivatives, Zh. vych. Mat. i mat. Fiz., 24, 10, 1497-1513, 1984. 8. KERIMOV M.K. and SKOROKBODOV S.L., Computing the complex zeros of a Bessel function of the 2nd kind and its derivatives, Zh. vych. Mat. i mat. Fiz., 25, 10, 1457-1473, 1985. 9. DELVES L.M. and LYNESS J-N., A numerical method.for locating the zeros of an analytic function, Math. Comput., 21, 543-560, 1967. 10. CARPENTER M.P. and DOS SANTOS A.F., Solution of equations involving analytic functions, J. Comput. Phys. 45, 2, 210-220, 1982. 11. DAVIESB., Locating the zeros of an analytic function, J. Comput. Phys., 66, 1, 36-49, 1986. 12. LI TIEN-YIEN. On locating all zeros of an analytic function within a bounded domain by a revised Delves-Lyness method, SIAM J. Numer. Analys., 20, 4, 865-871, 1983. 13. IOAKIMIDIS N.I. and ANASTASSELOU E.G., A modification of the Delves-Lyness method for locating the zeros of analytic functions, J. Comput. Phys., 59, 3, 490-492, 1985. 14. BO'ITENL.C., CRAIG M.S. and MCPBEDRON R.C., Complex zeros of analytic functions, Comput. Phys., Communs., 29, 3, 245-259, 1983. Translated by D.E.B.

U.S.S.R. Comput.Maths.Math.Phys.,Vol.27.No.6,ou.25-32.1987 __ Printed in Great Britain _

0041-5553/87 $lO.OC+O.CQ 01989 Pergamon Press plc

ITERATIVE REGULARIZATIONOF A METHOD FOR SOLVING OPTIMAL CONTROL PROBLEMS* S.V. SOLODQVA

A method of iterative regularization of the algorithm of constrained convex optimization is proposed for optimal control problems for linear systems of partial differential equations. Provision is made for "fanning out" the constraints-equationsat each iteration. Let t be the time, t=[O, Tj, where T>O fs given; x is the space variable, z~,y. where and denote by 1' the set in which X is a convex solid compacturnof R'. We put Q==XX(O, T], Given the weak compacturnU in the space of the boundary conditions are specified, %Q. controls UocLr(Q) (the weak compactum U can be given e.g., by the constraints u(L)=)' for any t=[o,Tl, where V is a compactum in Euclidean space). For each control u=U the trajectory y is defined as the solution of the linear system Ay(r.Q-/(2, t)+Bu(r.1).

(z,f)=Q\r.

(1)

and of the equations specifying the boundary conditions, Their form and number depend on the type of system and problem; for instance, for a parabolic system we may have the boundary condition (2) Y(r.O)'Y0(r)> t&Y. where y,=Lz(X) is given. In (l), A and B are given matrices of differential mappings, A: W-L,, WCL,. B:U-L:; f is a given function of Lz,see /l/. ~11 these equations will be written conventionally with the aid of a linear operator G:L,XU,XXX[O. T]--R as follows: G(y(.). cl(.), r,t)-o,

(r,t)Er.

(3)

To the set of controls u=u there corresponds the set of trajectories Y=Y, Y&(Q) (for the condition that a trajectory belongs to L:, see /l/). For instance, if

(4)

(the derivatives in (4) are understood in the where % are functions given in SX(0. T) generalized sense), which satisfy the conditions a,,fL”(XX(O.T)) and

YF,=R” :‘O, i: ?r,,(l.t)g,l,Zy(Ls,*i...i~,*).

i.,-l

lZh.vychfsl.Mat.mat.Fis.,27,11,1640-1650,1987