Numerical solution of a problem of analytic riemann conjugation

U.S.S.R+ Comput.Haths.Math.Phys.,Vo1.29,No.2,pp.39-45,1989

0041-5553/89 $lO.OO+O.OO 01990 Pergamon Press plc

Printed in Great Britain

NUMERICAL SOLUTION OF A PROBLEM OF ANALYTIC RIEMANN CONJUGATIONAL I.T. KHABI8ULLIN and A.G. SHAGALOV An approximate method is proposed for solving the regular Riemann problem for matrices on a circle. The method simultaneouslyyields a construction of the potential in the corresponding scattering problem. The degree to which it is well-posed and stable is presented. I. Introduction Riemann's factorization problem for a function r(li)deffned on a circle arises in many boundary-value problems of field theory, hydrodynamics, the theory of elasticity, etc. [1,21. Various approximate methods ware investigated in [l-31. We might mention that Riemann's problem can be rephrased in the language of singular integral equations of the Wiener-Hopf type, and the related approximate methods [3-6) essentially constitute one approach to the solution of Riemann's problem and are at present the most commonly used means to that end. Apart from the above-mentionedapplications, which are by now classical, there is a comparatively new field, connected with the inverse problem of quantum scattering theory and soliton theory [7-91. Two types of problem are particularly interesting, according as the conjugation contour is the real line or the unit circle I&l-l. In the first case, Riemann's problem is intimately connected with the construction of a "potential" q(x) in the Zakharov-Shabat system

[91

dyldx=i&y+q(x)

y

(1.1)

given the spectral scattering data as 1x1 -+m,in the second it is connected with the analogous problem for discrete equations of the following form (see [8], [lOI and 1111): (1.2) Y(~~l)~~(~~,~)Y(~). It can be shown I121 that Riemann's problem on the real axis is satisfactorily approximated by the problem on the circle (alternatively, (1.2) approximates (l.l)), and therefore one generally bases approximate methods of solution on the second type of problem. The latter may be phrased as follows: construct matrix-valued functions cp(J.)and $(A),hEC',havinga non-degenerate analytic continuation inside and outside the unit circle 1~1-f respectively, such that cp(h)r().)--g(h), IhI=% cp(O)--E, (1.3) where r(k)is a non-degenerate matrix-valued function on the unit circle, of orderIVXN,andE is the identity matrix. The specific feature of this new field of application of Riemann's problem is that in order to determine the potential fb, h) in (1.2), with support atm points, one has to solve m separate Riemann problems f8, 101. The technique of singular integral equations as applied in this context involves the solution of a sequence ofm Riemann problems and is therefore ineffective. In fact, however, there is no need to solve all m problems, since the solutions at successive points of the support satisfy recurrence rLlations. On the other hand, these recurrence relations may be used [131 to construct an algorithm solving the formal Riemann problem, which compares well in performance with the traditional method of integral equations. Since the recurrence relations are trivially related to (1.2), this enables us, while solving (1.3)‘ to construct a potential f(s,&)for the appropriate scattering problem; this is a major advantage of the method proposed in this paper. In particular, the method may serve as a basis for the numerical implementationof the "inverse problem" method for solving non-linear evolution equations (for alternative approaches see a4, 151). 2. Recurrence relations Let us assume that all the functions in (1.3), together with their inverses, belong to the Wiener algebra, i.e. they may be expanded in Fourier series which are absolutely convergent for IhI-1

with norm

:tZk.vyckis2.Mat.mat.Fiz.

r(h)=E&h'. -_

w 'P(h)=&+&", 0

,29,3,382-391,198S. 39

*' q(h)-_C*z "*

(2.1)

40

(2.2)

where Ir,lis the Euclidean norm of the matrix r,. Following the scheme developed in 1131, we define an integer parameter n by (2.3)

r(n,h)=~Pr(h)z-~, where z is the following diagonal matrix: z==diag(lrTI, . . . . hTn), yr> ...>mX.

(2.4)

We may assume without loss of generality that as=0. Consider Riemann's problem extended with respect to n: cp(n, JVr(n, I)-_g(n,

A),

(2.5)

cp(n. 0)--E.

IhI--l,

Assuming that there exists a solution to (2.5) at points n and n+l. we obtain q(n+lyh)zcp-'(n, h)==$(n+l,

h)~q-~(t~, h)--f(n,

A).

(2.6)

It can be deduced from the analytical properties of 'p-and rp-functions that f(n.h) is an entire function of h with a pole 7, at infinity. By Liouville's Theorem,f(n,h) is a polynomial in A of degree ~1. The structure of this polynomial is readily determined in the special case (2.7) whereE,andE,are

the identity matrices of orders m and N-m cp(n. k)-E+A(n)l+O(h*),

h-+0,

respectively. Defining

$(n, A)--B(n)+o(A-I),

(2.8)

h-c=, we substitute (2.8) into (2.6) and obtain

II’

hu (4 b(n)

where

E,+hv(n)u(n)

(2.9)

u(n)=A,,(n+l)=-B,,(n)B,,-l(n),

(2.10a)

v(n)--.4,,(n)=B1,(n+1)B,,-‘(nfl).

(2.10b)

Here A
hhh

hk’.

In the general case, the matrix z in (2.4) can be factorized: z=zIs?...z,,, wherez)(j=i,&...,pGy,)arefactors more integer variable:

of type(2.7) with E,.oforder mjXmj.

r(n,1.j)-ilzz.. . qr(n,

h)z,-I..

Let r(n.b)

depend on one

. zZ-*z,-l,

Obviously,r(n,h.O)=r(n,h)and r(% h,p)-r'(n+l,h). Consider the Riemann problem cp(n, h. j)r(lz.~. j)~~(n,h,j). As before, we obtain recurrence relations

cp(n,h,i+l)=fi(n,h)9(n,h,i)z,~,,. The product of p

j -steps is equivalent to one n-step:

~(n+i.X)=f(n.A)9(n, A)z-1 where f(n, h)=f,-,(n, 1.~ . ..fo(n. A). Similar relations are obtained for Q(n,h). The following result was proved in [13]:

(2.11)

Lemma Suppose there exists a solution of Riemann's problem (2.5) for scae n-n0 and the mGrer(1) is positive definite for all h, IhI==1: Then: a) Idet, r*‘(a,k) I>a,'a,'>O, j=l,2,..., N, where a,'are real numbers, det,ris the j-th principal minor of r, and B is defined in (2.8). It follows from part (c) that under the assumptions of the lemma the recurrence relations are well defined (i.e. the operations in (2.10) do not involve division by zero) and

31

make it possible to extend the solution to the pointsn=%*i.This fact provides a basis for a constructive proof of the following classical theorem (see the survey in [IS]: Theorem 1. If rer(h)>Ofor allh,Ihj-i, then Riemann's problem (1.3) has a solution in the class of absolutely convergent Fourier series. By Lemmas 3, 4 of [91 and part (a) of the above lemma, Riemann's problem (2.5) for sufficiently large (nIBI. provided all the exponents 71 in (2.4) are pairwise distinct. This implies that the minors of the matrices g*'are non-singular at i-m (lemma, part cc)). This condition, in turn, guarantees that the technique of continuation with respect to the parameter is applicable, and hence that there is a solution for n-0. This proves the theorem. Consider the scattering problem for a discrete equation of type (1.2), where the "potential"f(n,a) has the same structure in terms of h.as the function (2.6) and satisfies the condition +a, Proof

is

solvable

llf(~,h)--zltc=. z "_--00 Then, if there is no discrete spectrum, Eq. (1.2) has a solution Of the form cp(?z,h)z" for {Xi-cland of the form $(s,il)z' for jLl;ai.Thus, system (1.2) may be associated with a Riemann problem of type (2.3), (2.5) with matrixr(~)=~-*(O.~)~(O,~).~on the other hand, continuation of the solution of Riemann's problem from the domain(nl>ito all values of n via the recurrence relations (2.11) is equivalent to the determination of the potential in a certain scattering problem (1.2). As will be clear from the sequel, if n is sufficiently large in absolute value, an approximate solution of problem (2.5) can be explicitly determined. Therefore, the procedure described in the proof of Theorem 1 yields a method for solving Riemann's problem (1.3), a by-product of which is the potential of the corresponding scattering problem (1.2). For practical implementationof the method, we must prove that the procedure of "descending" from Isf~i to all n,implicit in the recurrence relations (2.11), is stable to initial perturbations of the solution. Theorem 2. Letg(s,h),Jj(n,X)be perturbed values of cp(n,h),g(n. A),sufficiently near them in the norm (2.2) and maintaining their analytic structure. Then for any e there exists fi implies such that !$(a,h)--cp(a, h)ll<6, II++, A)--g(n, a)&& Ijg(n+K, A.)-cp(n+K, hIlICe. ll$(n+K AI-$ (t&K’,

A)//+

uniformly

in K.

Proof. For sufficiently small perturbations, one can always ensure that the relation @(s,Vf(s,a)==~J+, a),

IaJ4,

(2.12)

holds with rei(h)>O, i.e. that the perturbed Riemann problem (2.12) is also solvable. Then there exists a constantc, independent of the perturbations, such that lti(s, a)-+,

a)lp3.

Using (2.3) and the fact that\jzX#=N for ihI=& we obtain ll%~+fi. a)-r(s+X;a)ll~~ll~(n,

a)-r(a,li)ll
uniformly in K. Thanks to the continuous dependence of the solution of Riemann's problem on the factorized function, there exists a constant C, such that IlW+K

a) -cp(n+K,

awC,ll~(n+~, a)-r(n+K, ~)lleqqvv.

An analogous estimate can be established for the function *(n+K,&). The proof by taking 6
now follows

3. Numerical method of solution We reduce Riemann's problem (1.3) to canonical form

lap-*,wt=4 bray, in which all principal minors of the conjugation matrix equal unity:

(3.1)

det,R(a)=i, j-1.2,..., N. This can be achieved through the transformation

R(awmwwa),

cp(abdfa)P(ahq(a)=-Yp.)Q(a).

(3.2)

The diagonal matrices P(h)-=diag(p 1,- . I, PN),

P(O)-E,

Q(a)=diag(q,,

. . . , qh.)

are analytic in the domainsjK/91 and j&l,i respectively, and their elements satisfy the relations pr(a)Pf(a) -da), P,(a)~detir(a)/detr-,r(a),

i--1, 2,. ..,N,

(3.3)

detar(a)=i.

Thus, reduction to canonical form requires the solution of N scalar Riemann problems. It is obvious that all results of Section 2 remain valid for (3.1), and the potential/(&h) in the recurrence relations for the function Q(n,a) and y(a,a) is unchanged.

42

It is well-known that the scalar Riemann problem has an exact analytical solution, but this approach is useless for numerical methods. Let us consider problems (3.3) (omitting the index i for simplicity) in the class of absolutely convergent series of type (2.1), and if (2.2). The lemma of Section 2 guarantees the validity of the estimate Ip(h)l-=m, rer(h)>Othen rep(A)>O, hence the scalar Riemann problems (3.3) are solvable. Truncated series P(h)-1+&h+ . ..+pYlin. 4(h)-q,+qJ.+ .. . +q&-=, will be called an approximate solution of (3.3) if they satisfy the relation (I+plh+. . . +PJ”) (P-‘fp-‘+,A+ . . . +p.,,kY+‘) =(QL+!&lh+. . . +qOh‘)+u(hJ(+L+o.

m

(3.4)

The solution of (3.4) is given by the equations (3.5a) n-1

4r -

z

Pi-nPn=-PA

P-k-Ilent

n-0

j-1,2,..., M,

Po=~, K-min(L-k,M),

k=o 91t..., L .

(3.5b)

It is evident from (3.5) that fland hLa define a Padd approximant [L/M] of the function h'p(h),where N P OJ = z p.h”. --L

(3.6)

Equations (3.5) are known as the Pad& equations [17]. Consequently, the well-developed methods of Padd approximation [17] can be used in order to find approximate solutions of the scalar Riemann problem. As L,M-+m the function (3.6) converges top(h).For the approximations 8, p to converge to the exact solutions P, q in the norm (2.2), it will suffice that p(h)20 for allh.lhl=land which is ensured by the condition p(h)>0 and part (b) of the lemma in indp(k)=O(see [31), Section 2. In addition, it was shown in 133 that this method is stable to perturbations of the Fourier coefficients of p(h) provided the perturbations are sufficiently small in the norm (2.2). If the matrix Riemann problem is reduced to canonical form, we can find an approximate solution of the extended problem: @(s,h)R(n,h)=Y((n,h)

(3.7)

for sufficiently large InlBl. Indeed, the matrixR(h)has a triangular factorization:

P-R=l.r+

(3.8)

where p- (or p+)are lower (upper) triangular matrices with ones along the diagonal. Multiplying (3.8) on the left by z" and on the right by P-",weobtain p-(W(s,V=p+(n), where p,(n)-Z"ptZ-'havecomponents

(111 cn))~J-~""l-"' (,b)U,

6 i-l,

2,. * . ,N.

(3.9)

The matrices p+ can be expanded in absolutely convergent Fourier series, and they may therefore be approximated with arbitrary accuracy by truncated series: .W

If ally are pairwise distinct, it follows fr% n~max(-(L+l)lr,-M/y},

(3.9) and (3.10) that when T-y2

(Ti-%),

p-(n) is a truncated series in positive powers of h. and p+(s) in negative powers. Consequently,p-(s) and p+(n)yield approximate solutionsQ,(n,h)andV(n,h)ofthe Riemann problem (3.7). Similarly, ifnamax (M/y,(L+1)Iy) the series p+-'(n) and p--'(n)also yield approximate solutlonsQ,(n,h)andY((n,li)of problem (3.7). Knowing an approximation solution of Riemann's problem for sufficiently large Inl, one can use the recurrence relations of Section 2 to determine a solution of the original problem (3.1) (correspondington-O). The process will be stable if perturbations of the solutionsQ,(n.i.),V(n, h)do not take the latter out of their analyticity class (see Theorem 2). This imposes certain restrictions on the numerical organization of the recurrent process: it must be organized in the space of sequences of Fourier coefficients. Thus, perturbations are applied only to coefficients of series (2.1) for functions UIand Ythat do not change their analytic structure. Organization of this type is also convenient,for computing f(n,A)

43

This method for solving matrix Riemann proolems of according to formulae (2.81, (2.9). arbitrary dimensionality NXIV has been programmed in FGRTRAN. The unknown potential f(n,h)is asymptotic to Z as Inl+aDwithYI% i,i-1, 2.. . ..N.This limits the class of potentials. The restrictions are due to the method of solving the extended Riemann's problem for Inl) l.Generalizlngthe transformation (3.2) to block-diagonal matrices P and Q,one can adapt the method to the case of equal%The algorithm outlined above is specially aimed at determining the potential f(s h) in t1.2) using Riemann’s problem. At the same time, it may also be used quite effectively to solve the formal Riemann problem (1.3) (or, what is the same, the corresponding system of convolution integral equations). When that is done the necessary operation count is -IV’JW (assuming L=M).Use of a standard projection method to solve the system of integral equations corresponding to (1.3) (cf. e.g. [3]) necessitates the solution of discrete WienerHopf equations with block-Toeplitz structure and also requires at least PM2 operations 1181. Below (Section 4) we shall compare the numerical results obtained by both methods. Returning to the inverse scattering problem, we remark that the determination of an I%$-point potential of Eq. (1.2) in our algorithm does not demand additional expenditure and can be accomplished in parallel with the solution of Riemann's problem, i.e. in Nsyz operations. However, if one chooses not to use the recurrence relations in determining the potential, one has to solve a series of M Riemann problems with total operation countNaP- a procedure which is certainly not efficient for large M'~alues. Table 1

T __

1

XI

-I

T -

-0.00026 -6.00030 ~~~~~ -0:00017

t%E

o:ooooo OdOOOO O.OOMO

O.OMOO 0.400# m2~ _

-!%%I+ 0.34281 0.28569 0.22886 0.17142 0.11428 0.05714

0.22857

0.17142 O.ii428 0.057i4 ~~

-0.39999 0.39993 -0.40000 OBQQQ3 -0.40660 0.39992 -0.09002 -0.00003 -0.OOc03 I;:E; -0.06605 -0.90005 -0.oooo9

~~ ~~ O&Xl0 O.ooooO 0.00666

-~~~~ -0&?023 -0.ooO18 -0.00912 -0.00023

Table 2

l-

T

M-8 I

II

O.oooOoO

I-

I

-0.OooO23 o.oocQoo 0.066085 0.060095 0.059809 0.059996 0.060019 -0.019921 -I6$M;; -0.02Oc91 -0.020059 0.02cOO0 0.020012 E%EE o:otXl33 -0:06#43 -0069630 ~:~ ~~~~ O.~i6 O.ooO655 0.~ ~~:~~ -0.665184 -0.006097 -0.906627 0.069225 ~:~ 0.066604 -0.066167 -0.990289 o.ooooo2 _ 8:EEil 0.000005 %E%Z -0.006051 :,0:pd; -0.cm9013 -0.006694 -0.06031i -0.000017 -0.06677T

4. Discussion of the numerical results Let us consider the inverse scattering problem for (1.2) in the case of 2 x 2 matrices. Suppose the support of the potential /(n,h) is concentrated at gfl points, i.e. f(n,A)Zzfor Kand /(n,h)=sotherwise. Then the scattering matrix for Eq. (1.2) is [8, 101 n=G, I,..., S(.Q=Z-~-‘~(K)~(K-~)

and the corresponding conjugation matrix [9] is

. .f(o),

44

In our numerical computations we took K-12. The exact values ofv(n).u(s)are presented in column I of Table 1; they were the basis for the calculation of S(h)and r(h).Theknown values of r(i) were then used to determine f(n,h) The computed values of v(n)and u(n)are listed in column II of Table 1. The parameter values were chosen as L=AI-i9.Comparison of the exact and computed values of v(n).Ir(s)clearly shows that the potential for the above parameter values can be determined to within -0.1%. As M increases the accuracy is improved. As already remarked, the method may also be used successfully to solve the formal Riemann problem (1.3). As an example, we solved the 3 x 3 problem with conjugation matrix r(h)-m-'(h)g(h), where the elements of the matrices q and $ are as follows: (p,,=6,,,f0.02[)I((m+in)+h2(irn-n)+hJ(im+n)+

(4.la)

+h'(m+i(n-Z))], ~,,=~~,,+0.02[h-'(m-n)+ih-2(m-n)+ +h-3(m+n-3)+h-'(im-n)],

(4.lb)

m, n- 1, 2,3.

The functionsq)(A) and*(h) are obviously an exact solution of the problem. Comparison of the exact solution (4.1) with the approximation obtained in numerical computation enables one to estimate the characteristic features of the method and its accuracy. All computations were run for two parameter values, M-9 and 15 (assumingL--M).As a typical example, Table 2 (column I) lists the values of the coefficients of the series (2.1) for the functionrecp,l(h).The rapid increase in accuracy asM.increases is obvious. Column II of the same table presents the solution of the same matrix Riemann problem, but by another method - the "projection method [3] (using the algorithms of [18]). The criterion for comparing the methods was the accuracy of the results for the same computer time. As is evident from Table 2, the method proposed here yields more accurate results. On the other hand, projection methods are generally reckoned to be among the most efficient direct methods for solving integral equations 1191, (see also [S], where the projection method was compared with a version of the quadrature method). It is a reasonable conclusion, therefore, that the performance of our method as applied to solving the formal Riemann problem (1.3) or the corresponding system of integral equations compares well with the class of direct methods. The author is indebted to A.B. Shabat for his interest and for useful discussions. REFERENCES 1. 2.

3. 4. 5.

6. 7. 8. 9. 10.

11. 12. 13. 14. 15.

NOBLE B., Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations. Pergamon Press, Oxford, 1958. SHESTAPALOV V.P., The Method of the Riemann-Hilbert Problem in the Theory of the Diffraction and Propagation of ElectromagneticWaves. Izd. Kharkov. Gos. Univ., Khar'kov, 1971. GOKHBERG I.TS. and FEL'DMAN I.A., Convolution Equations and Projection Methods for Solving Them, Nauka, Moscow, 1971. IVANOV V.V., Theory of Approximate Methods and its Application to the Numerical Solution of Singular Integral Equations, Naukova Dumka, Kiev, 1968. GABDULKHAEV B.G., Finite-dimensionalapproximations of singular integrals and direct methods for solving singular integral and integro-differentialequations. Itogi Mat. Tekhn. Ser. Mat. Anal., 18, Moscow, VINITI, pp. 251-307, 1980. BELOTSERKOVSKIIS.M. and LIFANOV I.K., Numerical Methods in Singular Integral Equations, Nauka, Moscow, 1985. ZAKHAROV V.E., MANAKOV S.V., NOVIKOV S.P. and PITAEVSKII L.P., Soliton Theory: Method of the Inverse Problem, Nauka, Moscow, 1980. TAKHTADZHYAN L.A. and FADDEEV L.D., The Hamiltonian Approach in Soliton Theory, Nauka, Moscow, 1986. SHABAT A.B., The inverse scattering problem. Differents. Uravn., 15, 10, 1824-1834, 1979. NOVOKSHENOV V.YU. and KHABIBULLIN I.T., Asymptotic forms of the solution of certain differential-differenceequations integrable by the method of the inverse scattering problem. In: Partial Differential Equations. Proceedings of S.L. Sobolev Seminar Novosibirsk, Inst. Mat. Sibirsk. Otdel. Akad. Nauk SSSR, pp.lO-53, 1981. ABLOWITZ M.J. and LADIK J.F., Nonlinear differential-differenceequations. J. Math. Phys., 16, 598-603, 1975. KHABIBULLIN I.T., Discrete approximation of the inverse scattering problem. In: Integrable Systems, BFAN SSSR, Ufa, pp. 68-87, 1982. KHABIBULLIN I.T., On the problem of linear conjugation on a circle. Mat. Zametki, 41, 3, 342-347, 1987. TAHA T.R. and ABLOWITZ M.J., Analytical and numerical aspects of certain nonlinear evolution equations. J. Comput. Phys., 55, 192-230, 1984. VYSLOUKH V.A. and CHEREDNIK I.V., Simulation of the self-action of microwave impulses in fibre light-conductorsby the inverse scattering problem technique. Dokl. Akad.

45

Nauk SSSR, 289, 2, 336-340, 1988. GOKHBERG I.TS. and KRBIN M.G., Systems of integral equations on a ray withg kernels dependent on the difference of the arguments. Uspekhi Mat. Nauk, 13, 2, 3-72, 1958. 17. BAKER G.A. and GRAVES-MORRIS P., Pad& Approximants. Addison-Wesley, Reading Mass,, 1981. 18. VOEVODIN V.V. and TYRTYSHNIKOV E.E., Computations with Toeplitz matrices. In: Computing Processes and Systems, No. 1, Nauka, Moscow, pp. 124-266, 1983. 19. VERNAL' A.F. and SIZIKOV V.S., Integral Equations: Methods, Algorithms, Programs, Naukova Dumka, Kiev, 1986. 16.

Translated by D.L.

U.S.S.R.Comput.~aths.Math.Phys.,Vo1.29,No.2,pp.45-54,1989. Printed in Great Britain

0041-5553189 $lO.OO+O.OO 01990 Pergamon Press plc

APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS WITH CONJUGATIONSig V.D. DIDBNKO and V.N. MATSKUL The approximate solution of equations with additive continuous operators is considered. Necessary and sufficient conditions are established for convergence inL,,+cp
-wf&)f-

B*(t) q(7) B*(t) cpq)od7+ f -dTfA,(t)rp(t)+ni p r-t ni pI 7--t

(1)

where-&,(t), B,(t)(m=O,l), f(t)aregiven functions andcp(t)isan unknown function. SIE of type (1) and their boundary-value problems in the theory of analytic functions are widely used to solve applied problems (see, e.g. [l]-[41). Various aspects of the approximate solution of SIE with conjugation (the case Al(t)=O& (t)-0, R,(t, z)=O) are discussed in [5-81. But if one of the coefficientsA,(t),B,(f),K((t,7ldoes not vanish on I',thenthe operator corresponding to Eq. (If does not remain homogeneous on multiplication by complex numbers. In connection with the difficulties that arise in this context, projection methods have been applied by some authors (see, e.g. [9-101) not to the equations themselves but to associated systems of SIE without conjugation. This approach produces an unjustified increase in the orders of the systems of algebraic equations that must be solved in the approximation process. In addition, the auxiliary systems of SIE obtained in this approach involve coefficients with partial indices, for whose determination no methods are as yet known. Silbermann has proposed a new approach to the study of projection processes for solving equations with continuous linear operators; this approach has formed the basis of the invest.igation Eli-141 of several methods for the approximate solution of SIE, Wiener-Hopf equations, SIE with simple Carleman shifts, etc. The present paper is devoted to an extension of the results of [ll-141 to additive continuous operators, which has made it possible to investigate the convergence of the reduction and collocation methods, applied directly to SIE with conjugation of type (1). The criteria thus established (see Theorems 4 and 5) state that these projection processes (see Theorems 4 and 5) are applicable if and only if a certain auxiliary operator and the operator corresponding to the original equation are invertible. It should be mentioned that the invertibility of such operators has been studied before 13-41, and the results, together with our Theorems 4 and 5, enable one to derive sufficient conditions for convergence (see Corollaries 1 and 21. Moreover, our methods, when used in numerical work, yield systems of algebraic equations (adequate to a prescribed degree of accuracy) whose orders are half those of the systems used in [9-101.