A N N A L E S S O C I E T A T I S M A T H E M A T I C A E P O L O N A E S e r i e s I : C O M M E N T A T I O N E S M A T H E M A T I C A E X V I I ( 1 9 7 3 )

K O C Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P E A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )

**Y. **^{K}^{o m k o v}**(Texas Tech.)**

Review oî the Kantorovic variational method

**This article reviews some recent advances in the development of **
**Kanatorovic technique with a particular application to the problem of **
**shafts of constant cross section.**

**0.1. Introductory disscusion. This is basically an expository article **
**reviewing the developments of the so-called Kantorovië method of obtaining ****solutions of equations which arise from a variational problem. Throughout **
**this article one particular problem will be used to illustrate the numerical **
**procedures. Perhaps the title of this article should have been changed **
**to : “Becent results in the development of Kantorovic method with a par**

**ticular application to the problem of pure torsion of elastic shafts of constant **
**cross sections”. My aversion to long titles is the only excuse I have to **
**offer.**

**Before we discuss the review of various generalizations of Kantorovic **
**procedure, because of the expository nature of this article, it is proper **
**to offer the basic concepts and definitions which will be used in the varia**

**tional problem discussed here.**

**The main physical problem illustrating the techniques of [5] and**
**[7] is closely related to Dirichlet’s problem: Let Q be a bounded (and for ****simplicity, simply connected) region of the Euclidean plane E****2****-,x,y the ****cartesian coordinates of E2. The boundary dQ of Q consists of a union ****of a finite number of smooth arcs. Q will denote Q u dQ. f(s) is a given ****function of the arc length s on dQ.**

**The Dirichlet’s problem consists of finding u(x, y), such that**

*d 2u * *d 2u*

*V2u*** =** **---+ — —**

*doc2 * *d y***2** **in Q,**

**wlao ' = /(«).**

**It is easy to reduce the problem of a pure torsion of a shaft to the Di**

**richlet’s problem.**

**106** У. K o m k o v

**We have**

**д2Ф ****д2Ф**

**дх2 ****ду2**

**Ф)дО —**** О**
**Replacing Ф by a function:**

**—2 ** **in Q ,**

**®2**** + У****2****х**
**f = ф -1---m****we have**

**(0.1)** ^{ьэ}**il**

**(0.2)** **v» =**

**where**

**^*2**

**-ir2 ** **on сШ, **
**ж2 + 2/2***

**The related variational problems consist of minimizing **
**(0.3, ** **r w = / J { g ) 2 + g ) !J « ,**

**subject to (0.2), or of minimizing**

**(0.4)** **! ( ? ) = / /** **дФ****\ 2****1дФ****\ 2****Л ** **,**

**\_\dx) + Ы**** “ И * " * ***
**subject to Ф = 0 on dû.**

**The following comments are immediately applicable to equations **
**of the type (0.1), (0.2).**

**Operator ( — V2) is positive definite, and therefore symmetric. For ****operators of this type the solution of the Dirichlet’s problem**

**Р2Ф = / ** **in Û,****Ф|да = 0 ****(on dû)****is unique.**

**(This is trivially proved by assumming to the contrary the existence **
**of two solutions Фх, Ф2, Фх ф Ф2. Then**

**р2фх_ Р2фа _ о ** **in **
**or**

**Р2(Ф****х****- Ф 2) = 0 ** **in Q,****Фх — Ф2 = 0 ****in d û ,****and**

**< - Г , (Ф1- Ф >) , Ф1- Ф , ) а = - / / ( ( Р ^ - Ф ^ ^ - Ф , ) ) ^ = 0 .**
**Ü**

*Kantorovië variational method* 107

**By positive definite property of the operator ( — F2) this implies****( Ф г - Ф 2) **^{= 0}

**is Q. Alternately, if we prefer classical analysis, we conld nse Green’s ****formula and reach the same conclusion, obtaining a contradiction.)**

**Before we make some glib statements about the solutions and the **
**implications of this classical theorem, we should make certain of what **
**is exactly meant by “a solution”. It is immediately apparent that if we **
**consider only the problem of (0.4) we need only to wory about the functions **
**Ф(х,у) which posesses weak derivatives of order one which are square ****integrable, that is of the Sobolev class W\, while in problem (**^{0}**.**^{1}**) we **
**seek a solution among functions posessing continuous second derivatives. **

**In each case we are dealing with a subset of the Hilbert space L****2****(Q), ****and in fact with a dense subset of L****2****(Q). If we follow Mikhlin [**^{8}**] and **
**define HA to be the completion of the space of twice differentiable ****functions in Q, which vanish on dD, we formulate both problems in S A. **

**The Hilbert space HA has the inner product****{ u , v } = f f [( — V****2****u) 'V]dxdy.**

**Q**

**While in the case discussed in papers [5], [7], [**^{1}**] the problems **

**«concerning Ъг convergence of (0.4) are trivial, in the general case, when ****Ф{х,у) is an arbitrary continuous function, and dQ is an arbitrary piecewise ****smooth Jordan curve, there exist classical examples of non-convergence **
**of the integral (0.4). Hadamard offered the first example of the function **
**which is harmonic in the unit disc, which coincides on the unit circle with **
**a continuous function, but which has a divergent Dirichiefs integral **
**(0.4) (see Mikhlin [**^{8}**], § 7), while Lebesgue gave an example of a region Ü, ****which can be used as a counterexample to the usual uniqueness and exi**

**stence theorems for the Dirichlet’s problem. (Obviously, the boundary **
**of such region cannot satisfy the hypothesis of being a bounded curve **
**consisting of a piecewise smooth arcs with bounded curvatures.)**

**For the elementary discussion of difficulties which may arise in the **
**exterior Dirichlet’s problem see for example [11], § 32.**

**A brief review of the Rayleigh-Ritz and Galerkin methods. It is well **
**known that the Bayleigh-Bitz method may be regarded as a special **
**case of the Galerkin’s method. (A more detailed explanation of this remark **
**will be given in this paragraph.) However, the Galerkin method is far **
**more general, although in some cases it coincides basically with the **
**Bayleigh-Bitz method.**

**First let us point out the differences between the two techniques.**

**Let I be a differential operator. If: L u —f = **^{0}** in Q, Ли **^{= 0}** on**

108 Y. К о т к о у

**d û , is the variational system resulting from the extremal problem****à [/(« )] = 0,**

**where I is a linear functional, then choosing some set of functions**
**Ф^х), Ф****2****(х) ... Фт(ос),**

**satisfying the homogeneous boundary conditions ЛФг — 0, i = 1, 2, ..., mf **

**\ve consider the linear form:**

**u(m){%) = а****1ф1**** (x) + сс****2****Ф****2****(х) + ... + атФт(х),**

**and seek the values of the coefficients cq, a2, ..., am, which extremize ****the functional I(um{x)). The obvious necessary conditions are:**

*d l _ d I * *__ * *d l _ _*

**даг ****da****2****dam**

**This is basically the Bayleigh-Bitz technique.**

**An analysis of convergence and a study of a number of important **
**cases can be found in the original, paper of Bitz, and in the papers of **
**Krylov and Bogoliubov.**

**In 1915 Galerkin proposed a technique which appears similar to **
**the Bayleigh-Bitz technique. However, it does not depend on the varia**

**tional formulation, and therefore can be used to obtain approximate **
**solutions of non-self adjoint systems. It was applied with good results **
**to non-conservative mechanical systems.**

**As in the Bayleigh-Bitz method we choose some functions Фг-(а?)г **
**i = 1 , 2 , ..., m, which satisfy the boundary conditions, and consider ****the linear form:**

*m*

**ит(я) =****г=1**

**We assume that the solution of the problem u(x), satisfying Lu —f ****is an element of a Hilbert space H, and so are the functions Ф*, the function ****Lum{x) and the functions f{x). <(/, g} will denote the inner product in H. **

**The Galerkin method can now be summarized as follows:**

**We choose a subspace G of И, which is spanned by elements ^****1****{x) ****... ipm(x) and determine the coefficients a{ from the system of m equations ****with m unknowns :**

**<Lum, ****= </, yq>,**

**(Lum^ ****1****р2У — 'Cfi tyi) ) • • • j ****ФтУ ~ (fi Wrn) •**

**The convergence of Galerkin’s method has been studied by Keldysh **
**(see [4]), where he proved strong convergence in the particular case of **
**Dirichlet’s problem.**

*Kantorovic variational method* 109

**2.0. ** **Some basic definitions related to iterative procedures. The concepts **
**given below are well known, but have sometimes been given varied inter**

**pretations. To avoid confusion we shall refer as much as possible to the **
**following basic sources: [2], [10], [14].**

**Let Q be a region of n-dimensional Euclidean space, let x denote ****a vector (a point) in Û. Let f(x) be a function whose domain contains Ü, ****and whose range is contained in Ü, and xx, x2, xz, . . . be an infinite sequence ****of points in Ü, such that:**

**X****2****f (xi) j • • • ? ****x n ****f *** i x n —* l ) ?

**• • • )****then we shall say that {ж J, i =**^{1}**,**^{2}**, . . . , is an iterative sequence defined ****by the procedure (xx,f). xx shall be called the starting, or initial choice. **

**The function /: Q -> Ü will be called the iterative function. If there exists ****a point xn, such that f( xn) = xn, it will be called a fixed point of the ****iterative function (or of the iteration).**

**D****efinitio n** **2 . 1**** (See Ostrowski [10]). A point ! 0e Ü is called a point ****of attraction for the iterative function /, if there exists an open region ****V a D, containing £0, such that for any starting point xxe V the cor**

**responding iterative sequence of points xx, x2 ... will converge to £0.**

**Obviously **^{£ 0}** must be a fixed point of the iterative function /.**

**D****e fin it io n** **2****.**^{2}**. A point £e Q is a point of repulsion of the iterative ****procedure /, if there exists a neighborhood F c Й, containing **^{£ 0}** such **
**that for any xx Ф **^{£ 0}** e V the corresponding iterative sequence xx, x2, ... **

**will fail to converge to | 0, unless for some integer n, xn = £0.**

**We shall make some routine observations which will be used in the **
**subsequent discussion. Let x{ be real (complex) numbers.**

**Ee ma r k ** ^{1}**. If f(x) is differentiable in some neighborhood of £0, ****and lim xn = £0, then**

* n*—>00

**/'d o ) = lim**

* n*—► OO

**X n J T****1**^{ — £ o}
**Xn ****£o**

**Ee ma r k 2. If f(x) is differentiable in some neighborhood V of i 0, ****and xn is an iterative sequence of points in V defined by (xx,f), then ****if I/'do) I < **^{1}**? the point **^{£ 0}** is a point of attraction, and if I/'**^{d o ) }**I > **^{1}** the **
**point f 0is a point of repulsion of the iteration (see [10], p. 39-40).**

**Ee ma r k **^{3}**. Let ****£0**** be a point of attraction of the iterative procedure. **

**Consider the replacement of the iterative function ** **#**
**f(x) ****by F{x) = x - f ( x ) , ****or by y(x) = (x - £****0****) - f ( x ).**

**We obviously have F'{x) = 'tp'(x) = l —f'(x). If |. f d 0)| < 1, then ****0 < F'(x) < **^{2}**, in some neighborhood of | 0.**

**I**

п о Y. K o m k o v

**Hence in that neighborhood F(x) is monotone increasing. (The for**

**mula for y>(x) appears to be useless, since in general we don’t known the ****value of | 0. However, if we use an approximate value, obtained by ac**

**celerating the original convergence it can be proved that ip(x) will again ****converge to £0). We consider the function [C-F(x)) with the choice of the ****constant G: 0 < C < ****to make certain that (CF(x)) is positive increasing, ****and 0 < (CF(x))' < 1. Hence zero is a point of attraction for the iterative ****function (C-F(x)) and the corresponding iterative sequence is monotone ****increasing.**

**We remark that our definition of a point of attraction coincides with **
**the definition of some authors of a point of partial attraction. We shall ****say that | 0 is a point of full atraction in Q if it satisfies definition 2.1 and ****the region V in that definition can be taken to be of all Q. A similar ****change in definition 2.2 will define a point of full repulsion in Q.**

**D****efinition**** 2.3 (Ostrovski). An iterative procedure (x , f ) is uncon**

**ditionally (conditionally) stable in Q if:**

**(i) / maps Q into Ü,**

**(ii) there exists only one (at least one) fixed point £0e Q of the iter**

**ative procedure {xl f f), for any starting point хг е Q,**

**(iii) the (each) fixed point | 0 is a point of full (partial) attraction **
**for the iterative function /.**

**3.3. Review of the Kantorovic method. We shall attempt a very basic **
**explanation of Kantorovic idea.**

**We wish to minimize a functional I (f(xx, x2, ..., xn)) subject to some ****constraints (or boundary conditions) imposed on / ( %. . . xn).**

**The Rayleigh-Bitz technique, and the Galerkin technique both start **
**by selecting certain collection of functions**

**Фг{х), ****Фт(х) ****(where x = {хг, x2, ... , xn))**

**and then optimizing with respect to the constants ai . the approximate ****solution of the problem of the form:**

*Ш*

**(1) ** **/m(y) = ^ Ч Фг(я),**

**г=1**

**while making certain that f m(x) obeys the required constraints and (or) ****boundary conditions. The functions Ф^х) are guessed a priori, and this- ****guess makes the application of this technique a matter of experience **
**and intuition.**

**Kantorovic has suggested that a certain degree of arbitrariness of **
**a priori choice of Фг{х) may be eliminated by assumming the coefficients ****alf a%, .. . , am to be functions rather than constants. There is of course**

*Kantorovic variational method* **111**

**a penalty i napoaed by this method. We have to solve a system of differ **
**ential equations instead of a system of linear algebraic equations, to **

**obtain the appropriate forms of ax(x ), a****2****(x) , . . . , am(x). Eor obvious reasons ****it is most convenient if at each step we only have to solve one ordinary **
**differential equation with one unknown function. Hence Kantorovic **
**imposed the condition ax = ****a****2**** = а****2****(хх), • ••> am = am{xi)- This****implies that the arbitrariness of choice has been eliminated in the direction **
**of the xx axis, but very little has been done to help the solutions in the ****directions of remaining axes.**

**A casual look at the form (1) convinces us that no greater generality **
**is gained by assumming Ф{{х) = Ф{(хх, x2, ****xn), then by assumming**

**= ** **2****> *^з? • • •} ^n)у ***** = 1, 2** **,** **m.**

**In the obvious generalization of the procedure, we would first optimize **
**a{(xi), i = **^{1}**, **^{2}**, ..., m, until a fixed point of the procedure is hopefully **
**obtained (within the round off error), then somehow we attack the variable **
**x2, until the result of the iterative procedure exhibits a fixed point beha**

**vior, and go on to finally establish a form:**

*m*

**fm M ****^ ****‘ ' “S ** *****

**г=1**

**We either quit at this point, or else this will be regarded as the comple**

**tion of the first cycle. (This is indicated by the superscript (1)). We iterate **
**the whole procedure once again by commencing to regard а^(х), j > **^{1}**, **
**as known (a prior chosen) functions, discard ali(x1) and optimize I [ f m(x)) ****with respect to ** **(xx) regarded as the new and unknown functions, esta**

**blish fi^(x), etc.**

**Various variants of such procedures have been suggested in a liter**

**ature, which includes a paper published in 1933 by T. E. Schunk (refer**

**ence [**^{1 2}**]) which somehow escaped the attention of the mathematical **
**community until quite recently (1966 is the first reference), and which **
**roughly suggests the above technique for dealing with the problem of **
**buckling of the cylindrical shells. In 1964, the author of this article has **
**published a report which suggested a procedure similar to Schunk’s but **
**with additional simplifying steps (see [7]).**

**Subsequently two papers of Kerr [5], and [**^{6}**], and a paper of An- **
**derssen [**^{1}**] have extended these ideas, and in particular the paper of **
**Anderssen has for the first time considered the problem of stability of **
**such procedures in the general case. In this paper we shall review one of **
**the particular procedures suggested in [1], [5], [**^{6}**] and [7].**

**First I would like to make a comment that neither of the papers**
**[5], [**^{6}**], [**^{7}**], attempted to settle the basic theoretical problems of stability**

**112** Y. K o m k o v

**of the method, or of the theoretical error bounds, or of the error propa**

**gation in case of the iterative procedures suggested by Kerr in [6].**

**All of the above papers contain some numerical results of experi**

**mentation with the method, and all use the same physical problem, namely **
**the pure torsion of a rectangular shaft. This choice is natural. We consider **
**the equation V%0 = constant in a'region Q c E2. Ф(х, у) is the Prandtl ****stress function (the exact equation is Р2Ф = —2), subject to the condi**

**tion Ф. = 0 on dû, where Q is some simply connected compact subset ****of the Euclidean plane, whose boundary consists of a union of a finite **
**number of smooth arcs. I have stated that the choice of this problem **
**for numerical experiments is natural, since this is the simplest form of **
**a partial differential equation of the elliptic type where the extension **
**of the Kantorovic method is non-trivial and where the accuracy of any **
**method can be immediately checked, since the exact solutions of the **
**problem are known for some simple choices of the region Q.**

**4. Modifications of Kantorovic technique. We shall consider the **
**classical variant of Kantorovic as presented in [3] and examine the iterative **
**procedures suggested by Komkov [7], Kerr [5], [6], Anderssen [1].**

**In [7] the author suggested the following iterative procedure:**

**Let us consider a variational problem of minimizing a functional **
**I(u, (x)) over a given class of real functions u(x), whose domain is Q <= En, ****u(x) satisfying some homogeneous boundary conditions on dû. Bather ****then contenting ourselves with generalities let us examine the procedure **
**on the specific case of minimization**

**I(u(x, y)) = j j****n****where Q is a rectangle:**

**dw\2 ** **I du **

**_\ dxj ****dy****àuj dxdy,****a ^ |ж[, ****b ^ \y\.**

**We guess an approximate solution to be u(x) = {y^ — b^X^x). Now we ****proceed in the spirit of the Galerkin method by first computing the residue **
**function r}(x,y) = F2u + ****2****, and then orthogonaliling r}(x,y) with respect ****to a chosen subspace of L Z(Q).**

**In [7] Komkov chooses this subspace to consist only of the original **
**function (y****2**** — b2), which reduces the result to the one which would have ****been obtained by the Bayleigh-Bitz technique:**

**+ь**

**(4.1) ** **' / { № - Ъ ' ) Х ,Дх) + 2 Х****1****{х) + Щу*-Ъ*)4у = 0.**

*- b*

**After preforming the integration we have:**

**(4.2)** **XAx)****5**

**262** **= 0.**

*Kantorovic variational method* 113

**The solution, which is supposed to satisfy the boundary conditions:**

**X x{x) = 0 ****at ж= ± a**
**is given by**

**cosh**

**X x(x) =**

## (¥)

о о Л ( - г )

**and the solution of the first step of the iterative process is:**

**Ui(x,y) ****(y****2****- b 2)**

**cosh****/ Xxx**

**\ b ~****cosh** **Xxa**^{- 1}**where**

**к - A -**

**At this instance we observe the symmetry of the original problem and **
**symmetrize the solution:**

**ux{ x, y) = ax{y****2****- b 2)**

**cosh**

## (¥)

**cosh**

## (¥)

^{+ a }

^{2}^{(x}^{2}^{ — a2)}**cosh** **К У****cosh** **Xxb****ax and a**^{2}** are now computed by the Eayleigh-Eitz method:**

**d l ( u x) ****d l { u x)**

**dax****даc****= **^{0}**.**

**If we wish to stop at this point it turns out that the solution is point- **
**wise within 7 **^{° / 0}** of the exact solution in the case a — b = 1. If we wish ****to continue the iteration we choose:**

**Щ{х,У) = ****y) + p****2****(y****2****- b****2****)****2****X****2****(x)**

**and obtain additional accuracy by repeating exactly the first iterative **
**step, mixing the Kantorovic-Galerkin and the Eayleigh-Eitz technique. **

**In [7] Komkov has shown only that the value of the functional ****1****{щ) ****is a monotone decreasing function of the index i independently of the ****original choice. He did not examine the strong convergence, or the uncon**

**ditional stability of the process. The problem of error propagation and **
**error estimate was also left unanswered. Numerical experimentation **
**with the technique gave extremely favorable answers. It appeared that**

8 — R o c z n i k i P T M — P r a c e M a t e m a t y c z n e X V D

114 . V. K o m k o v

**even a clumsy choice of the first guess resulted in an extremely good ap**

**proximation after only one iterative step.**

**A similar technique was suggested in [5] by Kerr, who examined the **
**same physical problem**

**Р2Ф + 2 = 0 in Q, ****Ф = 0 on dQ.**

**Again the same first guess for the rectangular shape Ü was:**

*У) = / И ( у 2- б 2).*

**This is treated by the Kantorovic-Rayleigh-Ritz technique, which now **
**leads to the minimization of:**

**+ b****3****f****2****(x) — 2 b****3****f(x)\dx.**

**by:**

**(4.3) **
**where**
**(4.4)**

**The solution of Euler’s equation for this variational problem is given**

*ш* = ^ y [ cosW' - cosh(Aif ) ] ’

**h = d**

**Hence the first approximation is the Kantorovic function:**

**(4.6)**
**where**

***i.o ***= f i ( v ) 9 o ( y ) ,*

**(4.6) ** **g****0****(y) = b2**

**This is rewritten in the form: **

**f l****,0**

**(4.7) Ф**1г0** =****— TcoshAi — cosh ** **]1 f l —**

**ii L ** **\ « /J L** **(/l,0**

**coshAj I ** **\ «ж /II ** **\b**

**and the optimizing function дг(у) is sought to minimize 1(ФП)****ф1Х = j^coshAi - cosh**

**We have to solve the Euler’s equation **
**d****2****9i ****/Гх'2**

**dy*****91 — -4.Ц (^i)***

*Kantorovic variational method* 115

**The solution is:**

**9i(V)****5**^{1 1}**c o sh (^ ^ l +j521sinh**
**where**

**b**

**9****10** =

**A**^{1}**[sinh(**^{2}**A1) — **^{2}**^ ]**

**+ **^{0 1 0}**)**

**1/2**

**4a**^{2}** I cosh**^{2}** ^**

**8**** a**^{2}** ( Xx cosh Xx — sinh Xx **

**^ (sin !^ ^ — ****2****Аг)****Bo **^{1}** =**

**sinh**^{2}**A1+ |**

**0****.**
**-Bn = -srio/coshy1}**

**The next assumption is:**

**Ф**^{21}** = M x ) ; 9 i ( y ) t****where /**^{2}**(ж) satisfies the Euler’s equation:**

**d2h**

**dx****2**^{(A2}**la)**** ^{/ 2}** —

**- ^ 2 1**( y i ) *

**The procedure now can be regarded as an iterative system**

**7 i = Щ ( V i ; « ) , ****^ = а К ^ Х ^ г ; Ь ).**

**The problem of convergence of the approximate (iterative) solutions **
**Фг-(а?, у) to the exact solution Ф(ж, у) of equation (0.1) was never settled ****in Kerr’s paper. However, he did prove that if the procedure converges, **
**then the final form of Ф(х, у ) will be unique, and independent of the ****initial choice of g****0****(y)-**

**In a recent paper of Anderssen [**^{1}**] the arguments summarized below **
**prove a lack of unconditional stability of the iterative procedure as sug**

**gested by Kerr. We follow Kerr’s technique and define the iterative pair:**

**7n = g{K) =**^{A 4}

**\ 2 ** **-|l/2**

**-)) ; ** **,**
**resulting from his equations:**

**(4.8)**

**(4.9)**

(A

**An+i ** **f ( y » )**

**yi(sinh2yt- — 2уг-)**

**г+ l /**

**yï**

**iV‘****cosh2^ —****3 sinh yi****éyi****Х{(&т]****12****Х{ — ****2****Х{**

**4 a****2**

**cosh**^{2}** A-** **ЗзтЬ2Яг-**
**4Я^**

116 V. K o m k o v

**Let the iterating function**

**h{a\ ****q****) be defined by the relation, ****h{o\ ****q****) = e2(sinh****2e — 2 g)/or(gcosh2g — f sinh2^ + £e).**

**Anderssen ****[ 1 ] ****has observed that his iterative process has ****oo ****as a fixed **
**point, and, moreover, that oo is a point of attraction of Kerr’s procedure.**

**If e (A) is differentiable, then a sufficient condition for Я = ****oo ****to be **
**a point of attraction is:**

**lim**

A—mo

**G < 1.**

**A straightforward differentiation of the composite function f(g{ty) ****and the use of l’Hôspital’s rule results in the conclusion:**

**(4.10)** **lim** **df-g(K)**

**dl n****= 0.**

**Hence +oo is a point of attraction (provided it is a fixed point), **
**and Kerr’s procedure can not be unconditionally stable.**

**To prove that +oo is a fixed point of the iterative procedure it is **
**sufficient to note that the maps:**

**fig(*n)) = 4 + i ****and ** **g-(f(yn)) = yn+i****are both continuous, that [0, ****oo) ****is maped into itself, and**

**f-(g( °°)) = g-(f(°°)) = °°***

**In a way one could anticipate the problems which arise in Kerr’s procedure, **
**or in any procedure inductively defined by:**

*j * *m*

**(4.11) ** *щ* ** = / 7** *ап ]ы* * f j* ** 4 - i***ы ,*
*k***= 1 ** *k**—**j**+1*

**(4.12)**

**since the ability to separate variables is the underlining assumption **
**of the method. In the case of the torsion problem the assumptions are **
**even stronger. Since at every step of the procedure we reduce the problem **
**to the solution of a second order linear differential equation with constant **
**coefficients it is clear that we have restricted ourselves to the class of so**

**lutions which can be written as products of exponential functions of each **
**variable. To offset this obvious disadvantage we would have to introduce **
**a variant of the method. A technique tried by the engineers (in somewhat **
**simpler situation) would introduce an iteration:**

**^ n +1) = < +1(%-i) W ,,/«?-i1)(^+i) + Cf V****+ 1 ( ® 1 1**** a?2, ..., CDj, oc}+2, ..., œn)},**

*Kantorovic variational method* **117**

**where q****)1+1**** is an a prori chosen function. In the corresponding Bayleigh- **
**Bitz technique Timoshenko chooses **^{93}** to be a polynomial in the remaining **
**variables, and optimizes C by the Bayleigh-Bitz method. This could be ****incorporated in a manner similar to [7] in an attempt to obtain some **
**functions which can not be written in the form (4.12).**

**References**

**[1] R. S. A n d e r s s e n , ****A ***stability analysis for the extended K antorovic method applied *
**to the torsion problem (to appear).**

**[2] P . H e n r ic i , Error propagation fo r num erical methods (SIAM Series), N ew York ****1966.**

**[3] L. V. K a n t o r o v i c and Y. L. K r y lo v , A pproxim ate methods in higher analysis, ****Noordhoff, Groningen, 1958.**

**[4] M. Y. K e ld y s h , O n B . G. GalerJcin’s method fo r the solution o f boundary value ****problems, Faraday Translations, Report F -1 9 5 , NASA, Langley, 1964, 56 p.**

**[5] A .D . K e rr , A n extension o f the Kantorovic method, Quart. Appl. Math. 26 (1968), ****p. 219-229.**

**[6] — and H. A le x a n d e r , A n application o f the extended K antorovic method to the ****stress analysis o f a clamped rectangular plate, Acta Mech. 6 (1968), p. 180-186.**

**[7] Y. K o m k o v , A n iterative method fo r obtaining decreasing sequences o f upper ****bounds o f functionals, Math. Res. Centre U niv. of W isconsin Tech. Report # 5 3 4 , **

**N ovem ber, 1964.**

**[8] S. G. M ik h lin , The problem o f a m in im u m o f a quadratic functional, Holden ****D ay, San Francisco 1965.**

**[9] ** **— The num erical perform ance o f variational methods, Noordhoff Groningen, 1969.**

**[10] A .M . O s t r o v s k i, Solutions o f equations and systems o f equations, N ew York 1966.**

**[11] I. G. P e t r o v s k i i , P artial differential equations, Sanders, Philadelphia 1967.**

**[12] T. E. S c h u n k , Z u r KnechfestigTceit schwach gékrum m ter zylindrischen Schalen, ****Ingeneur Archiv 4 (1933), p. 394-414.**

**[13] S. T i m o s h e n k o and J. N . G o o d ie r , Theory o f elasticity, second edition, N ew ****York 1951.**

**[14] J. F . T r a u b , Iterative methods fo r the solution o f equations, Prentice-H all, Series ****in A utom atic Computation, Englewood Cliffs, 1964.**