IMPROVEMENT OF AN APPROXIMATE SET OF LATENT ROOTS AND MODAL COLUMNS OF A MATRIX BY METHODS AKIN TO THOSE OF CLASSICAL PERTURBATION THEORY


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1 IMPROVEMENT OF AN APPROXIMATE SET OF LATENT ROOTS AND MODAL COLUMNS OF A MATRIX BY METHODS AKIN TO THOSE OF CLASSICAL PERTURBATION THEORY By H. A. JAHN {University of Birmingham) [Received 7 October 947] SUMMARY A method is described for simultaneously improving all the latent roots and modal columns of a given matrix, starting from a given complete set of approximate modal columns. It is considered that the method will be useful as a final step in any iteration process of determining these quantities. The method is illustrated by a numerical example. The modification needed when two or more of the latent roots are coincident, or nearly so, is very briefly indicated. The fundamental formulae are akin to those of classical perturbation theory, the corresponding formulae of which, for the special case of a Lagrange frequency equation, are given for convenience in the Appendix.. The fundamental equations LET a complete, linearly independent set of approximate modal vectors of a square matrix A of order n be given. Then operating on any x< r 0) by the matrix A gives a vector Ax^ which can be expressed as a linear combination of the vectors x^\..., zf \ If we write this linear combination as WK () n where 2' denotes 2 > then, since the a^0' are approximate modes, the AJ. ' 8 8= 8#r will be first approximations to the latent roots and the a*.*' will be small quantities. An improvement on the original mode will then bo given by
2 32 H. A. JAHN so long as none of the differences Aj. ' A^ are very small (see 6), for we have < ) 4 o) + 2' 3^B or.44 X> = A^^'f terms of second order. Having found this first approximation to the latent roots and the modes, we may now take into account the secondorder terms, writing giving A< 2) as the second approximation to the root, and (3) as the second approximation to the mode. Or, in general, for the pth approximation p* «2 g^w (5) " for the determination of A^ and 4 P) given 4"~ ) ( r > a ^,,n). 2. Determination L the coefficients. First method  To find the coefficients A,, a n at any stage we have the linear vector equation \.x r \^.'a n x a = Ax r, (7) i s or, in terms of components, 8 where (Axjf are the components of the vector Ax T. For a given value of r these are n linear equations for the n unknowns
3 Writing IMPROVEMENT OF A SET OF LATENT ROOTS 33 the solution of these equations may be written as (9) *ln X 2n X n 'nl ^ln X 2n x rl,n \A x r)n D.a T8 = () We see that to find the improved value of the rth latent root at any stage we take thematrix whose columns are the modes obtained from the preceding stage and replace the rth column by the components of the vector Ax r. Then the improved value of the root is the value of the determinant of this modified matrix divided by the determinant of the original modal matrix. The coefficients a^ in the expression (6), viz. + for the improved modes are obtained in the same manner by replacing the ath column of the matrix of the modes a^?" * by the components of Ax r, evaluating the determinant of the matrix so modified, and dividing again by the determinant of the original modal matrix. 3. The equations in matrix notation It may be noted that the essential computational element in the method described here is the calculation of the inverse, or adjoint, of the given approximate modal matrix. For if 7(P) (2) is the given approximate modal matrix and we write (3)
4 34 H. A. JAHN then comparison with equation () shows that a«) (4) i.e. A x is determined by (5) where X< > is the adjoint of a and D o = Having found A x and hence A*. ' (r = n) in this manner, we find the improved modal matrix a* ' from 3*» = a* 0^, (6) where J?! is derived from A x as follows: «2n as is seen by comparison with equation (2). Or, in general, (7) (8) (9) where B v is obtained from A p in the same manner as B x is obtained from A v i.e. This is illustrated in the following example. 4. Application to a symmetrical thirdorder matrix (Elementary Matrices (l)f, p. 30) By the matrix iteration method the following roots and modes of the matrix A shown below have been found: I" 8209] , x x = 2 5 llj [" r ' , x,= , 0 0 \ = , A 2 = 2652, A 3 = t See reference () at the end.
5 IMPROVEMENT OF A SET OF LATENT ROOTS 35 Suppose the following approximate modal matrix z< 0) had been found for which the adjoint X m, determinant D o, and Aaf are as shown: ro ll ' D o = \3f i\ = Then we have for the first approximation J *3 giving A^' = , A > = 2653, A^x) = , roots. For the improvement of the modes, with Aj»>A?> = 855, we find A$ A > = , J ] I Proceeding to the second approximation we find , J J l» = 80000, for the > A*" = 664, J = ,
6 36 Hence Pi 2) A H. A. JAHN r J giving for the second approximation to the roots \M = , A^ = , Ag» = , = Connexion with Rayleigh's principle and second method There is another method of calculating A,, a rs (s ^ r) from the vector equation Kx,+ 2'"r.x. = M. (?) 8 which brings out the connexion with Rayleigh's principle (2) for the determination of latent roots given approximate modes. Taking scalar products with the vectors x x (I = l,...,n) we obtain (Ax,, as,) = K(x r, x,)+ 2' «*(*.> *») («=!,...,»). (20) These, for a given value of r, are n linear equations for the n unknowns A,, a rs (s = l,2,...,r l,r+l,...,n). Putting for brevity and writing A rl = (2) (22) (23) That (22) and (23) are identical with (0) and () may be seen as follows. Since the S r, = (x T, x t ) are the scalar products of two vectors we have the matrix relation "8 U S 2.. S ln "l RCi, x 9.. x,»~]fx 2 x m ^ nl Zm.a;In where a; rl, aj^,..., a; rn are the components of the vector x r. Consequently A = D 2. (25) ii "nl (24)
7 IMPROVEMENT OF A SET OF LATENT ROOTS Similarly we have the relations 37 X /y < ^ f* If* * ( M *V \ It* XX *2 x n2 x ln x ll *2 *rl,l l' aa Wl x r+l,l X nl (Ax r ) n x r+hn.. x nn \ 8 2 Krl "In,.(26) J nl K,rX S nr + where (Ax r ) lt (Ax r ) 2,, (Ax r ) n are the components of the vector Ax r. So also #2 x s+l,x X nil X nn \ \.Xxn X 2n X 8X,n \AX r ) n X s+ln.. X n 82 $X,8X A r , 8, s n o l8+ (27) n2 8 ng _ A rn Consequently equations (0) and () may be obtained from (22) and (23) by dividing by D. The equations (2), (22), (23) have the advantage over the corresponding equations (9), (0), (), that, for a symmetrical matrix A (for which the exact modes are orthogonal), the matrix A will be approximately diagonal and also, so long as none of the latent roots are very small, the terms A rr = (Ax r,x r ) will be larger than the nondiagonal terms A rl = (Ax T,x { ). Neglecting the smaller terms we have then from (22) Arr. or " r t rr  (x r,x r )' which is Rayleigh's approximation for the latent roots. For the improvement of the modes, we have, for 8 < r, , 82 82,8+ (28) A.a rs = 8g+l,2 8, 8,2.. 8^.! A^ $rj+x Kr K Kx 8 n2.. 8^.! A rn 8 n^+.. 8 nr.. 8 n, (29)
8 38 H. A. JAHN where the large terms are underlined. Retaining only the largest terms in the" expansion of this determinant, we find Aa r8 =.4 rg 8 u S g _i,8i 8 8+i.8+i Kh+ Since to the first approximation we have we find, on division by A, A = SnS^. The same result is obtained for «> r. Thus in all cases A rs 8 rs (30) To this approximation we are able to improve the roots and the modes without the solution of linear equations. For the iteration process of approximation we have thus (3) (32) The improved mode is given by yiv) afpi). which becomes, since >_A</> = ^ 2 ^ _  g.<j» (6), (33) 2 > <*» 8g«P ' which gives the improvement in the modes in terms of the coefficients rr rr U(l>) (34) ( Sg" = (4" '. a*" *). (35) A'^" = (^x<"», «?"«), (36) derived from the modes 7ff~ x) of the preceding approximation. '
9 IMPROVEMENT OF A SET OF LATENT ROOTS 6. Application of the second method to a thirdorder matrix Applying these formulae to the example of 4 we have with p , I I ] , J 39 I" = giving XV = , = 2652, Ai, ' = , A*. ' = , off = r.tt) off = 8 2 2»8 8<»>8 8$ 8 From these we find, with 334 ( i2 8$' A (0) S(0) = , = , ^(0) g(0) % z ^(0) g(0) = , = , ol ax = , A(0) g(0) = A (0) 5(0) A^Aa ' = 854, *»*» = , the following improvement of the modes (see 3), Af> = 663, ro Lo J o ] I
10 40 H. A. JAHN It is seen, as was to be expected, that the method described in this paragraph converges slightly less rapidly than that employed in 4; it has, however, the distinct advantage that the solution of the set of n linear equations is avoided. 7. Application to the Lagrange frequency equation The approximate orthogonality of the approximate modes which form the basis of the method of 5 applies only when the matrix A is symmetrical. The method is, however, easily extended to a Lagrange frequency equation of the form (\M+K)x = 0 (37) or (M+U)x = 0, (38) where M is the kinetic energy matrix, K the potential energy matrix, and U = M~ l K, (39) if the usual extended definition of orthogonality in terms of the kinetic energy is made, i.e. x r, z a are denned to be orthogonal when For, starting from the equation (x r,mxj = o. (40) and taking scalar products with Mx lt we have \{x r,mx l )+ 2,'aJXvJUx,) = (Ux r,mx t ) = (M x Kx r,mxi) = (Kx r,x,). (42) The formulae of 5 will then hold, assuming again that none of the latent roots are very small, if we replace A rl = (Ax n x t ) by K Tl ={Kx n x l ) (43) and 8 ri = (x T,x,) by M,, = {x T,Mx t ) = (Mx r,x t ). (44) Consequently A*> is the improved value of the root, and k the improvement in the mode. These may be compared with the corresponding formulae of classical perturbation theory given in the Appendix.
11 IMPROVEMENT OF A SET OF LATENT ROOTS 4 8. The case of coincident roots The modification needed to the approximation procedure when two or more latent roots are coincident, or nearly so, is outlined briefly below. For the sake of simplicity take the case where the first two modes correspond to approximately equal roots. Let y^\ y 2 0) be the approximate modes corresponding to these roots, whilst x^0) (r = 3,..., n) are those corresponding to the other roots. Then we will have relations of the form I > (47) 8=3 I W, (48) = 3 4 ) 2/ ( 2 0) + I' <4V * s 0)  (49) s=3 Here the coefficients /4V, MV, /4V > /4V. M ' will au< ^e of the same order of magnitude, but the coefficients a will still be small quantities. As in classical perturbation theory (3), before proceeding farther, we first diagonalize the matrix Let the transformation of y^\ y^ which does thiti (at least to the second order of small quantities), be given by *i 0) = c 2/ o) +c 2 y 2 ) ) (50) 4 o) =Wi O) +c 222 /< >, (5) and let A^, 4 ' be the resulting diagonal elements and consequently the first approximation to the first two latent roots. We shall then have Improved modes may then be found as before, viz. 8=3 2 (52) = A^w+Ja&'a;" 0  (53) 8=3 (54) 8=3 n Am4 0). (55)
12 42 H. A. JAHN and the process repeated if necessary. The coefficients A,, a rs can be determined by any of the methods outlined in the preceding paragraphs. 9. Conclusions It is considered that the method of improving the latent roots and modal columns of a matrix described here will be useful as a final step in any iteration process of determining these quantities. The special method of 5 and 7 becomes invalid when one or more of the latent roots are very small and applies further in its present form only to a symmetrical matrix or to one which is the product of two symmetrical matrices. It should be noted that the fundamental formulae of this report are akin to those of classical perturbation theory (2,3), as shown in the Appendix. The method might in fact be described as that of perturbation theory in reverse, for in that theory one starts from the known latent roots and modal columns of a given matrix and derives those of a matrix differing slightly from the original matrix, whilst here one starts from approximate modal columns and latent roots of a given matrix and deduces improved values of these for the same matrix. This analogy formed the basis of the original derivation of the formulae given here. REFERENCES. FKAZEK, DUNCAN, and COLLAR, Elementary Matrices (Cambridge Univ. Press, 038). 2. RAYLEIGH, Theory of Sound. 3. COTXBANTHILBEBT, Methoden der Mathematischen Physik, Bd. (93). APPENDIX The Corresponding Formulae of Classical Perturbation Theory for the Case of a Lagrangian Frequency Equation Let M and K be the known kinetic energy and potential energy matrices of a given conservative system. Let x r, A, denote the known modal columns and latent roots of the system, so that \ = a>j, where a> r is the circular frequency. Then we have (\ r M + K)x r =0, (58) or Kx T = XrMx T. (59) Denoting the scalar product of two modal columns by &,>*.) = 2 x H x, it (60) where x ri are the components of the modal column or vector x r, then the orthogonality with rospecfr to the kinetic energy of the modes may be expressed by (Mx r,x,) = i^8 r,, (6) where S r, is the usual Kronecker symbol 8 r, = 0 for r jt s and = for r = s. (62)
13 IMPROVEMENT OF A SET OF LATENT BOOTS 43 The classical perturbation problem consists in rinding the changes A\, Ax r in thelatent roots and modes of the above system due to small changes AM, AK in the kinetic or potential energy matrices. Putting, for the modified system, A^^ + AA,, (63) K = *r+z'c t,x,, (64) t we have {(A r +AA r )(M4 AM)+K + AK}(x r + Jf c r,x,) = 0. (65) a Equating the small terms of first order to zero we find (\M\K) 2'c M x. + (A r AAfilfAA r + AiC)a:. = 0. (6b Taking scalar products with x r, we obtain from this vector equation, since 80 that (Mx r,x s )  Mrrh,,, (Kx r,x t )=^A r (Mx r,x 3 ) = X r M rr 8 ra = K rr S r,, (67) K n = \M rr, (68) A r (AM) rr iw; T AA r +(AA') rr = 0. (69) from which we have A r = A r + AA r j ^ jjt (7) since A,. = On the other hand, we have M' n K TT +(AK) rt f J M, r Thus the modified latent root is given by K' n _ A 'H; T  M TT +AM TT ' which may be compared with equation (45) of the text. The equivalent form A^= (W^jAJQg (74) AT Mrr AJT for the relative change in the latent root, derived from (70), is useful. The fact that the introduction of small coupling terms (AM) rl, (AK) T, between the different normal modes of the original system has no effect on the frequency to the first order of small quantities may be noted. To find the coefficients c TI determining the changes in the modes we take scalar products of the vector equation (66) with x, (s ^ r). This gives c r,(\ r M t,+k, i )\(AM) rt +(AK) r, = 0, (75), A r (AAr) r.+(alq r,, C " = KMK < 76 > or, since A, = (AK) n (AM), K rr M u K,,M rr Krr M u (77)
14 44 IMPROVEMENT OF A SET OF LATENT ROOTS giving, for the modification to the mode, which may be compared with formula (46) of the text. The relation < u M a K,, M, Ht KM,, M u may be noted (compare equation (33)), so that the c r, are of the form (78) (79) c T, = KK (compare equation (2)). The case of coincident latent roots of the unmodified system and the extension of the calculations to the second approximation (secondorder perturbations) will be found treated in ref. (3) or in any standard textbook of quantum theory, although the formulae are given there as applied to a complex hermitian matrix. (80)
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