ANALYTICAL MATRIX INVERSE CALCULATIONS, APPLYING PREDICTIVE COORDINATION

IFAC DECOM-TT 2004 Automatic Systems for Building the Infrastructure in Developing Countries October 3 - 5, 2004 Bansko, Bulgaria ANALYTICAL MATRIX INVERSE CALCULATIONS, APPLYING PREDICTIVE COORDINATION 1 T. Stoilov and K. Stoilova Institute of Computer and Communication Systems - Bulgarian Academy of Sciences Acad.G.Bonchev str. Bl.2, 1113 Sofia, tel. (359 2) 979 27 74; e-mail: todor@hsi.iccs.bas.bg Abstract: The research derived analytical relations between the components of a square matrix and it s inverse. The relations have been worked out, applying a hierarchical approach and non-iterative concept of coordination. The analytical relations have been assessed by computational performance. It has been proved that for large-scale matrices, these relations are computationaly efficient. Copyright © 2004 IFAC Key words: hierarchical systems theory, coordination, matrix inverse calculations 1. INTRODUCTION Some numerical algorithms involve solving a succession of linear systems Ax=b, each of which slightly differs from its predecessor in matrix A* . Instead of solving each time the equations from scratch, one can often update a matrix factorizations of A to find the new inverse A*-1 having A-1 . Thus the calculation of matrix A-1 , which is the inverse of a square matrix A , is a key stone in the implementation of real time control, optimization and management algorithms and online decision policies. Practically, three types of factorizations are under way by numerical calculations: LU factorizations (Fausett, 1999), QR decomposition and SVD-singular value decomposition (Flannery, 1997). The peculiarities of the LU, QR and SVD factorization techniques define the computational efficiency in finding A-1 . It is worth to find methods for evaluating an inverse matrices A* , which slightly differ from the initial one A. The new A* can skip one or several columns/rows of A, or to have only few new components aij . Thus the utilization of the inverse A-1 or several of its components in finding the new inverse A*-1 can speed a lot the computational efforts. Attempts in finding results for matrix inversion are given in (Strassen, 1969). For a given matrix A with a structure (a) where a ij could not be only scalar components, but 1 This research is supported in part by the Information and Communication Technology Development Agency, Ministry of Transportation, Republic Bulgaria, contract o ИД 14/1.7.2004. also appropriate matrices, the inverse matrix (b) can be calculated. a a α α12 (b) A = 11 12 (a) A −1 = α = 11 a21 a 22 α 21 α 22 If it is noted −1 R1 = a11 R 2 = a21 R1 R3 = R1 a12 (1) R 4 = a21 R3 R5 = R4 − a22 R6 = R5−1 the components αij of the inverse matrix A-1 =a are found according to the equalities: α12 = R3 R6 R7 = R3α21 α 22 = −R 6 (2) α21 = R6 R 2 α11 = R1 − R 7 In these relations the “inverse operator” occurs only twice on matrices, which have lower dimensions in comparison with the initial A. Thus it is worth to perform the inversion of A not by factorization, but applying relations (1) and (2), especially for the case of large scale N of A. This research presents an appropriate result in finding the inverse matrix of an initial one A with dimension NxN (N is a big value). Here A is a symmetric, A=A T . Analytical relations are derived between the components a ij of A and α ij of a=A-1 . These relations have been worked out, applying a hierarchical approach and non-iterative concept of coordination (Stoilov and Stoilova, 1999). The analytical relations have been assessed by computational performance. It has been proved that for large N the application of the derived relations is computationaly efficient. An optimisation problem, concerning a resource allocation problem, has to be solved 1 Q1 0 x1 x1  (3) min  x1T xT2 + R1T R2T  x 2 0 Q x x 2 2 2   i bi |mi xni ; Ci | m x1 ; i A = A1 0 i Q= Q1 0 0 Q2 0 i R= ; are (4) 0 a2 m0 xn1 A2 |( m0 + m1+ m2 ) xn2 = 0 b2 m1xn1 m2 xn1 b2 Q a m1 xn2 T = C1 m2 xn2 C2 ( −1 2 * A1Q R1 + A2 Q R2 + T ) * ) −1 −1 T 1 1 1 (7) −1 T 2 2 AQ A = A Q A + A2 Q A = a1Q1−1 a1T + a2 Q2−1aT2 m xm = b1Q1−1 a1T m1 xm0 b2Q2−1 aT2 m2 xm0 0 0 a1Q1−1b1T m xm a2 Q2−1b2T m xm b1Q1−1b1T m1 xm1 0m2 xm1 0m1 xm2 b2Q2−1b2T m2 xm2 0 It can be seen that the matrix symmetric one. 1 = α 11m0 xm0 α12 m0xm1 α13m0xm 2 a1Q1−1R1 + a2Q2−1 R2 + t α 22 m1xm1 α 23m1xm 2 α 31m2xm0 α 32m2xm1 α 33m2 xm2 = α 21m1xm0 b1Q1−1 R1 + C1 b2Q2−1R2 + C2 0 AQ −1 AT (8) 1 2 0 1 2 α determines that it is a symmetric one. After transformations the solution of the problem (3) is derived analytically x  a1Q1−1R1 + a2Q−2 1R2 + t   T T α1 1 α1 2 α1 3  = −Q R + Q  a1 b1 b1Q1−1R1 + C1  α α 0 2 1 2 2 − 1   b2Q2 R2 + C2   x opt 2   = −Q2−1R 2 + Q2−1  a2T   opt 1 −1 1 1 −1 1 (9) * Using (4) and (5) it follows T b1Q1−1R1 + C1 0 3.1. Matrix evaluation −1 * Comparing the notations in (7) and (8) it follows α 2 3 = 0, α 3 2 = 0 . The way of defining the matrix ) x o2 p t = −Q2− 1R2 + Q2− 1 A2T A1Q1− 1 A1T + A2 Q2− 1 A2T −1 1 0 m1 xm2 b 2 Q2−1 b2T m2 xm2 0 m2 xm1 α 31 α 32 α 33 ( m +m + m ) x ( m +m + m ) . t m0 xn2 Taking into account the notations R Q 0 ; x R = 1 ; A = A1 A2 ; x = 1 , (5) Q= 1 0 Q2 R2 x2 the analytical solution of (3) according to (Stoilov and Stoilova, 1999) is −1 x1o p t = −Q1− 1 R1 + Q1−1 A1T (A1Q1−1 A1T + A2 Q2−1 A2T ) * (6) ( −1 a1Q1− 1R1 + a 2Q2− 1R2 + t 3. ANALYTICAL SOLUTION BY GOAL COORDINATION ( b1Q1−1 b1T m1xm1 T 2 m 2 xm0 α 11 α 12 α 13 ( AQ −1 AT )−1 = α = α 21 α 22 α 23 This problem is solved by non-iterative goal coordination (Stoilov and Stoilova, 1999) and by predictive coordination (Stoilov and Stoilova, 2002), where the solution can be presented analytically. * A1 Q1−1 R1 + A2 Q2−1 R2 + T a 2 Q 2−1b T2 m0 xm2 where the notations are applied R1 ; R2 A2 |(m0 +m1 +m2 )x (n1xn2 ) ; a1 A1 |(m0 + m1 +m 2 )xn1 = b1 −1 2 , where the matrices dimensions for i=1,2, xi |n x1 ; Qi |n xn ; Ri |n x 1 ; ai |m xn ; t |m x1 ; i b1Q1−1 a1T m1 xm0 = a 1Q1−1 b1T m0 xm1 0 b2Q2−1 R2 + C2 b2 x 2 = C2 i (AQ −1 AT )− 1 ( AQ − 1 R + T ) = a 1Q1− 1a 1T + a 2Q 2−1 a T2 m0 xm 2. PROBLEM DEFINITION a1 x1 + a2 x2 = t b1 x1 = C1 3.2. Determination of ( AQ −1 AT ) −1 ( AQ −1 R + T ) b2T α 11 α 31 a1Q1−1 R1 + a2 Q2−1R 2 + t  α 12 α13  b1Q1−1 R1 + C1  0 α 33 −1  b2 Q2 R2 + C2  4. ANALYTICAL SOLUTION BY PREDICTIVE COORDINATION Due to the constraint a1x1 + a2 x2 = t it is not possible to perform a decomposition of the initial problem (3). Applying predictive coordination this constraint can be decomposed to the equations , (10) a 2 x2 = y2 a1x1 = y1 satisfying the condition of resource limitation (11) y1 + y 2 = t Applying (10) the initial problem (3) is decomposed to two lower scaling subproblems 1  1  min  x1T Q1 x1 + R1T x1  min  x2T Q2 x2 + R2T x2  2  2  2 is a a1 x1 = y1 a 2 x2 = y2 , b1x1 = C1 and additionally it holds y1 + y2 = t . b2 x2 = C 2 (12) The analytical solution of the first subproblem (12), using (Stoilov and Stoilova, 1999), is [ x1opt = −Q1−1 R1 − A1T ( A1Q1−1A1T ) −1( A1Q1−1R1 + T1) ] −1 T Ai Qi Ai , the To use the lower rank o f matrices definition of subproblems (12) has to be done by rejecting the zero rows in matrices A1 and A2 . Hence in the modified subproblems will present only the corresponding valuable components as follows a1 _ a ; A1 = b1 ⇒ A1 = 1 b1 0 a2 y1 _ y1 y1 = C1 ⇒ y1 = C1 0 = y2 _ y . y2 = 0 ⇒ y2 = 2 C2 C2 a2 ; A2 = 0 ⇒ A2 = b2 b2 = _ The modified subproblems (13) have lower dimension than (12), received by direct decomposition 1  1  min  x1T Q1 x1 + R1T x1  min  x2T Q2 x2 + R2T x2  2  2  __ _ _ A1 x1 = y1 _ A1 = a1 ; b1 _ A2 = _ A2 x2 = y2 _ y y1 = 1 ; C1 a2 ; b2 (13) _ y2 = y2 C2 The analytical solutions of subproblems (13), according to (Stoilov and Stoilova, 1999), are _ _ _ _ _  opt −1  −1 −1 xi = −Qi  Ri − AiT ( Ai Qi ATi ) −1 ( Ai Qi Ri + y i )   i = 1,2 or  x1 ( y1 ) = −Q R1 − a1T  −1 1 a Q −1a T b 1 1−1 1T b1Q1 b1 T 1 a1Q1−1b1T b1Q1−1a1T −1 a1Q1−1R1 + y1   b1Q1−1R1 + C1   It is used the notations a1Q1−1a1T 1 424 3 m0 xm0 a1 Q1−1b1T 1 424 3 m0 xm1 b1Q1−1 a1T 1 424 3 b1Q1−1 b1T 1 424 3 m1 xm0 m1 xm1 β 11 { =β = m0 xm0 β 21 { β 12 { m0 xm1 , β22 { m1 xm0 m1 xm1 β 11 b1T β 21  x2 ( y2 ) = −Q2−1 R2 + Q2−1  aT2  bT2 (14) β12 a1Q1− 1R1 + y1  (15)  β 22 b1Q1−1R1 + C1  γ 11 γ 12 a 2Q2−1 R2 + y2  (16)  γ 21 γ 22 a2 Q−21 R 2 + C 2  T where γ, accordingly, is a symmetric one, γ 12 = γ 21 γ = γ{ 12 m0 xm0 m0 xm2 m 2 xm0 m2 xm 2 γ{ 21 γ{ 22 a 2 Q2−1 aT2 a 2 Q2−1b2T 1 424 3 1 424 3 = m0 xm0 −1 T 2 1 2 m 0 xm2 −1 T 2 2 2 m2 xm0 m 2 xm2 bQ a bQ b 1 424 3 1 424 3 y 2opt is done by the 5. COORDINATING PROBLEM Applying substitutions of x1 ( y1 ) and x2 ( y 2 ) in the initial problem (3) the arguments of the optimization problems become the resources yi . Thus the coordinating problem is analytically derived 1 T  x ( y )Q x ( y ) + R1T x1 ( y1 ) +   2 1 1 1 1 1  min w( y ) = min   y∈S y 1 + x T ( y )Q x ( y ) + R T x ( y ) 2 2 2 2 2 2 2 2  2  , or S y ≡ y1 + y2 = t min {w( y) = w1( y1 ) + w2 ( y2 )} (17) y1 + y 2 = t The relation xi (y i ) is an inexplicit function and it can be approximated in Mac-Lauren series at point yi =0 x1 ( y1 ) n x 1 = x10 n x 1 + X 1n xm y1m x1 (18) 1 1 1 0 0 where β11 β12 a1Q1−1 R1 (19) β 21 β 22 b1 Q1−1 R1 + C1 xi10 = −Q1−1 R1 + Q1−1 a1T b1T T X 1n1xm 0 = Q1−1n1xn1 a{ 1 β 11m0xm 0 T b{ 1 β 21m1xm 0 n xm n1 xm 0 1 (20) 1 and x 10 is the solution of subproblem (13) for y1 =0. Respectively, for the II subproblem it holds (21) x2 ( y 2 )n x1 = x20 n x1 + X 2 n xm y 2 m x 1 2 2 0 0 where where the matrix β is a symmetric according to its definition and xi ( yi ), i = 1, 2 are γ{ 11 The calculation of y1opt , coordinating problem. 2 −1  x1 ( y1 ) = −Q1−1 R1 + Q1−1  a1T  If the optimal values y1opt , y2opt are known , after their substitution in (15)-(16), the solution of the initial problem (3) can be easily evaluated x1opt = x1 ( y1opt ), x2opt = x2 ( y opt ). 2 −1 . x2 0 = − Q2−1R2 + Q2−1 a2T T X 2 n 2 xm0 = Q2−1 n 2 xn2 a 2 { n 2 xm0 b2T γ 1 1 γ 1 2 a2 Q2−1 R2 γ 2 1 γ 2 2 b2Q2−1 R2 + C 2 (22) γ 11 m0 xm0 T b {2 γ 21 m xm n xm 2 0 2 2 Substituting (18) in (17) the component w1 (y1 ) of the coordination function is analytically determined 1 T w1 ( y1 ) = (x10 + y1T X 1T )Q1 ( x10 + X 1 y1 ) + R1T ( x10 + X 1 y1 ) 2 T Because the components x10 Q1 X 1 y 1 and T T y1 X 1 Q1 x10 are equal, the coordination function is w1 ( y1 ) ≡ or 1 T T y X Q X y + y T X T Q x + y T X T R (23) 2 1 1 1 1 1 1 1 1 10 1 1 1 w1 ( y1 ) ≡ 1 T y1 q 1 y1 + y1T r1 , 2 where q1 = X 1T Q1 X 1 ; Applying (32) and (33), relations (31) become y1opt = − q1−1 r1 + q1−1 ( q1−1 + q −2 1 ) −1 ( q1−1 r1 + q −2 1 r2 + t ) (34) r1 = X 1T Q1 x10 + X 1T R1 . The same relation holds for the 2d subproblem. The functions wi (yi ) can be expressed in the terms of the initial problem (3). Respectively T T q1 = β 11 β 21 T = β 11T β 21 a1Q1−1b1T β 11 = −1 T b1Q1 b1 β 21 a1Q1−1 a1T −1 T 1 1 b1Q a −1 1 −1 1 −1 1 1 −1 1 1 (24) a1 Q a β 11 + a Q b β 21 T 1 T 1 T 1 T 1 b1 Q a β 11 + b Q b β 21 Using the notations of matrix β according to (14): −1 T 1 1 −1 T 1 1 a a b 1Q 1Q 11 1 4 24 3 a1 4 24 3 β { m0 xm0 b1Q1−1a1T 1 424 3 m1 xm0 m 0 xm1 β 12 { m 0 xm0 1 = m 0xm1 b1Q1−1b1T β 21 1 424 3 m{ xm m1xm1 −1 β 22 { m1 xm1 0 I{ 0{ m0 xm0 m 0 xm1 0 { {I m1 xm0 m1xm1 , (25) From this matrix equation the following equations are valid (26) a1Q1−1 a1T β 1 1 + a1 Q1− 1b1T β 2 1 = I m xm 0 0 b1 Q1−1 a1T β 11 + b1Q1−1b1T β 21 = 0 m1 xm0 . After a substitution of (26) in (24) it follows T q1 = β 11 { m0 xm0 I{ , T m0 xm0 β =β 21 11 { { 0{ m0 xm0 m0 xm1 (27) m1xm0 and β11 is a symmetric and square. By the same way q2 = γ 11 (28) Using the notations (19) and (20) for ri it holds T T r1 = X 1T (Q1 x10 + R1 ) = β 11 β 21 a1 Q1−1 R1 b1Q1−1 R1 + C1 T r2 = X 2T (Q2 x 20 + R2 ) = γ 11 γ T21 (29) a 2 Q2−1 R2 y= y1 ; y2 q= q1 0 I m0 xm 0 0 q2 I m0 xm0 (30) r1 r2 ; It is a linear-quadratic optimization one and its solution can be found in an analytical form according to (Stoilov and Stoilova, 1999) o r (31) y opt = − q −1 r − AIT ( AI q −1 AIT ) −1 ( AI q −1r + t ) [ ] 5.1. Definition of the component AI q −1 ATI AI q −1 ATI = Im0 xm0 Im0 xm0 q1− 1 0 0 Im0 xm0 (32) = q1−1 + q 2−1 q2−1 Im0 xm0 5.2. Definition of the component AI q −1r + t AI q −1r + t = I m0 xm0 I m 0xm0 −1 1 q 0 y1opt = β 11−1 β 11T T β 21 a1 Q1−1 R1 + b1 Q1−1 R1 + C1 (36)  −1 T  a1Q1−1 R1 T +   β1 1 β 1 1 β 2 1 −1 b Q R + C  1 1 1 1  + β1−11 ( β1−11 + γ 1−11 )−1   −1 a Q R 2 2 2 + γ −1 γ T γ T + t 11 11 21 − 1   b2 Q2 R2 + C2   T y opt = γ 11−1 γ 11 γ T21 2 −1 a 2Q 2 R2 + b2 Q 2−1 R2 + C2  −1 T  a1Q1− 1R1 T + . β 11 β 11 β 21 −1 b1Q1 R1 + C1   + γ 11−1 (γ 11−1 + β11−1 ) −1   −1 a Q R 2 2 2 + γ −1 γ T γ T + t 11 11 21 −1   b 2Q2 R2 + C2 After substitution of the analytical relations of yoi p t , i=1,2 from (36) in the description of the relation x1 ( y1 ) from (15) , respectively in x 2 ( y 2 ) x1 ( y 1opt ) = − Q1−1R1 + Q1−1 aT1 m0 x2 m0 r= Substituting (29) and (35) in (34) the analytical descriptions of yiopt are x1 ( y1opt ) and x2 ( y opt are found 2 ) . 1 1   min w1 ( y1 ) + w2 ( y 2 ) = y1T q1 y1 + r1T y1 + y T2 q 2 y2 + r2T y2  2 2   ⇒ The values of y opt can be expressed in terms of the i components of the initial problem a i , b i , Qi , Ri , Ci , β , γ, i=1,2 . According to (27) and (28) q1 = β11 ⇒ q1−1 = β11−1 q2 = γ 11 ⇒ q −21 = γ 11−1 . (35) from (16) , analytical descriptions of b2 Q2−1 R 2 + C2 The coordinating problem becomes y1 + y 2 = t y o2 p t = − q2− 1r2 + q2−1 (q1− 1 + q2− 1 ) − 1 (q1− 1 r1 + q2− 1r2 + t ) 0 r1 (33) + t = q1−1r1 + q2−1r2 + t q2−1 r2 + Q1− 1 a1T a1Q1− 1 R1 + a 2Q −2 1 R2 + t ( β 1−11 + γ 1−11 ) −1 b1T 0 m0 x m1 (b1Q1−1R1 + C 1) + β 22 − β 21β 11−1β 21T b1T β 2 1β 1−11 (β 1−11 + γ 1−11 )− 1 Im 0xm 0 β 1−11 β 2T1 γ 1−11γ T2 1 b1Q1−1 R1 + C1 b2 Q2−1 R2 + C 2 Applying matrix transformations it follows −1 −1 x1 ( y 1 ) = −Q1 R1 + Q 1 a1 opt ( β11−1 + γ 11− 1 ) −1 * β 21 β 11− 1 ( β11−1 + γ 11− 1 ) −1 T T b1 * ( β11−1 + γ 11−1 ) −1 β 11− 1β T21 −1 −1 −1 −1 −1 T β 21 β 11 ( β11 + γ 11 ) β 11 β 21 + −1 ( β 11−1 + γ 11−1 ) − 1γ 11−1γ T21 β 21 β 11−1 ( β11−1 + γ 11−1 ) −1γ 11−1γ T21 * + β 22 − β 21 β 11 β 21 T a 1Q1− 1R 1 + a 2 Q −2 1R 2 + t * b1 Q1−1 R 1 + C1 b 2Q 2−1 R 2 + C 2 −1 −1 T x2 ( y opt 2 ) = −Q 2 R2 + Q 2 a 2 −1 11 −1 −1 11 (β + γ ) * b T2 * −1 11 (β −1 −1 −1 T + γ 11 ) β 11 β 21 −1 −1 −1 −1 −1 −1 −1 T γ 21γ 11 ( β11−1 + γ 11 ) γ 21γ 11 ( β11−1 + γ 11 ) β 11 β 21 −1 1 1 (37) −1 −1 −1 T ( β11−1 + γ 11 ) γ11 γ 21 −1 −1 −1 −1 T γ 21γ11−1 (β 11 + γ 11 ) γ 11 γ 21 + * −1 T + γ 22 − γ 21γ 11 γ 21 −1 2 2 a Q R1 + a 2 Q R + t b1 Q1−1 R1 + C1 b2 Q 2−1 R 2 + C 2 (38) The analytical relations xi ( yiopt ), i = 1,2 ((37) and (38)) are derived, applying the predictive coordination multilevel approach for the initial optimization problem (3). But analytical relations for the same optimization problem (3) have been derived, applying goal coordination multilevel approach, which resulted in relations (10). Because for the both cases the solutions x i ( y opt i = 1,2 and x iopt , i = 1,2 are the same, i ), the relations (10) and (37), (38) must be equal. Thus relations between the components of α and the components of matrices β and γ are found. According to (10) and (37) it follows T ; α11 = ( β11−1 + γ 11−1) −1 ; α12 = ( β 11−1 + γ 11−1 ) −1 β 11−1 β 21 −1 −1 −1 −1 T ; −1 −1 α 13 = (β 11 + γ 11 ) γ 11 γ 21 α 21 = β 21 β 11 ( β 11 + γ 11−1 ) −1 ; T ; α23 = β21β11−1( β11−1 + γ 11−1) −1γ 11−1γ 21 (39) T T ; α 22 = β 21 β 11−1 ( β 11−1 + γ 11−1 ) −1 β 11−1 β 21 + β 22 − β 21 β 11−1 β 21 −1 −1 −1 −1 α 31 = γ 21γ 11 ( β 11 + γ 11 ) ; −1 T ; α32 = γ 21γ 11 ( β11−1 + γ 11−1 ) −1 β11−1β 21 α 33 = γ 21γ 11− 1 ( β 11−1 + γ 11−1 ) −1 γ 11−1γ T21 + γ 22 − γ 21γ 11−1γ T21 Thus by putting on equality the analytical relations of the problem solutions of (3), derived by goal and predictive coordination approach of multilevel methodology, explicit analytical relations between the components of the inverse matrices α and the smaller matrices β and γ are derived. Because β , γ have lower dimensions it is useful to evaluate α by fewer calculations in comparison with the direct inversion of α . 6. ANALYSIS OF THE COMPUTATIONAL EFFICIENCY OF THE RELATIONS From computational point of view it is important to find analytical relations, defining the components of inverse matrix α with larger dimension, using the components of inverse matrices with lower dimensions β and γ . The notations and the correspondence between the matrices and their inverse are a1 Q1−1 a1T a1Q1−1 b1T c11 c12 1 424 3 1 424 3 { { m 0 xm1 0 c = m0 xm0 m0 xm1 = m0 xm − 1 T − 1 c21 c 22 b1 Q1 a1 b1Q1 b1T β { { 1 424 3 1 424 3 m xm m xm 1 d= 0 1 1 d{ 11 d{ 12 m0 xm0 m0 xm2 d{ 21 d{ 22 m2 xm0 m2 xm2 where c=( β -1 = m1 xm0 m1 xm1 −1 T a1 a32 2Q 24 4 2 −1 T a1 b3 2Q 24 2 4 2 m0 xm0 −1 T 2 1 2 m0 xm2 −1 T 2 2 2 b1 Q24 a3 4 b1 Q24 b3 4 m2 xm0 m2 xm2 = γ ) and d= ( γ ) are symmetric ones. -1 The matrix, corresponding to the inverse matrix α is noted by AL. It can be described as a composition of the components of matrices c and d by the following way c 11+d11 c 12 d 12 AL = c 21 d 21 c 22 0 0 d 22 The direct evaluation of the inverse matrix of AL requires a large amount of calculations. That is why it is useful to find the inverse matrices β and γ, which have lower dimensions in comparison with AL. Then by relations (39) the inverse Al-1 = α could be calculated. This will save computations and will increase the computational efficiency in comparison with the direct evaluation of the inverse α . 7. NUMERICAL EXPERIMENTS It is assumed that matrices c and d are given 1 3 5 6 c= ⇒β d= ⇒γ . 3 4 6 8 These matrices are used in defining the matrix AL, which has bigger dimension in comparison to c and d: AL = c11 + d 11 c12 c 21 c22 d 21 0 d12 1+ 5 3 6 0 = 3 4 0 6 0 8 d 22 It has to be evaluated the components of inverse matrices β , γ and α Solution: β is evaluated as an inverse to c β= β1 1 β21 β12 1 3 = c −1 = β22 3 4 −1 = − 0,8 0,6 0,6 0 ,2 The matrix γ is also evaluated as an inverse to d γ = γ 11 γ 12 5 6 = d −1 = γ 21 γ 22 6 8 The inverse matrix (39) and it follows α −1 = 2 − 1,5 . − 1,5 1 of AL is found according to α11 = ( β11− 1 + γ 11− 1 ) −1 = [( −0, 8) − 1 + ( 2 ) −1 ]− 1 = −1, 3333 α 12 = ( β 11−1 + γ 11− 1 ) −1 β 11− 1β 21T = α 11β 11− 1β 21T = − 1,3333 ( −0, 8) −1 ( 0,6 )T = 1 α1 3 = ( β 1−11 + γ 1−11 )− 1 γ 1−11γ 2T1 = α 1 1γ 1−11γ 2T1 = −1,3333(2) −1 (−1,5)T = 1 α 21 = β 21β 11−1 (β 11−1 + γ 11−1 ) −1 = β 21β 11−1α11 = 0,6.1.(− 1,3333) = 1 −1 −1 − 1 −1 T −1 T α 22 = β21 β11 (β11−1 + γ 11 ) β11 β 21 + β22 − β 21β11−1 β T21 = β21β11−1α 12 + β 22 − β 21β11 β21 = = 0,6.( −0,8) −1 .1 + 0,2 − 0,6( −0,8) −1 (0,6)T = −0,5 α 23 = β 21β 11−1 (β 11−1 + γ 11−1) −1 γ 11−1γ T21 = 0,6( −0,8) −1( −1, 3333). 0,5.(−1, 5) T = −0, 75 α 3 1 = γ 2 1γ 1−11 (β 1−11 + γ 1−11 ) −1 = − 1,5.0,5(−1,3333) = 1 −1 −1 −1 −1 T α 32 = γ 21γ 11 (β 11 + γ 11 ) β11−1 β 21 = −1,5. 0,5 (−1,3333)(−0 ,8) −1 (0 ,6) T = −0,75 α 33 = γ 21γ 11−1 ( β11−1 + γ 11− 1 ) −1 γ 11−1γ T21 + γ 22 − γ 21γ 11−1γ T21 = = (−1, 5)0,5( −1,3333 ).0, 5( −1, 5) T + 1 − (−1, 5).0, 5(1,5) T = −0,625 α 11 α 12 α = α 21 α 22 α 31 α 32 α is evaluated to α 13 − 1,3333 1 1 , α 23 = 1 − 0,5 − 0,75 α 33 1 − 0,75 − 0,625 which is found not by direct inverse calculations of AL . The computational workload of relations (39) are assessed, according to the matrices dimensions. The assessment is performed according to the number of the flow point operations (flops) done by the processor, evaluating the inverse matrix α. Two comparisons are done, applying two scenarios. Scenario 1: the inverse matrix α is calculated for given inverse matrices β and the computational workload is assessed. The flops are compared with the calculations for direct evaluation of α. Scenario 2: The inverse matrix α is calculated for given matrices c and d and the computational workload is assessed. The flops are compared with the calculations for direct evaluation of α. The computational efficiency is estimated as number of “flops” in MATLAB environment. The numerical data for the matrices c, d and AL is given bellow. The matrix ? is the following: 1 5 3 4 1 6 1 2 4 5 3 7 2 3 4 5 1 9 7 1 3 3 4 8 m0 =4 3 4 5 4 2 4 5 2 9 1 3 6 4 1 1 2 4 2 2 1 5 8 8 3 1 6 9 4 2 1 2 3 7 6 4 5 2 1 7 5 2 2 8 5 9 1 3 2 m 2=7 4 4 3 9 5 7 9 8 5 2 1 3 3 2 1 2 1 3 5 9 8 2 6 4 5 5 3 1 8 6 1 2 2 3 4 5 6 3 4 3 8 4 3 6 1 4 9 5 7 7 8 6 3 5 2 4 3 5 5 2 Changing the matrix dimensions m0 , m1 and m2 scenarios 1 and 2 are performed. The computational workload, expressed by flops for direct inversion of matrix AL using matrices c, d, α, β, γ is related to the dimension m1 of c, assuming constant scales of m0 and m2 , Fig.1. flops /dimension m1, m0=3, m2=7 11000 10000 9000 al 8000 7000 flops Thus the inverse matrix full 6000 5000 ff 4000 3000 nic 2000 1000 1 2 3 4 m1-dimension 5 6 7 Fig. 1. Relations between flops and the dimension m1 when m0 , m2 =cte (m0 =3, m2 =7) The matrix A L is composed from c and d as If the components of the inverse matrices β and γ are given, the amount of flops are always less than the direct calculation of inverse matrix α, noted as “nic” and “al” (fig. 1). If the components of inverse matrices β and γ are not given, the amount of flops for the direct calculation of the inverse matrix α (“AL-1 ”) are bigger for larger dimensions of the matrices, estimated to the value of m0 >3, the curve full . Hence applying two coordination strategies analytical descriptions of the components of inverse matrix are found, related to appropriate matrices with lower dimensions. These relations have potentiality in decreasing the computational workload, which is beneficial for on-line and real time control. REFERENCES The matrix d is given as Fausett L., Applied numerical analysis. Prentice Hall, NY, 1999, p.596. Flannery B., (1997). Numerical Recipes in C. The Art of Scientific Computing. Cambridge University press, William Press, Second edition, p.965. Stoilov T., K.Stoilova (1999). Non iterative Coordination in Multilevel Systems. Kluwer Academicians, Dordrecht. Stoilova K., T.Stoilov (2002). Predictive Coordination in TwoLevel Hierarchical Systems. IEEE Symposium “Intelligent Systems”, Varna, vol.I, p.332-337 . Strassen V., (1969). Numerische Mathematik, vol.13, 354-356.