The solution of linear algebraic systems

SHORT COMMUNICATIONS

THE SOLUTION OF LINEAR ALGEBRAIC SYSTEMS* V. N. KUBL~OVS~YA Leningrad

(Received 11 F~b~~~ 19’75; revised3 1 Mzrck 1975) THE NECESSARY and sufficient conditions for the solvability of the system Ax = f, where A has dimension tn X pt and the rank r G min (m, n). are given. Let Ax=f

(1)

be a linearly-algebraic system, A a real rectangular (mXn)-matrix of rank r
(2)

be the resolution of the matrix A into the product of the two matrices S and Q, of dimensions mXr and rXn respectively, of rank r, where QQ’=Ir,

(3)

and let the principal minor of order r of the matrix S be non-zero. Here I, is the unit matrix of order 1. The resolution (2) forms the basis of many computational algorithms for the solution of problems of linear algebra (see, for example, [l-3]). It can be obtained by multiplying the matrix A on the right by elementary orthogonal matrices (of rotation or reflection) and on the left by a permutation matrix, for example, by means of a normalization process [l] . In this case the rows of the matrix Q will be orthogon~, so that Eq. (3) will be satisfied. Moreover, the principal minor of order r of the matrix S is the greatest of all the minors of the same order. We use the expansion (2) to investigate the system (l), and in terms of the initial data and of the expansion (2) we formulate the solvability condition of the system and give a formula for its general solution. We introduce the notation: (4)

where S,, Q1 are square matrices of order r, Sz, Q2 are matrices of dimension (m-r)Xr and rX(n-r) respectively, and fi and f2 are the vectors of the first r and last m-r components of the free term f of the system (1) *Zh. vj2hisL Mat. mat. Fiz., 16,6, 1578-1579,1976.

192

193

Short communications

Theorem For the system (1) to be solvable it is necessary and sufficient that Szs,-lfi=fz. The

(5)

general solution of system (1) is given by the equation s=Q’(S,-‘f,-Qb),

(6)

where b is an arbitrary vector of dimension n. To prove this we consider the cell matrices R, =

II

0

-s,s,-’

0

Rnp=In-Q=Q,

I,_, II ’

(7)

of dimension mXm, nXm and nXn, respectively. Here we denote by 0 the zero matrices of the corresponding dimensions. We establish the validity of the equations R,A=O,

AA-A=A,

AR,,=O,

(8)

for which we represent the matrix A in cell form: (9)

Then, taking (3) into account, the validity of (8) is verified by direct substitution of (7) and (9) into (8). It is known (see, for example, [4]), that, firstly, for system (1) to be solvable it is necessary and sufficient that &f=O,

(10)

and secondly, the general solution of system (1) is given by the equation x=A-j+R,,b,

(11)

where R,, Rnp, A- -are any left and right annihilators, and the semi-inverse matrix of the matrix A respectively. Then the proof of the validity of Eqs. (5) and (6) follows from (10) and (1 l), if as R,, Aand R,, we take the corresponding matrices in the form (7). Corollary. For the solvability of the indeterminate system (1) (the matrix A has rank r=nc m) it is necessary and sufficient that AzAi-lg,=gz. Here @A=

A, II II ,

e/ =

‘42

IIg2 II gi

,

A, and A2 are matrices of dimensions nXn and (m-n)Xn, respectively, o is the resulting matrix of permutations of the rows of the matrix A, so chosen that det A,+O, and g1 and g2 are vectors of dimensions n and m-n respectively. Translated by J . Berry. REFERENCES 1.

FADDEEV, D. K., KUBLANOVSKAYA, V. N. and FADDEEVA, V. N. On the solution of linearlyalgebraic systems with rectangular matrices. 7%Matem in-ya Akad Nauk SSSR, %,76-92,1968.

K A. Morozov

194 2.

KUBLANOVSKAYA, V. N. Application of a normalized process to the solution of linear algebraic systems. Zh vj%hisL Mat. mat. Fiz., 12,5, 1091-1098, 1972.

3.

GOLUB, G. Numerical methods for solving linear least squares problems. Numer. Math, 7,3,206-216,196s.

4.

SOBOLEV, S. L. Introduction to the theory of cubature formulas (Vvedenie v teoriyu kubatumykh “Nauka”, Moscow, 1974.

formul),

NUMERICAL REALIZATION OF THE OPTIMAL DISCREPANCY METHOD* V. A. MOROZOV (Received 13 February 1975)

IT IS SHOWN that the solution of the conditional variation problem of the optimal discrepancy method for operator equations of the first kind can, subject to certain conditions, be found as the solution of a simpler parametric problem. This fact forms the basis of a numerical method of solving it. The determination of approximate solutions of an operator equation of the first kind in accordance with the optimal discrepancy principle, developed in papers [ 1,2] , is reduced to the solution of the following extremal problem. where A : H--F, L : H +G are linear U--u*)IlG+t}#0, Let the set U= {UED : IIh-fllF~hllL( operators acting from the Hilbert space Hinto the similar spaces F and G, with the linear domain of definition D=H; let h : Oth~h,, t : 0~ t
(1)

problem (1) is solved approximately, namely elements ueE c’, are found for which E=O, remains e>O. Thereby the question of the possibility of putting uninvestigated, that is, the question of the existence of solutions of problem (1). The question of the numerical realization of a solution of problem (1) was also not considered. Our note is devoted to the solution of these two problems. In [ 1,2]

m
1. We impose the following constraints on the operators A and L [3] . Firstly, we wilI regard them as collectively closed, that is, for every sequence u,,=D, u,,+ u. in H, Au “-+f. in F, La .-t go in G, it follows that tco=D, Auo=fo, Luo=go, and secondly we regard them as complementary, that is,

where 7 is some constant. Below we consider two cases: m = 0 and m>O. For convenience in the discussion we make the substitution Z=U--u*, u=D. Then obviously the set U={z=D

where f*=f-Au*,

: IIAz-f’llP~hllLzll,+t),

and problem (1) becomes the problem m = inf

ZEU

*Zh vychisL Mat. mat. Fiz., 16,6,1580-1583,1976.

llLZllG =llLz/G,

ZEU.’

(2)