The pivotal condensation method for second-order systems

THE PIVOTAL CONDENSATION METHOD FOR SECOND-ORDER SYSTEMS* P. 1. MONASTYRNYI Minsk (Received

20 December

A PIVOTAL condensation with the boundary are considered.

method,

conditions

1968, revised version 5 October

based on unitary

1970)

transformations,

(2) and (3) and the question

for the system (5)

of the stability

of the method

Abramov [l] proposed a pivotal condensation method, without exceptional cases, for solving the boundary value problem of one linear second-order equation. Speaking generally, the method was based on the idea of an orthogonal transformation of the unknown variables. With some modifications Abramov’s method was extended in [2] to the case of the selfconjugate second-order system of equations (1)

Y” =

p(t)

with the boundary

Y + f(L)

conditions

(2)

Aid(a)

(3)

&y’(B) +hy(P)

+Ag(a)

=

6

= b

on the assumption that P(t) is a Hermitian matrix of order n, f(t) is an n-dimensional column-vector, the coefficients of the system (1) are piecewise-continuous functions of t, a <

(4)

t <

p;

A 1, AZ and B1, B2 are complex square matrices of order n such that

Ap.42’ = fL4i*,

and the rectangular column-vectors.

B,B,* = BzB,=,

matrices (A,, A*) and (B, , B2) have rank n; c and b are complex

The purpose of this paper is to study the pivotal condensation of the form (5)

Y” =

P@)Y

*Zh. vjkhisl. Mat. mat. Fiz.,

-i- Q(t)Y

+ f(t)

11, 4, 925-933,

1971.

138

method

for a system

The pivotal condensation

with the boundary square

matrices

that for

conditions

(2) (3). It is here assumed that P(t) and Q(t) are arbitrary

of order n whose elements

P(t) s

0

139

method

are piecewise-continuous

the matrix Q(t) is not necessarily

removed. The remainder of the assumptions (2) and (3) are the same as in [2].

Hermitian.

on the quantities

Some of the results of [3-61 are used in the derivation transfer equations of the boundary conditions. Let y(t) be the unknown solution of the problem derivative. We introduce the auxiliary matrices W,(t), vectors

r(t),

s(t)

functions Condition

defining

of t, so (4) is also

the problem,

of the formulas

(5),

and the

(5), (2) and (3) and y’(t) its and the i = 1, 2, 3,4,

by the formulas

(6)

r(t)

=

%*(t)!/‘(t)

+ W,‘(l)y(t),

(7)

s(i)

=

%‘(l)y’(t)

+ w,*(t)y(2);

here the asterisks denote the conjugate Assuming that the matrices

matrix.

W,(t) satisfy the conditions

IJ’,(t)Wi*(t)

+ W,(t)Wz*(d)

=

E,

W,(t)W,*(t)

+ W,(t)W,*(t)

=

E,

wi(t)w,*(t)

+ rv,(t)Wk*(t)

=

0,

w,(t)w,*(t)

-I- W,(t)Wz*(t)

=

0,

(8)

we determine

y(t) and y’(t) from (6) and (7):

(9)

y’(t)

=

w,(t)r(t)

+ W,(t)s(t),

(IO)

Y(,t) =

w,(t)r(t)

+ w,(t)s(t).

It is easy to check that the matrix of the linear transformation (6) and (7) is unitary because of (8). We now apply the transformation (9) and (lo), also with a unitary matrix, to the system of equations (5). This leads to a new system of equations for the vector functions r(t) and s(t): r’ =

[(IV,“’

+ W,‘P(t)

+ W,*Q(t))W3]r+ + s’=

(IV,*‘+

[(WI*‘+

W,‘Q(t))W]s

[(w,*‘+w,‘P(t)

+W,*Q(t))W,]r-j+

+ W3*) WI +

(IV,“‘+

w,‘P(t)

+ W3*)W2 +

+ JV,*.i, +w;*)m-,+

[(W2*‘+W,*P(t)

W’,*Q(t))W,]s

(W3*’ +

+ W,*f.

(Iv,“+ +W,*)W,+

140

P. I. Monastyrnyi

Bearing in mind the problem of the reduction of the system (5) with the boundary conditions (2) and (3) to some problems with initial conditions for WI(~), r(t) and s(t), we select from all unitary transformations of the form (9) and (IO) one such that (Wi.‘f

(11)

W,‘P(2)

+ Iv,*)%

+ (Ws*‘+

W,*Q(t))W;

= 0.

+ %*)W,

+ (J+‘z’+

~~**~(~))~~

= G(t),

In addition to this we put 02)

(TV**‘+ W,*P(t)

(13)

(wz*'+ wiv,*P(t) + tt',*)W, + (%*'+W*Q(t))Ws = H(t),

(14)

(I&*'+wz*P(q+ W) TV, + (WI*'+WQ(t))%

= J(t)

and we choose the matrices G(t), H(t) and J(f) in such a way that the following relations are satisfied:

From (II), (12) and (15) we obtain

Ws = W,G*(t),

w,‘+P’(t)Wi+

(18) w/+Qyt)W

= W,G’(t),

where G”(t) = W,*W~ + W,*P(t)W,

+

Wi"Q(t)K

Similarly, from (13), (14) and (16), (17) we obtain

($9)

Iv,'+ P(l)W, + TV‘--W,P(~t)+ W,P(t), H'L'+ Q*(t)% = W,fP(t)+ ~~~(~),

where H*(t)

=

W,“(P(z)

+P*(t))Wz+ W,*(E+Q*(t))K+

+ wig@+ Q)W,

It follows from (18), (19) and (15) - (17) that any solution of these four equations which satisfies the “unitary” initial conditions, is “unitary” on the whole

The pivotal condensation

segment

fo, /3]. Taking

this into account,

initial values at the point

Yl

Z--

i Y where the solution

U!?)

I

i=

(a,, z(o))

(20)

I,2 ,‘..,

’

are the elements

of theaboundary

The boundary

we consider

I = cr for the matrices

value problem

condition =

the problem

w,(t),

W,(t)

of determining

and the vector r(t).

n,

of the matrix Ak, k = 1, 2, y is the unknown (S), (2) and (3), and y’ is its derivative.

(2) can now be written i =

CL,

i4E.

method

in the form

1, 2, . . . , n,

where the (pi are the c~~ordinates of the vector a. Similarly,

we write condition

(b,,s(P))

(21)

=

Pi,

(3) in the form 1, 2, . . . f n,

i =

where

the h 1:’ vector b.

are the elements

of the matrices &,

For our purposes it is convenient

the pi are the coordinates

to reduce the condition

form by carrying out the process of orthonormalization We have (9)

(Ui, s(a))

=

T#,

where

k = 0, 1, . . . , n - 1.

i=1,2

).“,

n,

of the

(20) to an equivalent

of the system of vectors ‘li.

the

P. I. Monastyrnyi

142

We notice that

t”i94 =

{iv:7 2 ,

9

i

k

,

where Vr and V, are square matrices of order we obtain

n

1 2

=

t

,...,

n, y =

.

(yi, yz, . . . , Y,)~. From (22)

KY’(U) + V$/(u) = y.

(23)

(6) for t = cx and (23), we obtain

Comparing

IV,* (a)

=

WSL (a)

V,,

=

‘Fla,

or (24)

W,(a)

(l?)

r(u)

=

=

IV,(a)

vi*,

=

Vz*

y.

It is easy to verify that the matrices Wr and Wz satisfy the condition (26)

w**(a)W,(a) Therefore,

+ w,*(a)w,(a)

integrating

=

the system of matrix equations

(241, we find on the whole segment a < t < and W3(t), since by (15) and (26) we have (27) for all

wi*(t)Wl(t)+ t E La, p].

E.

w,*(t)WJ(t)=

fl

(18) for the initial condition

the smoothly

IV,*(a)W,(a)+

We find the vector r(t) as the solution

varying matrices

W,(t)

Ws*(a)W~(u)=E of the Cauchy problem for

the equation

(28)

r’ =

G(t)r

+ IV,*/

with the initial condition (25) at the point t = a. This makes it possible to transfer the boundary condition (22), or what is the same thing, the boundary condition (2), from the point Q! to all the points of the segment [or,/3] , and at the point t = p we obtain

The pivotal condensation

143

method

lvi*(B)d(B) + Ws*(P)~(P) = r(P), or

(29)

(coz(B) ) =

et,

i=1,2

,...,

72,

where the ci are the vectors formed from the elements P?(B),

MiS-(P)lr, Together

the

6, are the coordinates

with (29) we consider the boundary

of the i-th column

of the matrix

of the vector r(P). condition

(3). Let the rank of the

matrix ]

w,*(P)

w,*(P)

B,

B*

be 2n. This condition

I

is equivalent

to the unique solvability

of the boundary

value

problem (S), (2) and (3). Relations (29) and (3) make it possible to determine the initial values for W,(t), lv,, (2 ) and s(t) at the point c = 0. Let the boundary condition (3) be written in the form (21). We apply to the vectors 6,, b,, . . . , b, a linear transformation that the transformed vectors together with the vectors cl, cZ, . . . . . , c, an orthonormal system of 2n vectors. After the completion of the orthogonalization can be written in the equivalent form

(30)

(Cn+k+l, z(P))

=

process the boundary

in order will form

condition

(3)

E?i+k+l,

where ?I+!4

n-r-k cMit1

=

en+,+i=[

[ bh+i-_C

b!++,-C

(b,,,,ci)ci]]r-: i=i

n+k

ntk

L+i-r,(b,+i,cJ&i]ll 1=*

k = 0, 1, . . . ) n If the vectors the orthogonalization

(b,+,,ct)cilJI i=,

1,

,

b,;,-~(b,,,,ci,cillIT:, 1=i ei =

6,

i=

1,2, . . . , n.

are almost linearly dependent a,, a?, . . . , a,, or b,, b2,. . . , b, process must be carried out by Wilkinson’s rule (see [7] , section 13).

144

P. I. Monastyrnyi

We note that the vectors We introduce the notation

cI, czt . . +, czn taken together form a unitary

where Vs and V, are square matrices of order n. in the new notation

W(B)

(31)

Comparing

matrix.

(30) has the form

+ V‘Y(B) = 6. (7) with f = fi and (31), we obtain

wz*(B) = v, w;*(fi) = v,,

(Z,

w,(p) = V,‘,

(33)

s(j3) =E.

W&3) =

v,*,

Therefore, the matrices Wa(t) and W,(t) can be found in the inversion of the pivotal condensation method as the solution of the Cauchy problem for the system of equations (19) with the initial condition (32). We find the vector s(t) from the differential equation

(341

s’=H(t)r+J(t)s+

with the initial condition By construction,

(35)

(33) at the point t = /3.

the matrices W3 and W, satisfy at the point t = /3 the condition

TJ%*(p)W,(@) By (16) this property [ 01,/?I, since

way

+ W~*(B)W~(B)

of the matrices

~~*(~)~~(~) + ~~*(~)~*(~)

E.

Wz and IV, is preserved on the whole segment

+ ~~*(~)~~(~) =

=

=

~~*(~)~~(~)

+

E.

Consequently, the matrices W2(t) and W,(t), like WI(t), W,(t), will also be smoothly varying matrices for all t E [a, p 1, It is easy to establish that the matrices IV+(t), = 1,. 2, 3, 4, for a < t < p also satisfy the condition (36)

IV** (I) VVz(t) + 1’v3*(t) IV&(t) = 0.

The above discussion enables us to indicate boundary value problem (S), (2) and (3).

the following

scheme for solving the

i =

145

The pivotal condensation method

1. determine

We orthonormalize

the boundary

condition

(2) and by formulas (24) and (25)

the initial values for Wr (t), IV,(t) and r(t). Then using the direct pivotal

condensation

method we find these matrices and the vector r(t) as the solutions

of the

Cauchy problems (18) and (24), (28) and (25), respectively. cr, cz, . . . , c,, and the scalar quantities

2. We write out the system of vectors

d., 82, . . . ( A,,. Using this system we orthonormalize the system of vectors b,, b,, . . . , b,, and reduce the boundary condition (3) to the form (30). In the orthonormalization process the question of the rank of the matrix wl*(P)

1

w3*(P) B3

B,

’

must be clarified. If the rank equals 2~2, we calculate W,(p),

W,(p)

formulas (32) and (33). Then using the inverse pivotal condensation by equations (19) and condition (32) the matrices (34) and condition (33) the vector r(t). 3. We find the unknown

solution r(t)

W,(t),

from formula

W,(t),

and

s (0)

by

method we find, and by equation

(lo), and v’(t) from formula

(9). It is established by a check that the vector function conditions (.5), (2) and (3).

v(t) found in this way satisfies

As in [4], it can be shown that if the problem (5), (2) and (3) is stable for small variations of the quantities determining it, the above method of solving this problem is also stable. Indeed, if

(jet

W,‘(P)

II

W3’03)

B,

B3

and consequently, the problem has a unique because of (27) (35) and (36) that ll~(t)ll’+

lls(t)ll*

-

+() II

solution,

II~(t)ll~+

it follows from (6) and (7)

W(t)‘/*.

That is, in the process of calculation we have to find functions which have the same order of smallness as the unknown solution of the boundary value problem and its derivative. Also, by (15) and (18) the matrices IV,(t), W,(t), obtained by the approximate integration of the equations, must for CL< t < /3 deviate little from the condition

146

P. 1. Monastyrnyi

~~*{t)~*(t)

+ ~s.(t)~s(t)

= E;

similarly, the matrices Wz(t), W,(t) must by (16) and (19) deviate little from the condition

Wz’(t)Wz(t)

+ w4’(t)w&(t)

by (17), (18) and (19) for a < the condition

t <

= E

i&3 all the matrices

zg t < p;

W&t) must deviate Iittle from

= E.

Wi*(t)Wz(t) -f- Ws*(t)W;(t) i=

a

for

We continue the proof, following [I] . We assume that all the equations for Wi (t) , 1, 2, 3,4, r(t) and s(t) are solved approximately. Then we will have w<(t),

r”(t),

g(t),

g(t),

g’(t),

obtained

from the formulas

FV,‘+P”(t)tVl

+ 57s =

w,‘+Q*(1)17,

=

tu,(n)

=

T&Y?(k)

+ E,(1),

FVv,C*(2) +-G(t),

W,(a) = Vz* + A?

Vi’ + A:’ ,

w,‘+P”(2)~~27,+1;Yr=GV,R*(2) w+Q*(t)w;-z=Wzff”(t)

+lTJ*(t)+E,(t),

+wJyq

-tE4(1).

W,*(B)= &+A:), r”‘= G(t)? + a,*f(t)

+ CL(t),

2 = R(t)r”+J(Qs”+

Wz”f(l) +02(t),

y”‘(t)

=,w,(t)r”(t)+

By direct substitution

,

WC(B)=

V,+

A:),

r”(a) = y + 6cC4,

lT+&)s”(l),

g (f3) = & + 8&

y”(l)=

m(t)j:(q+

w,(t)3(q-

it may be verified that the vector function

y(t) satisfies the

equation 6)

[G-‘(r)

((y”)‘-

+ &-‘(+$o)] -

L(t))]‘= (g)‘+

G~(~)~~-‘(~)~~(~}

[(P(r) [Q(t)

$-Ho(t)

+ C,(t))&-‘(t)

+&-‘(t)

(Fe’(t)

-~(~)~~-*(~)~*(~)~~

+ -&‘(r)&(t)) + f(t)

+JirO(t) -- (P(i) + G(t))%-‘(~)Jo(~), where to shorten the formulas we have introduced the notation E,(t) =B$-E,(t)W*” +&(t)+K*, F&)=

E,(t)W,*

G,(t) =E,(t)Wt* Jo(t) = W,ai(q

+ E,(t)W,‘,

-tE,(t)W,*, -f- W&o,(q,

H,(t) =&(1)tPz*+E&)RC fG(Q = WIG(t) + %a&),

+

The

pivotul

co~den~t~on method

147

and the boundary conditions

(vi + A!?)Gil (a) (y”(a))’ + [Vi + Ah2’ - (vi + A?) ZG1(a) x x Fe @)I ii (4 = y + &?+ P’f + A&“> G1(a) Jo (a), P, + Wt’ + Q$“‘>J&i’(B>((y”(8))’- Jo(6,) + -t

[B, + D,Af’ + D,Ai:’ -

(B, + D,A’,l’ + DIA~‘) x

x E,-‘(B) F,(fQ]g(fS) = b -IIA&'+

D 10, 6@)

where we have introduced the notation pl,, = B,W, (fi) + B,W2 (@). D, = B,W2 and A,(‘) are the errors in calculating the values of (f3) + &MZ (B) ; SP, A&‘) and wg( fi). Here we note that the boundary condition (2) is WL IWBf reduced to the form (23) by non-degenerate transformations, hence problem (S), (2) and (3) is replaced by the equivalent problem (S), (23) and (3).

Let IIEi tt) II7IIEi’ lt) 111 i = 2, 2,3,4, ]]ol(t)]\, jIoi’(t)lJ, i = 2, 2, be small on the segment b,Pl and I)Ah”11, I!A? 11, 11 AF’II, I{Af’lj, Ijd$?11, 11 A(f) II,11 Sb”11, IISj? 11. jl@‘Ij be small numbers. Then equation (5) is reduced to the form W’ = W) W+ WfS +r(% (5%) where on [a,@] the functions P(t), q(t), f(t) differ little from P(t) O(t), f(t); respectively; the boundary conditions are reduced to the form

(239

~,~~(a))'+~~~~~~= i?

t3*j

B:tfv"(p))'+mm

=K

where the matrices v,, Pz, Bt, BZ and the vectors v, 6 differ little from V,, Vz, 2% Bz and ‘y, b respectively. Consequently, the vector function y(t) is a solution of problem (S*), (23*) and (3*), approximating problem (S), (23) and (3). Hence the proposition formulated

above is established.

In conclusion we mention that the method applied here can be used by analogy (see ]2], section 2) with the solution of an eigenvalue problem. ~ans~te~ by J. Berry. REFERENCES 1.

ABRAMOV,

A. A. A variation of the pivotal condensation

method, Zh. vjkhisl. Mat. mat. Fiz

1,2,349-351,196l. 2.

LIDSKII, V. B. and NEIGAUZ, M. G. The pivotal condensation method in the case of a selfconjugate system of the second order, Zh. vj&zisZ. Mat. mat. Fiz. 2,1,161-164,1962.

3.

ABRAMOV, A. A. On the transfer of boundary conditions for systems of ordinary linear differential equations (a variant of the dispersive method). Zh. vjichisl. Mat. mat. Fiz.

1,3,542-545,196l.

P. I. Monastyrnyi

148

4.

MOSZYNSKI, K. A method of solving boundary value problems for systems of linear ordinary differential equations.Akorytmy 2, 3,25-43, 1964.

5.

TAUFER, J. On factorization

6.

MONASTYRNYI, P. I. Reduction of a multipoint problem for a system of differential equations to Cauchy problems by the method of orthogonal transformations, Zh. vjkhisl. Mat. mat. Fiz.

method.ApZ. Math. 6, 11,427-450,

1966.

7,2,284-295,1967.

I.

VOEVODIN, V. V. Numerical Methods ofA lgegebra(theory and algorithms) (Chislennye metody algebry (teoriya i algorifmy)), “Nauka”, Moscow, 1966.