- Email: [email protected]
On an iterative process for the numerical solution of systems of linear algebraic equations
232
L. L. Kosachevskaya, V. V, Romanovtsev and I. E. Shparlinskii
3. We make two remarks in conclusion. First, given poor scaling of the initial matrix, the growth of the conditionality number of the intermediate matrices occurs, not only in the optimal elimination method, but in all versions of Gaussian elimination which provide for normalization of the leading row by the leading element, and notably, in most versions of Jordan’s method. The reason is always the same: the expressions for the different coefficients of the k-th intermediate matrix have a different degree of homogeneity relative to the elements of the initial matrix.
Second, we must emphasize that all the matrices of our numerical example are diagonally dominant. The growth of conditionality observed for matrices A (a) for large and small a! at the intermediate steps of the optimal elimination method needs to be associated with the following fact, proved in [2] . If the optimal elimination method is performed in the modified form, which does not require division of the leading row by the leading element, then, for any k, we have the inequality (in the notation of Section 1) llCII~lIDllnr=condw(C)=411‘4IIMllBIIM. Here, therefore, the conditionality of the intermediate matrices is independent of the (uniform) scaling of matrix A and in essence, does not increase throughout the process. Translated by D. E. Brown
REFERENCES 1. VOEVODIN, V. V., Numerical methods of algebra (Chislennye metody algebry), Nauka, Moscow, 1966.
2. IKRAMOV, KH. D., On the conditionality of the intermediate matrices of the Gauss, Jordon, and optimal elimination, methods, Zh. vj&hisZ.Mat. mat. Fiz., 18, No. 3,531~545, 1978.
USSR Comput..hSzths.Math. Phys Vol. 22, No. 6, pp. 232-237, Printed in Great Britain
1982.
0041-5553/82307.50+,00 01984. Pergamon Press Ltd.
ON AN ITERATIVE PROCESS FOR THE NUMERICAL SOLUTION OF SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS* L L KOSACHEVSKAYA, V. V. ROMANOWSEV and I. E. SHPARLINSKII Moscow (Received
12 November
198 1)
AN INFINITE sequence of iterative parameters is chosen, whereby the corresponding iterative process is stable and has asymptotically best convergence. 1. Consider the system of linear algebraic equations Au=f, *Zh. vychisl Mat. mat. Fiz., 22,6,1504-1508,1982.
(1)
Short communications
233
where A is a symmetric square positive-definite matrix, having a complete system of eigenvectors and spectrum Sp (A), lying in the interval [m, M] , 0 < m
Uk=uk-I-a/,(AUk--l-f),
2, . . . .
(2)
where cu= (&k) is a sequence of iterative parameters. It is well known (see, for example, [l] ) that process (2) is optimal at the n-th step if we put ah =
2 l?‘f+?tL-((M-m)Tk
k=i, 2,. . . , n, ’
where ~1, . . . , rn is any permutation of the roots of the Chebyshev polynomial T, (x) = cos (?I arccos x).
There are many publications (see, for example, [l-5 ] and the references therein) in which an order is given for choosing 71, . . . , rn, such that iterative process (2) is stable, i.e. all the intermediate approximations ~1, . . . , Un have bounded norm (otherwise, rounding errors have a strong influence, and the computer may come to an emergency stop due to leaving the place mesh). In [5] an infinite sequence cr = (ok) was constructed, for which process (2) is stable and becomes optimal in an infinite sequence of steps ni. Below, we construct an infinite sequence 01= (ok), for which process (2) is asymptotically optimal at each step and is stable to computing errors. Given this property of the iterative scheme, we can manage without a priori estimates of the required number of iterative steps and the process can be stopped in accordance with the properties of the approximations obtained. One important criterion for stopping the process is given in Section 4 below. 2. Denote by p = (p(k)) the Van der Corput sequence (see [6] ), i.e. p(k) =Z-‘e1+2-2e2+ . . . where k=ei+2ez+. . . +27-Ler is the binary expansion of integer k (there are quite +2-‘er, economic algorithms for computing sequence p, see, for example [6]). We put ar -
2 M+m-(M-m)cosnp(k)
’
k==i, 2,. . . ,
where /.Iis the conditionality number of matrix A : p = M/m. We shall assume that each step of the iterative process involves an error of norm not exceeding TJ,i.e. instead of (2) we have h-h-1
-Ur(Aur-1-f)
+qr,
k--i, 2, . . . ,
Denote by E&= ](uk - u IIthe error of the k-th approximation.
(4)
L. L. Kosmhevskaya, V. C: Romunovtsev and 1: E. Shpariinskii
234 Theorem
Let the parameters in iterative process (4) be given by (3); then, n+o(n) + O(q),
n+-,
where the constants in the order symbols o and 0 depend only on matrix A. 3. The proof of the theorem is based on the following Lemma 1, showing that the Van der Corput sequence is “distinctly” uniformly distributed (see [7 I); the lemma is a consequence of Theorem 2 of [7] and Theorem 9, Chapter 2 (61. Lemma
1
Let the function P(X) be Riemann-integrable. Then, a function e(N)+O, N-+ m, exists such that, given any k = 1, 2, , , . , we have W-N
1
zrp(P(i)) 1c
cp(s)ds - +
IS
e(N).
i-k+1
0
Lemma 2
For all X E [m, M] , 1
In
0J
p’h- 1
211
I
l-
do =lIl-.
M+m-(M-m)cosno
I
Proof: After changing the variable: y=2/[M+m-(M-m) evaluated in [8, Section 41.
p”+i
cosno]
, we obtain the integral
We put (I-arA),
Pk,N@)-=
Ph.N’
II i=k+l
max
IPk,N(k)
1,
IsSy(A)
where the max is taken over all A of the spectrum of matrix A. Lemma 3
A function e(N)+O, N+ 00, exists, such that, for all k = 1, 2, . . . , we have Nff+~(N)l .
proofiLet Sp(d)={hr, . . . , A,). Then, by Lemma 1, for each At, I< c < s, there is a function .et(N)+O, N+ =, such that, for any k = 1,2, . . . , 1
IJ l In
0
l-
2h M+m-(M-m)cosno
I
do
235
Short communications
--
1 N
hi-N
c
lnll-afhtj
G et(N),
ll-aikl>etW).
I
4=&+1
Applying Lemma 2, we obtain
InIl-aiht]
Q
et(N),
I
i-A+1 A-kN
Nine+
h(i-afht(<
Nei0’).
p”+l
c iaA+l
Consequently, k+N
lnll-arhrl
IPk,N(ht) I- exp
) s; (s)
Nexp[Ner(N)].
i-A+1
Putting
we obtain the lemma. Coroky.
Constants c > 0 and 8 < 1 exist such that, for all integers k and IV,we have
(5)
PA, N
Lemma 4 We have n
enceOPt,n+q
c
Ph,?L-h.
h-1
For the proof see, for example, [5, Section 11. From (5) we have
since 8 < 1. From this and Lemmas 3 and 4 we obtain the theorem. 4. Let us give an example of an algorithm for finding the approximate solution IIu, - uII Q E of E$. (1) with an assigned accuracy E > 0. Assume that the computational error is negligible, i.e. 11Q E. Then, from the estimate proved for n)no(e). Hence, for n > no (e), in the theorem, no (e) exists such that ilu,-ull
(6)
L, L. Kosachevskuya, V. V. Romunovtsevand 1: E. Shparlinskii
236
Further, it follows from (2) that Ilu,-ull=IIA-‘(Au,-Au)ll=IIA-1(Au
n-f) II=II~-~a~-:i(u,-u,+~)II
(7)
G ~-ian-:*Il~a+i-unll. Hence, if a,+*>l/2n and inequality
(6) holds, then we find from (7) that ]I+ - u ]I< E.
We define NO-[x(2(p-l))‘~~]+~. condition
(8)
We shall now show that, from the instant of satisfying
(6) until (8) is satisfied, we have to perform at most No steps.
We shall show that, among the No consecutive
values
, at
least one satisfies
to take the case M > 2m. We define an
(8). We first note that ak > l/M, so that it is sufficient integers
Q,+~, . . . ,ak+N,
by the inequalities 28-1 < Jd
(
p-l 2
%
< 2”.
1
Since p-1
2%2n
-
2
(
‘h
= n[20,+1)
]‘“
1
then, among the numbers k + 1, . . . , k + No, there is a number t, divisible by 2s. The binary form of this t is t=2ae,+,+ . . . +2r-ie,, and hence p(t) =2-+*ea+i+ . . . +2-re,<2-+1+. . . +2-m<2-*. SinceM>2m,
thens>
l,p(t)<%,
c0s[~p(~)l={l-sin2[np(t)]}~~~1-sin2[np(t)]> >1-(n.2-“)*)1--=-,
at=2{M+m-(M-m) 22[M+m-(M-m)
2
P-3
p--l
p-1
I-[np(t)
I2
cos [rip(t))}-1 (p-3)/(p-I)]-‘=1/2m.
and Notice that, from (7) and the inequality ok > l/M, we have IIu~-uH
Short communimtions
231
REFERENCES 1. MARCHUK, G. I., Methods of computational mathematics (Metody vychislitel’noi matematiki), Nauka, Moscow, 1980. 2. LEBEDEV, V. I., and FINOGENOV, S. A,, Solution of the problem of ordering the parameters in Chebyshev iterative methods, Zh. vychisLMat. mat. Fiz., 13, No. 1, 18-33, 1973. 3. NIICOLAEV,E. S., and SAMARSKII, A. A., The choice of iterative parameters in Richardson’s method, Zh vychisl Mat. mat. Fiz., 12, No. 4,960-973, 1972. 4. SAMARSKII, A. A., Introduction to the theory of difference schemes (Vvedenie v teoriyu raznostnykh
skhem), Nauka, Moscow, 1971. 5. LEBEDEVLV.I., and FINOGENOV, S. A., On the use of ordered Chebyshev parameters in iterative methods, Zh vychisLMat. mat. Fir., 16, No. 4, 895-907,
1976.
6. SOBOL’, I. M., MultCdimensionalquadrature formulae and hbar functions (Mnogomernye kvadraturnye
formuly i funktsii Khaara), Nauka, Moscow, 1969. 7. SOBOL’, I. M., On best uniformly distributed sequences, Usp. mat. Nauk, 32, No. 2,231-232,1975. 8. VOROB’EV, YU. V., A random iterative process, Zh. vychisl Mat. mat, Fin, 4, No. 6, 1088-1093, 1964.
USSR. Cornput. Maths Math. Phys Vol. 22, No. 6, pp. 237-243, Printed in Great Britain
1982.
0041-5553/82$07.50+.00 01984. Pergamon Press Ltd.
ON THE STABILITY OF POSITIVE SOLUTIONS OF CONVERSE PROBLEMS OF HEAT CONDUCTION* E. A. GRIGOR’EV Moscow (Received 9 December 1980)
THE CONDITIONS
second boundary
isolating compact correctness
value problems for the equation
classes are stated for the time-reversed of heat conduction.
first and
These converse problems are
shown to be stable in a uniform metric in the set of positive solutions. 1. Consider the boundary
au -e
at
value problem for the equation
a2u -+...
ax,2
U~S==$J(~,t),
azu +-,
(x, t)
axn2
x=(x* ,...,
x,).
E
n>l,
of heat conduction
D =QX(O, Tl, u(x, O)=cp(x),
xeC2,
where S is the lateral surface of a cylinder D with base Sl. Let the domain 52, the initial function cp, and the boundary
function
JI, be such that the problem has a unique solution u (x, t) in A
Assume that the solution u (x, 7’) is known at an instant T > 0. The reverse-time problem for the equation of heat conduction is the problem of finding u (x, t) for I < T given the function f(x)
= u (x, T), x E S2, when the boundary
*Zh. v&hisl tit.
mat. Fiz., 22,6,1508-1513,
conditions
1982.
u IS = $ (x, t) are known.
L. L. Kosachevskaya, V. V, Romanovtsev and I. E. Shparlinskii
3. We make two remarks in conclusion. First, given poor scaling of the initial matrix, the growth of the conditionality number of the intermediate matrices occurs, not only in the optimal elimination method, but in all versions of Gaussian elimination which provide for normalization of the leading row by the leading element, and notably, in most versions of Jordan’s method. The reason is always the same: the expressions for the different coefficients of the k-th intermediate matrix have a different degree of homogeneity relative to the elements of the initial matrix.
Second, we must emphasize that all the matrices of our numerical example are diagonally dominant. The growth of conditionality observed for matrices A (a) for large and small a! at the intermediate steps of the optimal elimination method needs to be associated with the following fact, proved in [2] . If the optimal elimination method is performed in the modified form, which does not require division of the leading row by the leading element, then, for any k, we have the inequality (in the notation of Section 1) llCII~lIDllnr=condw(C)=411‘4IIMllBIIM. Here, therefore, the conditionality of the intermediate matrices is independent of the (uniform) scaling of matrix A and in essence, does not increase throughout the process. Translated by D. E. Brown
REFERENCES 1. VOEVODIN, V. V., Numerical methods of algebra (Chislennye metody algebry), Nauka, Moscow, 1966.
2. IKRAMOV, KH. D., On the conditionality of the intermediate matrices of the Gauss, Jordon, and optimal elimination, methods, Zh. vj&hisZ.Mat. mat. Fiz., 18, No. 3,531~545, 1978.
USSR Comput..hSzths.Math. Phys Vol. 22, No. 6, pp. 232-237, Printed in Great Britain
1982.
0041-5553/82307.50+,00 01984. Pergamon Press Ltd.
ON AN ITERATIVE PROCESS FOR THE NUMERICAL SOLUTION OF SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS* L L KOSACHEVSKAYA, V. V. ROMANOWSEV and I. E. SHPARLINSKII Moscow (Received
12 November
198 1)
AN INFINITE sequence of iterative parameters is chosen, whereby the corresponding iterative process is stable and has asymptotically best convergence. 1. Consider the system of linear algebraic equations Au=f, *Zh. vychisl Mat. mat. Fiz., 22,6,1504-1508,1982.
(1)
Short communications
233
where A is a symmetric square positive-definite matrix, having a complete system of eigenvectors and spectrum Sp (A), lying in the interval [m, M] , 0 < m
Uk=uk-I-a/,(AUk--l-f),
2, . . . .
(2)
where cu= (&k) is a sequence of iterative parameters. It is well known (see, for example, [l] ) that process (2) is optimal at the n-th step if we put ah =
2 l?‘f+?tL-((M-m)Tk
k=i, 2,. . . , n, ’
where ~1, . . . , rn is any permutation of the roots of the Chebyshev polynomial T, (x) = cos (?I arccos x).
There are many publications (see, for example, [l-5 ] and the references therein) in which an order is given for choosing 71, . . . , rn, such that iterative process (2) is stable, i.e. all the intermediate approximations ~1, . . . , Un have bounded norm (otherwise, rounding errors have a strong influence, and the computer may come to an emergency stop due to leaving the place mesh). In [5] an infinite sequence cr = (ok) was constructed, for which process (2) is stable and becomes optimal in an infinite sequence of steps ni. Below, we construct an infinite sequence 01= (ok), for which process (2) is asymptotically optimal at each step and is stable to computing errors. Given this property of the iterative scheme, we can manage without a priori estimates of the required number of iterative steps and the process can be stopped in accordance with the properties of the approximations obtained. One important criterion for stopping the process is given in Section 4 below. 2. Denote by p = (p(k)) the Van der Corput sequence (see [6] ), i.e. p(k) =Z-‘e1+2-2e2+ . . . where k=ei+2ez+. . . +27-Ler is the binary expansion of integer k (there are quite +2-‘er, economic algorithms for computing sequence p, see, for example [6]). We put ar -
2 M+m-(M-m)cosnp(k)
’
k==i, 2,. . . ,
where /.Iis the conditionality number of matrix A : p = M/m. We shall assume that each step of the iterative process involves an error of norm not exceeding TJ,i.e. instead of (2) we have h-h-1
-Ur(Aur-1-f)
+qr,
k--i, 2, . . . ,
Denote by E&= ](uk - u IIthe error of the k-th approximation.
(4)
L. L. Kosmhevskaya, V. C: Romunovtsev and 1: E. Shpariinskii
234 Theorem
Let the parameters in iterative process (4) be given by (3); then, n+o(n) + O(q),
n+-,
where the constants in the order symbols o and 0 depend only on matrix A. 3. The proof of the theorem is based on the following Lemma 1, showing that the Van der Corput sequence is “distinctly” uniformly distributed (see [7 I); the lemma is a consequence of Theorem 2 of [7] and Theorem 9, Chapter 2 (61. Lemma
1
Let the function P(X) be Riemann-integrable. Then, a function e(N)+O, N-+ m, exists such that, given any k = 1, 2, , , . , we have W-N
1
zrp(P(i)) 1c
cp(s)ds - +
IS
e(N).
i-k+1
0
Lemma 2
For all X E [m, M] , 1
In
0J
p’h- 1
211
I
l-
do =lIl-.
M+m-(M-m)cosno
I
Proof: After changing the variable: y=2/[M+m-(M-m) evaluated in [8, Section 41.
p”+i
cosno]
, we obtain the integral
We put (I-arA),
Pk,N@)-=
Ph.N’
II i=k+l
max
IPk,N(k)
1,
IsSy(A)
where the max is taken over all A of the spectrum of matrix A. Lemma 3
A function e(N)+O, N+ 00, exists, such that, for all k = 1, 2, . . . , we have Nff+~(N)l .
proofiLet Sp(d)={hr, . . . , A,). Then, by Lemma 1, for each At, I< c < s, there is a function .et(N)+O, N+ =, such that, for any k = 1,2, . . . , 1
IJ l In
0
l-
2h M+m-(M-m)cosno
I
do
235
Short communications
--
1 N
hi-N
c
lnll-afhtj
G et(N),
ll-aikl>etW).
I
4=&+1
Applying Lemma 2, we obtain
InIl-aiht]
Q
et(N),
I
i-A+1 A-kN
Nine+
h(i-afht(<
Nei0’).
p”+l
c iaA+l
Consequently, k+N
lnll-arhrl
IPk,N(ht) I- exp
) s; (s)
Nexp[Ner(N)].
i-A+1
Putting
we obtain the lemma. Coroky.
Constants c > 0 and 8 < 1 exist such that, for all integers k and IV,we have
(5)
PA, N
Lemma 4 We have n
enceOPt,n+q
c
Ph,?L-h.
h-1
For the proof see, for example, [5, Section 11. From (5) we have
since 8 < 1. From this and Lemmas 3 and 4 we obtain the theorem. 4. Let us give an example of an algorithm for finding the approximate solution IIu, - uII Q E of E$. (1) with an assigned accuracy E > 0. Assume that the computational error is negligible, i.e. 11Q E. Then, from the estimate proved for n)no(e). Hence, for n > no (e), in the theorem, no (e) exists such that ilu,-ull
(6)
L, L. Kosachevskuya, V. V. Romunovtsevand 1: E. Shparlinskii
236
Further, it follows from (2) that Ilu,-ull=IIA-‘(Au,-Au)ll=IIA-1(Au
n-f) II=II~-~a~-:i(u,-u,+~)II
(7)
G ~-ian-:*Il~a+i-unll. Hence, if a,+*>l/2n and inequality
(6) holds, then we find from (7) that ]I+ - u ]I< E.
We define NO-[x(2(p-l))‘~~]+~. condition
(8)
We shall now show that, from the instant of satisfying
(6) until (8) is satisfied, we have to perform at most No steps.
We shall show that, among the No consecutive
values
, at
least one satisfies
to take the case M > 2m. We define an
(8). We first note that ak > l/M, so that it is sufficient integers
Q,+~, . . . ,ak+N,
by the inequalities 28-1 < Jd
(
p-l 2
%
< 2”.
1
Since p-1
2%2n
-
2
(
‘h
= n[20,+1)
]‘“
1
then, among the numbers k + 1, . . . , k + No, there is a number t, divisible by 2s. The binary form of this t is t=2ae,+,+ . . . +2r-ie,, and hence p(t) =2-+*ea+i+ . . . +2-re,<2-+1+. . . +2-m<2-*. SinceM>2m,
thens>
l,p(t)<%,
c0s[~p(~)l={l-sin2[np(t)]}~~~1-sin2[np(t)]> >1-(n.2-“)*)1--=-,
at=2{M+m-(M-m) 22[M+m-(M-m)
2
P-3
p--l
p-1
I-[np(t)
I2
cos [rip(t))}-1 (p-3)/(p-I)]-‘=1/2m.
and Notice that, from (7) and the inequality ok > l/M, we have IIu~-uH
Short communimtions
231
REFERENCES 1. MARCHUK, G. I., Methods of computational mathematics (Metody vychislitel’noi matematiki), Nauka, Moscow, 1980. 2. LEBEDEV, V. I., and FINOGENOV, S. A,, Solution of the problem of ordering the parameters in Chebyshev iterative methods, Zh. vychisLMat. mat. Fiz., 13, No. 1, 18-33, 1973. 3. NIICOLAEV,E. S., and SAMARSKII, A. A., The choice of iterative parameters in Richardson’s method, Zh vychisl Mat. mat. Fiz., 12, No. 4,960-973, 1972. 4. SAMARSKII, A. A., Introduction to the theory of difference schemes (Vvedenie v teoriyu raznostnykh
skhem), Nauka, Moscow, 1971. 5. LEBEDEVLV.I., and FINOGENOV, S. A., On the use of ordered Chebyshev parameters in iterative methods, Zh vychisLMat. mat. Fir., 16, No. 4, 895-907,
1976.
6. SOBOL’, I. M., MultCdimensionalquadrature formulae and hbar functions (Mnogomernye kvadraturnye
formuly i funktsii Khaara), Nauka, Moscow, 1969. 7. SOBOL’, I. M., On best uniformly distributed sequences, Usp. mat. Nauk, 32, No. 2,231-232,1975. 8. VOROB’EV, YU. V., A random iterative process, Zh. vychisl Mat. mat, Fin, 4, No. 6, 1088-1093, 1964.
USSR. Cornput. Maths Math. Phys Vol. 22, No. 6, pp. 237-243, Printed in Great Britain
1982.
0041-5553/82$07.50+.00 01984. Pergamon Press Ltd.
ON THE STABILITY OF POSITIVE SOLUTIONS OF CONVERSE PROBLEMS OF HEAT CONDUCTION* E. A. GRIGOR’EV Moscow (Received 9 December 1980)
THE CONDITIONS
second boundary
isolating compact correctness
value problems for the equation
classes are stated for the time-reversed of heat conduction.
first and
These converse problems are
shown to be stable in a uniform metric in the set of positive solutions. 1. Consider the boundary
au -e
at
value problem for the equation
a2u -+...
ax,2
U~S==$J(~,t),
azu +-,
(x, t)
axn2
x=(x* ,...,
x,).
E
n>l,
of heat conduction
D =QX(O, Tl, u(x, O)=cp(x),
xeC2,
where S is the lateral surface of a cylinder D with base Sl. Let the domain 52, the initial function cp, and the boundary
function
JI, be such that the problem has a unique solution u (x, t) in A
Assume that the solution u (x, 7’) is known at an instant T > 0. The reverse-time problem for the equation of heat conduction is the problem of finding u (x, t) for I < T given the function f(x)
= u (x, T), x E S2, when the boundary
*Zh. v&hisl tit.
mat. Fiz., 22,6,1508-1513,
conditions
1982.
u IS = $ (x, t) are known.