I. INTRODUCTION
In the numerical analysis of many physical
problems, oftentimes the formulation will lead to
simultaneous, linear, algebraic equations. Field
problems governed by Laplace, Poisson, and bi-
harmonic equations are common examples. Nu-
merical solutions of the diffusion equation using
implicit representation offers another instance. In
general, they find their application to the solution
of various types of differential equations, both ordi-
nary and partial. Due to the advent of modern
high speed computers, seeking the solution of these
equations becomes, in many cases, a daily routine
even if the number of unknowns gets large.
The usual method for the solution of such linear
equations is the Gauss elimination procedure. How-
ever, difficulties arise when the true solution is
sensitive to round-off errors injected during compu-
tation. The latter may accumulate to such an
extent that the numerical answers become com-
pletely worthless. Besides, the solution may also
be sensitive to small errors in the coefficients and
quantities at the right hand side. In either case.
the determinant of the coefficient matrix is nearly
singular. Such equations are described as ill-
conditioned.
The occurrence of ill-conditioned equations in
physical problems are by no means random. As
pointed out by Turing,('" there is a large class of
problems which naturally give rise to highly ill-
conditioned systems. Head and Oulton(2' indicated
their appearance in problems associated with air-
craft design. In the evaluation of local tempera-
tures at the sliding interface between a cutting tool
and flowing metal chip, ill-conditioned systems
were likewise encountered.(':) While the subject has
received the attention of many people in recent
years and some means of measuring "ill-condi-
tionedness" have developed, one immediately dis-
covers the lack of adequate information when
attempting to solve such equations with high speed
computers.
* Numbers in sUilerseript refer io Referencts Cited.
II. LITERATURE SURVEY
Attempts which have been made at solving ill-
conditioned, simultaneous, linear, algebraic equa-
tions fall into two main categories. First, there is
the indirect method, in which the unknowns are
obtained by successive approximations. The second
is the direct approach, in which the solution is
obtained by a single application of the process.
An example of the indirect method is the widely
publicized relaxation procedure of Southwell.
Shaw,(4) Buckingham,(,) and Fox(6) gave detailed
accounts of the procedure when applied to ill-
conditioned systems. They all concluded that the
convergence might become too slow to be of any
practical value. In fact, to ensure convergence of
the process, the coefficient matrix must be positive
definite and symmetric. 7) While any matrix can
be converted into a positive definite and symmetric
form through multiplication by its transpose, the
degree of ill-conditionedness is always aggravated
as a consequence of such operation.(8) The relaxa-
tion procedure is also not suitable for machine
computation. (9)
Another indirect method is the escalator process
developed by Morris.0)' It consists of expressing
the solution as a linear combination of column
matrices or vectors which are orthogonal. A similar
technique has been described by Fox, Huskey, and
Wilkinson.(1) They are applicable only to sym-
metric matrices. The accuracy of Morris' method
has been examined by Neville,(12) who criticized its
lack of provision for estimating errors. Morris'
procedure has also been criticized as not being well
suited for automatic machine computation.(1")
Booth(31 proposed the method of steepest de-
scent for solving ill-conditioned equations. The
theory was based on positive definite and sym-
metric matrix form. Due to round-off errors, the
actual path of solution may oscillate.
A unified convergence criterion valid both for
iteration by total step (Jacobi) and iteration by
single step (Gauss-Seidel) has been given by Gei-
ringer.(04 Group iteration was also discussed. If
the degree of ill-conditionedness were severe, the
criterion would most probably not be met.
Direct methods of solution have also been ex-
amined. Buckingham(5) discussed the use of the
Gauss elimination procedure and the difficulty of
using it to solve ill-conditioned systems. Nielsen(' )
favored Crout's modification of the Gauss elim-
ination procedure. This modification will allow
measurement of the degree of ill-conditionedness
by the relative magnitude of the elements of the
main diagonal as compared to the rest of the ele-
ments in the derived matrix. A re-arrangement of
equations would be made if one of the diagonal
elements is either very small or large. Bodewig('0)
also discussed the possibility of suitably re-
arranging the rows and columns in order to reduce
inherent errors in computation. Neither scheme is
suitable for machine computation.
A method which combines iteration and elim-
ination has also been suggested.('" The approxi-
mate solution obtained from the computer is
substituted into the original equations and the
residuals calculated using double-precision compu-
tation. Corrections for the solutions arc then
calculated from these residuals. Forsythe(8) gives
a survey of the many methods available for the
solution of systems of linear, algebraic equations,
and includes a rather extensive bibliography. This
paper describes a new approach to the problem. It
is condensed and modified from a technical re-
port('9) to which two of the authors contributed. In
many cases, improved results are obtained. Further
work needs to be done to fully explore its merit.
III. A MEASURE OF ILL-CONDITIONEDNESS
In a two dimensional space, the ill-conditioned-
ness of linear equations may be readily visualized
geometrically. It is associated with the near
"parallelism" of the straight lines represented by
the equations. In an n-dimensional hyper-space,
the linear equations may be interpreted as hyper-
planes. The coefficients in the equations are pro-
portional to the components of vectors normal to
such planes. When the equations are ill-conditioned,
two or more of these normals are nearly in the
same direction.
Consider the set of simultaneous equations:
Y a;; x; = .f (i = 1, 2, . . . n) (3.1)
i'-
which in matrix form may be written:
AX = F (3.2)
where A is the coefficient matrix [a;i]; X and F are
column matrices. As pointed out by Booth,(13 the
set (3.1) will be ill-conditioned if the absolute value
of the determinant of the normalized coefficient
matrix is very small as compared to unity. That is,
IA.N << 1 (3.3)
Other quantitative measures of ill-conditioned-
ness have also been discussed in literature. Von
Neumann and Goldstine(20) proposed the use of
Amnx
the so-called condition number, defined by X.n ,
where Xmax and Xinin are respectively the largest
and smallest eigenvalues of the coefficient matrix.
The larger this number, the higher the degree of
ill-conditionedness is. Unfortunately, such a cri-
terion is of little practical value, since the numerical
determination of Xmax and Xmin may oftentimes be
as long a process as the evaluation of the solutions
of the original equations.(21)
Employing the concept of the condition num-
ber described above, Taussky(8) developed a useful
theorem concerning ill-conditioned matrices. If A
represents a real, non-singular matrix, and A' its
transpose, then their product AA' is more "ill-
conditioned" than A. This has been confirmed by
Hartree.'21' It seems that this important result
has not received the attention it deserves.
Turing(') suggested the use of other condition
numbers which also involve the multiplication of
the coefficient matrix by its transpose. In view of
Taussky's finding, such use has been discarded in
favor of the simple criterion suggested by Booth
(Eq. 3.3).
While the inequality (3.3) has been widely used
to ascertain ill-conditionedness, it is by no means
a sufficient condition under all circumstances. This
may be demonstrated by the following examples.
Consider the set:
4.011 xt + 4.012 .X2 = 0.001
4.012 x: + 4.014.x2 = 0.002
which has the exact solution, x, = -1 and x2 = +1.
The solution as obtained from the ILLIAC,1 using
the method of elimination and with round-off at
the 12th significant figure, has been found to be:
xt = -1.00000001819 and x2 = +1.00000001819.
If errors of +3 to 10 hundredths of one percent
are injected into the coefficients and quantities on
the right hand side, the system will read:
4.012 xt + 4.011 x2 = 0.001001
4.013 -xi + 4.013 x2 = 0.002002J
The corresponding ILLIAC solution is xi =
-1.0000022347 and x2 = +1.0005011134. These
agree with the exact solution to a high degree of
accuracy. On the other hand, when one computes
A.v of (a), it is found to be equal to 0.000705;
very small indeed as compared to unity.
Let us now consider a different set which has
the same coefficient matrix as that of (a). Thus,
4.011 x, + 4.012 x2 = 4010
4.012 x, + 4.014 x2 = 4010
' The digital computer at the University of Illinois.
BULLETIN 459. ILL-CONDITIONED LINEAR ALGEBRAIC EQUATIONS
which has the exact solution, xl = +2000 and
x2 = -1000. The ILLIAC solution reads xi =
+2000.00003641 and x2 = -1000.00003638. Again,
if errors of + 3 to 10 hundredths of one percent
are injected, we obtain
4.012 x1 + 4.011 x2 = 40111
(d)
4.013 x. + 4.013.. = 4012
The solution as obtained from the ILLIAC now be-
comes: x = +999.5016083 and x2= +0.0002492015.
These in no way resemble the actual solution.
Lastly, if errors are only introduced into the coeffi-
cients, i.e., if we seek the solution of the set:
4.012 x, + 4.011 x2 = 4010 (e)
4.013 x, + 4.013 x2 = 40101
we obtain: x1 = +1998.5048432 and x2 =
-999.2524135. The foregoing examples clearly
illustrate the inadequacy of describing the degree
of ill-conditionedness by ANv alone, without any
reference to the quantities on the right hand side,
namely, the fi's.
As has been mentioned earlier, the ill-condi-
tionedness of (a) is geometrically associated with
the near parallelism of the lines represented by the
equations. Changes in the quantities fe's will exe-
cute a parallel shift of these lines. A question then
naturally arises: Why is the ill-conditionedness of
the set (a) not significantly influenced by the
changes in f's as demonstrated, while the set (c) is?
The reason becomes obvious if one observes that,
in the latter case, the two f,'s are equal, and con-
sequently, in the process of computation, one en-
counters the difficulty of taking small differences
of two large and nearly identical numbers.
A mathematically rigorous analysis of ascer-
taining the ill-conditionedness of any set of linear
equations needs separate research.
IV. A NEW ITERATIVE PROCEDURE
We shall now describe how to modify a given
ill-conditioned matrix in order to make it a better
conditioned one. Then we shall apply a new itera-
tive procedure to the modified system to obtain a
solution. It should be noted that the method does
not require that the matrix be symmetric, positive
definite, or both.
Consider the ill-conditioned normalized matrix
of real numbers
(4.1)
AN = [ai]
Let us add to A . the diagonal matrix
71 0 . . . O
S= . . . . . .
_0 0 . 7,_
where the 7y's are, for the moment, arbitrary
numbers. Let us now form the modified matri
611-+71 (12
62t (22 -
AN + = .
J.,1 (n2
+-72
S . BiB+7aJ
aaa. ý
A measure of the improvement in the 'condi-
tion' of a system whose matrix of coefficients is
given by (4.3) may be expressed by the ratio of
the absolute values of the determinants of AN and
AN+F, namely,
S_ I[ A[ labs
= I AN + F Iabs
For, clearly, if 3<<1, the improvement will be
significant and the originally ill-conditioned matrix
will become a well-conditioned one. 0 is called the
conditioning index.
The problem now is one of selecting the y7's
appropriately so that
2 In a private communication, Dr. S. D. Conte of Space Tech-
nology Laboratories, California, recently informed the authors that the
proposed method was similar to the one used by Riley.'22) However,
Riley considered only the positive definite and symmetric matrix.
(4.4)
Unfortunately, this problem is extremely in-
volved and, instead, we solve a simpler one which
in practice is oftentimes adequate. We take 1 =
72= .. =7y,=7y and show that 7, under condi-
tions stated below, can always be chosen so that
(4.4) obtains.
For this case of equal 7y's (4.3) becomes AN+-l,
where I is the unit matrix, and, as is well known,
A., + 7-y[ I,,, = 7"y + pY"- + p27-2
+ . . . + P-i 7 + pI -- P (7) , (4.5)
wherein the pi's are constants depending only on
(4.2) the elements a,7 and the vertical bars on the right
side of the equal sign are now absolute value signs.
Clearly,
'real P (0) p,- = AN Aabs , 0
ix"2 since we are considering non-singular matrices.
Therefore, if pn-i#O, there will be no relative
maximum or minimum of P(y) at 7= 0. Hence, in
view of the continuity of P(y), there exists a c >0,
(4.3) such that
P (7) 1,- > AN Iabs,
P (7) ,=-, > A ,,b.,
(4.6)
(4.7)
In practice, one computes the determinants
AN +7y abs for 7= ±0 and selects the larger of
the two quantities.
If pni =0, which corresponds to a rare matrix
form, P(7) may have a relative maximum or mini-
mum, or an inflection point at 7 = 0. If a maximum
occurs, it is not possible to improve the matrix by
the present method.
Let us suppose that the ill-conditioned system
of equations
n
Z aix; = fi (i = 1, 2, . . . , n)
i=i
(4.8)
is modified according to the present method. Then
the improved system will read:
SA., + F' ,,,s > I A, I ,Ibs
BULLETIN 459. ILL-CONDITIONED LINEAR ALGEBRAIC EQUATIONS
(anl+yi) xi+a12 X2+. .+a, X1n =f1+y1 x
a21 ix+ (a22+7y2) x2+. . .+a2.x =f2 +y722 (4.9)
an, xl+a2 x2,+. . .+(a.n+yn) 1x =fn+y,y x,
Note: In what follows the restriction of equal yi's
will not be imposed because the analysis is per-
fectly general and does not require it.
Mathematically speaking, equations (4.8) and
(4.9) are equivalent. However, their behavior with
respect to arithmetical operations in machine com-
putation may be entirely different. This arises
from the fact that the coefficient matrix in (4.9)
is better conditioned.
To solve the system (4.9) we shall adopt the
following iterative procedure. In the first step we
delete the terms 7yxi(i= 1, 2, . . . , n) on the right
side of equations (4.9) and solve the resulting
system on the ILLIAC. The solution so obtained
we designate by
V(1) =
- (1)-
(1)
(4.10)
translation of coordinates from X to E(). We
observe that the coefficient matrix of (4.12) is
the same as that of (4.8), and hence it cannot be
solved directly on the computer. To solve (4.13)
we simply proceed as before, and write
(A + F) s(2) = F (1)
(4.14)
where again the term on the right side, namely,
pr(2) has been deleted. The system (4.14) is again
solved by the ILLIAC. Let us denote this solu-
tion by
- (2)
1;-) = (
(4.15)
The difference E(') -(2) is the error incurred in
calculating -(2) and we designate it by E(). There-
fore, E(2)=E - (2)=X- -(1) (2). If the fore-
going procedure is repeated m times, we obtain
E(-) = E(1-) - (V) = X - ';(i)
X= 2 2() + E(-)
j=1
(4.16)
If the exact solution of (4.8) is X, the difference
X-Zo( will then be the error in the first iterative
solution and, accordingly, we write
e2(1)
E( = X - E"> =
(4.11)
If the method converges, then the solution of the
original system is
X = E w
(4.17)
The condition under which convergence takes
place and the expression for the error term after
m iterations will be given later.
At this point we list the formulae for the deter-
mination of (i)'s, namely,
By comparing the original set of equations (4.8)
with the set satisfied by Er0, we obtain
an el1)+a 12 e2( +...+a . en(1) = 71 1)
a21 el(1)+a22 e2(1)+. . .+a2 e,(1) = 2 2() (4.12)
a., e1 )+a.2 e2(1) +. .. +an e,,(1 = y . )
or in matrix notation
AE(1) = p a ) (4.13)
Geometrically, we may interpret this as a
(A + F) (m") =
F when m = 1,
Fg(-I) when m > 1.
An important feature of this method is that the
coefficient matrix for the determination of succes-
sive iterations remains the same. This fact makes
it possible for the complete iterative computation
to be readily programmed on the ILLIAC.
As has been pointed out previously, in the
present method of calculation there is a displace-
ment in the origin of the coordinate axes for the
unknowns in the equations after every cycle of
iteration. Errors will thus likely be accumulated
1-1
IV. A NEW ITERATIVE PROCEDURE
in the sum Y 2"i as the process continues. This
i=1
source of error may be conveniently removed by
starting anew with (4.9) after the r-th cycle, but
this time with the terms 7aix (i= 1, 2, . . . , n) at
the right hand side of the equations not set equal
to zero, but replaced by the terms appearing in
the column matrix F Z i). The iteration pro-
i=1
ceeds in the usual manner and this artifice may
be repeated if necessary.
A. THE GEOMETRY OF THE NEW
ITERATIVE PROCESS
Before discussing the convergence criterion, it
is interesting and instructive to describe geomet-
rically the nature of the convergence of the new
procedure. For ease of illustration we shall con-
sider a system of two equations in two unknowns.
Suppose, therefore, that the given system is
al xl + a12 2 = ffl
a21 .rX -+ a22 2= 2
(i)
Modifying it for the first iteration, we have
(an + 71) ý1(1) + a12 2(1) = fl,
a21 (1) + (a22 + 72) 2(1) = f2,
(j)(1)
(ii) (1)
in which 71 and 72 are of the same sign as that of
a1n and a22. In Fig. 1, the original equations are
Fig. I
represented by lines (i) and (ii), while the modi-
fied set is represented by lines (i)(1) and (ii)().
The latter intersect at a larger angle and hence
they are better conditioned. Their point of inter-
section ( 2(1), 2() represents the result of the
first iteration as well as the origin of the new
coordinate system el1l) and e2('). The original equa-
tions, when referred to the new coordinates, are:
anl e(1) + a12 e1 = 71(1),
a21 e(1) a22 e2(1) = 72 2(1)
For the second iteration, the modified equations
become:
(an + 7_) ~1(2) + a12 ý2(2) = 71 i(1)
(i)(2)
a21n -2) + (a22 + 72) 22) = 72 2(1), (ii)(2)
which are represented by lines (i)(2) and (ii)(2).
Their intersection is the point (W1(2), ý2(2)). Provided
the method yields a convergent sequence, then,
by repeating the foregoing procedure a sufficient
number of times, the successive intersections tend
to the required solution (xi, x2).
B. THE CONVERGENCE CRITERION
AND ERROR ESTIMATION
On the basis of equations (4.18), we have
(1) = (A + r)-1 F = C = [ci]
E(") = (A + F)-1 r(m-1), m > 1
(4.19)
Let (A+r)-1'F-B=[bi]. Then for m>1, we
have
V(m--1) = BEV(m-2)
(3) = BI(2)
g(2) = Bg(1) = BC
Therefore,
(") = Bm-1C, m > 1
where B° = I, and
) = (I + + B2 + . . . + B-..+B 1) C.
i=1
Let us now assume that 5 Bi converges. Then,
j=0
in view of the fact that X = C+BX, we have
X = (I - B)-1 C = BC,
and therefore
and therefore
BULLETIN 459. ILL-CONDITIONED LINEAR ALGEBRAIC EQUATIONS
m /C m-1 \
X - B7) = fB - Bi C
j=l 0 j=o
= BC = (I - B)-1 B"C
= B" (I - B)-1C - E ). (4.20)
Now, lim B"'= 0 because T Bi< 0m.
Therefore, taking the limit of both sides of (4.20)
as m- c, we see that lim E(m) = 0
m--*cO
00
x = E S"
jyi
Consequently, we have the following Theorem.
If E Bi converges, then the series E converges
to the solution X. Furthermore, the error E(>')
committed after m iterations is given by either
(I-B)-1 BmC or B"(I-B)-IC.
In summary, it should be emphasized that the
method here discussed requires the satisfaction of
two conditions, namely, (i) the improvement of
the condition of the matrix and (ii) the conver-
gence of Z B'. The selection of I (or even -yl)
which will ensure the convergence of Z Bi is a
difficult problem. Even after this is done the com-
putation of E(m) can be formidable. Consequently,
we propose the following simple alternative; and
we seek an expression for an upper bound of error
instead of calculating the error itself. In practice,
this is what one usually needs.
Referring to (4.19), let us denote the inverse of
the matrix (A + ) by D. Thus,
where [yMu and (Imv-(' \ are the maximum values
of the y I's and the ,('-I 's (i= 1, 2, 3, . . . , n),
respectively. Since (4.22) holds for r = 1, 2, 3, . . . ,
n, it will hold if the left hand side is replaced by
ý-(m) I = max {f I '() , 12 w I) , n,(-) I}
Therefore, we may write
(ve") d | |--1- ,
-Y I i AI
r = 1, 2, 3, . . . , n.
(4.23)
Let d. = max {i d0 , d2; , , . , ldi ,
an= let usi= i=require that the product
and let us require that the product
Sy1 (IuM = K < 1,
or, in the case of equal y;'s,
S7 Im = K < 1.
(4.24)
(4.25)
K will be referred to as the convergence constant.
These conditions can always be satisfied by
choosing yM or - sufficiently small. To be sure,
small values of -y may result in only a slight im-
provement of the matrix condition. Consequently,
a compromise has to be made which depends on
the degree of ill-conditionedness of the original
set of equations, the round-off errors injected
during the computation, cost of machine time, and
the accuracy required in the final result.
Returning to (4.22) and (4.24), we have for
m >l,
(A + F)-1 = D = [d,;].
Then the r-th row of ("') (m> 1) will be given by
S= d 71 im-l) + d,2 Y2 2(m-1)
+ . . . + d, 7y ,( "-1),
r = 1, 2, 3, . . . , n,
for which the following inequality holds,
| (. I | dri y- y ( -1) + dr, 72 7 2 '"-,)
+ . . . + d,, , 7 ,n ,, -1) 1(
(4.22)
SY MI{ dil + drI +.
+ d, } I- ,
where it is assumed that ' 1m-1) 0. Therefore,
by a well-known theorem, the series Z ^u"') is
absolutely convergent. m
From the foregoing discussion it is seen that,
in some instances, it is possible to relax the restric-
tion imposed by (4.24) or (4.25), i.e., to allow
1- d.u to go slightly beyond unity, yet (4.26) is
still satisfied. Example 2 in the following section
is selected to demonstrate this point.
Consider now the expression for each individual
component of the matrix X, namely,
Xr = = )+ , ), r = 1, 2, 3, . . . , n.
)= )=1 i'=m+l
(4.21)
M-"-) K < 1,
ý_M1 I--1)1
(4.26)
IV. A NEW ITERATIVE PROCEDURE
The error committed after m it erations is given by
J=t I=
and
| ,<> 4u< | , r = 1, 2, 3, .. . V, n.
However, in view of the fact that |,9)| ( [< 0[
and that (4.26) obtains, we have
[(<,, < K ,(") (1 + K + K2 + ...)
= -- 1K l"c), j'(4.27)
r = I or 2 or 3 ... or n. Thus, the absolute values
of the errors committed after m iterations are
bounded by I(")J K/(l-K).
In conclusion, the authors wish to emphasize
the fact that the procedure proposed in the paper
does not exclude the use of any refined programs
of computer calculation, such as the double preci-
sion routine, when the modified set (4.18) is solved.
On the other hand, such refined computation pro-
cedure is not a substitute for the present method,
which hinges on an appropriate modification of the
coefficient matrix and iteration with progressive
displacements of the vector 7. Rather it supple-
ments the many known methods of solving linear
systems when the latter are ill-conditioned.
V. NUMERICAL EXAMPLES
We shall now demonstrate the usefulness of
the proposed method by two numerical examples.
In each case, improvement on the solution was
clearly indicated as compared with the well-known
"method of elimination."
Example 1:
Consider the set:
4.011 x, + 4.012 x2 = 1.0001
4.012 x, + 4.014 x2 = 2.1001
(5.1)
whose solution accurate to 10 significant figures is
x, = 1098.664654 and x2 = +1098.640407.
In order to examine the influence of round-off
errors on the solution we arbitrarily decree that all
calculations be carried out to 4 significant figures.
Using the method of elimination, one obtains:
x, = - 549.8
x2 = +550.0
These represent errors of approximately 50%.
Next, we solve (5.1) by the proposed iterative
method. For m = 1, the modified equations become:
(4.011 + -y) 11) + 4.012 ý2(1) = 1.000 1
(5.2a)
4.012 ,(i) + (4.014 + 7) 2(1) = 2.100
If y is selected to be 0.002, one finds #=0.044,
which is much less than unity. The elements in
the inverse matrix D as defined in (4.21) are:
[ 200.2 -200.0]
-200.0 -200.1_
from which one computes the convergence con-
stant :
K= yd m=0.002 {| 200.2 +1 -200.1 } =0.8006,
which is also less than unity. This ensures the
convergence of the method. With -y = 0.002, and
using also the method of elimination, (5.2a) has
the solution:
V(1) = -220.0
ý2(1) = +220.0
For m=2, the modified equations are:
4.013 (2) + 4.012 (2) = "(1) = -0.4400
(5.2b)
4.012 ^(I) + 4.016 2(2) = "2(1) = +0.4400
whose solution is:
(2) = -175.9
ý2(2) = +175.9
The process has been repeated and the results ob-
tained from the first fifteen cycles of computation
are tabulated below:
m 61(W
1 -220.0
2 -175.9
3 -140.7
4 -112.5
5 -90.00
6 -71.98
7 -57.60
8 -46.08
9 -36.86
10 -29.48
11 -23.58
12 -18.86
13 -15.09
14 -12.07
15 -9.656
Thus,
and
15
xl = E 1(])
i=1
2"0)
220.0
175.9
140.7
112.5
90.00
71.98
57.60
46.08
36.86
29.48
23.58
18.86
15.09
12.07
9.656
+ ie15) = - 1061 + eC5)
15
X2 = () + e2 15) = +1061 + e2(15)
i=1
Using (4.27) one finds le \(15) or e25) < 0.8006
1-0.8006
X 9.6561 = 38.77. This compares with actual errors
of I -1098.664654+ 10611 -37.66 and 1098.640407
-10611 37.64 respectively.
V. NUMERICAL EXAMPLES
Example 2:
Consider the severely ill-conditioned set:
-3.0000000000 xi + 2.9999999999 x2
+ 2.9999999999 x3
= 8.9999999998
-2.9999999999 x, + 3.0000000000 x2
+ 2.9999999999 x (5.3)
= 8.9999999998
-2.9999999999 xi + 2.9999999999 X.2
+ 3.0000000000 x3,
= 8.9999999998
which has the exact solution x,= -1, x2= 1 and
a3 = 1 as can be verified by direct substitution.
The ILLIAC solution with round-off error at the
12th decimal place has been found to be:
x, = -0.2857142857
X2 = 1.2857142856
xa = 1.4285714287
In this case, if the convergence criterion (4.25) is
to be satisfied, one has to select an extremely small
value of 7. This would result in an insufficient
improvement in the matrix condition. As pointed
out earlier in the text, one may attempt to relax
the restriction on 7 and allow \y'dM to assume a
value slightly higher than unity. Let us thus try:
1 = -0.1, 7.2 ='3 = 0.1. The corresponding modi-
fied set is,
-3.1000000000 ý() + 2.9999999999 63(1)
+ 2.9999999999 6(")
= 8.9999999998
-2.9999999999 1(t) + 3.1000000000 (1)
+ 2.9999999999 6(") (5.4)
= 8.9999999998
-2.9999999999 ý1(1) + 2.9999999999 2(1)
+ 3.1000000000 3(l)
= 8.9999999998
When the elements in the inverse matrix D are
evaluated and du computed therefrom, one finds
that -y dM = 1.33.
The numerical value of the conditioning index
3 can not be found since, due to the extreme ill-
conditionedness of the original set, it is not possible
to evaluate [IAv by the computer. This is imma-
terial, however, because it is obvious that 0<<1.
At this stage of calculation, there is no assur-
ance that the method will converge. We simply
proceed with the iteration in the usual way and
obtain the results of the first six cycles as follows:
m ^1tm
1 -0.989010989
2 -0.010868253
3 -0.000119433
4 -0.000001311
5 -0.000000013
6 -0.000000002
+0.989010989
+0.010868253
+0.000119433
+0.000001311
+0.000000013
+0.000000002
+0.989010989
+0.010868253
+0.000119433
+0.000001311
+0.000000013
+0.000000002
It is seen that for all m's listed, the ratio
( < -1)
Hence, it seems plausible that convergence will
ensue. From the above table one has,
x£ ' -' >3 = -1.000000001
f=1
x32 ~ Z 2 = +1.000000001
j--
xa -- [ 4(i) = +1.000000001
1=1
The improvement in the accuracy of the solution
over that obtained by the direct method of elimi-
nation is obvious. However, in this case, estima-
tion of error becomes difficult.
An ill-conditioned set involving equations of
ten unknowns has been successfully solved using
the present method. This arises in the evaluation
of sliding contact temperature distribution at the
tool-chip interface. Readers are referred to
References Cited(31 for details.