General Mathematics Vol. 18, No. 1 (2010), 25–32
Remarks on the cyclic reduction method
1
Ioana Chiorean
Abstract
A tridiagonal system of equations appears often by discretization
of differential equations. Several methods for solving such a system
are known. The aim of this paper is to compare, from computational
time point of view, the cyclic reduction method and an alternative,
which uses recurrence relations, by using parallel calculus.
2000 Mathematics Subject Classification: 65F05, 65M06, 65Y05.
Key words and phrases: Numerical analysis, differential equation,
parallel calculus.
1
Received 05 June, 2009
Accepted for publication (in revised form) 15 September, 2009
25
26
1
I. Chiorean
Introduction
A tridiagonal system of equations has the following form:


b1 c1
0




 a 2 b2 c2





..
.


a 3 b3


(1)


.. .. ..
.
.
.






.. ..

.
. cn−1 


0
a n bn
or
(2)
ai xi−1 + bi xi + ci xi+1 = di ,
i = 1, n
with a1 = 0 and cn = 0.
Due to the fact that such a system appears very often in all sort of problems, there are many approaches for solving it, both with serial and parallel
computers. It is well known the Thomas algorithm, which generates the
solution after O(n) operations (see [2]), but also algorithms which generate logarithmical time of execution (see [1]), which use different parallel
techniques, as cyclic even-odd reduction.
In what follows, we take into account the Thomas algorithm and show
that its coefficient can be computed also in logarithmical time.
2
Thomas Algorithm
It is also called ”the tridiagonal matrix algorithm”.
Remarks on the cyclic reduction method
27
Starting with a system of type (1), the method has the following steps:
Step 1. Modify the coefficients of
 c
1

,


 b1

(3)
c0i =


ci


,

bi − c0i−1 ai

d1


,



 b1
d0i =
(4)



di − d0i−1 ai


,

bi − c0i−1 ai
system (1) according with formulas:
i=1
i = 2, 3, . . . , n − 1
i=1
i = 2, 3, . . . , n
Step 2. The solution is obtained by back substitution:
xn = d0n
xi = d0i − c0i xi+1 , i = n − 1, n − 2, . . . , 1
It is clear that the solution of our system is obtained after O(n) opera-
tions.
3
The Cyclic Even-Odd Reduction Method
Another possibility for solving a system of type (1) is given by a technique
called Cyclic Even-Odd Reduction.
In the general case, considering a first order recurrence
(5)
xi = ai + bi xi−1
where ai and bi are known (i = 1, n) and x1 , . . . , xn are to be calculated,
with x0 = α, given, we set a0 = α and ai = 0 for i < 0 and bi = 0 for i ≤ 0.
Then (5) will also hold for i ≤ 0, with xi = 0 for i < 0.
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I. Chiorean
In (3) we replace i by i − 1 and we get:
(1)
(6)
(1)
xi = ai + bi (ai−1 + bi−1 xi−2 ) = ai + bi xi−2
Then, in (6) we replace i by i − 2 and we get
(1)
(7)
(1)
(1)
(1)
(2)
(2)
xi = ai + bi (ai−2 + bi−2 xi−4 ) = ai + bi xi−4
Continuing in this way, xi can be expressed successively in terms on xi−1 ,
xi−2 , xi−4 , xi−8 , . . . and after at most k steps, where k = dlog 2 ne, the index
(k)
(k)
i − 2k will be negative or zero, and therefore bi−2k = 0, then xi = ai−2k .
How can we apply this method to solve a system of type (1)? Like in
[3], consider the three equations involving xi :
(8)
ai−1 xi−2 + bi−1 xi−1 + ci−1 xi = yi−1
(9)
ai xi−1 +
(10)
bi xi + ci xi+1 = yi
ai+1 xi + bi+1 xi+1 + ci+1 xi+2 = yi+1
By subtracting suitable multiples of equations (8) and (10) from (9), we
can obtain an equation of the form
(11)
(1)
(1)
(1)
(1)
ai xi−2 + bi xi + ci xi+2 = yi
By similar manipulation of three equations of the form (11) we obtain
(12)
(2)
(2)
(2)
(2)
ai xi−4 + bi xi + ci xi+4 = yi
and so on. Setting xi = ai = ci = yi = 0 and bi = 1 when i is outside the
relevant range, after k = dlog2 ne steps, we reach the equation:
(13)
giving the value xi .
(k)
bki xi = yi
Remarks on the cyclic reduction method
4
Thomas Algorithm revisited
Let’s consider, again, the system (1)
b1 c1
a 2 b2 c2
a 3 b3
(14)
..
.
0
Then
and the determinant of the matrix
0 ..
.
.. ..
.
.
.. ..
.
. cn−1 a n bn det1 = b1
b1 c1 = b2 b1 − a2 c1 = b2 det1 −α1 det0 ,
det2 = a 2 b2 ..
.
for det0 = 1 and α1 = a2 c1
b1 c1 0 det3 = a2 b2 c2 = b3 det2 −β2 det1 with β2 = a3 c2 b1
0 a 3 b3 and so on.
Then
(15)
detn = bn detn−1 −αn detn−1 ,
with αn apriori computed.
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I. Chiorean
Having these observations in mind, we may observe that formulas (3)
becomes:
c0i =
(16)

c1


c01 =
,


det1

i=1



c deti−1

 c0i = i
, i = 2, 3, . . . , n − 1
deti
Formula (15) is a linear recurrence with two terms which can be perform,
in parallel, in the following way:


detn
detn−1


=

=
(17)

=
b n αn

0

b n αn

1
0

b n αn

1
1
0

detn−1
detn−2


bn−1 αn−1
1
bn−1 αn−1
0
detn−2

0
1


 = ··· =
detn−3



b α1
det1
... 1


1 0
det0

So, at every moment of time, we may compute det i , i = 1, n, according
with relation (17). The computation takes place only at matricial level, in
the following way: if we denote

Mi = 
b i αi
1
0

,
i = 1, n,
the matrix multiplication in (17) may be performed on a binary tree network
with 2n − 1 processors:
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Remarks on the cyclic reduction method
Mn
Mn−1
Mn−2
Mn−3
...
M2
M1
Mn Mn−1
Mn−2 Mn−3
M2 M1
...
...
Mn Mn−1 . . . M1
After dlog2 ne parallel steps, the final product will be in the root. The
parallel technique used here is called ”double recursive technique” (see [3]).
Concerning the coefficients d0i , i = 1, n in (4), a similar result may be
obtained, by considering the following determinants:
det01 = b1
b1
0
det2 = a2
b1
0
det3 = a2
0
..
.
b1
a2
det0n = 0
..
.
0
d1 d2 c1
b2
a3
d1 d2 d3 c1 . . . . . .
0
d1
b2
c2
...
0
d2
a3
b3
...
..
.
0
d3
0
. . . . . . b n dn
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I. Chiorean
Then
d0i =

d1


d01 =
, i=1


det1



0


 d0i = deti ,
deti
5
i = 2, 3, . . . , n
Conclusions
In this paper, we give an anternative to the cyclic even-odd reduction scheme
for solving a tridiagonal system of equations, by using Thomas algorithm
and the double recursive technique. Our approach generates also an execution time of order O(log2 n).
References
[1] I. Chiorean, Calcul paralel, Ed. Microinformatica, 1995.
[2] S.D. Conte, C. de Boor, Elementary Numerical Analysis, Mc Graw-Hill,
New York, 1972.
[3] J.J. Modi, Parallel Algorithms and Matrix Computation, Clarendon
Press, 1988.
Ioana Chiorean
Babeş-Bolyai University
Department of Applied Mathematics
Kogălniceanu 1, Cluj-Napoca, Romania
e-mail: robica chiorean@yahoo.com