# Convergent Homotopy Analysis Method for Solving Linear Systems

Authors: H. Nasabzadeh; F. Toutounian
Journal: Advances in Numerical Analysis
 (2013)

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## Abstract

By using homotopy analysis method (HAM), we introduce an iterative method for solving linear systems. This method (HAM) can be used to accelerate the convergence of the basic iterative methods. We also show that by applying HAM to a divergent iterative scheme, it is possible to construct a convergent homotopy-series solution when the iteration matrixG of the iterative scheme has particular properties such as being symmetric, having real eigenvalues. Numerical experiments are given to show the efficiency of the new method.


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## Body

## 1. Introduction

Computational simulation of scientific and engineering problems often depend on solving linear system of equations. Such systems frequently arise from discrete approximation to partial differential equations. Systems of linear equations can be solved either by direct or by iterative methods. Iterative methods are ideally suited for solving large and sparse systems. For the numerical solution of a large nonsingular linear system,(1)Au=b,
where A∈ℝn×n is given, b∈ℝn is known, and u∈ℝn is unknown, one class of iterative methods is based on a splitting (M,N) of the matrix A, that is,
(2)A=M-N,
where M is taken to be invertible and cheap to invert, which mean that a linear system with matrix coefficient M is much more economical to solve than (1). Based on (2), (1) can be written in the fixed-point form
(3)u=Gu+c,G=M-1N,c=M-1b,
which yields the following iterative scheme for the solution of (1):
(4)u(k+1)=Gu(k)+c,k=0,1,2,…,u(0)∈ℝnisarbitrary.A sufficient and necessary condition for (4) to converge to the solution of (1) is ρ(G)<1, where ρ(G) denotes spectral radius. Some effective splitting iterative methods and preconditioning methods were presented for solving the linear system of (1), see [1–9]. Recently, Keramati [10], Yusufoğlu [11], and Liu [12] applied the homotopy perturbation method to obtain the solution of linear systems and deduced the conditions for checking the convergence of homotopy series. In this work, we show how the homotopy analysis method may be regarded as an acceleration procedure based on the iterative method (4). We observe that it is not necessary that the basic method (4) be convergent. When ρ(G)>1, it is sufficient that the eigenvalues λi=Re(λi)+iIm(λi), i=1,…,n of iteration matrix G satisfy the relation Re(λi)<1, i=1,…,n, (or Re(λi)>1, i=1,…,n). When ρ(G)<1, by applying the homotopy analysis method to the basic iterative method (4), we can improve the rate of convergence of the iterative method (4). This paper is organized as follows. In Section 2, we introduce the basic concept of HAM, derive the conditions for convergence of the homotopy series, and apply the homotopy analysis method to the Jacobi, Richardson, SSOR, and SAOR methods. In Section 3, some numerical examples are presented to show the efficiency of the method. Finally, we make some concluding remarks in Section 4.


## 2. Basic Idea of HAM

The homotopy analysis method (HAM) [13, 14] was first proposed by S. J. Liao in 1992. The HAM was further developed and improved by S. Liao for nonlinear problem in [15].Here, we apply the homotopy analysis method (HAM) to the problem (3) for finding the solution of (1) when det(A)≠0. Consider (3), where u is unknown vector of (1) and G is the iteration matrix of an iterative method. Let w0 denote an initial guess of exact solution u, ℏ≠0 an convergence control parameter. Then, we can apply the homotopy analysis method and define H(u,p) by
(5)H(u,0)=F(u),H(u,1)=L(u)
and a homotopy as follows:
(6)H(u,p)=(1-p)F(u)-pℏL(u)=0,
where
(7)L(u)=(I-G)u-c,F(u)=u-w0,
and p is an embedding parameter. Hence, it is obvious that
(8)H(u,0)=u-w0,H(u,1)=(I-G)u-c.
And as the embedding parameter p∈[0,1] increases from 0 to 1, the solution of H(u,p) varies continuously from the initial approximation w0 to the exact solution u of the original equation L(u)=0. The homotopy analysis method uses the parameter p as an expanding parameter (see [16–18]) to obtain
(9)u=u0+pu1+p2u2+⋯,
and it gives an approximation to the solution of (3) as
(10)v=limp→1(u0+pu1+p2u2+⋯)=limp→1∑k=0∞‍pkuk=∑k=0∞‍uk.By substituting (9) in (6) and equating the terms with identical powers of p, we can obtain
(11)p0:u0-w0=0,p1:u1-ℏ[(I-G)u0-c]=0,pi:ui-ui-1-ℏ(I-G)ui-1=0,i=2,3,….This implies that(12)u0=w0,u1=ℏ[(I-G)u0-c],ui=[(ℏ+1)I-ℏG]ui-1,i=2,3,….Taking(13)Gℏ=(ℏ+1)I-ℏG
yields  that
(14)u1=ℏ[(I-G)u0-c],ui=Gℏui-1,i=2,3,…,
where u0 is an initial guess of exact solution u. Therefore,
(15)v=u0+∑i=1∞‍piGℏi-1u1.By settingp=1, we obtain
(16)v=u0+∑i=1∞‍Gℏi-1u1.It is obvious that ifρ(Gℏ)<1, then the series, ∑i=1∞Gℏi-1u1, converges and we have
(17)v=u0+(I-Gℏ)-1u1=u0+ℏ-1(I-G)-1u1=(I-G)-1c,
which is the exact solution of (3). A series of vectors can be computed by (14), and our aim is to choose the convergence control parameter ℏ≠0 so that ρ(Gℏ)<1. For improving the rate of convergence of iterative method, we present the following theorem.Theorem 1.
Suppose thatρ(G)<1, and let λi=Re(λi)+iIm(λi), and μi, i=1,2,…,n be the eigenvalues of G and Gℏ, respectively. Let αi=(1-Re(λi))2+Im(λi)2, βi=2(1-Re(λi)), and let gi(ℏ)=|μi|2-ρ(G)2=αiℏ2+βiℏ+1-ρ(G)2, i=1,2,…,n. If αi≠0 and βi-2αi≠0, i=1,2,…,n, then  (i)
the quadratic equationgi(ℏ)=0, i=1,2,…,n, has simple real roots γ1(i)<γ2(i)<0,(ii)
ℏ=-1 belongs to the interval [maxi=1nγ1(i),mini=1nγ2(i)] and ρ(G-1)=ρ(G)<1,(iii)
for eachℏ∈[maxi=1nγ1(i),mini=1nγ2(i)] and ℏ≠-1, the relation ρ(Gℏ)<ρ(G)<1 holds.Proof.
(i) We begin by defining two index setsN1={i∣|λi|=ρ(G)} and N2={i∣|λi|≠ρ(G)}. Since αi>0 and ρ(G)<1, for i∈N2, we have
(18)signgi(-∞)=1,gi(-1)<0,gi(0)>0.
So,gi(ℏ), i∈N2, has simple real roots
(19)γ1(i)<-1,-1<γ2(i)<0.
Fori∈N1, we have αi-βi<0 and
(20)gi(-1)=0,gi(αi-βiαi)=0.
So, by using the assumptionβi-2αi≠0, gi(ℏ), i∈N2, has also simple real roots γ1(i)<γ2(i)<0, defined as follows:
(21)γ1(i)=-1,γ2(i)=αi-βiαiifαi-βiαi>-1,γ1(i)=αi-βiαi,γ2(i)=-1ifαi-βiαi<-1.
(ii) From part (i), we observe that-1∈[maxi=1nγ1(i),mini=1nγ2(i)] and gi(-1)=0. This implies that ρ(G-1)=ρ(G)<1.
(iii) From part (i), we also observe that for eachℏ∈[maxi=1nγ1(i),mini=1nγ2(i)] and ℏ≠-1. We have gi(ℏ)<0, and the relation ρ(Gℏ)<ρ(G)<1 holds.The following theorem shows that by applying the HAM to a iterative scheme which is divergent, it is possible to construct a convergent homotopy-series vectors when the iteration matrixG has particular properties.Theorem 2.
Letλi=Re(λi)+iIm(λi), and μi, i=1,2,…,n, be the eigenvalues of G and Gℏ, respectively. Let αi=(1-Re(λi))2+Im(λi)2 and βi=2(1-Re(λi)). Suppose that αi≠0, i=1,2,…,n.  (i)
IfRe(λi)>1,  for  i=1,2,…,n, and ℏ∈(0,min1≤i≤n-(βi/αi)), then ρ(Gℏ)<1.(ii)
IfRe(λi)<1,   for  i=1,2,…,n, and ℏ∈(max1≤i≤n-(βi/αi),0), then ρ(Gℏ)<1.Proof.
From (13), we have μi=1+ℏ(1-Re(λi))-iℏIm(λi). So, it is sufficient to have
(22)|μi|2-1=αiℏ2+βiℏ<0,fori=1,2,…,n.
Under the hypothesis of part (i), we haveβi<0, i=1,2,…,n, and the relation (22) holds if ℏ∈(0,min1≤i≤n-(βi/αi)), and the proof of (i) is complete. A similar argument holds if Re(λi)<1,  for i=1,2,…,n, and part (ii) follows.When the assumption of part (i) (or part (ii)) of Theorem2 does not hold, the following theorem shows that, for certain cases, instead of (3) we can consider the equivalent equation
(23)u=G¯u+c¯,withG¯=(G-I)2+I,c¯=(G-I)c,
in which the iteration matrix G¯ has the eigenvalues with the desired properties.Theorem 3.
Letλi=Re(λi)+iIm(λi), and νi=Re(νi)+iIm(νi), i=1,2,…,n, be the eigenvalues of G and G¯, respectively.  If (1-Re(λi))2-(Im(λi))2>0 (or (1-Re(λi))2-(Im(λi))2<0) for i=1,2,…,n, then Re(νi)>1 (or Re(νi)<1) for i=1,2,…,n.Proof.
The proof immediately follows from the fact thatRe(νi)=((1-Re(λi))2-(Im(λi))2)+1.The following corollary shows that by using the modified linear equation (23) and the homotopy analysis method with the corresponding G¯ℏ=(ℏ+1)I-ℏG¯, we can construct a convergent homotopy series for linear system (1).Corollary 4.
IfG has only real eigenvalues, then there exists ℏ≠0 such that the series of vectors generated by
(24)u1=ℏ[(I-G¯)u0-c¯],ui=G¯ℏui-1,i=2,3,…,
converges to the exact solution of (1).Proof .
The proof immediately follows from Theorems2 and 3.This corollary establishes that the series of vectors generated by (24) always converges if the iteration matrix G is a symmetric matrix. When A is symmetric with diagonal elements positive real numbers, (1) can be written as follows: (D-1/2AD-1/2)(D1/2x)=(D-1/2b), where D is the diagonal of A. Denoting again by A, x, and b the expressions (D-1/2AD-1/2), (D1/2x), and (D-1/2b), respectively, it is obvious the new coefficient A is still symmetric and can therefore be written in the form A=I-L-LT. An immediate consequence of Corollary 4 and the above discussion is the following results.  (i)
The series of vectors generated by (24) converges when A is a symmetric matrix and the iterative method is the Richardson method
(25)u(k+1)=(I-A)u(k)+b.(ii)
The series of vectors generated by (24) converges when A=I-L-U is a symmetric matrix and the iterative method is the Jacobi method
(26)u(k+1)=(L+U)u(k)+b.(iii)
IfA=I-L-U is a symmetric matrix and the iterative method is SAOR method
(27)u(k+1)=Jr,wu(k)+c
with
(28)Jr,w=(I-rU)-1[(1-ω)I+(ω-r)U+ωL]×(I-rL)-1[(1-ω)I+(ω-r)L+ωU],c=ω(I-rU)-1[(2-ω)I-(ω-r)(L+U)](I-rL)-1b,
then the series of vectors generated by (24) converges if
(29)ω∈(0,2),ω+2-ωμmin<r<ω+2-ωμmax,
where μmin and μmax stand for the minimum and the maximum eigenvalues of B=L+U, respectively. This result follows from the fact that in this case the iteration matrix Jr,w of SAOR method has real eigenvalues (see Theorem 2 in [19]).(iv)
IfA=I-L-U is a symmetric matrix and the iterative method is SSOR method, then the series of vectors generated by (24) converges if ω∈(0,2). This result immediately follows from the fact that the SAOR method reduces to the SSOR method for r=ω.


## 3. Numerical Examples

For numerical comparison, we use some matrices from the University of Florida sparse matrix Collection [20]. These matrices with their properties are shown in Table 1. We determined the spectral radii of iteration matrices of the classical SOR, AOR, SSOR, SAOR, Jacobi, and Richardson methods (ρ(G)) as well as those of the corresponding Gℏ after the application HAM method with the experimentally computed optimal value of ℏ(ρ(Gℏopt)). In Tables 2–4, we list ρ(G), ρ(Gℏopt), the interval of convergence which introduced in Theorems 1 and 2, the experimentally computed optimal value of ℏ(ℏopt), and the spectral radius of iteration matrix ρ(G¯ℏopt) which introduced in Corollary 4.Table 1
Test problem information.
Matrix
Order
nnz
Symmetric
Positive definite
Condition number
cage5
37
233
No
No
15.4166
pivtol
102
306
No
No
109.607
pde225
225
1065
No
No
39.0638
Si2
769
17801
Yes
No
170.848
bfwb782
782
5982
Yes
No
18.0724Table 2
The basic method is convergent (Theorem1).
Matrix
Method
ρ
(
G
)
Convergence interval
ℏ
opt
ρ
(
G
ℏ
opt
)
pde225
SOR (w=r=1)
0.9776
(−1, −0.0743)
−0.7423
0.7768
pde225
AOR (r=1, w=0.8)
0.8053
(−1, −0.8096)
−0.9400
0.7739
pde225
SSOR (w=r=0.5)
0.8048
(−1.4441, −1)
−1.2941
0.7474
pde225
SAOR (r=0.5, w=1)
0.7773
(−1, −0.6038)
−0.8920
0.6673
cage5
SOR (w=r=1)
0.3388
(−1.2814, −1)
−1.1591
0.2314
cage5
AOR (r=1.2, w=1.9)
0.9240
(−1, −0.0589)
−0.68
0.3355
cage5
SSOR (w=r=0.5)
0.6427
(−1.7235, −1)
−1.5235
0.4557
cage5
SAOR (r=0.1, w=0.7)
0.5590
(−1.5609, −1)
−1.3756
0.3890
pivtol
SOR (w=r=0.1)
0.9487
(−3.9226, −1)
−2.7526
0.8588
pivtol
AOR (r=0.3, w=0.6)
0.9958
(−1, −0.0128)
−0.7300
0.7626
pivtol
SSOR (w=r=0.2)
0.8005
(−1.4850, −1)
−1.3450
0.7317
pivtol
SAOR (r=0.1, w=0.4)
0.9190
(−1, −0.2198)
−0.83
0.6943
bfwb782
SOR (r=w=1)
0.3732
(−1.2270, −1)
−1.1470
0.3024
bfwb782
AOR (r=1.2, w=1.9)
0.9942
(−1, −0.0045)
−0.65
0.3988
bfwb782
SSOR (w=r=0.5)
0.5984
(−1.6665, −1)
−1.4704
0.4103
bfwb782
SAOR (r=0.1, w=0.7)
0.5123
(−1.5123, −1)
−1.3423
0.3453
bfwb782
Richardson
0.999998
(
-
8.5201
×
1
0
4, -1)
-
8.0701
×
1
0
4
0.8952Table 3
The basic method is divergent (Theorem2).
Matrix
Method
ρ
(
G
)
Convergence interval
ℏ
opt
ρ
(
G
ℏ
opt
)
pde225
SOR(r=ω=1.5)
2.4937
(
-
0.4318, 0)
−0.3
0.7980
pde225
AOR(r=1, ω=1.5)
1.5542
(
-
0.6816, 0)
−0.5
0.7745
pde225
SSOR(r=2.01, ω=2.01)
1.1112
(
0, 17.9804)
12.0540
0.9388
pde225
SAOR(r=2, ω=3)
4.9844
(
0, 0.3408)
0.2510
0.7741
cage5
SOR(r=ω=2)
1.0150
(
-
0.9872, 0)
−0.4
0.7425
cage5
AOR(r=1.2, ω=3)
2.0370
(
-
0.6584, 0)
−0.43
0.3355
cage5
SSOR(r=2.2, ω=2.2)
1.8715
(
0, 2.2948)
2.0280
0.7681
cage5
SAOR(r=2, ω=3)
4.0002
(
0, 0.6666)
0.458
0.3748
pivtol
SOR(r=ω=1)
4.3002
(
-
0.3773, 0)
−0.33
0.7778
pivtol
AOR(r=0.467, ω=0.981)
1.5280
(
-
0.7566, 0)
−0.58
0.6766
pivtol
SSOR(r=ω=0.5)
2.1132
(
-
0.6424, 0)
−0.55
0.7306
pivtol
SAOR(r=0.1, ω=0.7)
2.6670
(
-
0.5454, 0)
−0.47
0.7235
bfwb782
Jacobi
1.0554
(
-
0.9731, 0)
−0.81305
0.6794
bfwb782
AOR(r=1.2, ω=3)
2.1398
(
-
0.6352, 0)
−0.4
0.3996
bfwb782
SSOR(ω=r=2.2)
2.1411
(
0, 1.7528)
1.5550
0.7748
bfwb782
SAOR(r=2, ω=3)
4.0000
(
0, 0.6667)
0.45
0.3570Table 4
The basic method is divergent (Theorem3).
Matrix
Method
ρ
(
G
)
Convergence interval
ℏ
opt
ρ
(
G
¯
ℏ
opt
)
Si2
SOR (r=w=1.3)
1.0993
(0, 0.8415)
0.8
0.9974
Si2
AOR (r=1.5, w=0.75)
1.0799
(0, 2.7964)
2.7
0.9942
Si2
SSOR (r=w=1.8)
1.24945
(0, 2.0007)
1.9
0.9909
Si2
SAOR (r=1.5, w=0.75)
1.1380
(0, 2.0949)
2
0.9916
Si2
Jacobi
1.3233
(0, 0.3705)
0.7
0.9999
Si2
Richardson
40.3813
(0, 0.0012)
0.0011
0.9999In Table2, we consider the convergent classical methods (ρ(G)<1). It is easy to verify that the numerical results are consistent with Theorem 1, and we observe that for the convergent classical methods by choosing suitable convergence control parameter ℏ, the rate of the convergence of the HAM method is faster than that of corresponding classical method.In Table3, we consider the divergent classical methods when Re(λi)>1 (or Re(λi)<1) for i=1,2,…,n. We can see that the numerical results are consistent with Theorem 2. These results show that by applying the HAM to a iterative scheme which is divergent, it is possible to construct a convergent homotopy-series vectors when the iteration matrix G has mentioned properties.In Table4, we report the results obtained for the symmetric matrix Si2 which has positive diagonal elements. For this example, the classical methods diverge and there exist i,j such that Re(λi)<1 and Re(λj)>1. We observe that the results are consistent with Theorem 3 and Corollary 4. The results show that the HAM method is convergent but the rate of the convergence is slow.Finally, Tables3 and 4 show that it is not necessary to choose the the parameters r and ω in the convergence interval of the classical methods. In the case of divergence, under the assumptions of Theorems 2 and 3, the application of the HAM can generate the convergent homotopy-series vectors for linear system (1).


## 4. Conclusion

In this paper, we proposed to apply the homotopy analysis method to the classical iterative methods for solving the linear system of equations. The theoretical results show the HAM can be used to accelerate the convergence of the basic iterative methods. In addition, we show that by applying the HAM to a divergent iterative scheme, it is possible to construct a convergent homotopy-series solution when the iteration matrixG of the iterative scheme has particular properties. The numerical experiments confirm the theoretical results and show the efficiency of the new method.



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