Download cg-type algorithms to solve symmetric matrix equations

CG-TYPE ALGORITHMS TO SOLVE SYMMETRIC
MATRIX EQUATIONS
Davod Khojasteh Salkuyeh
Department of Mathematics, Mohaghegh Ardabili University,
P. O. Box. 56199-11367, Ardabil, Iran
E-mail: [email protected]
Abstract
The global FOM and GMRES are among the effective algorithms to solve linear
system of equations with multiple right-hand sides. In this paper, we study these
algorithms in the case that the coefficient matrix is symmetric and extract two CGtype algorithms for solving symmetric linear systems of equations with multiple right–
hand sides. Then, we compare the numerical performance of the new algorithms with
some available methods. Numerical experiments are done on some test matrices from
Harwell-Boeing collection.
AMS Subject Classification : 65F10.
Keywords: Global FOM and GMRES algorithms; global CG and CR algorithms;
matrix equations; Krylov subspace.
1. Introduction
The Conjugate Gradient (CG) algorithm [4] is an extremely effective method to
solve symmetric positive definite (SPD) linear system of equations. This algorithm
can be extracted from the full orthogonalization method (FOM) [7] in the case that
the coefficient matrix is an SPD matrix. Also, the GMRES algorithm [7, 8] results
in the conjugate residual (CR) algorithm [7] to solve symmetric linear system of
equations not necessarily positive definite. It is well known that the CG and CR
algorithms are less cost effective than FOM and GMRES algorithms, respectively.
Consider the linear system of equations
AX = B,
1
(1)
where A is a symmetric matrix of order n and B = [b(1) , · · · , b(s) ] is an arbitrary rectangular matrix of order n×s with s of moderate size (s n) and X = [x(1) , · · · , x(s) ].
There are alternative strategies for solving (1) by iterative methods. In [6], the block
CG (Bl-CG) method has been presented when A is an SPD matrix. Another method
which is based on Krylov subspace methods has been proposed in [9] for linear system
of equations with general coefficient matrices. Recently, K. Jbilou et al. have proposed the global FOM (Gl-FOM) and global GMRES (Gl-GMRES) algorithms which
are based on block Krylov subspace methods [5]. These methods can be applied for
solving Lyapunov and Sylvester matrix equations [1, 3, 5] as well. In this paper, two
CG-type algorithms are extracted to solve (1) from Gl-FOM and Gl-GMRES in the
case that the matrix A is SPD and symmetric, respectively.
Throughout this paper, we use the following notations. Let E = Rn×s . For X and
Y in E, we define the inner product (X, Y )F = tr(X T Y ), where tr(Z) denotes the
trace of the square matrix Z and X T denotes the transpose of the matrix X. The
associated norm is the well-known Frobenius norm denoted by k . kF . For a matrix
V ∈ E the matrix Krylov subspace Km (A, V ) is defined as follows
Km (A, V ) = span{V, AV, · · · , Am−1 V },
which is a subspace of E. A set of members of E is said to be F -orthogonal if it is
orthogonal with respect to the scalar product (., .)F .
This paper is organized as follows. In section 2, we give a brief description of
Gl-FOM and Gl-GMRES algorithms. The new CG-type algorithms are presented in
section 3. Section 4 is devoted to numerical experiments. Some conclusion remarks
are given in section 5.
2. A brief description of Gl-FOM and Gl-GMRES algorithms
Let V ∈ E, we note that Z ∈ Km (A, V ) means that
Z=
m−1
X
αk Ak V.
k=0
The Modified global Arnoldi algorithm constructs an F -orthogonal basis V1 , V2 , · · · , Vm ,
i.e., tr(ViT Vj ) = 0 for i 6= j, i, j = 1, · · · , m, and tr(ViT Vi ) = 1, of the Krylov subspace
Km (A, V ). The algorithm can be written as following.
2
Algorithm 1. Modified global Arnoldi algorithm
1. Choose an n × s matrix V1 such that k V1 kF = 1.
2. For j = 1, 2, ..., m Do:
3. Compute W := AVj
4. For i = 1, ..., j Do:
5.
hij := tr(ViT W ),
6.
W := W − hi,j Vi
7. EndDo
8. hj+1,j :=k W kF .
9. Vj+1 := W/hj+1,j
10. EndDo
From this algorithm we have an upper Hessenberg matrix of order (m + 1) × m,
e
Hm , such that its entries are h(i, j) = trace(ViT AVj ) = (AVj , Vi )F , i = 1, · · · , m, j =
1 · · · , m + 1.
As in [5], we define some notations. By Vm we define the n × ms matrix Vm =
e m denotes the (m + 1) × m upper Hessenberg matrix whose
[V1 , V2 , · · · , Vm ]. Also, H
nonzero entries hi,j are defined by Algorithm 1 and Hm is the m × m matrix obtained
e m by deleting its last row. H.,j will denote the jth column of the matrix Hm .
from H
Next, we use the notation ∗ for the following product
Vm ∗ α =
m
X
αk Vk ,
(2)
k=0
where α = (α1 , α2 , · · · , αm )T is a vector in Rm and for a matrix Hm of dimension
m × m we define
Vm ∗ Hm = [Vm ∗ H.,1 , Vm ∗ H.,2 , · · · , Vm ∗ H.,m ].
(3)
It can be easily seen that
Vm ∗ (α + β) = Vm ∗ α + Vm ∗ β and (Vm ∗ Hm ) ∗ α = Vm ∗ (Hm α)
(4)
where α and β are two vectors in Rm .
Using these notations, we have the following results which will be used later.
Proposition 1. Let Vm be the matrix defined by Vm = [V1 , V2 , . . . , Vm ] where the
3
n × s matrices Vi , i = 1, . . . , m, are defined by the global Arnoldi algorithm. Then
we have
(5)
k Vm ∗ α kF =k α k2
where α is a vector of Rm .
Proof. See [5].
e m as defined before, then using the product ∗, the
Theorem 1. Let Vm , Hm and H
following relations hold:
(6)
AVm = Vm ∗ Hm + Zm+1 ,
where Zm+1 = hm+1,m [0n×s , . . . , 0n×s , Vm+1 ], and
em.
AVm = Vm+1 ∗ H
(7)
Proof. See [5].
In [5], K. Jbilou et al. proposed the global FOM and GMRES algorithms which
are based on the modified global arnoldi Algorithm, for solving linear systems with
multiple right-hand sides. In this section we review the Gl-FOM and Gl-GMRES
algorithms.
Let X0 be an initial n × s matrix guess to the solution X of the equation (1) and
R0 = B − AX0 its associated residual. At the mth iterate of Gl-FOM algorithm, a
correction Zm is determined in the matrix Krylov subspace Km (A, R0 ) = Km such
that the new residual is F -orthogonal to Km , i.e.,
Xm − X0 = Zm ∈ Km (A, R0 )
(8)
Rm = R0 − AZm ⊥F Km (A, R0 ).
(9)
The equation (8) yields Zm = Vm ∗ym , where ym is a vector in Rm . Hence the residual
Rm is given by
Rm = R0 − AVm ∗ ym ,
(10)
where R0 = B − AX0 . Suppose β =k R0 kF and V1 = V1 /β. Now, since Rm is
F -orthogonal to Km , we get
(11)
Hm ym = βe1
where e1 is the first canonical basis vector in Rm . As the FOM algorithm, these
relations result in the Gl-FOM algorithm. This algorithm can be written as following.
4
Algorithm 2. Gl-FOM algorithm
1. Choose X0 and compute R0 = B − AX0 , β :=k R0 k2 and V1 := R0 /β.
2. For j = 1, 2, ..., m for ym and construct the matrices V1 , V2 , ... by Algorithm 1
and set Vm = [V1 , V2 , · · · , Vm ].
3. Solve Hm ym = βe1 for ym and compute Xm = X0 + Vm ∗ ym .
It can be easily seen that
Rm = B − AXm = −hm+1,m eTm ym Vm+1 .
(12)
Any approximate solution Xm of the problem (1) in X0 + Km can be written as
Xm = X0 + Vm ∗ ym ,
(13)
where ym ∈ Rm . From the Eq. (10) and if V1 = R0 /β then we have
k Rm kF = k B − AXm kF
= k R0 − AVm ∗ ym kF
e m ym ) kF
= k Vm+1 ∗ (βe1 − H
e m ym k2 .
= k βe1 − H
from Theorem 1
from Proposition 1
e k y k2 , i.e.,
Now, vector ym is chosen the minimizer of k βe1 − H
e k y k2 .
ym = argminy∈Rm k βe1 − H
(14)
By using the above discussions the Gl-GMRES algorithm can be summarized as
following.
Algorithm 3. Gl-GMRES algorithm
1. Choose X0 and compute R0 = B − AX0 , β :=k R0 kF and V1 := R0 /β.
2. For j = 1, 2, ..., m for ym and construct the matrices V1 , V2 , ... by Algorithm 1
and set Vm = [V1 , V2 , · · · , Vm ].
3. Compute ym the minimizer of k βe1 −H m y k2 for y ∈ Rm and Xm = X0 +Vm ∗ym .
3. Global CG and global CR algorithms
5
In this section we study the Gl-FOM and Gl-GMRES in the case that the coefficient matrix A is symmetric. First, we state the following theorem.
Theorem 2. Assume that global Arnoldi’s method is applied to a real symmetric matrix A. Then the matrix Hm generated by the algorithm is tridiagonal and symmetric.
Proof. The matrix Hm is a Hessenberg matrix. On the other hand we have
hij = tr(ViT AVj ) = tr(VjT AVi ) = hji .
Hence, the matrix Hm is tridiagonal and symmetric.
This theorem shows that the matrix Hm can be written as following


α1 β2
 β2 α2 β3



.
.
.
.
Hm = 



βm−1 αm−1 βm 
βm αm
This leads the following form of the global modified Arnoldi’s method, namely
the global Lanczos algorithm.
Algorithm 4. Global Lanczos algorithm
1. Choose an n × s matrix V1 such that k V1 kF = 1. Set β1 = 0, V0 = 0.
2. For j = 1, 2, ..., m do
3.
Wj := AVj − βj Vj−1 (If j = 1 set β1 V0 ≡ 0)
4.
αj := tr(VjT Wi )
5.
Wj := Wj − αj Vj
6.
βj+1 :=k Wj kF . If βj+1 = 0 then stop.
7.
Vj+1 := Wj /βj+1 .
8. Enddo
By using the global Lanczos and FOM algorithms we can derive a new algorithm
for solving a symmetric linear systems with multiple right-hand sides.
Algorithm 5.
1. Compute R0 = B − AX0 , β :=k R0 kF and V1 := R0 /β.
2. At step m use the Laczos algorithm to get an F -orthogonal basis of Km .
6
3. Set Hm = tridiag(βi , αi , βi+1 ) and Vm = [V1 , . . . , Vm ].
−1
4. Compute ym = Hm
(βe1 ) and Xm = X0 + Vm ∗ ym
Now, let A be an SPD matrix. In this case the LU factorization of Hm exists and
can be written as Hm = Lm Um where



η1 β2
1

 λ2 1

η2 β3






.
.
1
λ
.
.
3
(15)
Hm = 
.

.
.



.
.
.
.


.
.
ηm−1 βm 
λm 1
ηm
Hence, the approximate solution Xm is given by
−1 −1
Xm = X0 + Vm ∗ (Um
Lm βe1 )
−1
= X0 + (Vm ∗ Um ) ∗ (L−1
m βe1 ).
from relation (4)
Letting
−1
Pm = Vm ∗ Um
,
(16)
zm = L−1
m βe1 ,
(17)
Xm = X0 + Pm ∗ zm ,
(18)
and
then
where Pm = [P1 , . . . , Pm ]. It can be easily seen that
Pm ∗ Um = Vm ,
(19)
[Pm ∗ (Um ).,1 , . . . , Pm ∗ (Um ).,m ] = Vm .
(20)
or
So the latter relation follows that
Pm ∗ (Um ).,m = Vm ,
(21)
βm Pm−1 + ηm Pm = Vm ,
(22)
and it follows that
or
Pm =
1
[Vm − βm Pm−1 ].
ηm
7
(23)
On the other hand from Eq. (14) we have
λm =
βm
,
ηm−1
(24)
ηm = αm − λm βm .
Now, let
zm =
zm−1
ζm
,
(25)
(26)
then we have ζm = −λm ζm−1 . Therefore we can conclude that
Xm = Xm−1 + ζm Pm ,
(27)
where Pm is defined by Eq. (23).
This gives the global D-Lanczos algorithm which refers to the direct version of
global Lanczos algorithm for solving Eq. (1).
Algorithm 6. Global D-Lanczos algorithm
1. Compute R0 = B − AX0 , ζ1 := β :=k R0 kF , and V1 := R0 /β
2. Set λ1 = β1 = 0, P0 = 0
3. For m = 1, 2, . . . , until convergence do:
4.
Compute W = AVm − βm Vm−1 and αm := (W, Vm )F
5.
If m > 1 then compute λm =
6.
ηm = αm − λm βm
7.
−1
Pm = ηm
(Vm − βm Pm−1 )
8.
Xm = Xm−1 + ζm Pm
9.
If xm has converged then Stop
βm
ηm−1
10.
W := W − αm Vm
11.
βm+1 :=k W kF , Vm+1 = W/βm+1
and ζm = −λm ζm−1
12. Enddo
To derive the Gl-CG algorithm from the Global D-Lanczos algorithm we state
and prove the following proposition.
Proposition 2. Let Rm = B − AXm , m = 0, 1, . . . , be the residual matrices and
Pm , m = 0, 1, . . . , be the auxiliary matrices produced by the global D-Lanczos algorithm. Then,
8
1. Each residual matrix Rm is such that Rm = σm Vm+1 where σm is a certain
scalar. As a result the Residual matrices are F -orthogonal to each other.
2. The auxiliary matrices Pm are A-conjugate set with respect to inner product
(., .)F , i.e. (APi , Pj )F = 0 for i 6= j.
Proof. It is obvious that the two algorithms 5 and 6 are mathematically equivalent.
Hence the first part of the proposition immediate consequence of the relation (12).
−1
in (16), then
For the secondP
part, let [δ1j , . . . , δjj , 0, . . . , 0]T be the jth column of Um
j
we have Pj = k=1 δkj Vk . Therefore,
(APi , Pj )F = (A
=
i
X
δli Vl ,
l=1
j
i
XX
j
X
δkj Vk )F
k=1
δlj δkj (AVl , Vk )F .
l=1 k=1
Since (AVl , Vk )F = 0, for l 6= k we conclude that (APi , Pj )F = 0, for i 6= j.
By using this proposition we can obtain the Gl-CG algorithm. From the global
D-Lanczos algorithm we have
Xj+1 = Xj + αj Pj .
(28)
Therefore the residual matrix must satisfy the recurrence
Rj+1 = Rj − αj APj .
(29)
From the F -orthogonality of Rj ’s we get (Rj − αj APj , Rj )F = 0 and as a result
αj =
(Rj , Rj )F
.
(APj , Rj )F
(30)
Also, it is known that the next search direction Pj+1 is a linear combination of Rj+1
and Pj (step 7 of the global Lanczos algorithm), i.e.,
Pj+1 = Rj+1 − βj Pj .
(31)
Thus, a consequence of the above relation is that
(APj , Rj )F = (APj , Pj − βj−1 Pj−1 )F = (APj , Pj )F ,
9
(32)
since APj is F -orthogonal to Pj−1 . Hence the relation (30) becomes
(Rj , Rj )F
.
(APj , Pj )F
αj =
(33)
From the F -orthogonality of Pj+1 to APj yields
βj = −
(Rj+1 , APj )F
.
(Pj , APj )F
(34)
1
(Rj+1 − Rj ),
αj
(35)
From the equation (29) we get
APj = −
and hence
1 (Rj+1 , Rj+1 − Rj )F
(Rj+1 , Rj+1 )F
=
.
αj
(APj , Pj )F
(Rj , Rj )F
The above relations result in the Gl-CG algorithm.
βj = −
(36)
Algorithm 7. Global CG algorithm
1. Compute R0 = B − AX0 , P0 := R0 .
2. For j := 0, 1, . . . , until convergence do
3.
αj := (Rj , Rj )F /(APj , Pj )F
4.
Xj+1 := Xj + αj Pj
5.
Rj+1 := Rj − αj APj
6.
βj := (Rj+1 , Rj+1 )F /(Rj , Rj )F
7.
Pj+1 := Rj+1 + βj Pj
8. Enddo
As we see the CG algorithm is an special case of Gl-CG Algorithm for SPD linear
system of equations with a single right-hand side. A new algorithm can be derived
from the Gl-GMRES algorithm when the coefficient matrix is symmetric (like Gl-CG
from Gl-FOM). In this case, we have
(APi , APj )F = 0,
for i 6= j,
(ARi , Rj )F = 0,
for i 6= j.
and
If we look for an algorithm with same structure as Gl-CG, but with these conditions
we get the Global Conjugate Residual (Gl-CR) algorithm.
10
Algorithm 8. Global CR algorithm
1. Compute R0 = B − AX0 , P0 := R0 .
2. For j := 0, 1, . . . , until convergence do
3.
αj := (Rj , ARj )F /(APj , APj )F
4.
Xj+1 := Xj + αj Pj
5.
Rj+1 := Rj − αj APj
6.
βj := (Rj+1 , ARj+1 )F /(Rj , ARj )F
7.
Pj+1 := Rj+1 + βj Pj
8.
APj+1 := ARj+1 + βj APj
9. Enddo
4. Numerical examples
All the numerical experiments presented in this section were computed in double
precision with some FORTRAN 77 codes. For all the examples the initial guess X0
was taken to be the zero matrix. The right hand side B was chosen such that the
exact solution X is a matrix of order n × s whose ith column has all entries equal
to one except the ith entry which is zero. The tests were stopped as soon as the
stopping criterion
k b(i) − Ax(i) k2
< 10−7 ,
max
1≤i≤n
k B kF
is satisfied.
For the first set of numerical experiments, some SPD matrices from HarwellBoeing collection [2] were used. These matrices with their generic properties are
given in Table 1. In this table, n and nnz stand for the order and the number of
nonzero entries of A, respectively. Table 2 also reports the numerical performance
of the Gl-CG algorithm and CG algorithm for these matrices applied to a sequence
of right-hand sides. In this table, we listed the number of iterations to converge for
s = 2, 4, 8, 16 and 32. We also give the CPU times to converge in parentheses.
For the second set of our numerical experiments some unsymmetric matrices from
Harwell-Boeing collection are chosen and their symmetrized part are considered, i.e.,
e were shown. In this table nnz
e := 1 (A + AT ). In Table 2, some properties of A
A
2
stands for the number of nonzero entries of symmetrized matrices. Table 2 also
11
presents the numerical results of the Gl-CR algorithm and CR algorithm for these
matrices applied to a sequence of right-hand sides.
Table 1 : Numerical results for SPD matrices
n
nnz
method
s=2
s=4
s=8
s = 16
s = 32
NOS5
matrix
468
5172
NOS6
675
3255
NOS7
729
4617
GR-30-30
900
7744
PDE900
900
7860
BCSSTK06
420
7860
BCSSTK08
1074
12960
BCSSTK09
1083
18437
BCSSTK10
1086
22070
BCSSTK11
1473
34241
BCSSTK19
817
6853
Gl-CG
CG
Gl-CG
CG
Gl-CG
CG
Gl-CG
CG
Gl-CG
CG
Gl-CG
CG
Gl-CG
CG
Gl-CG
CG
Gl-CG
CG
Gl-CG
CG
Gl-CG
CG
336(0.50)
368(0.77)
520(0.93)
572(1.32)
4009(5.44)
4347(7.63)
52(0.17)
52(0.22)
1316(2.86)
1251(3.24)
2397(3.35)
2601(4.84)
2392(6.48)
2595(9.23)
199(1.15)
198(1.32)
2356( 9.89)
2361(13.51)
2573(18.40)
2724(21.86)
1368(2.69)
1979(4.39)
365(1.37)
366(1.43)
494(1.54)
590(2.03)
4027(10.66)
4560(14.61)
52(0.44)
53(0.49)
1241(5.21)
1247(6.10)
1766(4.84)
2719(8.74)
2403(14.22)
2628(18.35)
199(1.92)
199(2.15)
2270(19.28)
2304(25.27)
2429(35.54)
2583(41.47)
979(3.79)
2097(8.73)
351(2.15)
357(2.41)
450(2.85)
547(3.79)
4023(26.75)
4744(34.38)
52(0.82)
52(0.94)
1241(12.19)
1306(13.46)
1686( 9.01)
2655(17.19)
2136(28.01)
2567(37.73)
218(3.90)
220(4.34)
2155(42.78)
2254(49.55)
2330(73.16)
2380(77.23)
852(7.63)
2560(24.39)
346(4.4)
345(4.5)
470(6.53)
521(7.41)
2784(42.34)
3780(59.82)
50(1.54)
51(1.65)
1226(25.81)
1303(27.85)
1713(21.04)
2699(36.14)
2068(58.77)
2775(84.70)
215(7.47)
214(8.35)
2099(82.06)
2094(96.40)
2299(144.13)
2304(154.34)
796(15.70)
2561(50.53)
328(8.46)
344(9.11)
454(13.29)
521(15.55)
2807( 92.06)
3780(127.92)
49(2.64)
49(2.80)
1221(53.61)
1300(58.44)
1692(44.21)
2601(75.08)
1815(104.25)
2744(172.30)
201(13.85)
210(15.76)
2025(164.78)
2061(190.54)
2170(273.25)
2300(310.77)
603(24.94)
2120(87.71)
Table 2 : Numerical results for symmetric matrices
n
nnz
method
s=2
s=4
s=8
s = 16
s = 32
CAVITY01
matrix
317
6589
CAVITY02
317
3851
CAVITY03
317
6627
CAVITY04
317
3851
CAVITY05
1182
26004
CAVITY06
1182
20026
SHERMAN1
1000
3750
SHERMAN4
1104
3786
GRE185
185
1425
GRE343
343
2277
west0132
132
686
Gl-CR
CR
Gl-CR
CR
Gl-CR
CR
Gl-CR
CR
Gl-CR
CR
Gl-CR
CR
Gl-CR
CR
Gl-CR
CR
Gl-CR
CR
Gl-CR
CR
Gl-CR
CR
486(1.37)
483(1.37)
87(0.11)
87(0.17)
3832(7.36)
4993(9.45)
263(0.55)
277(0.60)
970(9.67)
980(9.88)
160(1.59)
160(1.65)
389(1.32)
388(1.43)
127(0.49)
127(0.55)
204(0.22)
207(0.22)
531(0.94)
540(0.99)
2192(1.15)
2663(1.43)
486(2.20)
490(2.20)
87(0.33)
87(0.39)
4849(18.18)
4716(17.85)
260(1.15)
276(1.21)
799(16.09)
785(16.20)
159(2.80)
159(2.91)
377(2.36)
376(2.47)
126(1.27)
126(1.27)
209(0.33)
208(0.38)
533(1.53)
528(1.54)
2051(1.81)
2290(1.97)
494(4.32)
489(4.40)
86(0.77)
90(0.98)
3492(26.92)
5360(42.02)
277(2.03)
274(1.94)
710(28.35)
710(29.22)
157(5.39)
150(5.44)
366(4.95)
364(5.22)
124(2.14)
124(2.25)
208(0.71)
216(0.99)
553(2.69)
530(2.75)
1788(2.80)
2105(3.35)
485(8.57)
496(8.85)
86(1.49)
89(1.59)
2425(40.97)
3820(65.86)
266(3.62)
273(3.85)
580(48.71)
597(49.16)
156(10.44)
156(10.82)
357( 9.67)
354(10.27)
123(3.90)
123(4.28)
208(1.48)
215(1.54)
567(5.88)
524(5.76)
1506(4.45)
1951(5.71)
431(15.70)
446(16.36)
85(2.63)
87(2.81)
2314( 83.05)
3327(120.18)
272(7.47)
272(7.64)
537(86.18)
536(88.21)
155(20.43)
155(22.24)
347(19.50)
343(20.32)
122(7.74)
122(8.02)
210(2.91)
212(3.02)
556(12.8)
540(12.52)
1295(8.79)
1828(12.68)
12
Table 1 and 2 shows that the results of Gl-CG and Gl-CR algorithms is often
better than that the CG and CR algorithms applied to the sequence of right-hand
sides, both CPU times and iteration.
Numerical examples show that the Gl-CG is less cost effective than the Bl-CG
algorithm for solving SPD linear systems. To validate our claim we display the
convergence history in Figures 1 and 2 for the systems corresponding to the matrices
NOS5 and BCSSTK09, respectively, in the first set of our numerical examples.
5. Conclusion
In this paper, we have extracted the global CG and CR algorithms from the
global FOM and GMRES for solving matrix equations, respectively. Our numerical
examples show that the global CG and CR are often less cost effective than that
of the CG and CR algorithms applied to a sequence of right-hand sides. Numerical
results also show that the Bl-CG algorithm is more cost effective than global CG
algorithm for solving SPD linear matrix equations. We also note that the global CG
and CR algorithms can be directly applied to solve the symmetric Sylvester matrix
equation but the CG or CR algorithms can not.
References
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13
8
Gl−CG
Bl−CG
7
6
log10(||Rk||F)
5
4
3
2
1
0
0
200
400
600
Iterations
800
1000
1200
Figure 1: Convergence history of Gl-CG and Bl-CG for NOS5 with s = 4
9
8
Gl−CG
Bl−CG
7
log10(||Rk||F)
6
5
4
3
2
1
0
200
400
600
Iterations
800
1000
1200
Figure 2: Convergence history of Gl-CG and Bl-CG for BCSSTK09 with s = 4
14
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15