ETNA
Electronic Transactions on Numerical Analysis.
Volume 30, pp. 1-9, 2008.
Copyright  2008, Kent State University.
ISSN 1068-9613.
Kent State University
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SIMPLER BLOCK GMRES FOR NONSYMMETRIC SYSTEMS
WITH MULTIPLE RIGHT-HAND SIDES
HUALEI LIU AND BAOJIANG ZHONG
Abstract. A Simpler Block GMRES algorithm is presented, which is a block version of Walker and Zhou’s
Simpler GMRES. Similar to Block GMRES, the new algorithm also minimizes the residual norm in a block Krylov
space at every step. Theoretical analysis shows that the matrix-valued polynomials constructed by the new algorithm
is the same as the original one. However, Simpler Block GMRES avoids the factorization of a block upper Hessenberg matrix. In consequence, it is much simpler to program and requires less work. Numerical experiments are
conducted to illustrate the performance of the new block algorithm.
Key words. linear systems, iterative methods, block methods, GMRES, Simpler GMRES
AMS subject classifications. 65F10
1. Introduction. Block GMRES [13] and its variants [1, 6, 7] are effective for solving
large nonsymmetric systems with multiple right-hand sides of the form
!"#
where is a nonsingular matrix of order , and
and
are
rectangular matrices of dimension %$'& with &() . These block methods often promise
favorable convergence properties [8, 10], and may be effectively implemented on parallel
processors. In practice, if the initial block residual is nearly rank deficient, Block GMRES
should be implemented with initial deflation [2].
Of interest here is the original Block GMRES. Detailed descriptions can be found in
[8, 9, 11]. It is essentially identical to standard GMRES, except that operations
'* and + *,are-performed
/.012* ,
with multiple vectors instead of single vectors. Given
an
initial
guess
'
3
Block GMRES generates an approximate solution
over the block Krylov subspace
4 3 5 + * 6-7 &98 + * " + * ! 3,: + * <;
> @B3 , in which @63
3 = *?A
The approximate solution is of the form
tion problem
solves the minimiza-
* .U @ GH#I6C2JLDFKNE M?O PRQ"ST +
TV
(1.1)
T T V
the Frobenius norm.
X 3 X 3[Z \ , with each X^] being
Suppose that a block orthonormal sequence WYX
orthonormal, has been produced by the Block Arnoldi process (normally
based on a Modi* . Let
fied Block Gram-Schmidt3 procedure)
with
the
initial
block
residual
+
_ be the block
a X Y X 3 of 4 3 bc + * . Denote by e d 3 the
dimension of the basis `
block upper
` 3 ` 3LZ e d 3 , by f the &g$h& identity matrix,
Hessenberg
matrix
for
which
and by
i the zero matrix of order & . Furthermore, j,] k i f l i nm is the rectangu
o th block
lar matrix whose
element is f ,@ and
d+ *@ is the3,&pr9$
3 & matrix resulting from the r9QR
q
*
*
3
3 ,
X d+ . We can write
`
factorization +
with some _s&p$t& matrix
with
u
Received April 18, 2006. Accepted for publication October 15, 2007. Published online on March 14, 2008.
Recommended by M. Gutknecht.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016,
China( liuhl,zhbj @nuaa.edu.cn).
v
w
1
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H. LIU AND B. ZHONG
*1.U @ 3
3x
+
and +
squares problem
. Then the minimization problem (1.1) is equivalent to the block least
y 3x z6H C2{ JRDNE |~}| T j d+ *. e d 3,r T V ;
(1.2)
This problem can be solved recursively for each index € up to _ by updating
the QR factorec
y  T j d+ * . ecd  r  TV
ization of d (€‚(-_ ) with Givens rotations. The Frobenius norm
of size € >„ƒ &…$
& and
of
the
minimum
residual
of
such
a
least
square
problem
(with
j
>
ƒ
r  -r is dee d  of size €
&/$t€& now)
can then be computed even before the solution

termined, and this norm y is equal to the minimum norm in (1.1). After a certain number of
steps, which is normally denoted by _ , the algorithm is restarted if it has not yet converged.
Summarizing we can sketch the ordinary Block GMRES algorithm with restart after _
steps, and, for simplicity, based on Classical Block Gram-Schmidt, as follows.
A LGORITHM
(Block
* GMRES).
†.‡1 * . If + * ( TOL, accept * and exit; otherwise,
* 1.1
, set +
(i) Given
* : + T * TV X e ! .
compute a QR factorization
of
+
ƒ _ , do
(ii) Iterate: for € ˆ e ]  X ] m X  , o ƒ € . (These are the possibly nonzero block elements of
e d 3 , except for the last one on the subdiagonal.)

th
block
column
of
the
€
= .‡‹
;
ˆŠX ‰ 
ˆ
X
e 
]FΠX^] ]

  Z ec Z  ;
O
X
Compute a QR factorization of X ‰ : X ‰
ˆ Determine the Frobenius norm y  of the residual j d+ * . e5d  r  of the least square
problem
(1.2) with _ replaced by € . (There is no need to determine the solution
r  =r yet.)
ˆ If y  ( TOL, then go to (iv).
_ on exiting the loop.
(iii) Set €
r  r of the least square problem (1.2), and form the
(iv) Compute the solution  Ž2* > `  r  . If y  ( TOL, then accept  and exit;
approximate solution t*,
 and go to (i).
otherwise, restart: set
An essential component of Block GMRES
is the Block Arnoldi process in (ii). It usually
* . By shifting
the Block Arnoldi process to begin
starts with
the
initial
block
residual
+
*
*
with + instead of + , we obtain a Simpler Block GMRES that
QR
* X does
X require
3,: \ isthe
not
W
+
factorization
of
a
block
upper
Hessenberg
matrix.
In
this
case,
a
basis
4 3qb5 + *Y , and (1.2) is replaced by an upper triangular least squares problem, which can
of
be solved immediately.
In Section 2 the Simpler Block GMRES algorithm is formulated. In Section 3 an equivalence between Block and Simpler Block GMRES is established. The numerical stability of
the new algorithm is also discussed. In Section 4 the two algorithms are compared by using
two test matrices taken from the Matrix Market. The following notation is used. Subscripts
denote the iteration index and superscripts distinguish between individual columns in a block.
The symbols vec and  denote the vectorizing
operation and
product,
3“<” ,respec is denoted
the. Kronecker
is the
tively. The spectral 2-norm
of
a
matrix
by
Moreover,
’
T
~
T
‘
3
,
largest
one. The condition number of a matrix
3 smallest
3 .
b—˜singular
’ 3[“” value
,"™ ’ of3 ]F• ,band
— . ’ š d 3 ]F• and d+ the
is –
are the factors of a QR factorization of +
*
Block GMRES. Suppose that + is of* full rank.
Since is nonsingular,
+ * 2.is Simpler
›
"
full rank, and in the QR factorization +
the &2$& upper triangular
X
› also
! isofnonsingular.
matrix
X 3 be the block orthonormal vectors produced by _ steps of the Block
Let X
4 3 5 + * has block dimension _ . From
Arnoldi process. For simplicity, we assume that
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SIMPLER BLOCK GMRES
3œ › œ 3 œ
,
3
:
‹
X
the
Œ ›R3L3 Block Arnoldi process, we have X
is nonsingular. Hence,
with &0$L& matrices
3
œ
› 3
, where
4 3sb5 + *Y67 &98ž + *# X X 3,: </ 4 3qbc + * 67 &^8ž X X 3Ÿ<;
With square matrices ¡] of order & we can write
3 > X > > X 3 3 ;
(2.1)
+ * + %
Here we have
(2.2)
m
¢o (£_ <
X] + 3 i 
(2.3)
+ 3 + 3,: L. X 3 3 and
Define ¤
3 ¥
m 3 m m
m
3 X 3 + 3,: ;
. We can write (2.1) as
+ * + 3> ` 3 ¤ 3 (2.4)
¦§ ›!©ª› 3 «¬
and with
› 3 ¨
..
.
›
..
3[. 3
­® +
3 ž¯ 3 we get
+ *1.U @ + *1.…c + *ž X X 3,: r + 3 > ` 3q›R3—r„. ¤ 30<;
3 l¯
@ 3 4 3qbc + * or its coordinates r93
We want to determine
with respect to the
®‚°
3 such that ®
columns of `
@ 3-=±#²"³
C ¢J DFKNEM?O PRQ"ST + *1.… @ T V ;
(2.6)
GH#I62
(2.5)
Equation (2.5) can be written as
vec
+ *1.… @ 6
vec
+ 3Ÿ > f £` 3—
vec
b›R3,r„. ¤ 30;
Therefore, (2.6) is equivalent to the minimization problem,
C2z DNE T vec + 3— > f %` 3Ÿ vec ›R3—r„. ¤ 3— T ‘ ;
3 the space spanned by the column vectors of f %
7 Denote by &^8ž f £`
 ` 3
(2.7)
and (2.4), we have
. With (2.2)
+ 3 6´7 &^8ž f %` 3 vec + 3 6´7 &^8ž f gX 3 <;
r 3 › 3 : ¤ 3 solves (2.7). In consequence, @˜3 ` 3 r 3 is determined and
It 3 follows
that
+ + * .U @B3 is the residual.
vec
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H. LIU AND B. ZHONG
We can write (2.3) as
+ 3 6 vec + 3: µ. f £X 3 vec 3 ;
3Ÿ˜´=7 &98 f £X 3Ÿ , we derive from (2.8) that
Observing that vec +
3 3 3,: ‘ . f £X 3 vec 3 ‘ ;
T + T ‘V T vec + T ‘‘ T vec +
T‘ T
T‘
3
Since f gX
is a unitary matrix, we have
"¶ 3
¥
,
3
:
.
—
3
‘
‘
T+ TV
T vec +
T ‘ T vec T ‘ ‘
or
"¶
+T 3 T V T + 3,: T ‘V . T 3 T ‘V ‘
+ 3,: ƒ . 3 ™ + 3,: ‘ "¶ ‘
T + 3,: T V DFE b±#²!T ¸¸¹ T V 3 T ™ + T V 3,: ;
(2.9)
T
T Vq·
TV
· T TV T
(2.8)
vec
The last formula can be used to update the residual norm even if the block residual itself is
not updated.
Summarizing we obtain the following Simpler Block GMRES algorithm. Again, we
formulate it with restart after _ steps, and, for simplicity, based on Classical Block GramSchmidt instead of Modified Block Gram-Schmidt in the Block Arnoldi process.
A LGORITHM
(Simpler
2* 2.1
*2†Block
.‡12GMRES).
* . If + * ( TOL, accept 2* and exit; otherwise,
(i) Given
, set +
* T T *ŸV X › ! .
compute a QR factorization
of + : +
ƒ
(ii) Iterate:
€ . ƒ .
m € X  o  : ƒ _ , do
ˆ › ]  for
X
]
ˆ X ‰  = X  . ‹ ]FŒ X ] › ]  if €
º ƒ .
ˆ Compute a QR factorization of X ‰  : X ‰  X  › » .
ˆ Either compute  X  m +  : and +  +  : [. X   or compute T +  TV from
the recursion (2.9).
ˆ If T +  TV ( TOL, then go to (iv).
_ on exiting the loop.
(iii) Set €
›  r  ¤  for r  and form the approximate solution
(iv) Solve the triangular system
 *¢> + * X Y X  r 
 and exit; otherwise, restart: set * - 

If T + TV ( TOL, then accept
to (i).
*, + * ™ + *
R
EMARK 2.2. To improve the numerical stability, we may use +
T TV
*
of + in the practical implementation. This is explained in the following section.
and go
instead
3. A comparison with Block GMRES. The following theorem establishes an equivalence between the matrix-valued polynomials of Block GMRES and those of Simpler Block
GMRES.
T HEOREM 3.1. Suppose that _ steps of Block GMRES and _ steps of Simpler Block
GMRES have been taken, respectively. Then, the matrix-valued polynomials constructed by
the two algorithms3qare
same.
b½l the
3s½ž be the matrix-valued polynomials constructed by Block
Proof. Let ¼
and ¾
3
3
GMRES and Simpler Block GMRES, respectively. And let +c¿ÁÀ and +0ž¿?À denote the residuals of the two methods. We have
3,: Z Ä
+ 3¿?À ¼ 3 ,Ã + * + * . * ] + ~* Å ]
]NŒ
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5
and
3,: Ä ] Z *~Æ 3
*
*
b
—
Ã
.
+ 3¿?À ¾
+
+
+ ]
]FΠ*
Å
Æ
Ã
where ] and ] are &0$L& matrices, and where the notation is attributed to Gragg [3]. Thus,
3,: Z Ä
(3.1)
+ 3¿?À . + 3ž¿?À * ] + * Æ ] . Å ] ® 4 3 b5 + * <;
]FŒ
3,: * 3
*
*
¥
"
"
4
If we let
+
+
+ and Ç?] Æ ] . Å ] , we can write (3.1) as
3 . + 3ž¿ÁÀ ¢È f  b 4 3—" vec Ç 7 &^8ž f  b 4 3Ÿ
vec + ¿?À
®
¥
m
m
m
*
,
3
:
with Ç
Ç
Ç
. It is easily seen that
3 6´7 &98ž f  4 3 vec + 3ž¿?À 6´7 &98ž f  4 3 ;
vec + ¿?À
In consequence,
+ 3?¿ À . + 3ž¿?À 6´-7 &98 f  b 4 3 <;
3 . +03ž¿?À ˜É . Equivalently, f  b 4 3 " vec Ç 6-É . Since
We then have vec +Ÿ¿?À
Ê 8Ë f  4 3Ÿ¢ Ê 8žË f $ Ê 8žË b 4 3Ÿ<
f  b 4 3 is of full rank. Therefore, Ç É and hence Å ] Æ ] , indicating that the two
matrix-valued polynomials
are the same.
ƒ , the
Note that when &
theorem reduces to [5, Theorem 1].
Theorem 3.1 indicates that _ steps of Simpler Block GMRES is equivalent to _ steps
of Block GMRES. On the other hand, because no QR factorization of a blocki upper
Hessen
berg matrix is required, the new algorithm is easier to program and requires & ‘ _ ‘ fewer
vec
arithmetic operations than the original one. For Block GMRES, it is well known that the
number of iterations for termination is expected to decrease as the number of right-hand sides
increases. However, the QR factorization of a block upper Hessenberg matrix is time consuming. Thus the new algorithm offers improvements over the original one. It has been observed
[4, 12], however, that Simpler GMRES is, in general, less accurate than *GMRES.
X X In3fact,
: \.
Simpler GMRES is inherently unstable due to the choice of the basis W +
But, in practice, Simpler GMRES works well if we do not need very high accuracy and if
we restart frequently enough. This has also been observed in our experiments with Simpler
Block GMRES; see Section 4.
3 theorem is established. It indicates that the condition Ì
+ * InX the
following,
X 3,: cananother
during
the iteration process of Simpler Block GMRES.
* ™ + be
* controlled
* , and
Note that we use +
instead
of
therefore
+
T TV
Ì 3x¥ + *Y™ T + * T V X X 3,: <;
"¶ that Í abÎ2 j ‘ j 3 , where G is a _q&Ï$t& matrix. Then
L EMMA
3.2.
Suppose
L
ƒ
>
Î
T Í T‘ (
T T ‘‘ ‘ .
¥ Ð m Ð m3 m , with Т] being &/$ ƒ matrices. Let
Proof. Consider a unit vector Ð
: "¶ m m
m m
dÐ È ƒ . T Ð T ‘‘ ‘ i Ð ‘ Ð 3 ETNA
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H. LIU AND B. ZHONG
i
being a &/$
ƒ
dÐ is also a unit vector. Therefore,
¶
Î > ƒ . Ð ‘ ‘ dÐ ( Î Ð > ƒ . Ð ‘ "¶ ‘ dÐ ;
T ̈́РT‘ T Ð
T T‘
T‘ T T‘#T T‘
T T ‘ T T‘
¶
ƒ¢> Î ‘ ‘ . Consequently, Í
ƒ¢> Î ‘ "¶ ‘ ;
(
It follows that T ̈́РT ‘ (
T T‘
T T‘
T T‘
L EMMA 3.3. Let
¦§§ 3,: ÒÑ : i i «¬¬
§ d+ <Ñ : ¬
f i 3
Í
..
..
..
..
¨
­
3 ,: .<Ñ : i . . f .
with
zero matrix. It is easily seen that
3xÈ + *™ + * X X 3,: , where ÑÏ + * . Then – Ì 3Ÿ˜ – Í 3Ÿ .
T TV
T 3: T 5
3,: , with d+ 3,: of order
V š d 3,: d+ 3,: be a QR factorization
Proof. Let +
of +
& . By (2.1), we have
: : : Ì 3 + 3,: ÒÑ > X <Ñ > > X 3,: 3,: <Ñ X Y X 3,: <;
and Ì
It follows that
Ì 3 ¥ š d 3,: Y X X 3: Í 3 ;
3 : is unitary, – Ì 3 6 – Í 3 .
š 3: X X ,
Since d
+ * ™ + * X X 3,: and + 3,: ( + *
"¶ For
T HEOREM
3.4.
Ì 3: Ž
T
TV
T TV
3
3
* ™ T 3,TV : ;
have – Ì
(Ó& ‘ – +
T + TV T +
TV
3
Proof. By Lemma 3.3, we can alternatively consider the condition of Í
. Let
:
:
:
΄ d+ 3,: ÒÑ m Ñ m 3,: Ñ m m ;
: È Ît 3Ÿ
x
3
¥
2
Î
0
3
3
j ‘
j . It is easily seen that Í
j , where
Then Í
d j
:3, : m ~n. 3,: : m ~n. 3,: 3,: ‘ : m m ;
΄
b
Ñ
d
d+
d+
d+
By Lemma 3.2, we have
" ¶
ÍT 3 T‘ ( ƒ¢> T Î T ‘‘ ‘ (3.2)
and
: "¶ ;
¢
ƒ
>
Î
3
‘
T Í T‘ (
T d T‘ ‘
(3.3)
Let
ÎÈ
Since
(3.4)
¨
¦§§ . 3,: : Ñ
§ d+
f
« ¬¬
¬
..
3 : ( + * and Î d ( d+ 3,: : +T ,
TV T T V
T T‘ T
– Í 3 ( ƒ¢> Î ‘ ¶ ‘ ƒ¢> Ñ ‘ d+ 3,: : T T‘
T
.
Îtn.Ñ d+ 3,: : <;
­
f
Î Ñ
T ‘‘ T T ‘ ‘ , by (3.2) and (3.3) we have
: "¶
‘T ‘ T Î T ‘‘ ‘ ( ƒ¢> T Î T ‘‘ nÑ T d+ 3,: T‘ ;
, we
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SIMPLER BLOCK GMRES
It follows from (2.1) that
: : : : + *Ñ š d 3: d+ 3,: Ñ > X Ñ > > X 3,: 3,: Ñ ;
Equivalently,
: š 3,: Y Y 3: Î2;
+ *Ñ ¥
X
X
d
Therefore,
Î
T T‘
(3.5)
*+ Ñ : ( + * Ñ : ƒ ;
T T‘
T TV
On the other hand, we also have
: : 3,: "™ 3,: "¶ 3: !™ 3,: ;
3
,
(3.6)
T d+
T‘ – d+
T d+
T‘ (p& ‘ – +
T+
TV
3
¶ for
By (3.4), (3.5) and (3.6), an 3upper
is established:
– Í
(Ó<bound
‘ – + 3,: T + * TV ™ T + 3,: TV .
–
Í
&
3 6 – Í 3 , the proof is complete.
Since – Ì
3— is bounded by – + 3,: and + * ™ + 3,: . We may
By Theorem 3.4, – Ì
V T 30 T V
* ™ + 3,: ,T so T that
_
+
– Ì is controlled
choose a rather small
to
have
a
small
T
T
T
T
V
V
ƒ , the bound reduces to the one presented in [12, Lemma
well. When &
3.1] A detailed
explanation for normalizing the initial residual can also be found in [12].
4. Numerical experiments. In this section, Simpler Block GMRES is tested and compared with Block GMRES. For convenience, the two algorithms are denoted as SBGMRES(_ )
and BGMRES(_ ), respectively. The test matrices were taken from the Matrix Market
¹#³ * + [14].
3
Ô
All computations were carried out using Matlab. For each example, a plot shows
T TV
*
É
as a function of the number of iterations is presented. We take _ =10 and
. The righthand sides are chosen randomly.
E XAMPLE 4.1. The matrix is PSMIGR3, which comes from records containing counts of
persons by sex and age who migrated across counties in the USA between 1965 and 1970.
It is
Ó right:
a real unsymmetric matrix of size
N=3140
with
543162
nonzero
entries.
We
have
&
ƒ É iterations, and the convergence tolerance is › i Ð ƒ É »Õ .
hand sides, restart every _
Figure 4.1 shows that the two algorithms are almost equivalent with respect to the reduction
of the residual, until the residual norm comes near the convergence tolerance.
0
SBGMRES(10)
BGMRES(10)
−2.5
−5
−7.5
−10
−12.5
−15
0
25
50
75
100
125
Log. residual norm vs. the number of iterations
150
F IG . 4.1. Example 4.1
E XAMPLE 4.2. The matrix is JPWH991, which arises from computer random simulation of a circuit physics model. It is a real unsymmetric matrix of size N=991 with 6027
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H. LIU AND B. ZHONG
ă
ă
restart every _
iterations, and
nonzero entries. We have› & i
ƒ É : ‘ . It issides,
ƒ É the
right-hand
convergence tolerance is ƒ É Ð
seen from Figure 4.2 that SBGMRES( ) is
comparable to BGMRES( ). On the other hand, recall that Simpler Block GMRES is much
easier to program than Block GMRES.
2
SBGMRES(10)
BGMRES(10)
0
−2
−4
−6
−8
−10
−12
−14
0
20
40
60
80
100
120
140
Log. residual norm vs. the number of iterations
160
F IG . 4.2. Example 4.2
5. Conclusion. A Simpler Block GMRES algorithm for solving nonsymmetric systems
with multiple right-hand sides was presented and studied. It was shown that _ steps of
Simpler Block GMRES are equivalent to _ steps of Block GMRES. On the other hand,
the new algorithm does not require the factorization of a block upper Hessenberg matrix,
so it is easier to program and has lower computational cost per iteration. It works well in
practice despite the theoretical limitations on the accuracy and conceivable problems with
linear dependence of the block residuals, which, in theory, might require deflation.
Acknowledgement. We are grateful to Professor Martin Gutknecht for a careful reading of this manuscript, as well as for many helpful comments and suggestions that greatly
improved the quality of the paper. This work is supported by National Natural Science Foundation of China under Grant 60705014.
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ETNA
Kent State University
http://etna.math.kent.edu
SIMPLER BLOCK GMRES
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