ETNA
Electronic Transactions on Numerical Analysis.
Volume 23, pp. 5-14, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
CONDITION NUMBERS OF THE KRYLOV BASES AND SPACES ASSOCIATED
WITH THE TRUNCATED QZ ITERATION
ALEXANDER MALYSHEV AND MILOUD SADKANE
Abstract. We propose exact and computable formulas for computing condition numbers of the Krylov bases
and spaces associated with the Hessenberg-Triangular reduction of a regular linear matrix pencil.
Key words. condition number, Krylov spaces, QZ algorithm, generalized Arnoldi algorithm
AMS subject classifications. 65F35, 15A21
1. Introduction. The
algorithm [5] is the most popular algorithm for solving the
dense generalized eigenvalue problem
(1.1)
!
$%&
)*+,
#"
where
. It is a generalization of the
algorithm [2, Sec. 7.7], which solves the
standard eigenproblem
. The first step of the
algorithm reduces, via orthogonal
and , the pair
to
, where the matrices
and
matrices
are respectively upper Hessenberg and upper triangular:
'
(21 '
-3
(1.2)
(
455 798;: 8<798;: =
5 7 =B: 8<7 =B: =
6
..
.
.-/ " -0 (*1 '
" 455 H 8A: 8 H 8A: =I>?>J> H 8A: E?FF
7@8A: ?E FF
5
7
=?: F
H ?= : =I>?>J> H ?= : F >
" ..
..
..
..
6
G
.
7 : D .C 8 7 . : G
H .: >?>?>
>?>?>
The Hessenberg-Triangular reduction can be recast as
K -L
' ' ( #( " >
(1.3)
to construct
the reduction (1.3) iteratively, starting from a vector M with
N M N = ItPisO possible
N2N = stands
. The symbol
for the Euclidean vector norm. The following algorithm,
taken from [7], can be used to accomplish the iterative reduction. It is written in MATLAB
style.
A LGORITHM 1 (Generalized Arnoldi Reduction [7]).
Q
Q
Q
Q
L with N M N = O
ZN N = 8 [R
R `W _aY\T]V
8' SM R UM TVXW 8 b & M@V - 8 [R M (#VX8 1 Y b\TVc 8 W Vb " ( 8 - Y\8 T]V^V (
OU+f >J>?> do
d g eZ
N cih N = VjW cihJ_ g V
8
(Xhk [R (Xh W TV - h [R - h#V g9l h1 T]V
Choose
Set
Set
for
m Received January 3, 2004. Accepted for publication November 15, 2005. Recommended by D. Sorensen.
University of Bergen, Department of Mathematics, Johannes Brunsgate 12, N-5008 Bergen, Norway
(malyshev@mi.uib.no).
Université de Brest, Département de Mathématiques, 6 avenue Victor Le Gorgeu, CS 93837, F-29238 Brest
Cedex 3, France (sadkane@univ-brest.fr).
5
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6
A. MALYSHEV AND M. SADKANE
/n b #' 1 M@M n V M n WXoV M n 'Dhpb
VjY eO _ N!M n N = V
h
M SY M@R n V H " h b\Y@8V [R H q
' h ' h MUTjV/" hk
" h V YUTV
b 8 M V 7 ( h1 8Bk 7 8 b
-V 8 SR - 7
h T]V
c hk b ( hk V hk
Solve
end for
After
steps of Algorithm 1 we obtain a truncation of (1.3) in the form
r%sut
v
w x @' y (zy - y| {}c~y l 1y '@y z( ya"y
(1.4)
'*y 1 ' y € y (2y 1 (zy &€ y (2y 1 c~y &
OUƒ t ?OUƒ r and (zyqP( „O\ƒ t JO\ƒ r represent the first r columns
where the matrices '@y‚['
- o-…OUƒ r JO\ƒ r and "2y " „O\ƒ r ?O\ƒ r are the
of ' and ( respectively. The - matrices y
leading
rL†‡r submatrices of and " . The vector l y represents the r -th canonical vector
€
and y is the identity matrix of order r .
The form (1.4) is a generalization of the classical Arnoldi reduction [6, Chap. VI] and
can, for example, be used to compute eigenpairs of the problem (1.1), when the sizes of
and
are large (see [7]). In exact arithmetic, the sets
and
, form bases of the Krylov spaces
Š W 8 >?>?> W`yUŒ r S
O\;f! >?>?> t
Ž y  Ž y . C 8 M 8  Span Š M 8 + C
(1.5)


and
8 8 
C
)
$
Š 8 W
(1.6)
Span W
y  y
8 ' 8 ‘M and W 8 M 8 _ NB M
respectively, with M
ˆ‰y ‹Š M 8 >?>J> M\yDŒ
8 8 >J>?> i. C 8  y C 8 8
M
M Œ

y C 8 W 8 >J>?> J. C 8 y C 8 W 8 Œ
8N=.
.’ in a finite preUnfortunately, the generalized Arnoldi reduction of the matrix pair
cision - arithmetic will not generally satisfy the relations (1.4) because the computed matrices
' , ( , and " are subject to rounding errors. The effects of rounding errors are proportional
to condition numbers, which are usually determined in terms of infinitesimal perturbations.
The chief advantage of infinitesimal perturbations is that they allow one to neglect second
order terms.
Let
and
be infinitesimal perturbations of and respectively, and denote by
and
the corresponding Hessenberg and triangular
forms, where
and
are the computed orthogonal
matrices with
.
In the present paper we want to analyze the difference between the exact quantities , ,
,
and their computed counterparts , ,
,
under variations of the perturbations
and
. More precisely, we are interested in the condition numbers of the Krylov spaces
and
and corresponding bases
and .
We use the arguments similar to those from [1, 3] for the Hessenberg reduction by
Arnoldi’s method and those from [4] for the bidiagonal reduction by the Lanczos method.
We assume that the reader is familiar with the arguments and reasoning used in these references.
”“ 8
-3
“ (2“ 1 . S
{ ”“
(“ 8
M

“
Ž y y
” “ 8 ” “ =
Ž y
y
” =
= “ '“
•R W
M “8
8W
“
.
”
8
{ “ 3
' “ R 8 = ?> >?> = “" >J>?> (#“ W 1 T e
'
“
“
“M “ M
“M T

'“ (“ Ž “ y “ y
ˆ–y 
y
.€ { ™ “ ( ' “
' “ —)€ { ˜ “ € ' ˜ ( “ 3
€ { ™“
{ “
' (
2. Condition numbers. Following the strategy used in [1, 3, 4], perturbations of and
are sought in the multiplicative form, that is
and
. The
orthogonality of , , and implies the orthogonality of
and
. Since the
(“
'“ ' (“
(
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C ONDITION NUMBERS OF THE K RYLOV BASES ASSOCIATED WITH THE
˜“
™“
˜“
TRUNCATED
™“
QZ ITERATION
7
-—
“ (21 ) € }
{ ™ “ 1 B) {
and are infinitesimal, the matrices
and are indistinguishable from
perturbations
skew-symmetric matrices. Discarding quadratic terms in the identity
, we obtain the equation
” “ = B)€ {3˜ “ '
-š - “ {}( 1 ” “ = '{ ( 1 ™ “ 1 ( - {
- ' 1 ˜ “ ' z‘ >
(2.1)
If we set
˜
” = ( 1 ”“ = ' ˜ ' 1 ˜ “ ' ™
and
™ ( 1 ™“ ( then and are indistinguishable from skew-symmetric matrices and satisfy, up to the first
order, the equation
- -“ >
™ -›- ˜ ” = { š
(#1 . € œ
{ ™ “ 1 ?. {…” “ 8 ? .€ S
{ ˜“ '
In a similar way, another equation follows from " “
(2.2)
yields
, which
™ " " ˜ ” 8 {…" " “

” 8 (21$” “ 8 ' .
with

Ž
The Krylov spaces y and yŽ are the vector spaces spanned by the columnsS
of)'`
€ y { and
˜ “ (z' yy
“
respectively. Their perturbations “ y and “ y are spanned by the columns of '@
y
“ y [.€ { ™ “ (y . The Frobenius norm of the difference
and (
   ˜

(2.4)
 '@y '@“ y ?ž  ' ' 1 '@y ?ž [N ˜ OUƒ t ?OUƒ r JN ž
.¢C 8 * M 8 of the basis ˆ–y , which is defined
can be used to measure the conditioning Ÿ` z¡Ÿ (2.3)
as (see [1, 3])
v««
¹ ««
Ǽ
N
N
ž
w
± ²]³z' ´;y ²]µ¶ ' “ y ]² ³ µ ²µ¶ «« >
Ÿ S§©£¥¤!¨@¦ ª «x« N ” 8 ¬„N ­ž ®
]² ·‰²µ¶ { ²]¸^]² µ¶
s ¯
N” =Nž °
»
s ¯
°
«
In other words,
Ÿ@ is the smallest constant such that
N ” 8 N =ž N ” = N =ž >
N
„
\
O
ƒ
J
\
O
ƒ
?
N
˜
t r ž s¼Ÿ ¾½ NB¿N =ž { N?,N =ž
(2.5)
N ™ O\ƒ t ?OUƒ r JN ž can be used to measure the conditioning of the basis
Similarly, the quantity

y . Because of similarities between the bases ˆ‰y and 
y we will only analyze the conditioning
of ˆzy . To this end, we will give a computable estimate of the condition number ŸÀ .
in (2.3) and those below the subdi$Let
Á us take the components below the main diagonal
Á
agonal8„Â in (2.2). The operation of taking the components
=+Â below the main diagonal is denoted
by
and that of below the subdiagonal by
. Thus, from equations (2.3), (2.2) we
 Á
 Á
derive the system of linear equations
 Á 8„ ™ " " ˜ ^  Á 8 ” 8 9
(2.6)
= ™ -Ç- ˜ ^ = ” = >
(2.7)
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8
A. MALYSHEV AND M. SADKANE
˜
™
˜
and are equal to zero because the matrices
Note that the diagonal elements of
and are real skew-symmetric. Moreover, since
and
, we have
Ä' ÆÅi€ { ˜/“ Ç '
˜ 8 o and, therefore, the first column and row of ˜ are also equalÈ to8 zero.
the identity “ “M
˜ and ™ can be described with the help of vectors ȼX!C C É¡
The
of
O\ >?>Jstructure
> t f , and
ÊUË LDC Ë XÌ#SO\ >J>?> t °O , such that
455 E FF
455 Ê!81
>J>?> E FF
55 FF
55
FF
¾ 18
Ê!=1
55 .
F
5
FF
FF ™ 55
¾ =1
Ê!Î1
..
..
FF >
.
.
˜ 55 ..
F
5
..
..
..
..
..
. 8 =
>J>?.> .
8 Ê = Ê Î >?>J.> .
G
6
6 Í
Ê
G
™
“M
O\ >?>J> t f
ÈϼX!C
”c=
” 8
Similarly, with the help of vectors
, the matrices
and
455 †
5
†
55
” 8 5
8 M8
È É}ÃO\ >?>J> &O
DC Ë C 8 , ÌL
¼
t
,
, and Ð Ë
are written as
† ?E FFF
FF
F † G
455 †
5
†
† ?E FFF
FF
F >
†>
† G
† †
55
† † ...
5
”
=
..
. †
† . .> .
8
=
6 c c cÎ †
6
†
Ð8 Ð= ÐÎ
8 >J>?> DC = and Ê 8 >J>?> Ê DC 8 we determine the lower triangular
By the aid of the vectors
˜
™
parts of
and
..
†
:
455 .
>?>?> E FF
>?>?> FF
F
5
F >
5
™Ñ ˜Ñ ..
..
..
..
..
. 8 = . I . G
8 Ê = Ê Î. . G
6 Í
6
Ê
˜ ˜ Ñ ˜ Ñ 1 and ™ ™ Ñ ™ Ñ 1 .
Then
that
Á
 ItÁ is important now to observe
 Á
 Á
8„Â Å ™ Ñ Ç &
8 Š˜ Ñ Ç ‘
=+ Š™ Ñ - Ç & = Š- ˜ Ñ Ç & >
1"
1
1
" 1
55 >J>?> I
I
E FF
 Á
8„Â Ñ
 Á ř " "
=+ Š™ Ñ -Ã-
As a result,
(2.8)
(2.9)
Equation (2.8) is equivalent to the system
Ò
(2.10)
FF
455 55
 Á
ј Ç „8  ” 8 9  Á
ј Ç =+ ” = >
c 8 H 8A: 8 Ê 8 D C Ø C 8 ÛÚX¡f! >?>?> }OU
caÓ ÕÔ hÓ Ö 8 H h : Ó¥× DC Óh Ê h Ù
" Ó Ó
t
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Á
9
Á
Á QZ
Â
Â
Ø " ÜÚ { O\ƒ t „Ú { OUƒ t L D C Ó D C Ó and for Ú sd , × h oÝ) Ó h C Ó Â € Ó]Þ L Ó h
where " Ó
$Ú columns are equal to zero and the lastÓ Ú columns form the identity
d
is the matrix whose
first
Ú
matrix of order . Similarly, equation (2.9) is equivalent to
Ò 8 7@8A: 8 !DC C = 8 8 7
=?: 8 = Ð
×8 Ê {
Ê
!DC C h Á C 8 à- ÁØ C 8 ÛÚ^f >?>J> œf!
(2.11)
Ô
7
:
k
Ó
8
Ð\Ó
t
hÖ h Óß× Ó Ê h 8Â Ó Ó Â
- Ø &-….Ú { f!ƒ t Ú { O\ƒ t ‰ DC Ó C DC Ó .
where Ó
3O\;f! >J>?> t ¡O , we have a series of
Let us summarize the above derivations. For r
systems of linear equations
E FF 45 8 E F
455 c 8 E FF 5455 H 8;8A: : = 8 € DDC C =8 8 =B: = € =
FF 55 Ê = FF
5 c = F 55 H × DC H DC
F Ê
6 ... G
6 ... G
..
..
..
6
G
.
.
. !C 8
8
cy
Êy
H 8A: yA× DC y
H y C 8;: yA× !DC C y y k H y : y € DC y
455 E?FF
55 " Ø = FF 455 8 E?FF
5
FF 5 = F 5
.. ..
.
.
6 ... 8 G
.
..
G
6
yC
Ø"2y
455 Ð 8 E FF 455 7 8A: 8 × !DC C = 88
E FF 455 Ê 8 E FF
7 =?: 8 € DC = =
5 = F 5 7@8A: = !DC C Î
DDC C Î 7 Î : = € DC Î
F 5 = F
7
B
=
:
=
Ð
Ê
×
×
..
..
..
6
G
6
6 .8 G
=
. G
. !C 8
7 8;: y C 8 × DC 7 =B: y C 8 × !DC C
>J>?>
7 y : y C 8 € DC y
ÐyC
Ê
y
y
455 E FF y 455 8 E FF
F 5 = F
5 -Ø = ..
.
.
.
.
6
.
- Ø y C . 8 G 6 y .C 8 G
C ONDITION NUMBERS OF THE K RYLOV BASES ASSOCIATED WITH THE
TRUNCATED
ITERATION
which we can write in the compact form
Ò
á +8 8 Ê á 8= â
c (2.12)
á =+= >
Ð &á =A8 Ê œ
: ä and 7 y : y C 8 ä
Since Algorithm 1 proceeds if and only if H y yã
ã , the matrix á 88
nonsingular. It follows that the system (2.12) has the solution
&á…è,æ c Ê
ÐÕç
Ðeç
where
Ò áœå*[)á =+= œá =;8 á 8+C 8 8 á 8= ;C 8 Ýá =;8 á 8C 8 8 χ€?éXê
Þ
(2.14)
áœè2&á 8C 8 8 á 8„= áœå { Ý á 8C 8 8 j Þ >
Á
Á
€?é ê €?ë ê is the identity matrix of order ì y á y with ì y t r ¼Oi y y = k 8  { O
Here
á ât r y y = k 8Â .
and y
(2.13)
¿&áœå¢æ c
is
and
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10
´
N ˜ „ O\ƒ t J O\ƒ r ?N ž
NA y N == Þ µ Z
A. MALYSHEV AND M. SADKANE
NAXN = íÝ;NA 8 N == { >?>J> {
Using the identity
formula
N ˜ OUƒ t ?OUƒ r JN ž  áœå¢æ c 

Ð¡ç  =
 áœå¢æ N " N ž € ë ê


N Nž €ë ê
s  á å æ " (2.15)
 á å æ N N ž € ë ê
" (2.16)
s 
If we choose
 áœå¢æ

and
c
NB-uN ž € é ê ç æ

NB-uN ž € é ê ç  =

NB-uN ž € é ê ç  =
Ð
and such that
c
  áœå¢æ N N ž €?ë ê
"  = 
?
N
u
N ž €Béê ç
ÐÕç
 æ
N Nž
 cÀ_ NB" -uN ž
Ð _
ç
we arrive at the

c À_ N N?" -uN Nž ž  =
Ð
_
ç 
 æ
N Nž 
 cÀ_ N?" -uN ž  =
Ð
_
ç
=
N ” 8 N ž N ” = N =ž >
½ NB¿N =ž { NB,N =ž
  æ
N Nž
 =  cÀ_ NB" -uN ž
Ð _
ç

 =
 N ” 8 N =ž N ” = N =ž  = ½ ?N %N =ž { BN ’N =ž
then inequalities (2.15) and (2.16) become equalities (see [4] for a similar reasoning).
We thus obtain a series of condition numbers of the orthonormal bases (not spaces!) for
r OU+f >?>J> t }O

8 8   áœå‚æ N " N ž €?ë ê
C
.
:


?
N
u
N
B
€
é
ê
Ÿy M
ž

ç =
 Ýá =+= œá =;8 á +8 C 8 8 á 8 = Þ C 8 Ý;N " N ž á =A8 á 8+C 8 8 ‰N?-uN ž B€ éê Þ  = >
(2.17)
Ž
In order to derive condition numbers of the corresponding
Krylov spaces y we use the
P
)
€
˜
same arguments as in [4]. The main idea is to compare '@“ y
{•“ '@y with '@y\y instead
of '@y for all orthogonal matrices y of order r . The minimum
î £ï¤  ' “ y ' y  ž
L
y y
‘
#1j € y
is attained at
y Sðòñ)ð ó 1
tion of
(see, e.g., [2, p.582]). Since
'*y 1 ' “ y
, where
ð ñ 1 Å '*1 ' “ y Ç ð ó Pô
y
is the singular value decomposi-
€ õy {}' 1 ˜ “ ' y € õy { ˜ O2ƒ r J O#ƒ r ' y 1 '@“ y y
[ð/ñ€ y{ ˜ „O\ƒ r ?O\ƒ r is indistinguishable
from an orthogonal matrix, we can take ’y
ð
—
o
ô
Ã
€
and ó
In fact, the leading r‡†œr
˜ y . This leads to the˜ following interpretation.
submatrix of is cut off. The rest of represents
the
distance
between
the Krylov spaces,
i.e., only last t
r components of each vector Ó contribute to the difference between the
(2.18)
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C ONDITION NUMBERS OF THE K RYLOV BASES ASSOCIATED WITH THE
TRUNCATED
QZ ITERATION
11
ö y
, whose columns are the coordinate vectors
Krylov spaces. Let us introduce the matrix
corresponding to the contributing components. Then the condition numbers of the Krylov
spaces are determined for
as
lÓ
OU+f! >?>J> t °O
r
. C 8 * M 8 z  ö y á =+= á =;8 á 8C 8 8 á 8 = C 8 Ý+N " N ž á =;8 á 8+C 8 8 pNB-uN ž €BéXê Þ  =  >
(2.19) Ÿ y
The condition numbers of the Krylov bases  y and corresponding Krylov spaces y ,
r eO\;f! >?>?> t }O , can be derived analogously by the aid of the expression for Ê in (2.13):
 8 8   á è,æ N " N ž € ë ê
C
)
$
:


B
N
}
N
B
€
é
ê
(2.20)
Ÿy W
ž

ç =
 >
8 8   áœè’æ N " N ž €?ë ê
C
)
$

NB-uN ž €?éXê ç  =
Ÿy
W
(2.21)
ö y
3. Numerical examples. In this section we illustrate behavior of the condition numbers
of Krylov bases and subspaces given in (2.17) and (2.19) on some examples.
E XAMPLE 3.1. is the identity matrix, and is the upper-Hessenberg Toeplitz matrix
455 O
5 ø
÷¢
6
÷
÷
O
O
..
ø ZO
.
>J>?> O E FF
>J>?> O F
>
..
..
G
.
ø O.
The matrix
is singular if and only if
. Our aim here is to show the effect of conditioning of
on the computed basis and subspace condition numbers, when the parameter
varies. The Hessenberg and triangular forms obtained by Algorithm 1 are of order
.
The starting vector has all its components equal to before normalization. Table 3.1 summarizes the information obtained on the computed matrices , ,
and
for different
is also given.
parameters . The standard condition number
ø
O
ú .*÷@[N?÷j' N y = N\( . 2y ÷ " AC y 8 N = y
M
r oOpù
ø
TABLE 3.1
Numerical results from Algorithm 1 (Example 3.1)
ø
?N € y '#1 '@y N =
NB€ y (2y 1 ( y N =
NB- y (21y ' y N =
N " y (21 y ÷ ' y N =
NB ÷ ' y y ( y " y N =
BN ' y ( y - y c y l 1 N =
y
ú . ÷ ú " y
ú ðÏ÷@
û
> ù\ù,8OJC 8„8ü
f > ù ý OJC 8
f > ù ’OJC 8
ù > OJÿ’OJC
þ> ûUþ OJ C 88„ü
f > Oif,OJC =
f > ù ý Oi =
f > û ’Oi
ÿ >aý Oi 8
û
> ý\þ 4OJC 8„8ü
ý> f ý OJC 8
ý> O û OJC 8
O > þ ’OJC Î
O > Oif,OJ C 88„Îü
> O2OJC
û > f ý Oi Î
f > þ Oi Î
û > ÿ
OOi Î
ù > û f,2OJC 8„8ü
þ> f\f,OJC 8
f > OJC 8+8
O > \’OJC 8+8
O > \’OJ C
ù > \’OJC 8„ü
O > U,Oi ÿ > ûUû Oi
û > OOi û
> ù~ÿ,1.3OJC 8„8ü
þ> û OJC 8
f > ÿ ý OJC ü
O > ù þ Oi!C ü
O > ù û Oi C
û
> ,OJC 8 8„ü
f > ý\þ Oi 8+Î8
O > þ f,Oi
ý> f!O2Oi 8+8
( y
' y ( y - y {uc y l 1 ' y
÷'y ( y" y
y
÷
8
Ÿ Ý ¡Ÿ : y ݂C * M 8 ÞiÞ
û
> ù\f,0Oi!C
O > ù\f,Oi!C
O > û Oi!C
O > f!O2Oi!C
O > ý ÿ’Oi C
f > û O2Oi!C
f > \,OJ
f > ù~ÿ,OJ
The results show that the orthogonality of
and
is well maintained along with the
first equality of (1.4), i.e.,
. However, the relations
and
deteriorate, when
gets ill-conditioned. Figure 3.1 shows the behavior
of the condition numbers of the Krylov bases
and Krylov spaces
. We observe that the condition numbers of the bases increase with the
dimension of the bases and that the condition numbers of the corresponding spaces increase
and decrease but are always smaller than those of the bases.
(2y 1 ÷ ' y " y
8
Ÿ Ý Ÿ@y Ý ¢C * M 8 ÞiÞ
8
8
8„ü
8
8
8
8
8„ü
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12
A. MALYSHEV AND M. SADKANE
÷
ú . 2÷@
SO > þ
N" C 8N=
"2yy 
ú )á 8+8 ú ú " y" y clearly influences the computed condition numbers (see the
The ill-conditioning of
case
in Table 3.1). It can be more practical to monitor the condition number
instead of
. This quantity is always available. Since
,
large
means essentially that the matrix
in (2.12) is ill-conditioned. The computed
condition numbers might therefore be large or even inaccurate.
One may wonder whether the “non-normality” influences, in a way, the computed condenote a matrix of eigenvectors of
and define
dition numbers. Let
ø
N "2y N =
ú
á 8+8
ð‡÷
#÷
÷XN = N~)ð‡÷@;C 8 N = if ÷ is diagonalizable
ú )ð‡÷  K NJðÏ
>
otherwise
ð ÷ be used to quantify the departure from normality of ÷ . The smaller
The
ú ðÏfactor
÷@ , theú closer can
2÷ to a normal matrix. Table 3.1 and Figure 3.1 show that the computed
‡
ð
÷
factors ú
, Ÿ@ and Ÿ do not seem to be related.
Example 1, α = 8
4
10
Example 1, α = 4
5
10
κb
κb
3
4
10
10
condition number
condition number
κ
κ
2
3
10
10
1
10
2
2
4
6
8
10
dimension of the Krylov basis
12
14
10
16
Example 1, α = 2
9
10
2
4
6
8
10
dimension of the Krylov basis
14
16
Example 1, α = 1.3
15
10
κb
12
κb
14
10
8
10
condition number
7
10
13
κ
10
12
10
6
10
11
10
5
10
10
0
2
4
6
8
10
dimension of the Krylov basis
12
14
16
10
0
2
4
6
8
10
dimension of the Krylov basis
12
14
Example 1, α = 0
3
10
κb
κ
condition number
condition number
κ
2
10
1
10
2
4
6
8
10
dimension of the Krylov basis
12
14
16
F IG . 3.1. Condition numbers of the Krylov bases and spaces (Example 3.1).
16
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C ONDITION NUMBERS OF THE K RYLOV BASES ASSOCIATED WITH THE
QZ ITERATION
TRUNCATED
13
TABLE 3.2
Numerical results from Algorithm 1 (Example 3.2)
)*+,
þ
> OOi!C 888
f > ý Oi C
f > O OJ!C
û > û fOi!C 8=
f > OOi!C 8=
f > ÿUÿ,Oi!C 8 Î
ü
> OJ,OJ
BN € matrix
'*1 pair
N=
y
NB€ y ( y 1 (' y y N =
N?- y (21y '@y N =
N "2²y· ê (2y1 ê Àê '@² y N =
²]C ·‰ y ² µ µ
²]¸ ê C ê ^ê C ê ê ² µ
²¸^² µ
ú "2y . +,
þ > f*ª Oi!C 88
þ
> þUþ Oi C 8=
> f*Oi!C
û > ù þ Oi!C 8=
f > Oi!C 8=
ù > aû Oi!C 8ü
> O2OJ#
"
.*+ þ
> U’OJ!ª C 88=
O > þ f,OJ C 8
O > þUþ OJ!C ª
þ
> ÿ ý OJ!C 8=
8ü
> ùUù,OJ!C
f > ù OJ!C !8 O >ÿ ù
áe-%$ û O\
áe-%$
O
û
t
BN ’N = P
f > ù\f DOJ Î
=
OJ Î
NB¿N = f > OJÿ ú .,Ï
þ> ÿ\ÿ `OJ
ù~
r •
O
M
'y (
ú " y & N?,N = €
ª
BN ª N = û > O\O \Oi Î ú ) ª –f > O NB ª N = û > þ ÿ ú ª ) ª z{ f
û O&\
ú )2[ÿ > ù
E XAMPLE 3.2. The matrix is
and the matrix is
from the
&
MHD set1 . These matrices are of order
and arise in the modal(analysis
of
dissipative
'
)'
magnetohydrodynamics. The matrix is unsymmetric,
,
*' #
. The matrix
is symmetric,
,
. The Hessenberg
and triangular forms obtained from Algorithm 1 are of order
. As in the previous
example, the starting vector has all its components equal to before normalization. Table
3.2 summarizes the information obtained on the computed matrices ,
,
and
.
The condition number
is also given. For more comparisons, the table also shows the
results obtained
with the well-conditioned
matrices
and
.
+'
,
,
,
.
Example 2, matrix pair = (A , B)
15
10
κ
Example 2, matrix pair = (A0 , B)
13
10
b
κb
κ
14
10
- y
y " y
{ BN %N = €
12
10
13
10
12
11
10
condition number
condition number
10
11
10
10
10
9
10
10
κ
9
10
10
8
10
8
10
7
10
6
10
7
0
5
10
15
20
25
30
dimension of the Krylov basis
35
40
45
50
10
0
5
10
15
20
25
30
dimension of the Krylov basis
35
40
45
50
Example 2, matrix pair = (A , B0)
4
10
κb
κ
3
condition number
10
2
10
1
10
0
10
0
5
10
15
20
25
30
dimension of the Krylov basis
35
40
45
50
F IG . 3.2. Condition numbers of the Krylov bases # and spaces (Example 3.2)
From Table 3.2 and Figure 3.2, it is clear that the ill-conditioning of
ú "y , is responsible for the large condition numbers. We also see that
1 see http://math.nist.gov/MatrixMarket/
, also shown in
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14
(y 1 ( y , € y í- y-,
'`y , (yp"2y ,
leads to less accuracy in the approximation
- the less accuracy is more pronounced when both and are used.
In conclusion, the numerical experiments indicate that when the truncated reduction
is
good, i.e., when the relations (1.4) are accurately satisfied, the ill-conditioning of is re2( y 1 ' y the use of
- the use of
A. MALYSHEV AND M. SADKANE
leads to less accuracy in the approximations
sponsible for large basis and space condition numbers.
REFERENCES
[1] J.-F. C ARPRAUX , S. K. G ODUNOV, AND S. V. K UZNETSOV , Condition number of the Krylov bases and
subspaces, Linear Algebra Appl., 248 (1996), pp. 136–160.
[2] G. H. G OLUB AND C. F. VAN L OAN , Matrix Computations, Third editon, The Johns Hopkins Press, Baltimore, MD., 1996.
[3] S .V. K UZNETSOV , Perturbation bounds of the Krylov bases and associated Hessenberg forms, Linear Algebra Appl., 265 (1997), pp. 1–28.
[4] A. M ALYSHEV AND M. S ADKANE , Condition numbers for Lanczos bidiagonalization with complete reorthogonalization, Linear Algebra Appl., 371 (2003), pp. 315–331.
[5] C. B. M OLER AND G. W. S TEWART , An algorithm for generalized matrix eigenvalue problems, SIAM J.
Numer. Anal., 10 (1973), pp. 241–256.
[6] Y. S AAD , Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester,
1992.
[7] D. C. S ORENSEN , Truncated QZ methods for large scale generalized eigenvalue problems, Electron. Trans.
Numer. Anal., 7 (1998), pp. 141–162,
http://etna.math.kent.edu/vol.7.1998/pp141-162.dir/pp141-162.html.