Download Computations with semi-infinite quasi

Dario A. Bini, Stefano Massei, Beatrice Meini
Computations with semi-infinite quasi-Toeplitz matrices
University of Pisa
Scuola Normale Superiore
[email protected]
Como, 16 February 2017
Speaker: Stefano Massei
1 / 21
Motivation
The analysis of certain stochastic models requires to deal with computational
problems of the kind
solve the quadratic equation
A0 + A1 X + A2 X 2 = 0
(QBD processes),
compute the function
exp(A) =
∞
X
Aj
j=0
j!
(second-order fluid models),
where A0 , A1 , A2 and A are semi-infinite matrices of the form
T + E,
T = (tij )i,j∈Z+ ,
E = (eij )i,j∈Z+ ,
where
tij = aj−i ,
X
j∈Z
Speaker: Stefano Massei
|aj | < ∞,
X
|eij | < ∞.
i,j∈Z+
2 / 21
Motivation
That is, they are the sum
P of a semi-infinite Toeplitz matrix T (a) associated with a
Laurent series a(z) := k∈Z ak z k and of a semi-infinite matrix E having a finite
sum of the moduli of its entries. The function a(z) is called the symbol of the
Toeplitz matrix.
A typical example originated in the analysis of random walks in the quarter plane
involves matrices T + E of the kind




a0
a1
b0 b1 0 . . .
a−1 a0 a1





T =
E =  0 0 0 . . . .
,
a−1 a0 a1


..
.. .. . .
..
.
..
.
.
. .
. ..
.
Speaker: Stefano Massei
3 / 21
Motivation
We consider the slightly more general situation


X

X
a(z) ∈ W :=
ãj z j :
|ãj | < ∞


j∈Z
E ∈ F :=


Ẽ = (ẽij )i,j∈Z+ :

=⇒
j∈Z
X
i,j∈Z


|ẽij | < ∞

The set W equipped with the norm ka(z)kW :=
called the Wiener algebra.
=⇒
P
j∈Z
lim |aj | = 0,
j→±∞
lim
k(i,j)k∞ →∞
|eij | = 0.
|aj | is a Banach algebra
T (a) + E =
Speaker: Stefano Massei
4 / 21
Bi-infinite Toeplitz matrices and Laurent series
The space of bi-infinite Toeplitz matrices with summable diagonal entries is
isomorphic to the Wiener algebra, through the map


..
..
..
..
.
.
.
.


. . . a−1 a0
a1 a2 . . . . . .
X



T±∞ (a) := 
←→
aj z j .
. . . a−2 a−1 a0 a1 a2 . . .
. . . . . . a−2 a−1 a0 a1 . . .
j∈Z


..
.. .. ..
.
.
.
.
Given a(z) =
P
j∈Z aj z
j
and b(z) =
P
j∈Z
bj z j
T±∞ (a) + T±∞ (b) = T±∞ (a + b)
T±∞ (a) · T±∞ (b) = T±∞ (a · b)
T±∞ (a) is invertible ⇔ a(z) 6= 0 for |z| = 1 and T±∞ (a)−1 = T±∞ (a−1 )
This means that —for matrix operations— we can rely on power series arithmetic
(FFT).
Speaker: Stefano Massei
5 / 21
Semi-infinite Toeplitz matrices and Laurent series
The space of semi-infinite Toeplitz matrices with summable diagonal entries is not
isomorphic to the Wiener algebra, but we can still consider the bijection:


a0
a1 a2 . . .


a−1 a0 a1 . . . 
X


T (a) := 
←→
aj z j .

.
.
a−2 a−1 a0

.
j∈Z


..
..
.. ..
.
.
.
.
T (a) + T (b) = T (a + b)
In general T (a) · T (b)6=T (a · b),
 

1 1
1 1
1 1 1
 1 1
 


·
1 1
1
1

 
..
.. ..
.
.
.
e.g.:

2
2
 
 
=
 
1
..
.

1
1
..
.
1
..
.

2
3
1
2
1
2
..
.
3
..
.
2
..
.


.. 
.

..
.
In general, if T (a) is invertible T (a)−1 6=T (a−1 )
Speaker: Stefano Massei
6 / 21
Semi-infinite Toeplitz matrices and Laurent series
Things are simpler if we restrict to the class of upper (or lower) triangular
semi-infinite Toeplitz matrices with summable
P diagonal entries.
P This class is
isomorphic to the Banach algebra W + := { j∈Z+ ãj z j :
j∈Z+ |ãj | < ∞}
(or to W − defined analogously).
P
P
Given a(z) = j∈Z+ aj z j and b(z) = j∈Z+ bj z j
T (a) + T (b) = T (a + b)
T (a) · T (b) = T (a · b)
T (a) is invertible ⇔ a(z) is invertible in W + and T (a)−1 = T (a−1 )
Goal: Find a class of semi-infinite matrices closed by arithmetic operations and
containing the Toeplitz matrices with symbol in the Wiener algebra.
Question: In the general semi-infinite Toeplitz case, how much the outcomes of
multiplication and inversion differ from Toeplitz matrices?
Speaker: Stefano Massei
7 / 21
Theorem (Gohberg-Feldman)
Let a(z), b(z) ∈ W and c(z) := a(z) · b(z), then
T (a) · T (b) = T (c) − H(a− ) · H(b + )
where

a−1

a−2

H(a− ) := 
a−3

..
.
a−2
a−3
.
..
..
.
a−3
.
..
.
..
.
..

...
.
.. 

,
.
.
. 

.
..

b1

b2

H(b + ) := 
b3

..
.
b2
b3
.
..
..
.

b3 . . .
.
.
.. .. 

.
. . .
.
.
. 

.
.
.. ..
In words: Since |a−k | ↓ 0 and |bk | ↓ 0 as k → ∞, the product T (a) · T (b) differs
from a Toeplitz matrix by a correction which is located mostly in the upper left
corner.
Speaker: Stefano Massei
8 / 21
Theorem (Wiener-Hopf factorization)
P
P
Let a(z) ∈ W then ∃k ∈ Z, u(z) = j∈Z+ uj z j ∈ W and `(z) = j∈Z+ `j z j ∈ W
such that:
a(z) = u(z) · z k · `(z −1 ),
with u(z) 6= 0 and `(z) 6= 0 for |z| < 1.
If k = 0 the factorization is said canonical.
The matrix version of this result says that T (a) is invertible if and only if its
symbol does not vanish for |z| = 1 and admits a canonical factorization, in
particular


 `
0
u0 u1 u2 . . .
`1 `0


.. 


.
u0 u1
T (a) = 
 = T (u)T (`)t .
`
`
`


2
1
0


.. ..
.. . .
.. ..
.
.
.
.
.
.
Remark: T (a)−1 = T (`−1 )t T (u −1 ) together with the Gohberg-Feldman formula
tell us that T (a)−1 is Toeplitz plus a correction mostly located in the upper-left
corner.
Speaker: Stefano Massei
9 / 21
New result
Definition
A semi-infinite matrix of the form A = T (a) + E is said Continuosly Quasi
Toeplitz (CQT) if a(z), a0 (z) ∈ W and E ∈ F.
Theorem
The set CQT of CQT matrices equipped with norm
kAkCQT = kakW + ka0 kW + kE kF
is a Banach algebra
Speaker: Stefano Massei
10 / 21
Computational consequences
Any CQT matrix T (a) + E can be approximated to any desired precision
with a finite set of parameters
Since a(z) is represented by a bi-infinite sequence {aj } j ∈ Z having decay of
the coefficients, we may represent a(z) with a finite sequence
a ≈ (an− , . . . , a0 , . . . , an+ ) up to an arbitrarily small error.
Similarly, the matrix E is approximated with a matrix with finite support Ẽ
and represented by means of the pair of finite matrices (F , G) such that FG t
generates the non zero part of Ẽ . F and G have a number of columns given
by the numerical rank of Ẽ .
Speaker: Stefano Massei
11 / 21
Computational consequences
An approximated arithmetic can be defined for CQT matrices.
The complexity of the procedures depends on:
- The Toeplitz bandwidth t
- The dominant size of the correction m
- The rank of the correction k
Solving matrix equations by means of functional iterative methods
Computing matrix functions by means of power series
X
f (A) =
fj Aj
j∈Z
Computing matrix functions by means of integration
Z
f (A) = f (z)(zI − A)−1 dz
Γ
Speaker: Stefano Massei
12 / 21
CQT arithmetic
Addition:
A = T (a) + Fa Gat ,
B = T (b) + Fb Gbt ,
c(z) = a(z) + b(z),
Fc = [Fa , Fb ],
C = A + B = T (c) + Fc Gct
Gc = [Ga , Gb ]
Moreover, an SVD-based compression technique is performed on Fc Gct in order to
keep small the numerical rank.
Complexity: O(t + k 3 + mk 2 )
Speaker: Stefano Massei
13 / 21
CQT arithmetic
Multiplication:
A = T (a) + Fa Gat ,
B = T (b) + Fb Gbt ,
C = A · B = T (c) + Fc Gct
c(z) = a(z) · b(z),
−
Fc = [−H(a ), T (a)Fb , Fa ],
Gc = [H(b + ), Gb , T (b)Ga + Gb Fbt Ga ]
We need to compute products between Toeplitz and matrices with few columns
(FFT) and multiplications of finite/low-rank matrices.
Once again, the compression technique is performed on Fc Gct .
Complexity: O(kt log(t) + k 3 + mk 2 ) + compression of H(a− )H(b + )t
Speaker: Stefano Massei
14 / 21
CQT arithmetic
Inversion:
A = T (a) + Fa Gat ,
a(z) = u(z)`(z −1 ),
C = A−1 = T (c) + Fc Gct ,
c(z) = a(z)−1 ,
Fc = −[H(`−1 ), T (`−1 )t T (u −1 )Fa Y −1 ],
and
Gc = [H(u −1 ), T (u −1 )t T (`−1 )Ga ]
Y = I + Gat T (`−1 )t T (u −1 )Fa .
We need to
Compute the canonical factorization a(z) = u(z)`(z −1 )
O(t log(t))
Apply Sherman-Morrison-Woodbury formula
Compress Fc Gct
Complexity: O(kt log(t) + k 3 + mk 2 ) + compression of H(`−1 )H(u −1 )t
Speaker: Stefano Massei
15 / 21
Solving quadratic matrix equations
Equations of the kind
A0 + A1 X + A2 X 2 = 0
encountered in queuing models can be solved with Cyclic Reduction
(h)
(h)
(h)
(h)
(h)
(h)
(h+1)
= −A0 (A1 )−1 A0 ,
(h+1)
= −A2 (A1 )−1 A2 ,
(h+1)
= A1 − A0 (A1 )−1 A2 − A2 (A1 )−1 A0 ,
A0
A2
A1
(h)
(h)
(h)
(h)
(h)
(h)
(h)
b (h+1) = A
b (h) − A(h) (A(h) )−1 A(h) ,
A
1
1
0
1
2
(0)
(0)
(0)
(0)
b = A = A1 , A = A0 and A = A2 .
where A
1
1
0
2
For the minimal solution G it holds
b (h) )−1 A0 = G
lim (A
1
h→∞
and the convergence is quadratic (in the finite case).
Speaker: Stefano Massei
16 / 21
Solving quadratic matrix equations
Example: Ten instances of the two-node Jackson network from [Motyer & Taylor
2006], problems 1-10
A−1

(1 − q)µ2

=
qµ2
(1 − q)µ2

qµ2
..
.
..

,
.

−(λ1 + λ2 + µ2 )
λ1

−(λ1 + λ2 + µ1 + µ2 )
A0 =  (1 − p)µ1
..
.


λ1
pµ1 λ1

A1 = 
.
..
..
.
.

λ1
..
.
..

,
.
Motyer & Taylor. Decay rates for quasi-birth-and-death processes with countably
many phases and tridiagonal block generators, 2006.
Speaker: Stefano Massei
17 / 21
Solving quadratic matrix equations
Case
time
ResCQT
band
rows
columns
rank
1
2
3
4
5
6
7
8
9
10
2.61e+00
2.91e+00
2.88e-01
2.32e+00
4.77e-01
7.96e+00
2.90e+01
1.01e+00
3.01e-01
1.25e+00
5.98e-13
7.88e-13
2.67e-14
6.00e-13
1.07e-13
6.65e-13
6.87e-12
4.34e-13
2.48e-14
3.40e-14
561
561
143
463
233
455
1423
366
157
268
541
555
89
481
108
462
1543
348
81
241
138
145
66
99
148
153
247
40
86
107
8
8
8
9
9
10
13
6
8
8
Speaker: Stefano Massei
18 / 21
Solving quadratic matrix equations
0
−5
−10
−15
−20
−25
100
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
Figure : Log-plot of a portion of the solution G = T (g) + Eg for the problem 2
Speaker: Stefano Massei
19 / 21
Computing the exponential
We compute exp(T (a)) = T (exp(a)) + Eexp(a) with a(z) =
Pk
j=−1
k
time
band
rows
columns
rank
1
2
3
4
5
6
7
8
9
10
2.52e-02
3.22e-02
3.40e-02
3.77e-02
4.10e-02
4.34e-02
4.11e-02
4.52e-02
4.91e-02
5.16e-02
35
55
78
104
133
165
199
236
274
315
17
32
48
47
48
49
53
55
54
55
17
37
59
85
114
144
178
216
252
299
7
8
8
8
8
9
9
9
9
9
Speaker: Stefano Massei
zj.
20 / 21
Conclusions and research lines
The class of CQT matrices has been introduced; we proved that it is a
Banach algebra
CQT matrix arithmetic has been introduced and implemented
Applications to solving quadratic matrix equations and to compute matrix
functions have been given
All the above stuff for finitely large Toeplitz matrices, using the Widom
formula:
Tn (a)Tn (b) = Tn (ab) − Hn (a− )Hn (b + ) − Jn Hn (a+ )Hn (b − )Jn
Extend the approach to multilevel Toeplitz structures
Speaker: Stefano Massei
21 / 21