On the Newton method for the
matrix th root
Bruno Iannazzo
Dipartimento di Matematica
Università di Pisa
On the Newton method for the matrix
th root – p.1/18
Problem
Given a matrix , with no nonpositive real eigenvalues,
integer, find a solution of the equation
with
eigenvalues in the sector
S8
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Existence and uniqueness are guaranteed. We call this
principal th root of .
solution
On the Newton method for the matrix
th root – p.2/18
Available methods
#
&
!
) "'
+
!
*)('" $%
!
"
Schur decomposition method [Björk & Hammarling ’83, Smith ’03]
extremely good accuracy, high cost
.
taking times square roots requires
For
operations!
A desirable cost is
.
!
) "'
Newton’s method
per step, local
advantage: low cost,
quadratical convergence
drawbacks: instability and lack of global convergence.
Task
Removing these drawbacks
On the Newton method for the matrix
th root – p.3/18
Newton’s Method
0/
- .
which commutes with , one obtains
1 45
3
1
with an initial value
2 01
,
Applying Newton’s method to the equation
which generalizes the scalar iteration.
On the Newton method for the matrix
th root – p.4/18
Instability of the Newton method
7
5
10
0
10
Res
- - 9
(87
6
Let us consider a well-conditioned problem.
matrix having eigenvalues
Compute the th root of a
−5
10
−10
10
−15
10
0
5
10
Iter
15
20
On the Newton method for the matrix
th root – p.5/18
Nature of the instability
The Newton iteration suffers from instability near the
solution
where
<
1 1 ;
#
=
1 ;=
3
1 ;
1 :
is stable in a neighborhood
if the error matrices
<
: 21 21 ;
Def. An iteration
of the solution
satisfy
is linear and has bounded powers.
Reason: lack of commutativity.
On the Newton method for the matrix
th root – p.6/18
Illustration of instability
(unstable)
? >F
G
A @?>
B
C
D
E @?>
J?>
J?>
A J?>
B
C
D
E J?>
G
? >F
HIG
A @?>
B
C
D
E @?>
Numerically different iterations for the inverse of a matrix
(stable)
2
10
0
10
−2
10
−4
10
−6
10
−8
10
−10
10
−12
10
right
left
interpose
−14
10
−16
10
0
2
4
6
8
10
12
14
16
18
20
Interposition makes iterations stable!
It reduces effects of numerical noncommutativity.
On the Newton method for the matrix
th root – p.7/18
1 3
01
in the
21 How to get rid of monolateral multiplication by
Newton iteration?
Question
Solution: implicitly found on the known stable square root
algorithms.
On the Newton method for the matrix
th root – p.8/18
1 4
3
N
1 4K
1L
3
01
2 01
3
$
21 L
$
21 K
1K
3
1 4
1L
21 4
1L
$
2 01
M1
N
$
1 L
1
4
1 K
M1
1 4
$
21 1
Stable algorithms for the square root
Matrix sign function
Graeffe’s iteration
[Denman-Beavers]
[B. Meini]
On the Newton method for the matrix
th root – p.9/18
The iteration can be implemented with
) '
1 4K
RSQPRPP
21 K
1 K
3
1 4
1K
3
1 4
4
#
21 RPQPP
1 4K
3
01
O
N
1 4 4
1 K
N
21 1 45
3
1 Generalization to
matrix ops.
On the Newton method for the matrix
th root – p.10/18
Stability
Theorem. The iteration is stable in a neighborhood of the
solution.
Example. The iteration provides a stable algorithm for
computing the matrix th root
5
10
0
Res
10
−5
10
−10
10
−15
10
0
5
10
Iter
15
20
On the Newton method for the matrix
th root – p.11/18
Convergence
T
“The problem is to determine the region of the plane, such
that [initial point] being taken at pleasure anywhere within
one region we arrive ultimately at the point [a solution]”
Arthur Cayley, 1879
“J’espére appliquer cette théorie au cas d’une equation
cubique, mais les calculs sont beaucoup plus difficiles”
Arthur Cayley, 1890
:
“Donc, en general, la division du plan en régions, qui
conduisent chacune à une racine déterminée de
,
sera un problème impraticable.
Voilà la raison de l’échec de la tentative de Cayley”
Gaston Julia, 1918
On the Newton method for the matrix
th root – p.12/18
Choice of the initial value
Commute with
The initial value must
Converge to the principal th root
U
1V X
54
Y
1V
eigenvalue of .
W
with
SRP
/ V
21V
P
3
W
O
/0 A nice choice is
The problem of the convergence can be reduced to the
scalar iteration
On the Newton method for the matrix
th root – p.13/18
Question
W
W
Z
Which is the set
of for which the scalar iteration
converges to the principal th root
?
A set with fractal boundary.
[B
[B
]^\
[
in blue color the set of complex numbers for which the sequence converges to the principal
root
in red color the ones that generate sequences converging to secondary roots
and so
On the Newton method for the matrix th root – p.14/18
on...
\]
_
Main theorem
Re
- a
- a
`
The set
Z
`
cb`
is such that
for any .
Therefore convergence occurs for any matrix having
eigenvalues in .
On the Newton method for the matrix
th root – p.15/18
, the principal square root of
d
Compute
The algorithm
d=
f
e
d
e
=
Normalize:
so that
the matrix has eigenvalues in the set
convergence
e
$
is even compute the th root of
e
#
and set
and set
d=
e
g =
#
If
d=
e#
gh =
By means of the iteration proposed:
If is odd compute the
th root of
of
Convergence is guaranteed for any matrix
having a principal th root!
On the Newton method for the matrix
th root – p.16/18
Further results
High order rational iterations (König, Halley, Schröder).
The behavior and techniques are similar. Some of them
have very nice convergence regions (Halley’s method).
Scaling to reduce number of steps.
It is possible to provide a scaling to reduce the number
of steps to a fixed value. Proving this is work in
progress.
On the Newton method for the matrix
th root – p.17/18
Conclusions
Their cost is
!
*)('"
We have presented stable iterations for the th root
Convergence ensured whenever a solution exists
Further developments
Proving convergence for the Halley’s method and
designing a rigorous scaling procedure
On the Newton method for the matrix
th root – p.18/18