Download RESEARCH STATEMENT

RESEARCH STATEMENT
DAVID MILLER
My research interests fall into the general category of numerical analysis, but are more specifically contained in
structured matrices. The main objects that I am interested in studying are Leja ordering for Cauchy-Vandermonde
matrices and a fast inversion algorithm for a Laurent-Vandermonde system.
1. Introduction
We first consider the problem of inverting Laurent polynomial-Vandermonde matrices, which is an extension of
a subclass of Hessenberg-quaisiseparable-Vandermonde
matrices. For a set of n distinct nodes {xk }nk=1 , the classical
h
i
Vandermonde matrix V (x) = xj−1
is known to be invertible, provided the nodes are distinct. One can generalize
i
the structure by evaluating a basis other than the monomials at the nodes. That is, for a set of n polynomials
R = {rk (x)}n−1
k=0 , the matrix of the form


r0 (x1 ) . . . rn−1 (x1 )
 r0 (x2 ) . . . rn−1 (x2 ) 


(1.1)
VR (x) =  .

..
 ..

.
r0 (xn ) . . . rn−1 (xn )
is called a polynomial-Vandermonde matrix. In the simplest case where R = {1, x, x2 , . . . , xn−1 } is the monomial
basis, the matrix VR (x) reduces to a classical Vandermonde matrix and the inversion algorithm is due to Traub [9].
We are concerned in the case where R = {ψk (x)}n−1
k=0 is a system of Laurent polynomials.
We also consider the problem of evaluating a fast system solver for a Cauchy-Vandermonde matrix. Linear
systems with Cauchy and Vandermonde matrices,




1 x1 . . . xn−1
1
1
1
. . . x1 −y
x1 −y1
n
1 x2 . . . xn−1 
2




..
..
..
(1.2)
V (x1:n ) =  .

..
..  C(x1:n , y1:n ) = 
.
.
.

 ..
.
.
1
1
. . . xn −yn
xn −y1
1 xn . . . xn−1
n
are classical. They are encountered in many applied problems related to polynomial and rational function computations. Favorable numerical properties are understood for Vandermonde and related matrices (see, for example
[13, 16, 17, 18, 22, 14, 15].) We consider a fast system solver for a Cauchy-Vandermonde matrix:


1
1
...
1 x1 . . . x1n−`−1
 x1 − y1

x1 − y`


.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
C(x1:n+` , y1:` ) = 
(1.3)
.
.
.
.
.
.
.



1
1
...
1 xn . . . xnn−`−1
xn − y1
xn − y`
2. Traub-like algorithm for Laurent-polynomial system
2.1
The Traub Algorithm for classical Vandermonde matrices
We first consider the numerical inversion of Vandermonde matrices. Traub proposed the algorithm in [9], which says
that the inverse of a classical Vandermonde matrix is given by
1
Research Statement
David Miller
V (x)−1

qn−1 (x1 ) . . .
qn−2 (x1 ) . . .

=
..

.
q0 (x1 )

qn−1 (xn )
n qn−2 (xn )
1

,
 diag
..
P 0 (xk ) k=1

.
...
(2.1)
q0 (xn )
where P (x) is the master polynomial defined by
P (x) :=
n
Y
(x − xk ) = xn +
k=1
n−1
X
Pk · xk .
(2.2)
k=0
and {qk (x)} are the associated (or Horner) polynomials, defined by
qk (x) = xk + Pn−1 xk−1 + · · · + Pn−k−1 x + Pn−k .
2
This algorithm is fast in the sense that it computes all n entries of V
favorably with the complexits O(n3 ) flops of general algorithms.
2.2
−1
(2.3)
2
in only 6n flops, which compares
Green’s matrices and Laurent Polynomials
There has been much discussion on fast inversion of polynomial-Vandermonde systems for various polynomial systems. Recent work has concentrated on the inversion of (H, 1)-quasiseparable-Vandermonde matrices in [2] and
(H, m)-quasiseparable-Vandermonde matrices in [3]. We use this previous work and look at a subclass of (H, 1)quasiseparable matrices, called Green’s matrices (first introduced in [8]), and relate these matrices and their polynomials with Laurent polynomials.
Definition 2.1. A strictly upper Hessenberg matrix G is called Green’s (H,1)-qs (or simply Green’s matrix) if
max rank G(1 : i, i : n) = 1.
1≤i≤n
It is discussed in [8] that a Green’s matrix is (H, 1)-qs, which implies that it has a generator definition. A strictly
upper Hessenberg matrix G is Green’s (H, 1)-qs if it can be represented in the form


τ̂0 τ1 τ̂0 σ1 τ2 τ̂0 σ1 σ2 τ3
...
...
τ̂0 σ1 . . . σn−1 τn
 σˆ1
τ̂1 τ2
τ̂1 σ2 τ3
...
...
τ̂1 σ2 . . . σn−1 τn 


 0
σˆ2
τ̂2 τ3
...
...
τ̂2 σ3 . . . σn−1 τn 



..
.
..
..
..
..
(2.4)
G=

 ..
.
.
.
.
.



 .
..
..
 ..
.
.
σ̂n−2 τ̂n−2 τn−1
τ̂n−2 σn−1 τn 
0
...
...
0
σ̂n−1
τ̂n−1 τn
where {σk , τk , σ
bk 6= 0, τbk } are called the generators of G.
Definition 2.2. Let G be a Green’s matrix. Then it has decomposition
G = Θ0 Θ1 . . . Θn
where

Θ0 =
τb0
In−1

, Θk = 

(2.5)

Ik−1
τk
σ
bk
σk
τbk

 , Θn =

In−1
τn
(2.6)
In−k−1
Definition 2.3. A system of polynomials R = {rk (x)}nk=0 is called a system of Green’s polynomials if it is related
to a Green’s matrix G via
r0 (x) = λ0 , rk (x) = λ0 λ1 . . . λk det(xI − Hk×k ).
(2.7)
with λk = 1/b
σk .
2
Research Statement
David Miller
It was shown in [8] that if a system of polynomials are Green’s polynomials, then they have the recurrence relation
1 σ
fk (x)
bk σk − τbk τk τbk
fk−1 (x)
=
(2.8)
rk (x)
−τk
1 x · rk−1 (x)
σ
bk
where {fk (x)} are auxilary polynomials. In [8] it was shown that Green’s polynomials and the auxilary fk (x)’s define
Laurent polynomials:
Definition 2.4. Let J = (j1 , j2 , . . . , jn ) be a sequence of binary digits. We define a sequence of Laurent polynomials Ψ = {ψk (x)}n−1
k=0 as
(
Pk+1
x− m=1 jm rk (x) if jk+1 = 0
Pk+1
ψk (x) =
.
(2.9)
x− m=1 jm fk (x) if jk+1 = 1
It was also shown in [8] that these Laurent polynomials are related to a Twisted Green’s matrix:
Definition 2.5. Let J = (j1 , j2 , . . . , jn ) be a sequence of binary digits. We define twisted Green’s matrices GJ
by the recursion
(
Gk−1 Θk if jk = 0
G0 = Θ0 ,
Gk =
(2.10)
Θk Gk−1 if jk = 1
Finally, it was shown in [8] that the twisted Green’s matrices in (2.5) represent multiplication operators in the
frame of Laurent polynomials (2.8):
φ0 (x) φ1 (x) φ2 (x) . . . GJ = x · φ0 (x) φ1 (x) φ2 (x) . . .
Definition 2.6. Let Ψ be a system of Laurent polynomials as in (2.8). Then VΨ (x), given by


ψ0 (x1 ) . . . ψn−1 (x1 )
 ψ0 (x2 ) . . . ψn−1 (x2 ) 


VΨ (x) =  .

..
 ..

.
ψ0 (xn ) . . .
(2.11)
(2.12)
ψn−1 (xn )
is called a Laurent-Vandermonde matrix.
We look to use the properties of the Traub-algorithm and the multiplication operator (2.5) to find a fast inversion
algorithm for (2.11).
2.3
Main Results.
We begin by proving
ˆ T ˆ T
CΨ
b = G · I · CΨ · I · G ,
(2.13)
where G is a permutation matrix and CΨ is the multiplication operator of Ψ:
x · ψ0 (x) . . .
ψn−1 (x) = ψ0 (x) . . .
ψn−1 (x) CΨ
(2.14)
b = {ψb0 (x), . . . , ψbn (x)}. Using
The relation (2.12) gives a recurrence relation of the horner-Laurent polynomials, Ψ
these polynomials, we have the equation for the inverse:
T
VΨ−1 (x) = Iˆ · VΨ
b (x) · diag(c1 , . . . , cn ).
3
(2.15)
Research Statement
David Miller
3. Cauchy-Vandermonde matrices
3.1
Related Facts
The Björk-Pereyra algorithm for Vandermonde systems. The Björk-Pereyra algorithm is based on the
decomposition of the inverse of a Vandermonde matrix into a product of bidiagonal factors,
−1
−1
V −1 (x1:n ) = U −1 . . . Un−1
L−1
n−1 . . . L1 .
(3.1)
This description allows one to solve the associated linear systems in only O(n2 ) operations, which is by an order of
magnitude less than the complexity O(n3 ) of general methods. Moreover, the algorithm requires only O(n) locations
of memory.
The BKO Algorithm on general Cauchy systems. A fast Björck-Pereyra-type algorithm was established
in [11]. Again, it is based off the decomposition the inverse of a Cuachy matrix into a product of bidiagonal factors,
C −1 (x1:n , y1:n ) = U1 . . . Un−1 DLn−1 . . . L1 .
(3.2)
This description again allows one to solve the associated linear systems in only O(n2 ) operations. Moreover, the
algorithm requires only O(n) locations of memory. Also in [11], it was shown that the sparsity of the Lk , Uk and D
factors in 3.2 results in favorable upper bounds when computing forward stability of the algorithm.
New algorithm for Cauchy matrices. In [12], a new LU-factorization of Cauchy matrices was introduced.
This new algorithm is similar to the Björck-Pereyra algorithm, but used a “lambda” shape rather than the bidiagonal
factorization. Again, the Cauchy matrix was factored as in 3.2, but with the Lk , Uk of the form


 (k)
 (k)
u11
l11




..
..




.
.







(k)
(k) 
(k)
(3.3)
Lk = 
 Uk = 
ukk . . . ukn 
lkk




..




..
..
.
.




.
(k)
(k)
(k)
unn
lnk
lnn
Cauchy-Vandermonde Matrices. A Cauchy-Vandermonde matrix of order n is a matrix W of the form
W =[ C
V ],
(3.4)
where the first ` (1 ≤ ` ≤ n) columns form a Cauchy matrix and the last n − ` columns form a Vandermonde matrix:


1
1
...
1 x1 . . . x1n−`−1

 x1 − y1
x1 − y`


.
.
.
.
.
.
.
.

..
..
..
..
..
..
..
(3.5)
W (x1:n , y1:` ) = 



1
1
...
1 xn . . . xnn−`−1
xn − y1
xn − y`
Cauchy-Vandermonde matrices are encountered in applied problems related to rational-polynomial interpolation.
Ordering of nodes. It is important to note that the BKO algorithm is not invariant to permutations of the points
defining the Cauchy matrix. Different configurations of {x1:n } yield different decompositions for C(x1:n , y1:n )−1 ,
though all of the form 3.2. A reordering of the nodes:
|x1 | = max |xj |,
1≤j≤n
k−1
Y
i=1
|xk − xi | = max
k≤j≤n
k−1
Y
|xj − xi | for 2 ≤ k ≤ n
(3.6)
i=1
was proposed in [18] and in [23] it was called Leja ordering. Through numerical examples in [11], it was shown that
Leja ordering of the nodes (3.6) improves the backward stability of the BKO algorithm. This implies the need to
investigate the affects of Leja ordering on all BKO-type algorithms. In [21], a fast, BP-type algorithm is introduced
for solving a Cauchy-Vandermonde system. In addition, a numerical example is given to indicate the stability of the
algorithm, however the affect of the ordering of the nodes was not shown.
4
Research Statement
David Miller
The ordering (3.6) mimics the row interchange that would occur when Gaussian elimination with partial pivoting
is applied. In addition to (3.6), a similar pivoting method for Cauchy matrices was introduced in [12]. Here, it
is implied that partial pivoting on a Cauchy matrix, C(x1:n , y1:n ), is equivalent to successive maximization of the
quantities
Qi−1
Qi−1
j=1 (xi − xj )
j=1 (yi − yj )
(3.7)
|di | = ,
Qi−1
Qi−1
(xi − yi )
j=1 (xi − yj )
j=1 (xj − yi )
for i = 1, . . . , n. In [12] this ordering is called predictive partial pivoting, or rational Leja ordering by analogy with
Leja ordering of [18].
3.2
Main Results
New algorithm. In this paper we an alternative to the BKO-type algorithm, called the MO algorithm, based on
the factorization
W −1 (x1:n , y1:` ) = U1 . . . Un−1 DLn−1 . . . L1
(3.8)
where the L factors have the “lambda” shape as in (3.3). The new algorithm is provided to facilitate the computation
of the upper bounds when performing error analysis.
New ordering of nodes. After deriving the MO algorithm, we use the well known equation for the determinant
of a Cauchy-Vandermonde matrix to maximize the det(W1:k,1:k ) by maximizing the quantities

Qi−1
Qi−1

j=1 (xi − xj )
j=1 (yi − yj )


, for i = 1, . . . , `
Q
Q

i−1
i−1


(x
−
y
)
(x
−
y
)
(x
−
y
)
i
i
j
j
i
 i
j=1
j=1
|di | =
(3.9)
Qi−1




j=1 (xi − xj ) 

,
 Q`
j=1 (xi − yj )
for i = ` + 1, . . . , n − 1
Error analysis. After deriving the BKO-type algorithm for Cauchy-Vandermonde matrices, we provide nice
upper bounds on the unit roundoff error for the computed solution â.
Numerical experiments. Finally, we present several numerical experiments on the algorithm. In these experiments, we test different orderings of the nodes, including monotonic and (rational) Leja, to see their affect on the
stability of the algorithm.
References
[1] T. Bella, Y. Eidelman, I. Gohberg, I. Koltratch, V. Olshevsky, A fast Bjorck-Pereyra like algorithm for solving
Hessenberg-quasiseparable-Vandermonde systems, submitted to SIAM Journal of Matrix Analysis, (2007)
[2] T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky, E. Tyrtyshnikov, Fast inversion of polynomial-Vandermonde
matrices for polynomial systems related to order one quasiseparable matrices, Advances in Structured Operator
Theory and Related Areas, 237, (2013), 79-106.
[3] T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky, E. Tyrtyshnikov, P. Zhlobich, A Traub-like algorithm for
Hessenberg-quasiseparable-VNdermonde matrices of arbitrary order, (2010).
[4] D. Calvetti and L. Reichel, Fast inversion of Vandermonde-like matrices involving orthogonal polynomials, BIT
(1993)
[5] I. Gohberg and V. Olshevsky, Fast inversion of Chebyshev-Vandermonde matrices, Numerische Mathematik,
67, No. 1 (1994), 71-92
[6] J. Maroulas and S. Barnett, Polynomials with respect to a general basis. I. Theory, J. of Math. Analysis and
Appl., 72 (1979), 177-194.
5
Research Statement
David Miller
[7] V. Olshevsky, Associated polynomials, unitary Hessenberg matrices and fast generalized Parker-Traub and
Bjorck-Pereyra algorithms for Szego-Vandermonde matrices invited chapter in the book ”Structured Matrices:
Recent Developments in Theory and Computation”, 67-78, NOVA Science Publ., USA. (2001).
[8] V. Olshevsky, G. Strang, P. Zhlobich, Green’s matrices, Linear Algebra and its Applications, (2010).
[9] J. Traub, Associated polynomials and uniform methods for the solution of linear problems, SIAM Review, 8,
No. 3, (1966), 277-301.
[10] T. Boros, T.Kailath and V.Olshevsky, Fast Björck-Pereyra-type algorithm for parallel solution of Cauchy linear
equations, Linear Algebra and Its Applications, 302-303 (1999), p. 265-293
[11] T. Boros, T.Kailath and V.Olshevsky, A Fast Parallel Björk-Pereyra-type Algorithm for Solving Cauchy Linear
Equations
[12] T. Boros, T.Kailath and V.Olshevsky, Pivoting and Backward Stability of Fast Algorithms for solving Cauchy
Linear Equations
[13] A. Björck and V. Pereyra. Solution of Vandermonde systems of equations. Math. Comp, 24:893-903, 1970.
[14] D. Calvetti and L. Reichel. A Chebyshev-Vandermonde solver. Linear Algebra and its Applications, 172(219-229),
1992.
[15] D. Calvetti and L. Reichel. Fast inversion of Vandermonde-like matrices involving orthogonal polynomials. BIT,
1993
[16] N.J. Higham. Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems. Numerische
Mathematik, 50:613-632, 1987.
[17] N.J. Higham. Fast solution of Vandermonde-like systems, involving orthogonal polynomials. IMA J. Numerical
Analysis, 8:473-486, 1988.
[18] N.J. Higham. Stability analysis of algorithms for solving confluent Vandermonde-like systems. SIAM, 11:23-41,
1990.
[19] N.J. Higham. Accuracy and stability of numerical algorithms. SIAM, 1996
[20] J.J. Martinez, J.M. Peña, Factorizations of Cauchy-Vandermonde Matrices
[21] J.J. Martinez, J.M. Peña, Fast algorithms of Björck-Pereyra-type for solving Cauchy-Vandermonde linear systems
[22] L. Reichel and G. Opfer. Chebyshev-Vandermonde systems. Math Comp., 57:703-721, 1991.
[23] L. Reichel. Nerwton interpolation at Leja points. BIT, 23-31, 1990
6