.
.
.
.
..
.
Double ansatz spaces and barycentric
rootfinding
Alex Townsend
Oxford University
NLA group, Manchester University
6th of November 2012
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Global rootfinding
Find all x ∈ Ω ⊂ C such that
p(x) =
n
∑
ak φk (x) = 0,
ak ∈ C.
k=0
Examples:
.1. Monomials: φk (x) = x k .
.2. Chebyshev (of the first kind): φk (x) = cos(k cos−1 (x)).
.. Degree-graded: φk (x) is of degree k.
3
.
.
.
...
...
4
5
Lagrange: φk (x) = Lk (x).
( )
Bernstein: φk (x) = kn x k (1 − x)n−k .
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Global eigenvalue finding
Characteristic polynomial: p(x) = det (xI − A), A ∈ Cn×n .
Polynomials
of degree N
N x N matrix
conjugacy classes
Many, many eigenvalue and generalised eigenvalue problems.
Specht (1957), Barnett (1975), Feidler (2003), Corless (2010),
Mackey, Mackey, Mehl and Mehrmann (2005).
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Double ansatz spaces
Part I: Double Ansatz Spaces
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Companion form (of the first kind)



C1 (x) = 


an


x

1
..
.

an−1
 −1

+

an−2
..
...
a0
−1
0
.
1



.

It satisfies the following
 n−1  

x
p(x)
n−2
x
  0 

 

C1 (x)  .  =  . 
 ..   .. 
0
1
and the column shift sum relation
 

0
an
0
 0

1
0
 


..  +  ..
..

.
. .
0
1 0
an−1
−1
an−2
..
...
a0
−1
0
.
.
Alex Townsend
.



>
 = e1 ⊗ a .

.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Companion form (of the second kind)
The companion form of the second

an

1


..
C2 (y ) = y 
.


1
kind:
 
an−1
 an−2
 
  ..
+ .
 
 
1
It satisfies the following
( n−1
y
y n−2
...
0
0
...
0
0
.
a0
0
−1



.

−1
0
0
0
...
−1
..
a0
.
Alex Townsend
−1
..
)
(
1 C2 (y ) = p(y )
and the row shift sum relation
 

0
an
 an−1

1
 

 an−2

..
 

.
+ .

  ..

1
 


1 

−1
.
...
0
0
)





>
  = a ⊗ e1 .


−1
0
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Generalising the companion form
The companion forms motivate finding pencils L1 (x) = Ax + B,
A, BRn×n such that
[A|0] + [0|B] = v ⊗ a> ,
and pencils L2 (x) = Ax + B such that
[ ] [ ]
A
0
+
= a ⊗ w >,
0
B
v ∈ Rn×1
w ∈ Rn×1 .
These linearisations form vector spaces L1 (p) and L2 (p). Of
particular interest is their intersection:
DL(p) = L1 (p) ∩ L2 (p).
These spaces are studied in the M4 paper.
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Thinking in a continuous mode
Let’s replace matrices with functions:


an−1,n−1 an−1,n−2 . . . an−1,0
an−2,n−1 an−2,n−2 . . . an−2,0 


A=
..
.. 
..
..

.
.
.
. 
a0,n−1
a0,n−2
...
a0,0
Column-shift in a continuous mode is

an−1,n−1 an−1,n−2 . . . an−1,0
an−2,n−1 an−2,n−2 . . . an−2,0


..
..
..
..

.
.
.
.
a0,n−1
a0,n−2
...
a0,0
f (x, y ) =
n−1 ∑
n−1
∑
aij y i x j .
i=0 j=0
multiplication in ‘x’:

0
( n−1 n−1
)
0
∑∑

i j
f
(x,
y
)x
=
a
y
x
x
.. 
ij
.
i=0 j=0
0
Hence,
[A|0] + [0|B] = v ⊗ a>
[ ] [ ]
A
0
+
= a ⊗ w>
0
B
f (x, y )x + g (x, y ) = v (y )p(x).
yf (x, y ) + g (x, y ) = p(y )w (x).
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. L1 and L2 in a continuous mode
In a continuous mode this is:
L1 (p) = {f (x, y )x + g (x, y ) : f (x, y )x + g (x, y ) = v (y )p(x)} .
L2 (p) = {yf (x, y ) + g (x, y ) : yf (x, y ) + g (x, y ) = p(y )w (x)} .
Notice these definitions are basis independent.
.
Double Ansatz Space
.
..
Choose w (x) and v (y ) as polynomials of degree n and find f (x, y )
and g (x, y ) such that
f (x, y )x + g (x, y ) = v (y )p(x)
and
yf (x, y ) + g (x, y ) = p(y )w (x).
.
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
.
..
.
. A single ansatz space
.
Lemma (Double Ansatz space is a single ansatz)
..
We
. must choose v (x) = w (x).
..
.
Proof.
..
.
.
.
.
Hence, w (x) and v (x) are continuous and equal up to set of
measure
zero ⇒ v (x) = w (x).
.
..
This in a technical appendix of the M4 paper.
.
Alex Townsend
.
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
p(x)w (x) = xf (x, x) + g (x, x) = f (x, x)x + g (x, x) = v (x)p(x)
.
. Eigenvalue Exclusion Theorem
.
.
Theorem (Eigenvalue Exclusion (‘only if’ direction))
.
..
If p(x) is a regular polynomial and L(x) is in DL(p) with ansatz
v (y ) then L(x) is a linearisation of p(x) only if v (y ) does not share
a. root with p(x).
..
.
.
Proof.
.
..
yf (x, y ) − f (x, y )x = p(y )v (x) − v (y )p(x)
yg (x, y ) − g (x, y )x = yv (y )p(x) − p(y )v (x)x.
If p(y0 ) = v (y0 ) = 0 then
and
y0 g (x, y0 ) − g (x, y0 )x = 0.
For all x, det(L(x)) = det(p(x)) = 0 which contradicts p(x) being
regular.
.
..
.
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
y0 f (x, y0 ) − f (x, y0 )x = 0
.
. Multiplication matrices
A column and row shift can be written as

>


1
0
1
0
[ ]
X


 ..

.
.
.
..  = [X |0]
..
..  X =
X
.

.
0
1 0
1 0
|
{z
}
|
{z
}
M>
M
M represents multiplication.
For the Chebyshev basis it is:
1
1
2 0
2
1
1

0

2
2

.. ..

.
.
M=

1

2

..
.
..
.
1



 ∈ Rn×n+1 .


1
2
0
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Constructing DL
We have
yf (x, y ) − f (x, y )x = p(y )v (x) − v (y )p(x)
yg (x, y ) − g (x, y )x = yv (y )p(x) − p(y )v (x)x.
These become Sylvester matrix equations:
] [ 0 ]
[
] [ 0 ] [
>
>
0|M X −
= 0|av −
XM
va>
[
] [ 0 ] [ 0 ] [
]
>
>
0|M Y −
=
−
0|av
M
.
YM
M > va>
These can be solved by the Bartels–Stewart algorithm while taking
into account the extra degrees of freedom.
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Example:
For the cubic polynomials in the Chebyshev basis, i.e.
p(x) = AT3 (x) + BT2 (x) + CT1 (x) + DT0 (x) we have:
v
DL Linearisation
 
1
0
0

2A
x 0
0
 
0
1
0

 
0
0
1
0
2A − 2C
−2D
0
x 2A
0

0
x 0
2A
2A
2B
2A
0
4A
2B
 
0
B
−2D  + C − A
A−C
D
 
0
−A
2A  +  0
B −D
−A
0
C − 3A
D −B
 
2A
0
2B  + −2A
C +A
0

−A
D − B
−A

0
−2A 
D −B
−2A
−2B
−2A
.
Alex Townsend

D
C − A
D
C −A
2D
C −A
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Barycentric rootfinding
Part II: Barycentric rootfinding
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Barycentric rootfinding
The roots of the polynomial
p(x) =
n
∑
∏n
p(xk )`k (x),
k=0
j=0,j6=k (x − xj )
`k (x) = ∏n
j=0,j6=k (xk − xj )
are the eigenvalues of the generalised eigenvalue system associated
to




0
0 p0 p1 . . . pn
 −1

w0 x0









−1
w
x
1
1
B(x) = 
x
+


.




.
.
.
.
.
.



 .
.
.
−1
wn
xn
Exactly two eigenvalues are spurious. Lawrence and Corless (2011).
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Barycentric nodes and weights
Barycentric nodes: x0 , x1 , . . . , xn are the roots of φn+1 (x).
Barycentric weights:
wk =
const.
φ0n+1 (xk )
Golub–Welsch algorithm (1969) can be used to compute the
barycentric (and quadrature) weights in O(n2 ) operations.
Glaser–Lui–Rokhlin (2007) algorithm requires O(n)
operations.
Newton’s method has been forgotten! T. & Hale (2012)
Do not want to solve two eigenvalue problems!
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Newton’s method
Newton’s method in θ-space, x = cos(θ).
Newton’s method in θ−space
We will need:
.1. Fast and accurate evaluation of Legendre polynomial →
asymptotic formulae.
.. Fast evaluation of derivative → recurrence relations.
2
.
.
...
3
Good initial guesses → asymptotic approximations.
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. What does a Legendre polynomial look like?
Boundary Region: Bessel−like
Interior Region: Trig−like
cos(π/6)
cos(5π/6)
We use two asymptotic expansions: interior and boundary.
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Interior asymptotic formula
φn (cos θ) = Cn
M−1
∑
cos(αn,m )
hn,m
+ Rn,M (θ)
m+ 12
(2
sin
θ)
m=0
(
)
(
)
1
where αn,m = n + m + 2 θ − m + 12 π2 and
{
}
max |cos θ|−1 , 2 sin θ
|Rn,M (θ)| < Cn hn,M
.
1
M+ 2
(2 sin θ)
Absolute error of interior asymptotic for n=200
0
Convergence of Interior Asy formula
0
10
10
n=10
n=20
n=50
n=100
n=200
n=1000
−20
10
−5
10
Max Absolute error
Log error
−40
−10
10
10
−60
10
−80
10
−15
10
−100
10
−20
10
−1
−120
−0.8
−0.6
−0.4
−0.2
0
x
0.2
0.4
0.6
0.8
10
1
0
50
100
.
Alex Townsend
150
Terms
.
200
.
250
300
.
.
Double ansatz spaces and barycentric rootfinding
.
. Boundary asymptotic Formula
√
φn (cos θ) ∼
θ
sin(θ)
where ρ = n +
1
2.
(
M
M−1
∑
∑ Bm (θ)
Am (θ)
J0 (ρθ)
+
θJ
(ρθ)
1
ρ2m
ρ2m+1
m=0
)
,
m=0
Only the first few terms are known explicitly.
 (
)
π
c
θO n−2M− 23
n ≤θ ≤ 2
(
)
error =
θ3 O n−2M+ 12
0 ≤ θ ≤ nc .
Absolute error of boundary asymptotic for n=200
0
10
−5
Log error
10
−10
10
−15
10
−20
10
−1
−0.8
−0.6
−0.4
−0.2
0
x
0.2
0.4
0.6
0.8
1
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Initial guesses
Sufficiently good: Chebyshev points work! Petras (1998)
Initial guesses come from the asymptotic formulae. Analogously,
there are interior and boundary initial guesses. Lether & Wenston,
(1995).
Max error in the initial guesses
−4
10
Caused by the 5th
node changing from
interior to boundary
−6
10
−8
Max Error
10
−10
10
−12
10
−14
10
n = 5000
−16
10
−18
10
1
10
2
3
10
4
10
10
n
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Gauss–Legendre nodes and barycentric weights
Gauss–Legendre nodes and barycentric weights.
Relative error in Barycentric weights for 1000
Error in nodes for degree 1000
GLR
Asy
GW
−14
10
GW
GLR
Asy
−10
10
−12
10
−16
10
−14
10
−18
10
−16
10
200
400
600
800
1000
0
200
400
600
800
1000
This method can also compute the Gauss quadrature rules.
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Timing comparison
Timing comparison with GW, GLR and Asy.
Comparison of execution time
2
10
GW
GLR
Asy
1
Execution Time
10
0
10
−1
10
−2
10
0
2000
4000
6000
8000
10000
n
This asymptotic algorithm is default for Chebfun’s legpts and
jacpts command.
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.
. Conclusions:
...
DL literature can be simplified by thinking in continuous
mode.
...
Almost all the results from the M4 paper holds for
degree-graded bases.
...
...
3
Polynomial rootfinding can be done in the Lagrange basis.
4
Newton’s method with asymptotic evaluation can compute
the barycentric weights.
1
2
Thank you!
.
Alex Townsend
.
.
.
.
Double ansatz spaces and barycentric rootfinding
.