Orthogonal Rational Functions: Quadrature, Recurrence and...

Orthogonal rational functions Quadrature Recurrence Rational Krylov Future research Publications

Orthogonal Rational Functions: Quadrature,

Recurrence and Rational Krylov

Karl Deckers

Department of Computer Science, Katholieke Universiteit Leuven,Heverlee, Belgium.

Supervisor: Adhemar Bultheel.

February 4, 2009

Karl Deckers ORFs: Quadrature, Recurrence and Rational Krylov 1/40


Outline

1 Orthogonal rational functions

2 Quadrature

3 Recurrence

4 Rational Krylov

5 Future research



Outline


2 Quadrature

3 Recurrence

4 Rational Krylov

5 Future research



Rational functions

rational function = numerator polynomialdenominator polynomial

–200

–100

0

100

200

–15 –10 –5 5 10 15x

Figure: r3(x)=(x−0)(x−3)(x−(−2))

(1− x(−10) )(1−

x(−3) )(1−

x1 )

.

polynomial case results when all the poles are at infinity.



Rational functions

–15–10 –5 0 5 10 15–10–5

05

10–200

–100

0

100

200

ℜ{x}ℑ{x}

–15–10 –5 0 5 10 15–10–5

05

10–800–600–400–200

0200400600800

ℜ{x}ℑ{x}

Figure: Left: ℜ{r3(x)}. Right: ℑ{r3(x)}.



The case of the interval [−1, 1]

[−1, 1]

–3

–2

–1

0

1

2

3

–4 –3 –2 –1 1 2 3 4ℜ{x}

ℑ{x}

Polesα1, α2, α3, . . .

Arbitrary complex or infinite,but outside [−1, 1]

Fixed in advance

Function spaces

Ln = space of rational functions with poles among{α1, α2, . . . , αn}C =: L0 ⊂ L1 ⊂ L2 ⊂ . . . ⊂ Ln ⊂ . . . ⊂ L :=

⋃∞n=0 Ln



The case of the complex unit circle

Complex unit circle

–3

–2

–1

0

1

2

3

–4 –3 –2 –1 1 2 3 4ℜ{z}

ℑ{z}

Complex numbers

β1, β2, β3, . . .

|βk | < 1

Fixed in advance

Function spaces

Ln = space of rational functions with poles among{1/β1, 1/β2, . . . , 1/βn}C =: L0 ⊂ L1 ⊂ L2 ⊂ . . . ⊂ Ln ⊂ . . . ⊂ L :=

⋃∞n=0 Ln



Orthonormality

A weight function w(·) or w(·) defines an inner product 〈· , ·〉Orthonormalize canonical basis for L or L with respect to 〈· , ·〉,

Figure: canonical basis (left) and orthonormal basis (right)

we obtain an orthonormal basis:

{ϕ0(x), ϕ1(x), ϕ2(x), . . .} on L,with ϕk ∈ Lk \ Lk−1 and ϕk ⊥ Lk−1

{φ0(z), φ1(z), φ2(z), . . .} on L,

with φk ∈ Lk \ Lk−1 and φk ⊥ Lk−1



Orthonormal rational functions (ORFs)

ORFs on [−1, 1]

〈f , g〉 =

∫ 1

−1f (x)g(x)w(x)dx , f , g ∈ L.

〈ϕk , ϕl 〉 =

{

0, k 6= l1, k = l

.

–0.2

–0.1

0

0.1

0.2

0.3

0.4

0.5

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

Figure: Left: 〈ϕk , ϕl〉 = 0. Right: 〈ϕk , ϕk 〉 = 1.



Orthonormal rational functions (ORFs)

ORFs on the complex unit circle

〈f , g〉 =

∫ π

−π

f (z)g(1/z)w(θ)dθ, f , g ∈ L, z = e iθ, i2 = −1.

〈φk , φl〉 =

{

0, k 6= l1, k = l

.

–1.5

–1

–0.5

0

0.5

1

1.5

–2 –1 1 2



Outline


2 Quadrature

3 Recurrence

4 Rational Krylov

5 Future research



Quadrature

Quadrature formulas on [−1, 1]

Numerical approximation of theform:

∫ 1

−1f (x)w(x)dx ≈

n∑

k=1

λn,k f (xn,k)

{xn,k}nk=1 ⊂ [−1, 1] are called the

interpolation points or nodes

{λn,k}nk=1 are called the weights



Quadrature

Rational interpolatory quadrature formulas on [−1, 1]

Determine the weights so that the approximation is exact forevery gn−1 ∈ Ln−1, i.e.,

∫ 1

−1gn−1(x)w(x)dx =

n∑

k=1

λn,kgn−1 (xn,k)

Find gn−1 ∈ Ln−1 so that gn−1(xn,k) = f (xn,k) fork = 1, . . . , n

Approximate f (x) by gn−1(x):

∫ 1

−1f (x)w(x)dx ≈

∫ 1

−1gn−1(x)w(x)dx



Rational interpolatory quadrature formulas on the interval

Example




Example




Para-orthogonal rational functions (para-ORFs)

Generally, the zeros of ORFs are not all real,therefore we use para-ORFs Qn,τ (x):

Qn,τ ⊥ Ln−1(αn) = {f ∈ Ln−1 : f (αn) = 0}not unique → parameter τ ∈ C

if |τ | = 1, then zeros of Qn,τ (x) are all real

for certain values of τ , the zeros are all in [−1, 1]⇒ zeros can be used as nodes in quadrature formula

if αn real, then Qn,τ ⊥ Ln−1



Rational Gaussian quadrature formulas

Increasing the accuracy without increasing n

Let the

nodes xn,k be the zeros of the para-ORF Qn,τ (x)

weights be given by λn,k = 1/∑n−1

j=0 |ϕj (xn,k)|2.→ rational Gaussian quadrature formulas:

are exact for every f ∈ Ln−1 · Lcn−1, i.e., the space of rational

functions with poles among {α1, α1, α2, α2, . . . , αn−1, αn−1}have maximal domain of validity



Rational Gauss-Chebyshev quadrature

w(x) =

1/√

1 − x2√

(1 − x)/(1 + x)√1 − x2

Computing the nodes and weights

Approach:

derive explicit expressions for Chebyshev (para-)ORFs on[−1, 1]

obtain equations for the nodes→ have to be computed numerically

obtain equations for the weights→ computation is straightforward

[K. Deckers, J. Van Deun, and A. Bultheel, Math. Comp. 77(262)]




Equation for the nodes: Fn(θ) = (k − d/2)π, k = 1, . . . , n, withx = cos θ and θ ∈ [0, π]

Example

0

2

4

6

8

10

12

14

16

18

0.5 1 1.5 2 2.5 3θ




Newton’s method

Determine a set of initial values

based on thorough analysis of Fn(θ)[K. Deckers, J. Van Deun, and A. Bultheel, proc. ECT2006][K. Deckers, J. Van Deun, and A. Bultheel, Advances inEngineering Software]two methods based on whether exact solution is close to asteep gradient of Fn(θ)[J. Van Deun, K. Deckers, A. Bultheel and J.A.C. Weideman,ACM Trans. Math. Software 35(2)]

Use Newton iterations to obtain an accurate result

Implemented as MATLAB-function rcheb

Complexity = O(mn)



Rational Szego quadrature

numerical approximation:∫ π

−πf (z)w(θ)dθ ≈ ∑n

k=1 λn,k f (zn,k) , z = e iθ

nodes zn,k are zeros of para-ORF Qn,τ (z), where

Qn,τ ⊥ Ln−1(βn) = {f ∈ Ln−1 : f (βn) = 0}weights λn,k = 1/

∑n−1j=0 |φj(zn,k)|2

maximal domain of validity: Ln−1 · L(n−1)∗, i.e., the space ofrational functions with poles among{β1, 1/β1, β2, 1/β2, . . . , βn−1, 1/βn−1}



Rational Szego quadrature

w(θ) =

1 − cos θ1 + cos θsin2 θ

Computing the nodes and weights

Approach:

derive explicit expressions for Chebyshev (para-)ORFs on thecomplex unit circle

obtain equations for the nodes

obtain equations for the weights

[K. Deckers and A. Bultheel, J. Approx. Theory][A. Bultheel, R. Cruz-Barroso, K. Deckers, and P. Gonzalez-Vera,Math. Comp. 78(266)]



Outline


2 Quadrature

3 Recurrence

4 Rational Krylov

5 Future research



Recurrence and the Favard theorem

Theorem (Recurrence)

ORFs on [−1, 1] with poles outside [−1, 1] satisfy a three-termrecurrence relation:

ϕn = fr (ϕn−2, ϕn−1, αn−2, αn−1, αn,Cn,Dn,En) , n ≥ 1. (1)

Initial values (for n = 1):

ϕ−1(x) ≡ 0, ϕ0(x) ≡ 1,α−1 ∈ (R ∪ {∞}) \ [−1, 1], α0 ∈ (C ∪ {∞}) \ [−1, 1].




Theorem (Favard)

Let {fn}∞n=0, with fn ∈ Ln \ Ln−1, be a sequence of rationalfunctions generated by the 3-term recurrence (1). Under certainmild conditions, the fn form an orthonormal system with respect toa Hermitian positive-definite inner product.

Hermitian positive-definite

Hermitian: 〈f , g〉 = 〈g , f 〉 for every f , g ∈ LPositive-definite: 〈f , f 〉 > 0 for every f ∈ L with f 6= 0

Conditions

Cn = f (En,En−1,Dn,ℑ{αn−1})F (Dn, |En| , |En−1| ,ℑ{αn−2},ℑ{αn−1},ℑ{αn}) = 0




Dn

–2

–1

1

2

3

–3 –2 –1 1 2 3ℜ{Dn}

ℑ{Dn}

Midpoint and radius depend on |En| , |En−1| ,ℑ{αn−2},ℑ{αn−1}and ℑ{αn}[K. Deckers and A. Bultheel, J. Math. Anal. Appl. (submitted)]



Associated rational functions (ARFs)

ORFs:0 1 ϕ1(x) ϕ2(x) ϕ3(x) ϕ4(x) . . .

α−1 α0 α1 α2 α3 α4 . . .

C1,D1,E1 C2,D2,E2 C3,D3,E3 C4,D4,E4 . . .

ARFs of order 1:0 1 ϕ

(1)2\1(x) ϕ

(1)3\1(x) ϕ

(1)4\1(x) . . .

α0 α1 α2 α3 α4 . . .

C2,D2,E2 C3,D3,E3 C4,D4,E4 . . .

ARFs of order 2:0 1 ϕ

(2)3\2(x) ϕ

(2)4\2(x) . . .

α1 α2 α3 α4 . . .

C3,D3,E3 C4,D4,E4 . . .



Associated rational functions (ARFs)

Favard theorem for ARFs of order k.

Relation ARFs of different order.

If all poles are real:

Representation in terms of functions of the second kind.ϕ′

n(x) in terms of ARFs of different order.

[K. Deckers and A. Bultheel, proc. WCE2008][K. Deckers and A. Bultheel, IAENG Internat. J. Appl. Math.38(4)]



Asymptotic behavior

Relation ORFs on [−1, 1] and on the complex unit circle

Let

x = 12

(

z + 1z

)

,

αk = 12

(

βk + 1βk

)

for every k ≥ 1,

w(θ) = w(cos θ) |sin θ|.–1.5

–1

–0.5

0

0.5

1

1.5

–2 –1 1 2

Then, ϕn ∈ Ln is related with φ2n ∈ Ln · Lcn, i.e., the space of

rational functions with poles among{1/β1, 1/β1, 1/β2, 1/β2, . . . , 1/βn, 1/βn}.[K. Deckers, J. Van Deun, and A. Bultheel, J. Math. Anal. Appl.334(2)]



Asymptotic behavior

Asymptotics for ORFs on [−1, 1] when n tends to infinity based onknown results for the case of ORFs on the complex unit circle:

Ratio asymptotics ϕn+1(x)ϕn(x) , under the conditions that the

sequence of poles is bounded away from [−1, 1] and w(x) > 0almost everywhere on [−1, 1].

Asymptotics for the recurrence coefficients.

Strong convergence result for ϕn(x), under the conditionsthat the sequence of poles is bounded away from [−1, 1] and

∫ 1

−1

log w(x)√1 − x2

dx > −∞.

[K. Deckers and A. Bultheel, IMA J. Numer. Anal. 29(1)]



Outline


2 Quadrature

3 Recurrence

4 Rational Krylov

5 Future research



Rational Krylov subspaces

Suppose

A = n × n real symmetric positive-definite matrix with ndistinct eigenvalues xn,j , j = 1, . . . , n

q ∈ Rn \ {0}

{α1, . . . , αn} ⊂(

R−0 ∪ {∞}

)

rational Krylov subspace = space generated by the vectors bk(A)q,k = 0, . . . ,m − 1, with m ≤ n,

b0(A) = I and

bk(A) =Ak

(I − α−11 A)(I − α−1

2 A) · . . . · (I − α−1k A)

.




Orthonormalize vectors bk(A)q with respect to

〈a , b〉 =n

∑

j=1

ajbj ,

we obtain orthonormal vectors qk

= ϕk(A)q, with ϕk ∈ Lk .

The ϕk form an orthonormal system with respect to

〈f , g〉 =n

∑

j=1

f (xn,j)g(xn,j )w(xn,j)

3-term recurrence and Favard theorem for the vectors qk

Link: zeros of mth ORF ↔ mth Ritz values




Application: finding numerical approximations to u := f (A)q.

For large n, u is too expensive to compute. Therefore,

let xm,k denote the zeros of ϕm(x)find gm−1 ∈ Lm−1, with m ≪ n, so that gm−1(xm,k ) = f (xm,k )for k = 1, . . . , mu ≈ gm−1(A)q =: u(m)

We want an accurate approximation for u with m as small aspossible ⇒ need to determine an optimal set of poles.




Example: Time-periodic linear differential problem

d

dtu(t) + (A + 0.6I)u(t) = e−0.4tq, t ∈ [0,T ],

u(0) = u(T ), T = 0.01 .

u := u(0) = f (A)q, with

f (x) =e−0.4T [1 − e−(x+0.2)T ]

(x + 0.2)[1 − e−(x+0.6)T ].

α ≈ −1/T = −100 is optimal value in case of a multiple pole.

f (x) has a singularity in x = −0.6.




Relative error e = ‖u(m) − u‖/‖u‖ for the case of α = ∞ (solidline), respectively α = −100 (dashed line), with n = 2500.

0 20 40 60 80 100 12010

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100




Relative error e = ‖u(m) − u‖/‖u‖ for the case of α = −0.6 (solidline), respectively α = −100 (dashed line), with n = 2500.

0 5 10 15 20 25 3010

−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100




Relative error e = ‖u(m) − u‖/‖u‖ for the case of α1 = −0.6 andα2 = . . . = α7 = −100 (solid line), respectively α = −100 orα = −0.6 (dashed line), with n = 2500.

0 5 10 15 20 25 3010

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100



Outline


2 Quadrature

3 Recurrence

4 Rational Krylov

5 Future research



Future research

Rational Gauss-Radau and Gauss-Lobatto quadrature withrespect to Chebyshev weights on [−1, 1]

Computation of the nodes and weights based on 3-termrecurrence relation

Characterization of rational interpolatory quadrature formulaswith positive weights

Weak-star convergence for ORFs on [−1, 1]

...



Publications

Rational quadrature formulas

K. Deckers, J. Van Deun, and A. Bultheel. “RationalGauss-Chebyshev quadrature formulas for complex polesoutside [−1, 1],” Mathematics of Computation77(262):967–983, 2008.

K. Deckers, J. Van Deun, and A. Bultheel. “Computingrational Gauss-Chebyshev quadrature formulas with complexpoles,” in Proceedings of the Fifth International Conferenceon Engineering Computational Technology, B.H.V. Topping,G. Montero, and R. Montenegro, editors, paper 30,Civil-Comp Press, Stirlingshire, 2006.

K. Deckers, J. Van Deun, and A. Bultheel. “Computingrational Gauss-Chebyshev quadrature formulas with complexpoles: The algorithm,” Advances in Engineering Software,2008. (Accepted)



Publications

Rational quadrature formulas

J. Van Deun, K. Deckers, A. Bultheel, and J.A.C. Weideman.“Algorithm 882: Near best fixed pole rational interpolationwith applications in spectral methods,” ACM Transactions onMathematical Software, 35(2):14:1–14:21, 2008.

K. Deckers and A. Bultheel. “Orthogonal rational functionsand rational modifications of a measure on the unit circle,”Journal of Approximation Theory, 2008. (Accepted)

A. Bultheel, R. Cruz-Barroso, K. Deckers, and P.Gonzalez-Vera. “Rational Szego quadratures associated withChebyshev weight functions,” Mathematics of Computation78(266):1031–1059, 2009.



Publications

Three-term recurrence relation

K. Deckers and A. Bultheel. “Recurrence and asymptotics fororthogonal rational functions on an interval,” IMA Journal ofNumerical Analysis, 29(1):1–23, 2009.

K. Deckers and A. Bultheel. “Orthogonal rational functionswith complex poles: The Favard theorem,” Journal ofMathematical Analysis and Applications, 2008. (Submitted)

K. Deckers and A. Bultheel. “Orthogonal rational functions,associated rational functions and functions of the secondkind,” in Proceedings of the World Congress on Engineering2008, S.I. Ao, L. Gelman, D.W.L. Hukins, A. Hunter, andA.M. Korsunsky, editors, volume 2 of Lecture Notes inEngineering and Computer Science, pp. 838–843, NewswoodLimited, International Association of Engineers, London, 2008.



Publications

Three-term recurrence relation

K. Deckers and A. Bultheel. “Associated rational functionsbased on a three-term recurrence relation for orthogonalrational functions,” IAENG International Journal of AppliedMathematics 38(4):214–222, 2008.

K. Deckers, J. Van Deun, and A. Bultheel. “An extendedrelation between orthogonal rational functions on the unitcircle and the interval [−1, 1],” Journal of MathematicalAnalysis and Applications 334(2):1260–1275, 2007.



Publications

Rational Krylov sequences

K. Deckers and A. Bultheel. “Rational Krylov sequences andorthogonal rational functions,” Report TW499, Department ofComputer Science, K.U.Leuven, August 2007.

3iemes Journees Approximation 2008, Lille.

The 13th International Congress on Computational andApplied Mathematics 2008, Ghent.


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