Gauss Tchebychev Lobatto

7/29/2019 Gauss Tchebychev Lobatto

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Vandermonde systems on Gauss-LobattoChebyshev nodes

A. Eisinberg, G. Fedele

Dip. Elettronica Informatica e Sistemistica,Universita degli Studi della Calabria,

87036, Rende (Cs), Italy

Abstract

This paper deals with Vandermonde matrices Vn whose nodes are the Gauss-LobattoChebyshev nodes, also called extrema Chebyshev nodes. We give an analytic fac-torization and explicit formula for the entries of their inverse, and explore its com-putational issues. We also give asymptotic estimates of the Frobenius norm of bothVn and its inverse and present an explicit formula for the determinant of Vn.

Key words: Vandermonde matrices, Polynomial interpolation, Conditioning

1 Introduction

Vandermonde matrices defined by Vn(i, j) = xi1j , i , j = 1, 2..., n; xj C arestill a topical subject in matrix theory and numerical analysis. The interestarises as they occur in applications, for example in polynomial and exponen-tial interpolation, and because they are ill-conditioned, at least for positiveor symmetric real nodes [1]. The primal system Vna = b represents a mo-ment problem, which arises, for example, when determining the weights fora quadrature rule, while the matrix Vn = VTn involved in the dual systemVnc = f plays an important role in polynomial approximation and interpo-lation problems [2,3]. The special structure of Vn allows us to use ad hocalgorithms that require O(n2) elementary operations for solving a Vander-monde system. The most celebrated of them is the one by Bjorck and Pereyra[4]; these algorithms frequently produce surprisingly accurate solution, evenwhen Vn is ill-conditioned [2]. Bounds or estimates of the norm of both Vnand V1n are also interesting, for example to investigate the condition of thepolynomial interpolation problem. Answer to these problems have been givenfirst for special configurations of the nodes and recently for general ones [5].

Preprint submitted to Appl. Math. and Comp. 12 March 2005


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Polynomial interpolation on several set of nodes has received much attentionover the past decade [6]. Theoretically, any discretization grid can be used toconstruct the interpolation polynomial. However, the interpolated solution be-tween discretization points are accurate only if the individual building blocksbehave well between points. Lagrangian polynomials with a uniform grid suf-

fer for the effect of the Runge phenomenon: small data near the center of theinterval are associated with wild oscillations in the interpolant, on the order2n times bigger, near the edges of the interval, [7][8]. The best choice is touse nodes that are clustered near the edges of the interval with an asymptoticdensity proportional to (1 x2)1/2 as n , [9]. The family of Chebyshevpoints, obtained by pro jecting equally spaced points on the unit circle down tothe unit interval [1, 1] have such density properties. The classical Chebyshevgrids are [10]:

Chebyshev nodes

T1 =

xk = cos2k 1

2n

, k = 1, 2,...,n

(1)

Extended Chebyshev nodes

T2 =

xk = cos

2k12n

cos

2n

, k = 1, 2,...,n (2)

Gauss-Lobatto Chebyshev nodes (extrema)

T3 = xk = cos k 1n 1 , k = 1, 2,...,n (3)

In [11] it is proved that interpolation on the Chebyshev polynomial extremaminimizes the diameter of the set of the vectors of the coefficients of all possi-ble polynomials interpolating the perturbed data. Although the set of Gauss-Lobatto Chebyshev nodes failed to be a good approximation to the optimalinterpolation set, such set is of considerable interest since the norm of corre-sponding interpolation operator Pn(T3) is less than the norm of the operatorPn(T1) induced by interpolation at the Chebyshev roots [12].

This paper deals with Vandermonde matrices on Gauss-Lobatto Chebyshevnodes. Through the paper we present a factorization of the inverse of such ma-trix and derive an algorithm for solving primal and dual system. We also giveasymptotic estimates of the Frobenius norm of both Vn and its inverse and anexplicit formula for det(Vn). A point of interest in this matrix is the (relative)moderate growth, versus n, of the condition number 2(Vn), [13][3]. Figure 1shows the 2 comparison between the Vandermonde matrix on the Chebyshevnodes (Vn(T1)), Chebyshev extesa nodes (Vn(T2)) and Gauss-Lobatto Cheby-shev nodes (Vn(T3)).

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0 10 20 30 40 50 60 70 80 90 100

0.7

0.75

0.8

0.85

0.9

0.95

1

n

2(V

n(T

3)/

2(V

n(T

1))

2(V

n(T

3)/

2(V

n(T

2))

Fig. 1. Plot of the ratios 2(Vn(T3))2(Vn(T1))

and 2(Vn(T3))2(Vn(T2))

.

2 Preliminaries

Let Vn be the Vandermonde matrix defined on the set of n distinct nodesXn = {x1,...,xn}:

Vn(i, j) = xj1i , i, j = 1,...,n (4)

In [14] the authors show that the inverse of the Vandermonde matrix Vn,namely Wn, is:

Wn(i, j) = (n, j)(n,i,j), i, j = 1,...,n (5)

where the function (n, i, j) is defined as:

(n, i, j) = (1)i+jnir=0

(1)rxrj(n, n i r), i, j = 1,...,n (6)

and the functions (m, s) and (m, s) are recursively defined as follows:

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(m, s) = (m 1, s) + xm (m 1, s 1) , m, s integer

(m, 0) = 1, m = 0, 1, ...

(s < 0) (m < 0) (s > m) (m, s) = 0

(7)

(m + 1, s) = (m,s)xm+1xs

, m integer; s = 1,...,m

(m + 1, m + 1) =mk=1

1xm+1xk

(2, 1) = (2, 2) = 1x2x1

(8)

By (5), taking into account the (6), Wn can be factorized as:

Wn = S P F (9)

where:

S(i, j) = (1)i+j+1(n, n + 1 i j), i = 1,...,n;j = 1,...,n + 1 i

(10)

P(i, j) = (1)jxi1j , i, j = 1,...,n (11)

F = diag {(n, i)}i=1,2,...,n (12)

Note that:

Sm(x) =mi=1

(x xi) =mr=0

(1)r (m, r) xmr (13)

S

m(xk) = (1)m+k1

(m, k), k = 1,...,m (14)

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3 Main results

We start by noting that, for some sets of interpolation nodes, explicit expres-sion for and may be found in [15]. We consider the set of Gauss-Lobatto

Chebyshev nodes (Xn = T3) and give the proof of some properties useful inthe sequel.

Lemma 1

(n, 2s) = (1)s 12n2

n2

q=1

n12q1

qs

, s = 0, ...,n

2

(n, 2s + 1) = 0, s = 0, ...,n2

(15)

where notations and denote the floor and ceiling functions, respectively[16].

Proof. It is easy to show that (13) can be rewritten as:

Sn(x) =1

2n2(x 1)(x + 1)Un2(x) (16)

where

Um(x) =sin [(m + 1) arccos(x)]

sin [arccos(x)]

is the m-order Chebyshev polynomial of the second kind.

But [17]:

Un2(x) =

n2

q=1(1)q+1

n 12q 1x

n2q(1

x2)q1 (17)

by substituting the (17) in (16) one has:

Sn(x) =

n2

q=1

qs=0

(1)s

n 12q 1

q

s

xn2s (18)

and, therefore, the (15) follows.

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Lemma 2

Sn(xk) =n 12n2

(1)n+k (1)nk,1 + k,n

(19)

Proof. The (16) can be rewritten as:

Sn(x) =1

2n2(x 1)(x + 1) sin [(n + 1) arccos(x)]

sin [arccos(x)]

therefore, by standard algebraic manipulations:

Sn(xk) =1

2n2(n

1)cos[(n

k)]

1

2n2

cos

k1n1

1 cos2 k1n1

sin[(n k)]

Noting that:

limk1

cosk1n1

1 cos2

k1n1

sin[(n k)] = (1)n(n 1)

limkn

cos k1n1 1 cos2

k1n1

sin[(n k)] = (n 1)

the (19) follows.

By substituting the (19) in (14), one has:

(n, k) =

2n3

n1k = 1, n

2n2

n1k = 2,...,n 1

(20)

Lemma 3 An alternative formulation of (15) is:

(n, 2s) = (1)s 122s

n s

s

n2 n 2s

(n s 1)(n s) , s = 0, 1, ...,

n

2

(21)

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Proof. By the recurrence properties of the second-kind Chebyshev polynomials[18], one has:

Sn(x) xSn1(x) +1

4Sn2(x) = 0

therefore

n2 s=0

(n, 2s)xn2sxn12 s=0

(n1, 2s)xn2s1 + 14

n22 s=0

(n2, 2s)xn2s2 = 0(22)

must holds. The (22) can be proved by standard algebraic manipulations whenn is both odd and even.

By rearranging (10), (11) and (12), one has:

S(i, j) = (1)i(n, n + 1 i j), i = 1,...,n;j = 1,...,n + 1 i

P(i, j) = cos j1n1 i1 , i, j = 1,...,nF(i, i) = (1)i(n, i) i = 1,...,n

(23)

Following the same line in [19], the matrix P can be factorized as:

P = D U H (24)

where:

D(i, i) = 1

2i2, i = 2,...,n

D(1, 1) = 1

(25)

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U(2i 1, 1) =

2i3i1

, i = 1, ...,

n2

U(2i, 2j) = 2i1ij , j = 1, ...,n2 , i = j, ..., n2 U(2i 1, 2j 1) =

2i2ij

, j = 2, ...,

n2

, i = j, ...,

n2

(26)

H(i, j) = cos

(i 1)(j 1)

n 1

, i = 1, ..., n, j = 1,...,n (27)

If one defines the matrix Q as:

Q(i, j) = 2ni1 [S D U] (i, j), i, j = 1,...,n (28)

the (9) becomes:

Wn =1

n

1

K Q H F (29)

where

K = diag{2i1}i=1,2,...,n (30)

F(1, 1) = 12

F(i, i) = (1)i, i = 2,...,n 1

F(n, n) = (1)n 12

(31)

We present here an efficient scheme for the computation Q. It can be shownthat Q can be build by the following equalities:

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Q(1, n 2) = 2

Q(i, n + 1 i) = (1)i i = 1, 2,...,n

Q(1, n 2j 2) = Q(1, n 2j), j = 1, 2, ...,n42

Q(i, n + 1 i 2j) = Q(i, n + 3 i 2j) Q(i 1, n + 2 i 2j), i = 2, 3,...,n;j = 1, 2,...,j

Q(i, 1) = Q(i, 1)/2, i = 1, 2,...,n(32)

where

j =

ni

2 n even

n1i2

n odd

4 The Frobenius norm of Vn and Wn

Proposition 1 The Frobenius norm of Vn is

VnF =

n +n 122n3

+2

(n 1)

n + 12

(n)

(33)

where (x) is the gamma function [20].

Proof.

Vn2F =n

i=1

ns=1

cos

s 1n 1

2i2(34)

But

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cos

s 1n 1

2i2=

1

22i2

2i 2i 1

+

1

22i2

i2k=0

2

2i 2

k

cos

2(i 1 k)(s 1)

n 1

(35)

then (34) becomes:

Vn2F =n

i=1

ns=1

1

22i2

2i 2i 1

+

ni=1

ns=1

i2k=0

2

22i2

2i 2

k

cos

2(i 1 k)(s 1)

n 1

(36)

By using the identity

ni=1

ns=1

1

22i2

2i 2i 1

=

2

n

n + 12

(n)

(37)

and by standard algebraic manipulations the (33) follows.

Proposition 2 The Frobenius norm of Wn is given by

Wn2F = 12(n 1) +22n4

n 1

1(n) +1

n 12(n)

(38)

where

1(n) =n

k=1

nk2 r=0

nk2 s=0

(1)n+k+r+s 1

2

n k r s

(2r)(2s) (39)

and

2(n) =n

k=1

nk2

r=0

nk2

s=0

1

2(2r)(2s) (40)

Proof. The (38) follows from standard algebraic manipulations.

Taking into account only the term 1(n) in (38) and using the facts

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20 30 40 50 60 70 80 90 1001

2

3

4

5

6

7

8

9

10x 10

3

n

Fig. 2. Relative error estimating ||Wn||F.

nk=1

nk2 r=0

nk2 s=0

[] =n12 r=0

n12 s=0

n2max(r,s)k=1

[]

q

s=0 p

32

s 1q

s = p + q

32

q 1 we give the following conjecture.

Conjecture 1

WnF 2

n 1n21

p=1

n21

q=1

n 12p 1

n 12q 1

p + q 3

2

q 1

, n (41)

Figure 2 shows the accuracy of the estimate of the Frobenius norm of Wn interm of relative error for n in the interval [20, 100] by Eq. 41.

5 The determinant of Vn

The next proposition gives the value of the determinant of Vn.

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Proposition 3

det(Vn) = 2

(n 1)n2n(n2)

(42)

Proof. By the definition of the Vandermonde determinant we have

det(Vn) =

1i


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of both our and Bjorck-Pereyra algorithm. A set of experiments has been run,for n = 3 10, 20, 30, 40, 50, 100. We have generated the right-hand sides fand b with random entries uniformly distributed in the interval [1, 1]. Tables1 and 2 shows maximum and mean value of (43) and (44) over 10000 runs, thefraction of trials in which the proposed algorithms (EF) give equal or more

accurate result than Bjorck-Pereyra ones (BP) and also the probability thatc and a is less or equal than 10nu where u = 2

53 is the unit roundoff. Asto the computational cost the EF algorithms require 3n2 + O(n) while BPalgorithms cost 2.5n2 + O(n) flops. EF algorithms seem to perform betterthan the Bjorck-Pereyra ones in terms of numerical accuracy and stability asit can be seen for high value of n. Same results are obtained by computingthe approximate solutions c and a in Matlab package and then by migratingthe output in Mathematica in order to compare it with the exact one. ForMatlab code refer to Appendix A.

n BP EF s.r.

max mean max mean EF vs BP p(c 10nu)3 2.34-13 3.07-16 2.15-15 4.02-17 0.98 0.99

4 1.73-12 2.53-15 4.03-13 1.09-15 0.75 0.99

5 9.31-12 5.82-15 4.65-12 1.51-15 0.93 0.98

6 1.43-11 1.40-14 1.54-12 2.41-15 0.94 0.97

7 2.47-11 2.35-14 6.60-12 4.36-15 0.96 0.97

8 3.24-10 7.90-14 5.67-12 4.69-15 0.99 0.95

9 6.20-11 6.12-14 1.12-12 2.94-15 0.99 0.96

10 1.56-10 1.98-13 9.00-12 6.66-15 0.99 0.95

20 1.17-06 4.10-10 5.47-11 2.54-14 1.00 0.92

30 2.10-03 9.79-07 2.22-09 3.86-13 1.00 0.91

40 5.68+00 3.77-03 1.91-10 1.01-13 1.00 0.92

50 7.61+03 1.38+01 4.04-11 9.49-14 1.00 0.90

100 8.52+20 1.69+18 1.68-09 7.45-13 1.00 0.88

Table 1. Dual problem. Maximum and mean value of c. Success rate of EF algorithm over 10000 runs.

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n BP EF s.r.

max mean max mean EF vs BP p(a 10nu)3 1.70-13 2.90-16 5.46-13 3.26-16 0.79 0.99

4 1.06-10 1.33-14 3.25-11 3.99-15 0.74 0.985 3.96-11 8.87-15 7.94-13 1.32-15 0.86 0.97

6 6.69-11 2.68-14 3.68-12 2.57-15 0.91 0.97

7 2.93-11 1.66-14 4.69-12 2.91-15 0.95 0.97

8 6.00-11 3.52-14 2.48-12 3.15-15 0.98 0.96

9 9.70-11 4.33-14 6.00-12 3.16-15 0.96 0.96

10 8.44-11 7.77-14 3.06-11 7.37-15 0.98 0.95

20 1.12-08 3.25-12 4.49-11 1.98-14 1.00 0.94

30 1.89-07 9.62-11 2.43-11 2.81-14 1.00 0.93

40 1.22-05 4.26-09 2.13-10 4.33-14 1.00 0.95

50 1.52-05 3.18-08 1.84-11 2.56-14 1.00 0.94

100 3.88+02 1.68-01 1.71-10 7.30-14 1.00 0.94

Table 2. Primal problem. Maximum and mean value of a. Success rate of EF algorithm over 10000 runs.

7 Conclusion

In this paper we derived an explicit factorization of the Vandermonde matrixon Gauss-Lobatto Chebyshev nodes. Such factorization allows to design anefficient algorithm to solve Vandermonde systems. The numerical experimentsindicate that our approach is more stable compared with existing Bjorck-Pereyra algorithm. Starting from these theoretical results we are working witha conjecture on discrete orthogonal polynomials on Gauss-Lobatto Chebyshevnodes and its proof. The operation count and the accuracy obtained in theexperiments on least-squares problems seems to be very competitive.

Appendix A - Matlab code

function c=glc(f);

n=max(size(f));

nf=floor(n/2);

f(1)=f(1)/2;

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f(n)=f(n)/2;

for i=1:n

f(i)=(-1)^i*f(i);

end

% Matrix H

%--------------------------------------------------------H=zeros(n);

H(1,1:nf)=ones(1,nf);

H(1:nf,1)=ones(nf,1);

if rem(n,2)==0

start=1;

else

for j=1:ceil(n/2)

H(nf+1,2*j-1)=(-1)^(j+1);

end

H(:,nf+1)=H(nf+1,:);start=2;

end

for i=2:nf

for j=i:nf

H(i,j)=cos(rem((i-1)*(j-1),2*n-2)*pi/(n-1));

H(j,i)=H(i,j);

end

end

for j=1:nf

if rem(j,2)==0H(nf+start:n,j)=-flipud(H(1:nf,j));

else

H(nf+start:n,j)=flipud(H(1:nf,j));

end

end

for i=1:n

if rem(i,2)==0

H(i,nf+start:n)=-fliplr(H(i,1:nf));

else

H(i,nf+start:n)=fliplr(H(i,1:nf));

end

end

%--------------------------------------------------------

% Matrix Q

%--------------------------------------------------------

Q=zeros(n);

for i=1:n

Q(i,n+1-i)=(-1)^i;

end

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Q(1,n-2)=2;

for j=1:ceil((n-4)/2)

Q(1,n-2*j-2)=-Q(1,n-2*j);

end

for i=2:n

if rem(i,2)==0jmax=floor((n-i)/2);

else

jmax=ceil((n-1-i)/2);

end

for j=1:jmax

Q(i,n+1-i-2*j)=-Q(i,n+3-i-2*j)-Q(i-1,n+2-i-2*j);

end

end

Q(:,1)=Q(:,1)/2;

%--------------------------------------------------------aux=H*f;

c=zeros(n,1);

for i=1:n

for j=rem(n+i,2)+1:2:n+1-i

c(i)=c(i)+Q(i,j)*aux(j);

end

end

for i=1:n

c(i)=2^(i-1)*c(i);

endc=c/(n-1);

References

[1] Gautschi, W., Inglese, G., Lower bounds for the condition number ofVandermonde matrix, Numer. Math., 52 (1998), 241-250.

[2] Golub, G. H., Van Loan, C. F., Matrix Computation, third ed., Johns HopkinsUniv. Press, Baltimore, MD, 1996.

[3] Higham, N. J., Accuracy and Stability of Numerical Algorithms, SIAM,Philadelphia, PA, 1996.

[4] Bjorck, A., Pereyra, V., Solution of Vandermonde systems of linear equations,Math. of Computation, 24 (1970), 893-903.

[5] Tyrtyshnikov, E., How bad are Hankel matrices, Numer. Math., 67 (1994), 261-269.

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[6] Meijering, E., A Chronology of Interpolation: From Ancient Astronomy toModern Signal and Image Processing, Proc. of IEEE, 90 (2002), 319-342.

[7] Bjorck, A., Dahlquist, G., Numerical Methods, Prentice-Hall, Englewood Cliffs,NJ, 1974.

[8] Henrici, P., Essentials of Numerical Analysis, Wiley, New York, 1982.

[9] Berrut, J. P., Trefethen, L. N., Barycentric Lagrange interpolation, SIAM Rev.46(2004), 501-517.

[10] Brutman, L., Lebesgue functions for polynomial interpolation - a survey, Annalsof Numerical Mathematics, 4 (1997), 111-127.

[11] Belforte, G., Gay, P., Monegato, G., Some new properties of Chebyshevpolynomials, J. Comput. Appl. Math., 117 (2000), 175-181.

[12] Brutman, L., A note on polynomial interpolation at the Chebyshev extrema

nodes, Journal of Approx. Theory, 42 (1984), 283-292.

[13] Gautschi, W., Norms estimates for inverses of Vandermonde matrices, Numer.Math., 23 (1974), 337-347.

[14] Eisinberg, A., Picardi, C., On the inversion of Vandermonde matrix, Proc. ofthe 8th Triennial IFAC World Congress, Kyoto, Japan, 1981.

[15] Eisinberg, A., Fedele, G., Polynomial interpolation and related algorithms,Twelfth International Colloquium on Num. Anal. and Computer Science withAppl., Plovdiv, 2003.

[16] Knuth, D. E., The Art of Computer Programming, vol. 1, second ed., Addison-Wesley, Reading, MA, 1973.

[17] Gradshteyn, I. S., Ryzhik, I. M., Table of Integrals, Series and Products, thirded., Academic Press, New York, 1965.

[18] Rivlin, T. J., The Chebyshev Polynomials, John Wiley & Sons, New York, 1974.

[19] Eisinberg, A., Franze, G., Salerno, N., Rectangular Vandermonde matrices onChebyshev nodes, Linear Algebra Appl., 338 (2001), 27-36.

[20] Gatteschi, L., Funzioni Speciali, UTET, 1973.

[21] Wolfram, S., Mathematica: a System for Doing Mathematics by Computers,Second. ed., Addison-Wesley, 1991.

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Gauss Tchebychev Lobatto