ETNA
Electronic Transactions on Numerical Analysis.
Volume 9, 1999, pp. 65-76.
Copyright  1999, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
ORTHOGONAL POLYNOMIALS AND QUADRATURE∗
WALTER GAUTSCHI†
Abstract. Various concepts of orthogonality on the real line are reviewed that arise in connection with quadrature rules. Orthogonality relative to a positive measure and Gauss-type quadrature rules are classical. More recent
types of orthogonality include orthogonality relative to a sign-variable measure, which arises in connection with
Gauss-Kronrod quadrature, and power (or implicit) orthogonality encountered in Turán-type quadratures. Relevant
questions of numerical computation are also considered.
Key words. orthogonal polynomials, Gauss-Lobatto, Gauss-Kronrod, and Gauss-Turán rules, computation of
Gauss-type quadrature rules.
AMS subject classifications. 33C45, 65D32, 65F15.
1. Introduction. Orthogonality concepts arise naturally in connection with numerical
quadrature, when one tries to optimize the degree of precision. The classical example is the
Gaussian quadrature rule, which maximizes the (polynomial) degree of exactness. Closely
related quadrature rules are those of Radau and Lobatto, where one or two nodes are prescribed. In Gauss-Kronrod rules about half of the nodes are prescribed, all within the support
of the integration measure, which gives rise to orthogonality relative to a sign-variable measure. Quadrature rules with multiple nodes lead naturally to power (or implicit) orthogonality.
The classical example here is the Turán quadrature rule.
In the following, these interrelations between orthogonal polynomials and quadrature
rules, as well as relevant computational algorithms, are discussed in more detail.
2. Quadrature rules and orthogonality. We begin with the simplest kind of quadrature
rule,
Z
(2.1)
R
f (t)dλ(t) =
n
X
λν f (τν ) + Rn (f ),
ν=1
where the integral of a function f relative to some (in general positive) measure dλ is approximated by a finite sum involving n values of f at suitably selected distinct nodes τν .
The respective error is Rn (f ). The support of the measure is usually a finite interval, a halfinfinite interval, or the whole real line R, but could also be a finite or infinite collection of
mutually distinct intervals or points.
The formula (2.1) is said to have polynomial degree of exactness d if
(2.2)
Rn (f ) = 0 for all f ∈ Pd,
where Pd denotes the set of polynomials of degree ≤ d. Interpolatory formulae (or NewtonCotes formulae) are those having degree d = n − 1. They are precisely the quadrature rules
obtained by replacing f in (2.1) by its Lagrange polynomial of degree ≤ n − 1 interpolating
f at the nodes τν . We denote by
(2.3)
ωn (t) =
n
Y
(t − τν )
ν=1
∗ Received
November 1, 1998. Accepted for publicaton December 1, 1999. Recommended by F. Marcellán.
of Computer Sciences,
Purdue University,
West Lafayette,
IN 47907-1398
(wxg@cs.purdue.edu).
† Department
65
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the node polynomial associated with the rule (2.1), i.e., the monic polynomial of degree n
having the nodes τν as its zeros.
We see that, given the n nodes τν , we can always achieve degree of exactness n − 1. A
natural question is how to select the nodes τν and weights λν to do better. This is answered
by the following theorem.
T HEOREM 2.1. The quadrature rule (2.1) has degree of exactness d = n − 1 + k, k ≥ 0,
if and only if both of the following conditions are satisfied:
(a) the formula (2.1) is interpolatory;
(b) the node polynomial ωn satisfies
Z
(2.4)
ωn (t)p(t)dλ(t) = 0 f or all p ∈ Pk−1 .
R
It is difficult to trace the origin of this theorem, but Jacobi [11] must have been aware of
it.
We remark that k = 0 corresponds to the Newton-Cotes formula, which requires no
condition other than being interpolatory. The requirement (2.4), accordingly, is empty. If
k > 0, the condition (2.4) is a condition that involves only the nodes τν of (2.1); it imposes
exactly k nonlinear constraints on them, in fact, orthogonality (relative to the measure dλ)
of the node polynomial ωn to the space of all polynomials of degree ≤ k − 1. Once a set
of nodes τν has been determined that satisfies this constraint, the condition (a) then uniquely
determines the weights
R example, as the solution of the linear (Vandermonde)
Pnλν in (2.1), for
system of equations ν=1 λν τνµ = R tµ dλ(t), µ = 0, 1, . . . , n − 1.
2.1. The Gaussian quadrature rule. If the measure dλ is positive, then k ≤ n, since
otherwise (2.4) would have to hold for k = n + 1, implying that ωn is orthogonal to all
polynomials of degree ≤ n, hence, in particular, orthogonal to itself, which is impossible.
Thus, k = n is optimal, in which case ωn is orthogonal to all polynomials of lower degree,
i.e., ωn is the (monic) orthogonal polynomial of degree n relative to the measure dλ. We
express this by writing
ωn (t) = πn (t; dλ).
(2.5)
The interpolatory quadrature rule
Z
n
X
G
G
(2.6)
f (t)dλ(t) =
λG
ν f (τν ) + Rn (f )
R
ν=1
corresponding to this node polynomial is precisely the Gaussian quadrature rule for the measure dλ. It was discovered by Gauss in 1814 ([4]) in the special case of the Lebesgue measure
dλ(t) = dt on [−1, 1]. For general dλ, its nodes are the zeros of the orthogonal polynomial
πn ( · ; dλ), and its weights λν , the so-called Christoffel numbers, are obtainable by interpolation as above. (See, however, Theorem 3.1 below.)
2.2. The Gauss-Radau quadrature rule. If the support of dλ is bounded from one
side, say from the left, it is sometimes convenient to take a = inf supp dλ as one of the
nodes, say τ1 = a. According to Theorem 2.1, this reduces the degree of freedom by one,
and the maximum possible value of k is k = n − 1. If we write ωn (t) = ωn−1 (t)(t − a), the
polynomial ωn−1 having the remaining nodes as its zeros must be orthogonal to all lowerdegree polynomials relative to the measure dλa (t) = (t − a)dλ(t), i.e.,
(2.7)
ωn−1 (t) = πn−1 (t; dλa ).
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The corresponding interpolatory quadrature rule,
Z
n
X
R
R
(2.8)
f (t)dλ(t) = λR
f
(a)
+
λR
1
ν f (τν ) + Rn (f ),
R
ν=2
is called the Gauss-Radau rule for dλ (Radau [19]).
There is an analogous rule in the case b = sup supp dλ < ∞, with, say, τn = b. Indeed,
both rules make sense also in the case a < inf supp dλ resp. b > sup supp dλ. They are
special cases of a quadrature rule already considered by Christoffel in 1858 ([3]).
2.3. The Gauss-Lobatto quadrature rule. If the support of dλ is bounded from both
sides, we may take τ1 = a ≤ inf supp dλ and τn = b ≥ sup supp dλ, thereby restricting the
degree of freedom once more. The optimal quadrature rule becomes the Gauss-Lobatto rule
for dλ ([15]),
Z
(2.9)
R
f (t)dλ(t) =
λL
1 f (a)
+
n−1
X
L
L
L
λL
ν f (τν ) + λn f (b) + Rn (f ),
ν=2
whose interior nodes τν are the zeros of
πn−2 ( · ; dλa,b ), dλa,b = (t − a)(b − t)dλ(t).
(2.10)
It, too, is a special case of the Christoffel quadrature rule, which has an arbitrary number of
prescribed nodes outside or on the boundary of the support of dλ.
2.4. The Gauss-Kronrod rule. This quadrature rule has also prescribed nodes, but they
all are in the interior of the support of dλ, and it therefore transcends the class of Christoffel
quadrature rules. In trying to estimate the error of the Gauss quadrature rule, Kronrod in 1964
([12], [13]) indeed constructed a (2n+1)-point quadrature rule of maximum algebraic degree
of exactness that has n prescribed nodes — the n Gauss nodes τνG — in addition to n + 1 free
nodes. It thus has the form
Z
n
n+1
X
X
G
K
K
(2.11)
f (t)dλ(t) =
λK
λ∗K
ν f (τν ) +
µ f (τµ ) + Rn (f ).
R
ν=1
µ=1
Here, the node polynomial is
(2.12)
∗
∗
(t), πn+1
(t) =
ω2n+1 (t) = πn (t; dλ)πn+1
n+1
Y
(t − τµK ),
µ=1
and, according to Theorem 2.1 with n replaced by 2n + 1, the quadrature rule (2.11) has
degree of exactness d = 2n + k if and only if (2.11) is interpolatory and
Z
∗
πn+1
(t)p(t)πn (t; dλ)dλ(t) = 0 for all p ∈ Pk−1 .
R
∗
is orthogonal to all polynomials of
The optimal value of k is k = n + 1, in which case πn+1
lower degree, i.e.,
(2.13)
∗
(t) = πn+1 (t; πn dλ).
πn+1
∗
is the (monic) polynomial of degree n + 1 orthogonal relative to the oscillating
That is, πn+1
∗
always exists uniquely, there is no assurance
measure dλ∗n (t) = πn (t; dλ)dλ(t). While πn+1
that all its zeros are inside the support of dλ, or even real (cf. [5]).
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3. Computation of Gauss-type quadrature rules. All quadrature rules introduced in
§2 can be computed via eigenvalues and eigenvectors of a symmetric tridiagonal matrix. Part
or all of this matrix is made up of the coefficients in the three-term recurrence relation satisfied
by the (monic) orthogonal polynomials πk ( · ) = πk ( · ; dλ) relative to the (positive) measure
dλ,
πk+1 (t) = (t − αk )πk (t) − βk πk−1 (t), k = 0, 1, 2, . . . , n − 1,
π−1 (t) = 0, π0 (t) = 1,
R
where αk = αk (dλ) ∈ R, βk = βk (dλ) > 0 depend on dλ and β0 = R dλ(t) by convention.
The tridiagonal matrices involved are, respectively, the Jacobi, the Jacobi-Radau, the JacobiLobatto, and the Jacobi-Kronrod matrix.
(3.1)
3.1. The Gauss quadrature rule. The Jacobi matrix of order n is defined by
√


β1 √
0
√α0
 β1 α1

β2




.
.
.
..
..
..
.
(3.2)
JnG = JnG (dλ) = 




p
.
.
..
..

βn−1 
p
0
βn−1
αn−1
The Gauss formula (2.6) can be obtained in terms of the eigenvalues and eigenvectors of JnG
according to the following theorem.
T HEOREM 3.1. (Golub and Welsch [10]) The Gauss nodes τνG are the eigenvalues of
JnG , and the Gauss weights λG
ν are given by
(3.3)
G 2
λG
ν = β0 [uν,1 ] , ν = 1, 2, . . . , n,
G
G
where uG
ν is the normalized eigenvector of Jn corresponding to the eigenvalue τν (i.e.,
G T G
G
[uν ] uν = 1) and uν,1 its first component.
Thus, to compute the nodes and weights of the Gauss formula, it suffices to compute
the eigenvalues and first components of the eigenvectors of the Jacobi matrix JnG . Efficient
methods for this, such as the QR algorithm, are well known (see, e.g., Parlett [18]). Since the
G
first components uG
ν,1 of uν are easily seen to be nonzero, the positivity of the Gauss rule can
be read off from (3.3).
3.2. The Gauss-Radau formula. We replace n by n + 1 in (2.8) so as to have n interior
nodes,
Z
n
X
R
R
f (t)dλ(t) = λR
f
(a)
+
λR
0
ν f (τν ) + Rn+1 (f ),
R
(3.4)
ν=1
R
(P2n ) = 0.
Rn+1
The Jacobi-Radau matrix of order n + 1 is then defined by
G
√
J√n (dλ)
βn e n
R
R
Jn+1 = Jn+1 (dλ) =
(3.5)
,
βn eTn
αR
n
where JnG (dλ) is as in (3.2), eTn = [0, 0, . . . , 1] is the nth coordinate vector in Rn , and
(3.6)
αR
n = a − βn
πn−1 (a)
,
πn (a)
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with πk ( · ) = πk ( · ; dλ) as before. One then has the following theorem analogous to Theorem 3.1.
T HEOREM 3.2. (Golub [9]) The Gauss-Radau nodes τ0R = a, τ1R , . . . , τnR in (3.4) are
R
, and the Gauss-Radau weights λR
the eigenvalues of Jn+1
ν are given by
(3.7)
R 2
λR
ν = β0 [uν,1 ] , ν = 0, 1, 2, . . . , n,
R
R
where uR
ν is the normalized eigenvector of Jn+1 corresponding to the eigenvalue τν (i.e.,
R T R
R
[uν ] uν = 1) and uν,1 its first component.
R
, is again
As previously for the Gauss formula, the QR algorithm, now applied to Jn+1
the method of choice to compute Gauss-Radau formulae. Their positivity follows from (3.7).
Theorem 3.2 remains in force if a ≤ inf supp dλ. If the right end point is the prescribed node,
R
R
= b, or if τn+1
= b ≥ sup supp dλ, a theorem analogous to Theorem 3.2 holds with
τn+1
obvious changes.
3.3. The Gauss-Lobatto formula. We now replace n by n + 2 in (2.9) and write the
Gauss-Lobatto formula in the form
Z
n
X
L
L
L
f (t)dλ(t) = λL
f
(a)
+
λL
0
ν f (τν ) + λn+1 f (b) + Rn+2 (f ),
R
(3.8)
ν=1
L
(P2n+1 ) = 0.
Rn+2
The Jacobi-Lobatto matrix of order n + 2 is defined by
√
 G
Jn (dλ)
βn e n
√

T
L
L
βn e n
αn
= Jn+2
(dλ) = 
Jn+2
(3.9)

q
L
0T
βn+1

q0

L
βn+1
,

αL
n+1
L
with JnG (dλ) and en as before, and αL
n+1 , βn+1 the solution of the 2×2 linear system
L
αn+1
aπn+1 (a)
πn+1 (a) πn (a)
(3.10)
.
=
L
βn+1
πn+1 (b) πn (b)
bπn+1 (b)
We now have
L
= b in
T HEOREM 3.3. (Golub [9]) The Gauss-Lobatto nodes τ0L = a, τ1L , . . . , τnL , τn+1
L
L
(3.8) are the eigenvalues of Jn+2 , and the Gauss-Lobatto weights λν are given by
(3.11)
L 2
λL
ν = β0 [uν,1 ] , ν = 0, 1, 2, . . . , n, n + 1,
L
L
where uL
ν is the normalized eigenvector of Jn+2 corresponding to the eigenvalue τν (i.e.,
T L
L
]
u
=
1)
and
u
its
first
component.
[uL
ν
ν
ν,1
Also Gauss-Lobatto formulae are therefore computable by the QR algorithm, now apL
, and by (3.11) we still have positivity of the quadrature rule. Theorem 3.3
plied to Jn+2
holds without change for a ≤ inf supp dλ and b ≥ sup supp dλ.
3.4. The Gauss-Kronrod formula. It has only recently been discovered by Laurie that
an eigenvalue/eigenvector characterization similar to those in Theorems 3.1–3.3 holds also
for the Gauss-Kronrod rule (2.11). The Jacobi-Kronrod matrix of order 2n + 1 now has the
form
√


JnG
βn e n
0
p
√
K
K
βn+1 eT1  ,
= J2n+1
(dλ) =  βn eTn p αn
(3.12)
J2n+1
0
βn+1 e1
Jn∗
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where eT1 = [1, 0, . . . , 0] ∈ Rn and Jn∗ is a real symmetric tridiagonal matrix which has
different forms depending on whether n is odd or even:
p
"
#
G
Jn+1:(3n−1)/2
β(3n+1)/2 e(n−1)/2
(3.13)
,
Jn∗ odd = p
∗
β(3n+1)/2 eT(n−1)/2
J(3n+1)/2:2n

(3.14)
G
Jn+1:3n/2

Jn∗ even =  q
∗
β(3n+2)/2
eTn/2

q
∗
β(3n+2)/2
en/2

.
∗
J(3n+2)/2:2n
G
is the principal minor matrix of the (infinite) Jacobi matrix J G having diagoHere, Jp:q
∗
are symmetric tridiagonal matrices, and
nal elements αp , αp+1 , . . . , αq , and Jb(3n+2)/2c:2n
∗
β(3n+2)/2 an element yet to be determined.
∗K
∗
T HEOREM 3.4. (Laurie [14]) Let λK
ν > 0 and λµ > 0 in (2.11). Then Jn in (3.12) has
G
G
K
K
the same eigenvalues as Jn . Moreover, the nodes τν and τµ are the eigenvalues of J2n+1
,
and the Gauss-Kronrod weights are given by
K 2
∗K
K
2
(3.15) λK
ν = β0 [uν,1 ] , ν = 1, . . . , n; λµ = β0 [uµ+n,1 ] , µ = 1, . . . , n, n + 1,
K
K
K
where uK
1 , u2 , . . . , u2n+1 are the normalized eigenvectors of J2n+1 corresponding to the
G
G K
K
K
K
K
eigenvalues τ1 , . . . , τn , τ1 , . . . , τn+1 , and u1,1 , u2,1 , . . . , u2n+1,1 their first components.
Conversely, if the eigenvalues of JnG and Jn∗ are the same, then (2.11) exists with real
nodes and positive weights.
We remark that according to a result of Monegato [16] the positivity of λ∗K
µ implies the
reality of the Kronrod nodes τµK and their interlacing with the Gauss nodes τνG .
∗
if n is even, are
Once the trailing tridiagonal blocks in (3.13), (3.14), and β(3n+2)/2
known, the Gauss-Kronrod formula can be computed in much the same way as the Gauss
K
.
formula in terms of eigenvalues and eigenvectors of the symmetric tridiagonal matrix J2n+1
This in fact is the way Laurie’s algorithm proceeds. Note, however, that when the nodes τνG
are already known, there is a certain redundancy in this algorithm in as much as these Gauss
nodes are recomputed along with the Kronrod nodes τµK . This redundancy is eliminated in a
more recent algorithm of Calvetti, Golub, Gragg, and Reichel, which bypasses the computation of the trailing block in (3.12) and focuses directly onto the Kronrod nodes and weights
of the Gauss-Kronrod formula.
3.4.1. Laurie’s algorithm. ([14]) Assume that the hypotheses of Theorem 3.4 are fulfilled. Let, as before, {πk }nk=0 denote the (monic) polynomials belonging to the Jacobi matrix
JnG (dλ), and thus orthogonal with respect to the measure dλ, and let {πk∗ }nk=0 be the (monic)
orthogonal polynomials belonging to the matrix Jn∗ in (3.12) and measure dµ∗ (in general
unknown). Define “mixed” moments by
(3.16)
σk,` = (πk∗ , π` )dµ∗ ,
where ( · , · )dµ∗ is the inner product relative to the measure dµ∗ . Although the measure dµ∗
is unknown, a few things about the moments σk,` are known. For example,
(3.17)
σk,` = 0 for ` < k,
which is an immediate consequence of orthogonality. Also,
(3.18)
σk,n = 0 for k < n,
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again by orthogonality, since πn = πn∗ by Theorem 3.4. It is easy, moreover, to derive the
recurrence relation
σk,`+1 − σk+1,` − (α∗k − α` )σk,` − βk∗ σk−1,` − β` σk,`−1 = 0,
p
elements of Jn∗ and { βk∗ }n−1
where {α∗k }n−1
k=0 are the diagonal
k=1 the elements on the side di√
n−1
,
{
β
}
are
the
analogous
elements
of JnG . Some of the elements
agonals, while {α` }n−1
`
`=0
`=1
∗
∗
αk , βk are known according to the structure of the matrices in (3.13) and (3.14). Indeed, if
we assume for definiteness that n is odd, then
(3.19)
α∗k = αn+1+k for k ≤ (n − 3)/2, βk∗ = βn+1+k for k ≤ (n − 1)/2.
R
Using the facts that σ0,0 = R dµ∗ (t) = β0∗ = βn+1 and σ−1,` = 0 for ` = 0, 1, . . . , n −
1, σ0,−1 = 0, and σk,k−2 = σk,k−1 = 0 for k = 1, 2, . . . , (n − 1)/2, one can solve (3.19)
for σk,`+1 and compute the entries σk,` in the triangular array indicated by black dots in Fig.
3.1 (drawn for n = 7), since the α∗k and βk∗ required are known by (3.20) except for the top
element in the triangle. For this element the α∗k for k = (n − 1)/2 is not yet known, but, by
good fortune, it multiplies the element σ(n−1)/2,(n−3)/2 , which is zero by (3.17). This mode
of recursion from left to right has already been used by Salzer [21] in another context.
(3.20)
k
X
0
X
X
0
n -1
0
0
(n -3)/2
0
0
X
X
X
0
0
X
X
X
0
X
X
0
X
0
0
0
0
0
0
0
0
0
0
0
( n =7)
l
0
n
F IG . 3.1. Computation of the mixed moments
At this point, one can switch to a recursion from bottom up, using the recurrence relation
(3.19) solved for σk+1,` . This computes all entries indicated by a cross in Fig. 3.1, proceeding
from the very bottom to the very top of the array. For each k with (n − 1)/2 ≤ k ≤ n − 1,
the entries in Fig. 3.1 surrounded by boxes are those used to compute the as yet unknown α∗k ,
βk∗ according to
(3.21)
α∗k = αk +
σk,k+1
σk−1,k
σk,k
−
, βk∗ =
.
σk,k
σk−1,k−1
σk−1,k−1
This is the way Sack and Donovan [20] proceeded to generate modified moments in the
modified Chebyshev algorithm (cf. [6, §5.2]). In the present case, crucial use is made of the
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property (3.18) of mixed moments, which provides the zero boundary values at the right edge
of the σ-tableau. Once in possession of all the α∗k , βk∗ required to compute Jn∗ , one proceeds
as described earlier.
3.4.2. The algorithm of Calvetti, Golub, Gragg, and Reichel. ([2]) Here we assume
for simplicity that n is even. There is a similar, though slightly more complicated, algorithm
when n is odd.
The symmetric tridiagonal matrix Jn∗ in (3.12) determines its own set of orthogonal polynomials, in particular, an n-point Gauss quadrature rule relative to some (unknown) measure
dµ∗ . Since by Theorem 3.4 the eigenvalues of Jn∗ are the same as those of JnG , the Gauss
rule in question has nodes τνG and certain positive weights µ∗ν , ν = 1, 2, . . . , n. Likewise, the
G
of order n/2 in (3.14) determines another Gauss rule whose nodes
(known) matrix Jn+1:3n/2
and weights we denote respectively by τ̃κ and λ̃κ , κ = 1, 2, . . . , n/2. Let v = [v1 , v2 , . . . , vn ]
and v ∗ = [v1∗ , v2∗ , . . . , vn∗ ] be the matrices of normalized eigenvectors of JnG and Jn∗ , respectively. The algorithm requires the last components v1,n , v2,n , . . . , vn,n of the eigenvectors
∗
∗
∗
, v2,1
, . . . , vn,1
of the eigenvectors v1∗ , v2∗ , . . . , vn∗ .
v1 , v2 , . . . , vn and the first components v1,1
The latter, according to Theorem 3.1, are related to the Gauss weights µ∗ν by means of the
relation
(3.22)
∗ 2
] = µ∗ν , ν = 1, 2, . . . , n,
β0 [vν,1
and can therefore be computed as the positive square roots of µ∗ν /β0 . It can be easily shown,
on the other hand, that the weights µ∗ν are computable with the help of the second Gauss rule
above as
(3.23)
µ∗ν
=
n/2
X
`ν (τ̃κ )λ̃κ , ν = 1, 2, . . . , n,
κ=1
where `ν are the elementary Lagrange polynomials associated with the nodes τ1G , τ2G ,
. . . , τnG .
With these auxiliary quantities computed, the remainder of the algorithm consists of the
consolidation phase of a divide-and-conquer algorithm due to Borges and Gragg ([1]). The
spectral decomposition of Jn∗ is
(3.24)
Jn∗ = v ∗ Dτ [v ∗ ]T , Dτ = diag (τ1G , τ2G , . . . , τnG ).
Define

v
V :=  0
0
(3.25)

0 0
1 0 ,
0 v∗
a matrix of order 2n + 1. We then have from (3.12) that

(3.26)
Dτ − λI
√
K
− λI)V =  βn eTn v
V T (J2n+1
0
√
βn v T e n
p αn − ∗λ T
βn+1 [v ] e1

p 0 T ∗
βn+1 e1 v  ,
Dτ − λI
where the matrix on the right is a diagonal matrix plus a Swiss cross containing the elements
of eTn v and eT1 v ∗ previously computed. By orthogonal similarity transformations involving
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a permutation and a sequence of Givens rotations, the matrix in (3.26) can be reduced to the
form


Dτ − λI
0
0
K
,
(3.27)
− λI)Ṽ = 
0
Dτ − λI
c
Ṽ T (J2n+1
T
c
αn − λ
0
where Ṽ is the transformed matrix√V and c p
a vector containing the entries in positions n + 1
to 2n of the transformed vector [ βn eTn v, βn+1 eT1 v ∗ , αn ]. It is now evident from (3.27)
K
is {τ1G , τ2G , . . . , τnG }. The remaining eigenvalues are
that one set of eigenvalues of J2n+1
those of the trailing block in (3.27). To compute them, observe that
I
0
Dτ − λI
c
c
Dτ − λI
=
,
αn − λ
cT (Dτ − λI)−1 1
−f (λ)
cT
0T
where in terms of the components cν of c,
f (λ) = λ − αn +
(3.28)
n
X
c2ν
.
τG − λ
ν=1 ν
K
are the zeros of f (λ), which can be seen to interlace
Thus, the remaining eigenvalues of J2n+1
G
K
K
with the nodes τν . The first components of the normalized eigenvectors uK
1 , u2 , . . . , u2n+1
needed according to Theorem 3.4 are also computable from the columns of V by keeping
track of the orthogonal transformations.
4. Quadrature rules with multiple nodes. We now consider quadrature rules of the
form
Z
(4.1)
R
f (t)dλ(t) =
r−1
n X
X
(ρ)
λ(ρ)
(τν ) + Rn (f ),
ν f
ν=1 ρ=0
where each τν is a node of multiplicity r. The underlying interpolation process is the one
of Hermite, which is exact for polynomials of degree ≤ r · n − 1. We therefore call (4.1)
interpolatory if it has degree of exactness d = r · n − 1. As before, the node polynomial is
defined by
ωn (t) =
(4.2)
n
Y
(t − τν ).
ν=1
In analogy to Theorem 2.1, we now have the following theorem.
T HEOREM 4.1. The quadrature rule (4.1) has degree of exactness d = r · n − 1 + k,
k ≥ 0, if and only if both of the following conditions are satisfied:
(a) the formula (4.1) is interpolatory;
(b) the node polynomial ωn satisfies
Z
(4.3)
[ωn (t)]r p(t)dλ(t) = 0 f or all p ∈ Pk−1 .
R
The case k = 0 corresponds to the Newton-Cotes-Hermite formula, which requires no
extra condition beyond that of being interpolatory. If k > 0, the condition (b) is again a
condition involving only the nodes τν , namely orthogonality of the rth power of ωn to all
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polynomials of degree ≤ k − 1. It is immediately clear from (4.3) that for positive measures
dλ the multiplicity r must be an odd integer, since otherwise we could not have orthogonality
of ωnr to a constant, let alone to Pk−1 . It is customary, therefore, to write
r = 2s + 1, s ≥ 0.
(4.4)
It then follows that k ≤ n, since otherwise k = n + 1 would imply orthogonality of ωnr
to all polynomials of degree ≤ n, in particular, orthogonality to ωn , which, r being odd, is
impossible. Thus, k = n is optimal; the corresponding interpolatory quadrature rule is called
the Gauss-Turán rule for the measure dλ ([22]). For it, the polynomial ωn2s+1 is orthogonal
to all polynomials of degree ≤ n − 1, the polynomial ωn thus the so-called s-orthogonal
polynomial of degree n. There is an extensive theory of these polynomials, for which we
refer to the book of Ghizzetti and Ossicini [8, §§3.9,4.13]. We mention here only the elegant
result of Turán [22], according to which ωn is the extremal polynomial of
Z
(4.5)
[ω(t)]2s+2 dλ(t) = min,
R
where the minimum is sought among all monic polynomials ω of degree n. From this, there
follows in particular the existence and uniqueness of real Gauss-Turán formulae, but not
necessarily their positivity. Nevertheless, it has been shown by Ossicini and Rosati [17] that
(ρ)
λν > 0, ν = 1, 2, . . . , n, if ρ ≥ 0 is even.
5. Computation of Gauss-Turán quadrature rules. (Gautschi and Milovanović [7])
The computation of the Gauss-Turán formula
Z
(5.1)
R
f (t)dλ(t) =
2s
n X
X
λν(σ)T f (σ) (τνT ) + RnT (f ),
ν=1 σ=0
T
Rn ( 2(s+1)n−1 )
P
= 0,
involves two stages:
(i) The generation of the s-orthogonal polynomial πn = πn,s , i.e., the polynomial πn satisfying the orthogonality relation
Z
(5.2)
[πn (t)]2s+1 p(t)dλ(t) = 0 for all p ∈ Pn−1 ;
R
the zeros of πn are the desired nodes τνT in (5.1).
(σ)T
(ii) The computation of the weights λν .
Since the latter stage requires knowledge of the nodes τνT , we begin with the computation of
the s-orthogonal polynomial πn,s .
We reinterpret power orthogonality in (5.2) as ordinary orthogonality with respect to the
measure
dλn,s (t) = [πn (t)]2s dλ(t),
(5.3)
which, like dλ, is also positive,
Z
(5.4)
πn (t)p(t)dλn,s (t) = 0 for all p ∈ Pn−1 .
R
Since the measure dλn,s involves the unknown πn , one also talks about implicit orthogonality.
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Orthogonal polynomials and quadrature
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The measure dλn,s being positive, it defines a sequence πk (t) = πk (t; dλn,s ), k =
0, 1, . . . , n, of orthogonal polynomials, of which only the last one for k = n is of interest to
us. They satisfy a three-term recurrence relation (3.1), with coefficients αk , βk given by the
well-known inner product formulae
R
tπ 2 (t)dλn,s (t)
, k = 0, 1, . . . , n − 1,
αk = RR 2k
π (t)dλn,s (t) R
ZR k
π 2 (t)dλn,s (t)
(5.5)
,
dλn,s (t), βk = R R 2 k
β0 =
R
R πk−1 (t)dλn,s (t)
k = 1, . . . , n − 1.
This constitutes a system of 2n nonlinear equations for the 2n coefficients α0 , α1 , . . . ,
αn−1 ; β0 , β1 , . . . , βn−1 , which can be written in the form
φ = 0, φT = [φ0 , φ1 , . . . , φ2n−1 ],
(5.6)
where, in view of (5.3),
(5.7)
Z
φ0 := β0 − πn2s (t)dλ(t),
Z
R
2
(t) − πν2 (t)]πn2s (t)dλ(t), ν = 1, . . . , n − 1,
φ2ν := [βν πν−1
RZ
φ2ν+1 := (αν − t)πν2 (t)πn2s (t)dλ(t), ν = 0, 1, . . . , n − 1.
R
Here, each πk is to be considered a function of α0 , . . . , αk−1 ; β1 , . . . , βk−1 by virtue of
the three-term recurrence relation (3.1) which it satisfies. Since all integrands in (5.7) are
polynomials of degree ≤ 2(s + 1)n − 1, the integrals in (5.7) can be computed exactly by an
(s + 1)n-point Gaussian quadrature rule relative to the measure dλ. Any method for solving
systems of nonlinear equations, such as Newton’s method or quasi-Newton methods, can now
be applied to solve (5.6). (For Newton’s method, in this context, see [7].)
We now turn to the computation of the quadrature weights. Here we note that
Y
(t − τµT )2s+1 ,
ωρ,ν (t) = (t − τνT )ρ
(5.8)
µ6=ν
ρ = 0, 1, . . . , 2s; ν = 1, . . . , n,
are polynomials of degree ≤ (2s + 1)n − 1, hence, by (5.1), RnT (ωρ,ν ) = 0. This can be
written as
2s
n X
X
(5.9)
(σ) T
λκ(σ)T ωρ,ν
(τκ ) = bρ,ν ,
κ=1 σ=0
with
Z
bρ,ν =
(5.10)
R
ωρ,ν (t)dλ(t).
(σ)
By virtue of the definition (5.8) of ωρ,ν , we have ωρ,ν (τκT ) = 0 for κ 6= ν, so that the system
(σ)T
(5.9) breaks up into n separate linear systems for the weights λν , σ = 0, 1, . . . , 2s,
(5.11)
2s
X
σ=0
(σ) T
λν(σ)T ωρ,ν
(τν ) = bρ,ν , ρ = 0, 1, . . . , 2s.
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W. Gautschi
(σ)
Each of these systems further simplifies since ωρ,ν (τνT )
we obtain the upper triangular system


(2s)
0
(τνT ) · · · ω0,ν (τνT )
ω0,ν (τνT ) ω0,ν


(2s)
0
(τνT ) · · · ω1,ν (τνT )  
ω1,ν



(5.12) 
..
..

.


.
(2s) T
ω2s,ν (τν )
= 0 for ρ > σ. That is, for each ν,
(0)T
λν
(1)T
λν
..
.
(2s)T
λν


b0,ν


b

1,ν
 = 
 ..

 .

b2s,ν



.

The matrix elements can easily be computed by linear recursions (cf. [7]) and the elements
of the right-hand vector by (s + 1)n-point Gauss quadrature as before.
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