on computing gauss-kronrod quadrature formulae* · kautsky and elhay [15] and elhay and kautsky...

mathematics of computationvolume 47, number 176october 1986, pages 639-650

On Computing Gauss-Kronrod Quadrature Formulae*

By Franca Calió, Walter Gautschi, and Elena Marchetti

Abstract. We discuss the use of Newton's method for computing Gauss-Kronrod quadrature

formulae from modified moments. The underlying nonlinear maps are analyzed from the

point of view of numerical condition. A method is indicated of computing the polynomial

whose zeros are the Kronrod nodes. Examples include Gauss-Kronrod formulae for integrals

with a logarithmic and algebraic singularity at one endpoint. Pertinent numerical results are

tabulated in the supplements section at the end of this issue.

1. Introduction. Given a positive measure da(t) on the real line, whose support

contains infinitely many points and all of whose moments exist, we call

(1.1) / f(t)do(t) = i ajir,) + £a;/(r;) + Rn(f)■'» v=l ft-1

a Gauss-Kronrod quadrature formula if t„ = T„(n) are the Gaussian nodes associated

with do and the nodes t* and weights av, a* are chosen so as to maximize the

degree of exactness of (1.1). Since there are 3n + 2 unknowns, we can achieve degree

of exactness d = 3n + 1, i.e., R„(f) - 0 whenever / = P3n+1. It is well known, in

fact, that the nodes t* must be the zeros of w„*+1, the (monic) polynomial of degree

n + 1 satisfying the orthogonality property

(1.2) / wn*+x(t)irn(t)tido(t) = 0, 1 = 0,1,...,«,•'r

where tt„(-) = "n„(-;do) is the nth degree orthogonal polynomial belonging to the

measure da. Note that the measure da*(t) = ir„(t; da)da(t), with respect to which

v*+i *s orthogonal, changes sign. We cannot expect, therefore, that 7Tn*+1 has all its

zeros necessarily real.

We are interested here only in Gauss-Kronrod formulae (1.1) with real nodes t*,

all contained in [a, b]—the smallest interval containing the support of da—and

with positive weights a*. Then the interlacing property holds (Monegato [17,

Theorem 1]),

(1.3) a < t„*+1 < r„ < r* < t„_, < ••• < T, < T* < b.

Our concern is with the actual computation of these nodes and the corresponding

weights (provided they exist), given the integer n > 1 and the positive measure da.

Received May 7, 1985; revised September 24, 1985.

1980 Mathematics Subject Classification. Primary 65D32, 65A05; Secondary 65H10.

* The work of the first and third author was supported by the Ministero della Pubblica Istruzione and by

the Consiglio Nazionale delle Ricerche, while the work of the second author was supported by the National

Science Foundation under grant DCR-8320561.

639

©1986 American Mathematical Society

0025-5718/86 $1.00 + $.25 per page

640 FRANCA CALIÓ, WALTER GAUTSCHI, AND ELENA MARCHETTI

Constructive methods for this problem have been considered before. For the

Legendre measure da(t) = dt on [-1,1], Kronrod [16] in his original work obtains

7T„*+i in power form by solving (1.2). Patterson [22] expands n*+x in Legendre

polynomials, while Piessens and Branders [23] use Chebyshev polynomials, the nodes

t* each time being computed by an appropriate rootfinding procedure. The weights

can be obtained by interpolation. Monegato [17], [19], Baratella [1], and Dagnino

and Florentino [3] use similar procedures to compute tr*+ x for Gegenbauer measures

and other classical measures. Kahaner et al. [13], in the case of the Laguerre

measure, and assuming * > n + 1 Kronrod nodes t* in (1.1), expand Lnmk* in

Laguerre polynomials L . All these methods require three distinct phases to obtain

(1.1): The computation of the appropriate polynomial, say 7Tn*+1, finding the zeros of

tt*+x, and computing the weights ar, a*. An entirely different approach is taken by

Kautsky and Elhay [15] and Elhay and Kautsky [5], who compute the nodes t* as

eigenvalues of a certain matrix derived by matrix decomposition methods—an

approach which extends the well-known method of Golub and Welsch [12] for

ordinary Gaussian quadratures and its extension by Golub and Kautsky [11]. The

weights are then obtained using general methods for constructing interpolatory

quadrature rules [14], [6].

Here we propose to compute (1.1) directly by solving a set of nonlinear equations

that express the exactness of (1.1) for a given set of 3« + 2 polynomials. These

polynomials are chosen so as to generate a well-conditioned problem. The respective

system of nonlinear equations is solved by Newton's method, careful attention being

given to the choice of initial approximations and to monitoring the progress of the

iteration. We make no claims for our method to be superior in any way to other

methods, but merely demonstrate its feasibility and stability. It would indeed be

interesting to undertake a detailed comparative study of the various methods now in

use.

While aiming directly at the unknown quantities is certainly a virtue if our method

is successful, it is less than satisfactory otherwise, since no information about ir*+ x is

generated when the method fails. Nevertheless, if one wishes to examine the

polynomial it*+x, one can express it in terms of the orthogonal polynomials

{•nk(-;da)} and obtain the coefficients by solving a triangular system of linear

equations.

The implementation of Newton's method for computing the Gauss-Kronrod rule

(1.1) is discussed in Section 2. In Section 3 we study the condition of the underlying

problem. Section 4 deals with the computation of the polynomial it*+x. Examples

will be given in Section 5, and numerical results for da(t) = taln(l/t)dt on [0,1],

a = 0, + j, are tabulated in the supplements section at the end of this issue.

For additional questions related to Gauss-Kronrod quadrature we refer the reader

to surveys by Monegato [20], [21].

2. The Computation of Gauss-Kronrod Rules by Newton's Method. The Gaussian

nodes t„ for the measure da can be computed by well-known methods; see, e.g., [8,

p. 290]. We assume therefore that they are available. For a given system of monic

polynomials (pk(-)}, with degpk = k, k = 0,1,2,..., we further assume that the

GAUSS-KRONROD QUADRATURE FORMULAE 641

first 3« + 2 modified moments of da,

(2.1) mk= f pk(t)da(t), k = 0,l,...,3n + l,JR

are known. The system of nonlinear equations defining the Gauss-Kronrod formula

(1.1) then is

« « + i

(2-2) L^PkU)+ L%*Pk{r;) = mk, k = 0,1,...,3« + 1.p = i ii=i

We propose to solve this system for the 3n + 2 unknowns r*, av, a* by Newton's

method. A number of practical issues need to be addressed.

First the choice of polynomials pk. We assume that they satisfy a three-term

recurrence relation,

(23) P-M = 0, /»„(/)-1,

/»* + i(0 = it-ak)pkit)-bkPk_x(t), ¿ = 0,1,2,...,

with known coefficients ak, bk. This makes the evaluation of these polynomials, and

of their derivatives, which is required in the computation of the left-hand side of the

system (2.2) and its Jacobian, easy and straightforward. If, in addition, the poly-

nomials {pk} are orthogonal with respect to some given measure ds(t), then this

computation is also numerically stable. If da is one of the classical measures, the

simplest choice is pk(-) = irk(-; da), the corresponding recursion coefficients ak =

ak(da), bk = ßk(da) then being explicitly known; the modified moments become

(2.4) mk= f nk(t;da)da(t) = ß080k, k = 0,1,2,...,•'R

where ß0 = jKda(t) and 80 k is the Kronecker delta, 8QQ = 1, 80 k = 0 if k > 0.

For nonclassical measures da, the choice of pk is usually dictated by the necessity of

being able to compute the modified moments (2.1).

The next important issue is the computation of the initial approximations for the

Kronrod nodes t* and for the weights a„, a*. Since we are interested only in

Gauss-Kronrod formulae with nodes in [a, b] and with positive weights a*, we can

assume the interlacing property (1.3). This suggests as initial approximations t*,

2 < ft < n, the midpoints

(2-5) t; = Kvi + T,)' f» = 2,3,...,«,

between the Gaussian nodes t , and •y The choice of the initial approximations tx*

and t*+ x is less obvious. If the endpoint b is finite, we select a number of trial values

tx* equally spaced between tx and b. If a is also finite, we do the same for t„*+1 in

the interval (a, rn) and let both f„*+1 and if move inward simultaneously, or, if

necessary, let them move independently from one another. If an endpoint, say b, is

infinite, we select t* = rx + (tx ' — r2), or try a number of values rx* equally spaced

between tx and, say, t, + 2(tj - t2). For each set of initial approximations f * we

compute corresponding approximations r)„, à* to the weights by solving the first

2n + 1 equations in (2.2), where t* is replaced by r*. Since the matrix of this

system forms part of the Jacobian matrix used in the first Newton step, the only

overhead of this computation is the solution of the linear system.


Each Newton step is monitored as to the location of the iterates of t*. If at any

stage one of these iterates falls outside the interval [a, b] (if either a or b is finite),

the iteration is terminated and restarted with a new set of initial approximations.

The same action is taken if the number of Newton iterates exceeds a preset limit. If

none of the initial approximations leads to a convergent process, the attempt of

computing (1.1) is declared a failure. This is usually an indication (not a proof!) that

the desired Gauss-Kronrod rule does not exist.

If da(t) = da(-t) is an even measure and the support of da is symmetric with

respect to the origin, then both the Gaussian nodes t„ and the Kronrod nodes t * are

located symmetrically with respect to the origin, and weights corresponding to

symmetric nodes are equal. As a result, (2.2) is trivially true if pk(t) is an odd

polynomial. If we choose for { pk} a system of polynomials satisfying

(2.6) pk(-t) = i-l)kpkit), k = 0,1,2,...,

the system (2.2), therefore, is equivalent to the system

«/2 «/2 l

E °„Plki%) + E ^Plki^*) + 9°(*/2) + l/>2*(0) - 9 ™2*>(2.7e) .=i n=i ¿ ¿

k = 0,1,..., 3n/2,

if n is even, and to the system

[n/2] 1 l«/2] + l j

£ o,/>2*(T,) +2a[«/21 + i^(0)+ E o*p2k(r*) = -^m2k,

(2.7o) " = 1 "=1

fc = 0,l,...,(3« + l)/2,

if n is odd. This in effect reduces the size of the problem by a factor of 2.

3. The Condition of the Underlying Problem. Let mT = [m0, mx,..., m3n+x] be the

vector of modified moments, and yT = [ax,.. .,an,ax*,...,a*+x,tx*,.. . ,t„*+1] the

vector of the weights and Kronrod nodes of the Gauss-Kronrod formula (1.1). The

procedure of Section 2 is an attempt of carrying out the nonlinear map

G„: R3"+2 - R3n+2 m -> y,

where the Gauss-Kronrod formula is assumed to have real nodes. We now wish to

examine the sensitivity of the map Gn to perturbations in the modified moments.

The development parallels the treatments given for Gaussian formulae in Gautschi

[8], [9]; see also Gautschi [10, Section 5].

We assume that the polynomials {pk} defining the modified moments are

orthogonal on the real line with respect to some measure ds. The support of this

measure normally coincides with the support of da, but does not have to. We define

normalized modified moments by

(3.1) mk = d:lmk, d2=( p2(t)ds(t),JR

and consider, in place of G„, the map

(3.2) G„: R3n+2 -> R3n+2 m - y,

where mT = [m0,mx,...,m3n+x\. We analyze the sensitivity of Gn by computing the

Frobenius norm of the Jacobian matrix, J¿ , of the map G„.


The basic equations can be written in the form

(3.3) 0(y) = m,

where

(3.4) *k{y) = d-A E ovPk(rv) + E^(t;)),\v=l ii-1 )

k = 0,1,2,. ..,3n + 1.

Since the map G„ amounts to solving (3.3) for y, the Jacobian of Gn is the inverse of

the Jacobian 90/tfy of 0,

(3.5) Jôm = (30/3Y)-1.

An elementary computation shows that

(3.6) 90/3y = DXP2*,

where D = diag(d0, dx.d3n+x), 2* = diag(l,..., 1,1,..., 1, af,..., a*+1), and

(3.7)

/>o(Ti) ■■■ PoM Po(t*) ••• />o(TAt) Po(T*) •• Poitf+i)

Pi(T\) ■■■ PiM pAt*) •■• PiiT*+i) /i(t*) ••■ p'iiiï+i)

P3»+l(n) "■ />3» + l(T,.) />3„ + l(T*) •'■ P3»+i(t„*+i) />3„ +1 ( V ) •" P3»+l(T»*+l)

Therefore,

(3.8) Jô> = (S*)'1?"1/).

For the inversion of P, define g„, /i^, A:^ to be the elementary Hermite interpola-

tion polynomials of degree 3« + 1, belonging to the nodes t„ and r*, defined by

£,(tx) = S„x, X = l,2,...,n \

g,(V) = °> «:(V)-°' M = l,2,...,« + l/'

*- 1,2,...,«,

MTx)-°. a = 1,2,...,«

MT*)"V h'Á<) = ̂ r-l,2,...,n + l

MTx)-°' À = 1,2,...,«

M1--*)"0' *;(T*)-V r-l,2,...,» + lj

/x = 1,2,...,« + 1.

Writing

3« + 2 3n + 2 3« + 2

«,(0 = I Vp-i(')- MO- E ¿Vp-i(0, MO- I Vp-i(').p=l p = l p=l

(3.9)

it is easily seen that

(3.10) Px = , A = [aJ, 5 =[*>,„], C=[cJ.


By a computation similar to the one in [8, pp. 303-304] one finds from (3.8) and

(3.10) that the Frobenius norm of J¿ is given by

H+l

Es,2(0+ E M0 + 4ïW1/2

ds(t)(3.11) i/a- /R

Its magnitude, therefore, is critically influenced by the magnitude of the polynomial

(3.12) /„(o= tg,2(0 + "lÍM0 + 4iM09-1 H-l\ %

on the support of ds. The degree of /„ is 6« + 2. The integral in (3.11) can therefore

be computed exactly (up to rounding errors) by a (3n + 2)-point Gaussian quadra-

ture rule belonging to the measure ds.

Explicit forms of the polynomials g„, h^, k^ can easily be given in terms of the

fundamental Lagrange polynomials

1 + 1

ao-n tA = l

/;(') = n tX = l V

belonging to the nodes t„ and t*, respectively. One obtains

?,(') =',(0%\lit)

«tf+iK)

(3.13) .(0-^[W]a(i-(*-v)

y = 1,2,...,«,

+ wM

*n(0

jtt = 1,2,...,« + 1,

,2,...,« + 1,

where tt„(-) = "t„(-; da) and n*+x(t) = nx=i(í _ t*)- We also note the following

properties of /„, which follow directly from (3.9) and (3.12),

/„(/)>0 for all r g R,

(3.14) /„(Ü = l, /„'(t,) = 2g'v(jv), v=l,2,...,n,

/.(tf)-i, /«(t;) = °> f»-i.2,...,» + i.

These conditions, of course, are not sufficient to determine the polynomial /„. There

are 2« + 1 degrees of freedom left, which allow/, considerable room for movement.

4. Computation of the Polynomial m*+,. Expressing the polynomial tt*+ x in terms

of the orthogonal polynomials iTk(-) = trk(-; da),

(4.1) 77*+1(/) = W„ + X(t) + C0Í7„(0 + Cxir„_x(t) + ■■■ +C„770(f),

and replacing the powers t' in (1.2) by the polynomials 7T,(i), one obtains the

conditions

/.Vn(0 + E cktr„_k(t)

k=0

trn(t)ni(t)da(t) = 0, i = 0,1,...,«,


hence the linear system

«

(4.2) ¿Zalkck = b„ 1-0,1,...,«,k = 0

for the coefficients ck, where

a/* = / */(0*',,-*(0M0<M0. i,fc = 0,l,...,n,(4.3)

*,■= - ( »,-(0»„+i(0M0<M0. / = o,i,...,«.•'R

By orthogonality, aik = 0 if i < k, so that the matrix A = [aik] is lower triangular,

and ah■ = jRir2(t)da(t) > 0, so that A is nonsingular. The solution of (4.2),

therefore, can be effected by forward substitution.

The coefficients aik and b¡ = -a¡ _x satisfy a two-dimensional recursion relation,

which could be used for their computation (see [2]). Noting, however, that the

integrands in (4.3) are polynomials of degree at most equal to 3« + 1, we can also

use w-point Gauss-Christoffel quadrature relative to the measure da, with m =

[(3« + 3)/2], to compute aik and b¡. This might be preferable, since it requires

nothing beyond standard software.

While this procedure of generating tt*+ x is similar to Kronrod's original method,

it is significantly more stable, since the use of powers as a polynomial basis is

completely avoided.

5. Examples. It is known that for da(t) = (1 - t2)x~1/2dt on [-1,1] all Gauss-

Kronrod formulae exist if 0 < X < 2. Furthermore, the interlacing property (1.3)

holds and a* > 0 for ju = 1,2,..., « + 1 (Monegato [17]). If 0 < X < 1, one has in

addition o„ > 0 for v = 1,2,..., n (Monegato [18]). Our first example deals with the

case X = \, i.e., with the Legendre measure. All computations reported were done

on the CDC 6500 computer in single precision (machine precision * 3.55 X 10~15),

unless noted otherwise.

Example 5.1. da(t) = dt on [-1,1].

Taking advantage of symmetry, we apply Newton's method to the system (2.7),

using for p2k the (monic) Legendre polynomials of degree 2k. The required modified

moments are then given by (2.4). We had no difficulty with convergence. By

symmetry, only one "trial" initial approximation, f*, is needed, which was pro-

grammed to move in nine equal steps of length h = (1 - t,)/10 from 1 - h to

t, + h. Convergence was invariably achieved for the first choice of tx*. Moreover,

the problem, suitably scaled, turns out to be extremely stable. In the first four

columns of Table 5.1 we report on the number of iterations required for 12 decimal

place accuracy, the maximum of fn(t) (cf. Eq. (3.12)) on [-1,1] and the value of

\\Jg)\f (cf- Eq- (3.11)) for n = 5, 10, 20, 40, 80. The maximum of f„(t)~an even

function of /—is typically assumed between t2 and t2*, if « is even, there being a

couple of smaller maxima on either side of it. If n is odd, the maximum seems to

occur at / = 1. Through most of the interval (-1,1), however, /„ remains < 1.


Table 5.1

Performance and stability characteristics of Newton's method for generating the

(2n + l)-point Gauss-Kronrod formula with da(t) = dt on [-1,1].

#iter. H/, H. ||/*cond, cond2

(scaled) (scaled)

5 6 1.126 1.293 1.4(3) 3.3(4) 1.2(1) 1.4(1)

10 6 2.456 1.397 1.5(6) 5.2(8) 1.8(1) 2.3(1)20 6 2.520 1.333 1.5(12) 5.3(17) 4.1(1) 4.8(1)40 6 2.535 1.310 1.6(24) 5.8(35) 6.6(1) 7.7(1)

80 6 2.540 1.302 — 1.3(2) 1.7(2)

The linear systems of equations for determining the initial approximations r}„, a*

for the weights, as well as the Jacobian matrices in Newton's method, appear to

become rapidly ill-conditioned as n increases. Typical condition number estimates

(furnished by the LINPACK routine SGECO; cf. [4, Chapter 1]) for the former are

shown in the fifth column of Table 5.1, while those for the latter are shown in the

sixth column. (Numbers in parentheses indicate decimal exponents.) In spite of the

large condition numbers, numerical difficulties were not observed, except in the case

« = 80, when the computation was aborted due to an arithmetic error. We believe

that the apparent ill-conditioning is caused by the use of monic polynomials p2k in

(2.7); their L2-norm goes to zero rather quickly,

22k(2k)\2 I 2 /-„ „^^^-JÄkyTiÄkTl-^2 as/c-oo,

thereby introducing a systematic diminution of the rows down the matrices. If the

row involving p2k is scaled by dividing it by 22/c(2/c)!2/(4A:)!, the condition numbers

indeed become much more reasonable (the solutions remaining the same); they are

shown in the last two columns of Table 5.1.

Example 5.2. da(t) = ln(l/i) dt on [0,1].

It appears that this measure also admits Gauss-Kronrod formulae for all n,

satisfying the interlacing property (1.3) and having all weights positive. A summary

of our numerical experience with Newton's method is given in Table 5.2; it contains

information analogous to the one given in Table 5.1 for Example 5.1.

Table 5.2

Newton 's method for Gauss-Kronrod formulae with da(t) = ln(l/r) dt on [0,1].

#iter. \\fn\\x \\Jà\\F cond, cond-.cond¡ cond2

(scaled) (scaled)

5 6 2.90(4) 12.14 2.0(6) 6.9(9) 5.8(1) 2.7(3)10 6 4.09(5) 24.35 2.2(12) 7.5(18) 2.8(1) 1.6(4)

20 6 5.77(6) 47.43 2.4(24) 8.2(36) 9.8(2) 1.0(5)40 6 8.86(7) 94.59 — 3.8(3) 6.4(5)


We have used modified moments with respect to the (monic) shifted Legendre

polynomials, pk(t) = [k\2/(2k)\]Pk*(t); they are known to be (cf., e.g., Gautschi

[7])

-o=l, ^-r^OMl/OA-^TT^TT. *-1.2,....

Row scaling of the matrices was performed through division by k\2/(2k)\ (of the

row numbered k + 1).

The initial approximations f,* and fn*+1 were programmed to first move inward

symmetrically in nine steps of length h = (1 - t^/IO and W = t„/10, respectively,

and if this did not work, tx* was varied independently over the same set of points for

each fixed t*+x. Convergence, in general, was achieved for the very first choice of tx*

and f,*+1, i.e., for fx* = 1 — h and f„*+1 = h', except when n is small, for example,

« = 1 and « = 3, in which cases convergence was realized when fn*+1 = h' and

f * = 1 - Ah (for « = 1), t* = 1 - 2A (for « = 3).

The polynomial /„(f) invariably assumes its global maximum at t = 1, has a few

much smaller relative maxima between the first few neighboring nodes t* and t„,

and then settles down to magnitudes around 1 for the remaining portion of the

interval [0,1]. The condition of the problem, though slightly worse than in Example

5.1, is still remarkably good.

Numerical results for the nodes and weights of the (2n + l)-point Gauss-Kronrod

formula for « = 5(5)25 can be found in Table S.l of the supplements section at the

end of this issue. They have been computed in double precision to an accuracy of 25

decimal places after the decimal point. As the results are displayed in D-format,

some of the end figures may not be reliable in those numbers that are much smaller

than 1.

We used Example 5.2 to further experiment with alternative choices of initial

approximations. In particular, we examined how inaccuracies in individual initial

approximations affect the speed of convergence. To obtain a basis for meaningful

comparison, we first obtained "reference" values for the number of iterations

required when all initial approximations are at the same level of accuracy. This was

achieved by imposing on the "exact" results for av, a*, t* (computed to 12 decimal

digits) a random perturbation at level e, i.e., by taking 6„ = a„(l + rve), à* =

a*(l + r*e), t* = ^(1 + s*e), where rv, r*, s* are random numbers from [-1,1].

The results for e = 10"2, 10-5, 10"8, and 10"11 are shown in Table 5.3. (For

e = 10"2 and « = 40, Newton's method did not converge within 20 iterations.)

Table 5.3

The number of iterations required for initial approximations at accuracy level e.

e n = 5 « = 10 « = 20 « = 40

10"2 6 6 5

10"5 3 3 3 4

10"8 2 2 2 2

10-" 2 2 2 2


Table 5.4

The number of iterations required when two pairs of initial approximations

(with indices n andn + 2 — ¡i) are inaccurate.

: 5 „ = 10 „ = 20 « = 40 n = 5ji = n/4

10 n = 20 n = 40 n = 511 = n/2

10 „ = 20 ■ 4»

io-2io-5

IO-8

io-11

We now contrast this with the case in which all initial approximations are at the

accuracy level ¿10~12 of the initially computed results, except for two pairs of

Kronrod nodes and weights (situated symmetrically with respect to the midpoint of

the interval [0,1]), which are randomly perturbed at level e. Choosing the inaccurate

pairs i*, a* to be those corresponding to ju. = 1, \i = n/4, and /x ~ n/2 (and to the

symmetric indices n + 2 - ¡i), we observed the results shown in Table 5.4.

Comparing Tables 5.3 and 5.4, we note some improvement, particularly for

/* == n/2, in the case e = IO"2, when only two pairs of initial approximations are

inaccurate (though for « = 20 there are two instances of deterioration), but in all

other cases the performance of Newton's iteration is practically the same. This seems

to suggest that it is the maximum relative error in the initial approximations (usually

associated with a Kronrod or Gauss node near the end of the interval) which

determines the speed of convergence.

The choice of initial approximations proposed in Section 2 leads to initial

(relative) errors that are reasonably small for "interior" nodes, but comparatively

larger near the "boundary". Specifically, if we regard the three Kronrod nodes

nearest to each of the endpoints of [0,1], and the two Gauss nodes between them, as

belonging to the "boundary", and all others to the "interior", then the relative

errors of the initial approximations in our scheme range for « = 10, 20, 40

respectively from 2.0(-3) to 9.5(-2), 2.4(-5) to 5.5(-2), and 4.3(-6) to 2.9(-2) in the

interior, and from 4.3(-3) to 5.4(-l), 1.3(-3) to 5.6(-l), and 3.3(-4) to 5.7(-l),

respectively, at the boundary. The relatively large number of 6 iterations reported in

Table 5.2 appears to be due to the large maximum error of the initial approxima-

tions in the boundary zones.

Attempts to improve the initial approximations for a„ by using, for example,

o„ = \a¡¡n),** where a„(n) are the Christoffel numbers for da, and by obtaining the

remaining initial approximations a* from a reduced system (2.2) of n + 1 linear

equations, do not speed up Newton's iteration (in fact, require 7 iterations, instead

of 6, when « = 5 and n = 10), precisely because of the initial approximations in the

boundary zones remaining at the same low accuracy level.

Example 5.3. da(t) = taln(l/t)dt on [0,1], a = ± \.

Modified moments with respect to the shifted Legendre polynomials are again

available (Gautschi [7]) and suggest the same scaling as in Example 5.2. Results for

**This approximation was proposed to us by the referee.

gauss-kronrod quadrature formulae 649

Table 5.5

Newton 's method for Gauss-Kronrod formulae with da(t) = tx/1 ln(l//) dt on [0,1].

#iter. ||/. H. ||.fecondj cond2

(scaled) (scaled)

5 6 1.03(4) 7.542 3.8(3) 7.8(4)10 6 1.35(5) 14.64 7.3(4) 1.0(6)20 6 2.06(6) 29.60 5.2(5) 1.4(7)

40 6 3.40(7) 61.04 1.4(7) 3.0(8)

Table 5.6

Newton ys method for Gauss-Kronrod formulae with da(t) = /~1/2 ln(l/f) dt on [0,1].

#iter. II/IL \\JÔ\\Fcond! cond2

(scaled) (scaled)

4 7 9.78(6) 272.21 1.3(2) 9.0(5)

8 9 3.66(8) 889.01 2.0(3) 1.3(7)16 9 1.52(10) 2961.5 6.3(3) 2.0(8)32 — 6.56(11) 9907.2 8.8(3) 3.1(9)

a = { are summarized in Table 5.5. (The computation was done in double precision,

the number of iterations referring again to 12 decimal place accuracy.) The perfor-

mance of Newton's method is similar as in Example 5.2, except for the (scaled)

matrices now being more ill-conditioned. The ill-conditioning of the Jacobian

matrices is still worse in the case a = - \, as can be seen from Table 5.6. For this

value of a the independent variation of the starting approximations ff, t*+ x proved

to be rather essential, since convergence was never achieved for the first choices of

these starting values. This is in contrast to the case a = \, where the first choice of

t* and t*+x always worked.

When a = — \, Newton's method could not be made to converge for « = 32,

using the implementation described in Section 2. The difficulty, we believe, is caused

by the smallest Kronrod node being almost equal to zero. Computing (in double

precision) ir3*3 by the procedure of Section 4, and applying Newton's method to tr3*3,

we find t3*3 = 3.05867... XlO"9. Using all zeros of it3*3 computed in this way as

initial approximations to Newton's method, and lowering the accuracy requirement

to 20 decimal places, indeed restores convergence and yields the data for « = 32 in

Table 5.6 after 1 iteration.

Noncon vergence (in the case a= - \) was also observed for odd values of «, this

time because of the presence of negative Kronrod nodes. When « = 1, for example,

one computes directly ir2*(t) = t2 - (198/343)/ - (3671/117649), which has the

zeros tx* = .627023... and t2* = - .0497636.... We verified that for all odd

« < 32 the polynomial w*+ x has exactly one negative zero, while all other zeros are

between 0 and 1.

Numerical results for a = \-, n = 5(5)25, computed in double precision, are given

in Table S.2 and for a = - \, n = 4(4)24, in Table S.3 of the supplements section

at the end of this issue.


Dipartimento di Matemática

Politécnico Milano

Piazza Leonardo da Vinci, 32

20133 Milano, Italy

Department of Computer Sciences

Purdue University

West Lafayette, Indiana 47907

Dipartimento di Matemática

Politécnico Milano

Piazza Leonardo da Vinci, 32

20133 Milano, Italy

1. P.Baratella, "Un' estensione ottimale della formula di quadratura di Radau," Rend. Sem. Mat.

Univ. Politec. Torino, v. 37,1979, pp. 147-158.2. F. Calió, E. Marchetti & G. Pizzi, "Valutazione numérica di alcuni integrali con singolarità di

tipo logarítmico," Rend. Sem. Fae. Sei. Univ. Cagliari, v. 54, 1984, pp. 31-40.

3. C. Dagnino & C. Fiorentino, "Computation of nodes and weights of extended Gaussian rules,"

Computing, v. 32,1984, pp. 271-278.

4. J. J. Dongarra et al., UNPACK Users' Guide, SIAM, Philadelphia, Pa., 1979.

5. S. Elhay & J. Kautsky, "A method for computing quadratures of the Kronrod Patterson type,"

Proc. 7th Australian Computer Science Conf., Austral. Comput. Sei. Comm., v. 6, no. 1, February 1984,

pp. 15.1-15.9. Department of Computer Science, University of Adelaide, Adelaide, South Australia.

6. S. Elhay & J. Kautsky, IQPACK: Fortran Subroutines for the Weights of Interpolator Quadra-

tures, School of Mathematical Sciences, The Flinders University of South Australia, April 1985.

7. W. Gautschi, "On the preceding paper 'A Legendre polynomial integral' by James L. Blue," Math.

Comp., v. 33,1979, pp. 742-743.

8. W. Gautschi, "On generating orthogonal polynomials," SIAM J. Sei. Statist. Comput., v. 3, 1982.

pp. 289-317.9. W. Gautschi, "On the sensitivity of orthogonal polynomials to perturbations in the moments,"

Numer. Math., v. 48,1986, pp. 369-382.10. W. Gautschi, "Questions of numerical condition related to polynomials," in MAA Studies in

Numerical Analysis (G. H. Golub, ed.), Math. Assoc. America, Washington, D. C, 1984, pp. 140-177.

11. G. H. Golub & J.Kautsky, "Calculation of Gauss quadratures with multiple free and fixed

knots." Numer. Math., v. 41,1983, pp. 147-163.

12. G. H. Golub & J. H. Welsch, "Calculation of Gauss quadrature rules," Math. Comp., v. 23,1969,

pp. 221-230.13. D. K. Kahaner, J. Waldvogel & L. W. Fullerton, "Addition of points to Gauss-Laguerre

quadrature formulas," SIAM J. Sei. Statist. Comput., v. 5,1984, pp. 42-55.

14. J. Kautsky & S. Elhay, "Calculation of the weights of interpolatory quadratures," Numer. Math.,

v. 40,1982, pp. 407-422.

15. J. Kautsky & S. Elhay, "Gauss quadratures and Jacobi matrices for weight functions not of one

sign," Math. Comp., v. 43, 1984, pp. 543-550.

16. A. S. KRONROD, Nodes and Weights for Quadrature Formulae. Sixteen-place Tables, Izdat.

"Nauka", Moscow, 1964. (In Russian.) [English Translation: Consultants Bureau, New York, 1965.]

17. G. Monegato, "A note on extended Gaussian quadrature rules," Math. Comp., v. 30, 1976, pp.

812-817.

18. G. Monegato, "Positivity of the weights of extended Gauss-Legendre quadrature rules," Math.

Comp., v. 32, 1978, pp. 243-245.

19. G. Monegato. "Some remarks on the construction of extended Gaussian quadrature rules," Math.

Comp., v. 32,1978, pp. 247-252.

20. G. Monegato, "An overview of results and questions related to Kronrod schemes," in Numerische

Integration (G Hämmerlin, ed.), Internat. Ser. Numer. Math., vol. 45, Birkhäuser, Basel, 1979, pp.

231-240.

21. G. Monegato, "Stieltjes polynomials and related quadrature rules," SIAM Rev., v. 24, 1982, pp.

137-158.

22. T. N. L. Patterson, "The optimum addition of points to quadrature formulae," Math. Comp., v.

22, 1968, pp. 847-856. Loose microfiche suppl. cl-cll. [Errata: ibid., v. 23,1969, p. 892.]23. R. Piessens & M. Branders, "A note on the optimal addition of abscissas to quadrature formulas

of Gauss and Lobatto type," Math. Comp., v. 28, 1974, pp. 135-139. Suppl., ibid., pp. 344-347.

mathematics of computationvolume 47, number 176october 1986, pages s57-s63

Supplement to

On Computing Gauss-Kronrod Quadrature Formulae

By Franca Calió, Walter Gautschi, and Elena Marchetti

Table S.l. Nodes and weights for (2n+l)-point Gauss-Kronrod quadrature

with respect to the weight function ln(l/t) on [0,1], n=5(5)25.

nodes

9.6 70317011313184162899526d-Oi8.9477136103100828363888620-017.946457674432365150600413 0-01

6.77314174582 8203807018027O-015.4742415722 72107646167938a-014.11702520284 90204317493190-012.8339097129822170420424410-011.739772133208976287011397 0-018.7 93120246479704019290778 0-022.9134472151972 05330372676 0-023.0 55453450374047608522276O-03

weights

1.8176 74998509004844923280 0-039.8205147104594648494Ü1820O-032.52 81432798508611737501200-02

4.8359486624192653084139940-028.Ill987879942632270559929O-021.192 8870631016403584921680-01

1.515254284520667107440880a-011.71520396732757562B055522O-011.785532970704892670281547O-011.5186243751630582212233880-016.0850745987120544229083 960-02

nooes

9.9040145663466103992281040-01 19.688479887186335393915045O-01 8

9.375650909249013043766844O-01 28.9816109121900353816695320-01 4

8.5030626998372514460951420-01 8

7.941904160119662173585093O-01 17.3198728660817158247497020-01 26.657752055164245972223813O-01 2

5.9611711699600020923177600-01 35.2379231797184320116116380-01 44.510940209787955261216721O-01 53.8021253960933233397234130-01 63.1193271934195560332374330-01 72.470524162871598242225454O-01 8

1.875727i30810663812513927a-01 91.3531182463925O7748702321O-01 9

9.068916719091722376416700O-02 95.397126622250062950420123O-02 9

2.640564624946856773849341O-02 89.042630962199650636946628O-03 5

1.283562335805386114259342O-03 2

weights

533 569184207643623003683O-04487068240706659310736070O-0428455683274 5782346462B22O-036 71211724894484356 713219 0-03

4672 73 36990677 9154553426O-03

373507422471821554874215a-020098742972641137012981600-0276432 52 82485899850930559O-026865953 5228178105859621BO-027183 82784133641587909658 0-0273461316917431556072 5749O-027334 9195447 9115487650944O-02784758916696079582 379503O-02748258852663594360502 868O-0238182 691070552 392 23561890-026B5544 515771952 988656812O-02

79444646162853 683897 9348O-02456636029965041400178269O-02

19090031692 6699549095493O-02774125174 710110186 821323 0-02

52020199163 792 4753 93 4347O-02

S57

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S62 SUPPLEMENT

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on computing gauss-kronrod quadrature formulae* · kautsky and elhay [15] and elhay and kautsky...

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