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Iterative refinement for Neville eliminationP. Alonso a; J. Delgado a; R. Gallego a; J. M. Peña b

a Departamento de Matemáticas, Universidad de Oviedo, Campus de Viesques, Gijón, Spain b Departamentode Matemática Aplicada, Universidad de Zaragoza, Zaragoza, Spain

Online Publication Date: 01 February 2009

To cite this Article Alonso, P., Delgado, J., Gallego, R. and Peña, J. M.(2009)'Iterative refinement for Neville elimination',InternationalJournal of Computer Mathematics,86:2,341 — 353

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International Journal of Computer MathematicsVol. 86, No. 2, February 2009, 341–353

Iterative refinement for Neville elimination

P. Alonsoa*, J. Delgadoa, R. Gallegoa and J.M. Peñab

aDepartamento de Matemáticas, Universidad de Oviedo, Campus de Viesques, Gijón, Spain;bDepartamento de Matemática Aplicada, Universidad de Zaragoza, Zaragoza, Spain

(Received 18 December 2007; accepted 27 December 2007)

Neville elimination is an elimination procedure that is very useful when dealing with totally positivematrices. We provide a sufficient condition for the convergence of the iterative refinement using Nevilleelimination.

Keywords: iterative refinement; Neville elimination; total positivity

2000 AMS Subject Classification: 65F05; 65F10

1. Introduction

Let Ax = b be a linear system where A is a nonsingular matrix. Then, solving this systemwith some direct method, in floating point arithmetic, we get an approximation x(0) to the exactsolution. Iterative refinement is a well-established and studied technique to improve the accuracyof the computed solution to the linear system Ax = b. This process computes the residual, r (k),corresponding to a computed solution, x(k), of Ax = b, that is, r (k) = b − Ax(k), then solves thesystem Ay(k) = r (k) and, finally updates the solution x(k+1) = x(k) + y(k) as many times as itwould be necessary to get an accurate enough solution.

The usual method to solve a linear system of equations Ax = b is Gaussian elimination. Con-sidered in the literature has been the application of iterative refinement using Gaussian eliminationfrom several points of view: convergence [7,16,20,21], stability [13,18] and error analysis [15,21].

Neville elimination is an alternative procedure to Gaussian elimination to transform a squarematrix A into an upper triangular matrix U . Roughly speaking, Neville elimination makes zerosin a column of the matrix A by adding to each row a multiple of the previous one. It has beenproved that this elimination process is very useful with totally positive (TP) matrices, sign-regularmatrices and other related types of matrices (see [8] and [3]). A real matrix is called TP if all itsminors are nonnegative. This kind of matrix appears in several areas of mathematics, statisticsand economics [8], as well as in computer aided geometric design (CAGD) [17]. In [6,9–11] it isshown that Neville elimination is a very useful alternative to Gaussian elimination when working

*Corresponding author. Email: palonso@uniovi.es

ISSN 0020-7160 print/ISSN 1029-0265 online© 2009 Taylor & FrancisDOI: 10.1080/00207160802044134http://www.informaworld.com

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342 P. Alonso et al.

with TP matrices. In addition, there are some studies that prove the high-performance computingof Neville elimination [2].

As far as we know, no study of the convergence of the iterative refinement exists using Nevilleelimination. One of the goals of this work is to perform this task when working with any realmatrix. We obtain a sufficient condition which ensures the convergence of the process. In addition,taking into account the convenience of using Neville elimination with TP matrices, we also analysethe convergence of iterative refinement in the particular case where the coefficients matrix of thesystem is TP. In this case we obtain a sufficient condition, which depends on the condition numberof the matrix, weaker than that of the general case.

In Section 2, we give a brief explanation of Neville elimination and show some of its mostremarkable properties. Section 3 presents an iterative refinement and shows a general conditionfor its convergence through Neville elimination. Then, using this general condition, we providea weaker condition when the coefficients matrix A is TP. Finally, in Section 4 we carry out somenumerical experiments which confirm the theoretical results.

2. Neville elimination

Neville elimination is an alternative procedure to Gaussian elimination to make zeros in a columnof a matrix by adding to each row a multiple of the previous one. If A is a square matrix oforder n, this elimination procedure consists of at most n − 1 successive major steps, resulting ina sequence of matrices as follows:

A = A(1) −→ A(1) −→ A(2) −→ A(2) −→ · · · −→ A(n) = A(n) = U,

where U is an upper triangular matrix.On the one hand, A(t) is obtained from the matrix A(t) by moving to the bottom the rows with

a zero entry in column t below the main diagonal, if necessary, to get that

a(t)it = 0, i ≥ t =⇒ a

(t)ht = 0, ∀ h ≥ i. (1)

On the other hand, A(t+1) is obtained from A(t) making zeros in the column t below the maindiagonal by adding an adequate multiple of the ith row to the (i + 1)th for i = n − 1, n − 2, . . . , t

according to the following formula

a(t+1)ij =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

a(t)ij , if 1 ≤ i ≤ t,

a(t)ij − a

(t)it

a(t)i−1,t

a(t)i−1,j , if t + 1 ≤ i ≤ n and a

(t)i−1,t �= 0,

a(t)ij , if t + 1 ≤ i ≤ n and a

(t)i−1,t = 0(=⇒ a

(t)ij = 0),

for all j ∈ {1, 2, . . . , n}.If A is nonsingular, the matrix A(t) has zeros below its main diagonal in the first t − 1 columns.

Let us notice that in this process one has A(n) = A(n) = U , and that when no row exchanges areneeded, then A(t) = A(t) for all t . This happens, for example, when A is a nonsingular TP matrix(see Corollary 5.5 of [9]).

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International Journal of Computer Mathematics 343

The element

pij = a(j)

ij , 1 ≤ j ≤ i ≤ n,

is called the (i, j) pivot of Neville elimination of A and the number

mij =

⎧⎪⎪⎨⎪⎪⎩

a(j)

ij

a(j)

i−1,j

= pij

pi−1,j

, if a(j)

i−1,j �= 0,

0, if a(j)

i−1,j = 0(=⇒ a(j)

ij = 0),

the (i, j) multiplier. Observe that mij = 0 if and only if pij = 0 and that by Equation (1)

mij = 0 =⇒ mhj = 0, ∀ h > i.

Observe that the computational cost of Neville elimination coincides with that of Gaussianelimination.

Let us illustrate the Neville elimination process with the following matrix A. In this case, norow exchanges are used and we also present matrices A(2) and U :

A = A(1) =⎛⎝1 1 1

2 4 84 16 64

⎞⎠ , A(2) =

⎛⎝1 1 1

0 2 60 8 48

⎞⎠ , U = A(3) =

⎛⎝1 1 1

0 2 60 0 24

⎞⎠ .

Observe that the multipliers have been m31 = 2, m21 = 2, m32 = 4.Now let us consider the important case in which Neville elimination can be performed without

row exchanges. Matrices satisfying this condition will be referred to as matrices satisfying theWR condition (Without Row exchanges). We denote by Ei(α), 2 ≤ i ≤ n the bidiagonal lowertriangular matrix whose (r, s) entry (1 ≤ r, s ≤ n) is given by

⎧⎪⎨⎪⎩

1, if r = s,

α, if (r, s) = (i, i − 1),

0, otherwise.

In Theorem 2.5 of [10], it has been proved that a nonsingular matrix of order n satisfies the WRcondition if and only if it can be factorized in the form

A = [En(mn1)En(mn−1,1) · · · E2(m21)][En(mn2) · · · E3(m32)] · · · En(mn,n−1)U, (2)

with the m′ij s satisfying

mij = 0 =⇒ mhj = 0, ∀ h > 1; 1 ≤ j ≤ n − 1, i ≥ j + 1.

By this Theorem, if the matrix A satisfies this condition the factorization is unique and mij is the(i, j) multiplier of Neville elimination of A.

Observe that Equation (2) provides the same LU factorization as in Gaussian elimination, butwith the matrix L decomposed in a different way, as a product of elementary bidiagonal matrices.

The factorization (2) arises in a natural form when we carry out Neville elimination of A

column by column, as explained above. However, as it was seen in ref. [1] this factorization can

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344 P. Alonso et al.

be reordered in the following way

A = En(mn1)[En−1(mn−1,1)En(mn2)] · · · [E2(m21)E3(m32) · · · En(mn,n−1)]U,

which can be also written in the form

[En(−mn,n−1) · · · E2(−m21)] · · · [En(−mn2)En−1(−mn−1,1)]En(−mn1)A = U.

This means that Neville elimination of A can be done by subdiagonals, that is making zerofirst the (n, 1) entry, then the (n − 1, 1) and (n, 2) ones, and so on, finishing making zeros the(2, 1), (3, 2), . . . , (n, n − 1) entries.

Denoting

Fi := En−(i−1)(mn−(i−1),1)En−(i−2)(mn−(i−2),2) · · · En(mni) (3)

Neville elimination of a matrix A satisfying the WR condition, performed by subdiagonals, canbe described in the following form

A[1] = A,

Ft−1A[t] = A[t−1], 2 ≤ t ≤ n − 1,

(4)

with Ft−1 defined by Equation (3) and

A[t] =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a[t]11 a

[t]12 · · · · · · · · · · · · a

[t]1n

......

...

a[t]n−(t−1),1 a

[t]n−(t−1),2 a

[t]n−(t−1),n

0 a[t]n−(t−2),2 a

[t]n−(t−2),n

0 0. . .

...

.... . .

. . ....

0 · · · 0 a[t]nt · · · a [t]

nn

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

with

a[t]ij := a

[t−1]ij , 1 ≤ j ≤ n, 1 ≤ i ≤ n − t + 1,

a[t]ij := a

[t−1]ij − mi,i−n+t a

[t−1]i−1,j , n − t + 2 ≤ i ≤ n, i − n + t ≤ j ≤ n.

Remember that this elimination process is just a reordering of the one explained at the beginningof this section with the same multipliers.

By Corollary 5.5 of [9], TP matrices can be characterized by Neville elimination: a nonsingularmatrix is TP if and only if, when Neville elimination is applied, it is not necessary to perform rowexchanges and, in addition, all the pivots are nonnegative. So a nonsingular TP matrix satisfiesthe WR condition.

If A is a matrix satisfying the WR condition, and the multipliers are nonnegative and, for suffi-ciently high-finite precision, Neville elimination can also be computed out without row exchangeswe will say, for brevity, that we are under WR+ conditions.

Since nonsingular TP matrices have multipliers nonnegative, taking into account the definitionof matrices Fi’s in formula (3) and formula (4), we can deduce that Ft ≥ 0 and A[t] ≥ 0. We shallassume that, for sufficiently high-finite precision, one has

Ft ≥ 0 and A [t] ≥ 0 for all t. (5)

In addition, in this case let us observe that L and U are nonnegative matrices.

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International Journal of Computer Mathematics 345

3. Iterative refinement

Iterative refinement is a well-established and studied technique to improve the computed solutionx(0), in floating point arithmetic to the linear system Ax = b. If the coefficient matrix A is not wellconditioned, then the computed solution x(0) cannot be accurate enough. Iterative refinement is aprocess that allows us to get a sequence x(1), x(2), . . . that, under certain conditions, will convergeto the exact solution of the linear system of equations Ax = b.

Given an approximation x(0) to the exact solution of Ax = b, we can compute the correspondingresidual r (0) = b − Ax(0). Then, we solve the system Ay(0) = r (0) to obtain y(0) and improve theapproximation to the solution of Ax = b by making x(1) = x(0) + y(0). This process is repeated asmany times as necessary to get an accurate enough solution to the system Ax = b. This iterativemethod can be summarized in the following steps:

ALGORITHM 1 Iterative refinement

Input A, b, nT ol, xT ol

Compute an approximation to the solution of Ax = b to get x(0)

k = 0; x(−1) = ∞While k ≤ nT ol and ‖x(k) − x(k−1)‖ ≥ xT ol do

Compute the residual r(k) = b − Ax(k) to get r (k)

Solve the system A y(k) = r (k) to get y(k)

Update the solution x(k+1) = x(k) + y(k)

k = k + 1End-WhileOutput x(k)

In order to improve the solution computed initially with the iterative refinement it is advisableto compute the residuals with extra precision to avoid the errors produced by cancellation ofsignificant figures.

Usually, iterative refinement is used with Gaussian elimination, and residuals are computedin extended precision before being rounded to working precision. The behaviour of iterativerefinement can be summarized as follows: if double precision is used in the computation of theresidual, and the matrix is not too ill conditioned, then the iteration process produces a solutioncorrect to working precision and the rate of convergence depends on the condition number of thematrix. For more details on this technique see, for example [7,13].

Nevertheless, iterative refinement will not always converge to the solution of the correspondingsystem. As far as we know, no study of the convergence of the iterative refinement through Nevilleelimination exists. A complete analysis of iterative refinement is very technical and complicated,and some important aspects can be eclipsed by minor details. So, since in the previous algorithmthe residual r (k) is computed with extra precision before rounding it to working precision, wewill assume that its computation is exact, and that, for simplicity, the computation of the updatedsolution is also exact. Then, on the one hand, in Subsection 3.1 we perform a study of theconvergence of iterative refinement by Neville elimination with any matrix to which it can beapplied. While, on the other hand, in Subsection 3.2, using the results of the previous subsection,we provide a sufficient condition for the convergence of iterative refinement by Neville eliminationwhen working with TP matrices.

3.1 A general condition for the convergence

In this subsection, we study the convergence of iterative refinement with the approach introducedby Ortega in ref. [16] but adapted for Neville elimination.

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346 P. Alonso et al.

In Theorem 3.1 of [1], Alonso et al. proved that the computed solution y(k) of a linear system ofequations A y(k) = r (k), where A is a n × n nonsingular matrix, through the factorization LU of A

provided by Neville elimination is the exact solution of a perturbed system (A + Hk) y(k) = r (k)

with the perturbation matrix Hk depending on the exact solution of the system. Hence, the vectorsy(k) in the algorithm of iterative refinement satisfy, in exact arithmetic, equations of the form

(A + Hk) y(k) = r (k).

Then the vectors x(k) verify

x(k+1) = x(k) + (A + Hk)−1r (k), k = 0, 1, . . . (6)

assuming that the inverses of the matrices A + Hk exist.In our analysis, we shall assume that residuals are computed exactly since, in practice, they are

computed in extra precision. Then, taking into account that

r (k) = b − A x(k),

Equation (6) can be rewritten as

x(k+1) = x(k) + (A + Hk)−1(b − A x(k)) = (A + Hk)

−1 (Hk x(k) + b),

and so

x(k+1) = (A + Hk)−1 Hk x(k) + (A + Hk)

−1 b, k ≥ 1,

that is,

x(k+1) = Bk x(k) + ck, k ≥ 1, (7)

denoting

Bk := (A + Hk)−1Hk and ck := (A + Hk)

−1b.

Remark 1 Iterative refinement, considered as an iterative method for the resolution of a systemof linear equations Ax = b, is consistent with the exact solution x = A−1b of the system fromthe classical point of view. That is, we can prove that if the iterative sequence converges to x∗,then x∗ satisfies Equation (7) and x∗ is the exact solution.

Remark 2 Given A = (aij )1≤i,j≤n, from now on, let us denote by ‖A‖ any matrix norm and by|A| the matrix (|aij |)1≤i,j≤n.

It is well known that, if the iterative process in Equation (7) is consistent with x = A−1b and‖Bk‖ ≤ α < 1 for all k ≥ 0 then limk→∞ x(k) = x for any x(0) (see [16]).

Now we shall get a sufficient condition which ensures the convergence of iterative refinementby Neville elimination expressed in Equation (7). First, we will need another well-known result,the Banach Lemma:

LEMMA 3.1 (cf. [16, Lemma 2.1.1, p. 32]) If A is a nonsingular n × n real matrix and E is ann × n real matrix such that ‖A−1‖ ‖E‖ < 1, then A + E is nonsingular and

‖(A + E)−1‖ ≤ ‖A−1‖1 − ‖A−1‖ ‖E‖ .

The following result states a sufficient condition for the convergence of iterative refinementthrough Neville elimination.

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International Journal of Computer Mathematics 347

PROPOSITION 3.2 If A is a nonsingular n × n matrix satisfying

‖Hk‖ ‖A−1‖ <1

2for all k ≥ 0

then the matrices A + Hk , k ≥ 0, are nonsingular and the iterative refinement in Equation (7)

converges to x = A−1b for any x(0).

Proof By the Banach Lemma, taking into account that ‖Hk‖ ‖A−1‖ < 1 for all k ≥ 0 we deducethat the matrices A + Hk are nonsingular and that

‖(A + Hk)−1‖ ≤ ‖A−1‖

1 − ‖A−1‖ ‖Hk‖ for all k ≥ 0.

Then, by the previous formula we have that

‖(A + Hk)−1Hk‖ ≤ ‖(A + Hk)

−1‖ ‖Hk‖ ≤ ‖A−1‖ ‖Hk‖1 − ‖A−1‖ ‖Hk‖ for all k ≥ 0.

From the previous formula and taking into account the hypothesis, we conclude that

‖(A + Hk)−1Hk‖ ≤ ‖A−1‖ ‖Hk‖

1 − ‖A−1‖ ‖Hk‖ <(1/2)

(1 − (1/2))= 1 for all k ≥ 0

and hence the iterative refinement converges. �

In Proposition 3.2, we have proved that if the perturbation matrices Hk , k ≥ 0, satisfy thecondition ‖Hk‖ ‖A−1‖ < (1/2) then the iterative refinement converges. In Theorem 3.1 of [1],as commented before, Alonso et al. have bounded the perturbation matrix Hk in the case of theNeville elimination process for any matrix, with the particularity that the bound is independentof the exact solution of the system.

PROPOSITION 3.3 Let A be a nonsingular n × n matrix and assume that L := F1F2 . . . Fn−1 andU := A [n] have been computed by Neville elimination of A under the WR+ condition, with unitroundoff u. If L and U are used to compute the solution x of the system Ax = b, then one has

(A + H)x = b

with

|H | ≤ u

n−1∑j=1

|F1| |F2| · · · |Fj | |A [j+1]| + γn|L| |U |

and

γn := n u

1 − n u.

The following result follows straightforward from Propositions 3.2 and 3.3.

THEOREM 3.4 Let A be a nonsingular n × n matrix and assume that L := F1F2 . . . Fn−1 andU := A [n] have been computed by Neville elimination of A under the WR+ condition, with unitroundoff u. If L and U are used to compute the solution x of the system Ax = b, then one hasthat if ⎛

⎝u

n−1∑j=1

‖F1‖ ‖F2‖ · · · ‖Fj‖ ‖A [j+1]‖ + γn‖L‖ ‖U‖⎞⎠ ‖A−1‖ <

1

2,

then the iterative refinement in Equation (7) converges to x = A−1b for any x(0).

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348 P. Alonso et al.

3.2 Convergence with TP matrices

TP matrices arise in a natural way in many areas of mathematics, statistics, economics, etc.Especially, their application to approximation theory and CAGD is very interesting. For example,coefficient matrices of interpolation or least square problems with a lot of representations inCAGD (the Bernstein basis, the B-spline basis, etc.) are TP. Some recent applications of suchkinds of matrices to CAGD can be found in refs. [5,8,14,17]. For applications of TP matrices toother fields see [3] and [8]. In [6,9–11] it has been proved that Neville elimination is a very usefulalternative to Gaussian elimination when working with TP matrices. In addition, studies whichprove the high-performance computing of Neville elimination have been carried out [2].

Now, using the results in Section 3.1 we undertake a study of the convergence of iterativerefinement through Neville elimination for TP matrices. It is convenient to observe that all entriesof any TP matrix A are nonnegative and so, |A| = A.

In Theorem 4.1 of [1], taking into account Proposition 3.3 (that is, Theorem 3.1 of [1]), Alonsoet al. bounded the perturbation matrix H in the particular case when A is a nonsingular TP matrix:

THEOREM 3.5 Let A be an n × n(n ≥ 2) nonsingular TP matrix such that the matrices computedin the Neville elimination process, with sufficiently high precision, satisfy Equation (5). Then thesolution of Ax = b computed by Neville elimination x, satisfies (A + H)x = b, where

|H | ≤ (γn−1 + γn (1 + γn−1)) A (8)

and

γk := k u

1 − k u.

A bound for the perturbation matrix H in the case of performing Gaussian elimination on aTP matrix was presented by de Boor and Pinkus in ref. [4] assuming that γn ≤ (1/4). We willget a bound for the perturbation matrix H in the case of working with Neville elimination on TPmatrices keeping the assumption γn ≤ (1/4) in order to compare it with the one obtained by deBoor and Pinkus.

THEOREM 3.6 Let A be a n × n nonsingular TP matrix. Let γn = (nu/(1 − nu)) and let usconsider the unit roundoff small enough to have that γn ≤ (1/4) and Equation (5) holds. Thenthe solution x(0) of the system Ax = b computed by Neville elimination satisfies the equation(A + H)x(0) = b, where

|H | ≤ 9

4γnA. (9)

Proof Taking into account that γn−1 ≤ γn, from Equation (8) we deduce that

|H | ≤ (γn + γn(1 + γn)) A = γn (2 + γn) A,

and, since γn ≤ (1/4), we can conclude that

|H | ≤ γn

(2 + 1

4

)A = 9

4γnA,

and the result follows. �

Remark 3 De Boor and Pinkus consider in ref. [4] that the solution x of Ax = b computedthrough Gaussian elimination without pivoting verifies exactly an equation (A + H)x = b with

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International Journal of Computer Mathematics 349

|H | ≤ γn (2 + γn) |L| |U |. In [12, p. 144] Higham reminds us of this result. However, Highamstates that de Boor and Pinkus, in fact, prove that |H | ≤ γn (3 + γn) |L| |U |, remarking that, in thecase of the sharper bound, de Boor and Pinkus cite the original 1972 German edition of ref. [19]for a proof of the result. In addition, in the case of performing Gaussian elimination withoutpivoting with a TP matrix A, de Boor and Pinkus prove that

|H | ≤ γn (2 + γn) |L| |U |1 − γn

|A|,

and, when γn ≤ (1/4) and, L and U are nonnegative, they show that

|H | ≤ 3 γn |A|.

Let us notice that the bound obtained in Theorem 3.6 for Neville elimination with TP matricesimproves the previous bound for Gaussian elimination with TP matrices.

Let us be reminded that although the matrices H depend on the exact solution of thecorresponding systems, their bounds do not depend on such solutions.

The following result provides a sufficient condition for the convergence of iterative refinementby Neville elimination when working with TP matrices.

THEOREM 3.7 Let A be an n × n nonsingular TP matrix. Let γn = (nu/(1 − nu)) and let usconsider the unit roundoff small enough to have that γn ≤ (1/4) and Equation (5) holds. Then, if

9

4γn‖A‖‖A−1‖ <

1

2,

the iterative refinement through Neville elimination converges.

Proof Then, taking (absolute) norms in Equation (9), we deduce that

‖H‖ ≤ 9

4γn‖A‖.

By this last expression and the condition for convergence in Section 3.1, we can conclude fromthe hypothesis that

‖H‖ ≤ 9

4γn‖A‖ <

1

2‖A−1‖ ,

and then the iterative refinement through Neville elimination converges by Proposition 3.2. �

4. Numerical tests and conclusions

In this section, we include numerical tests for checking the usefulness of iterative refinement andthe sufficient condition for Neville elimination obtained in Theorem 3.7.

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350 P. Alonso et al.

The collocation matrix of a system (u0(t), . . . , un(t)) of univariate functions at t0 < · · · < tnin R is given by

M

(u0, . . . , un

t0, . . . , tn

):= (uj (ti))0≤i,j≤n.

Let (Bn0 , . . . , Bn

n ) be the basis of the space of polynomials of degree at most n formed by theBernstein polynomials of n degree given by

Bni (t) =

(n

i

)t i(1 − t)n−i , for all i ∈ {0, 1, . . . , n}.

In our numerical experiments we have considered linear systems of equations of the form

M

(Bn

0 , . . . , Bnn

tn0 , . . . , tnn

)x = b,

where tni = (i/n) for all i ∈ {0, 1, . . . , n}, the components of the solution x are generatedrandomly in the interval [0, 1] according to a uniform distribution and the vector b is com-puted straightforwardly as Ax for n = 9, 14 and 29. The matrices of these linear systems arenonsingular TP matrices [17] and play an important role in CAGD. These systems are solvedby iterative refinement through Gaussian elimination without pivoting and Neville eliminationwithout pivoting.

Remark 1 In the numerical tests, following the notation used in ref. [13], we will denote

ω|A|,|b| := maxi

|ri |(|A||x| + |b|)i .

In fact, for simplicity, we will use ω instead of ω|A|,|b|. The backward error can be computedexactly from the previous formula.

In the numerical tests, we perform all computations in IEEE single precision, except thecomputation of the residuals which is performed in IEEE double precision.

For n = 9 we have that

9

4γ10‖A‖∞‖A−1‖∞ = 1.8337739 × 10−3 <

1

2.

So, by Theorem 3.7 iterative refinement through Neville elimination converges. This behaviourcan be observed in Table 1.

Table 1. Iterative refinement through NE and GE for the Bernstein matrix of order 10.

Iteration Method ‖x(i) − x(i−1)‖∞ ‖x(i) − x‖∞/‖x‖∞ ω

0 GE 3.9920401 × 10−6 3.9629082 × 10−8

NE 2.0330428 × 10−5 5.7782287 × 10−8

1 GE 9.3281269 × 10−6 6.2295139 × 10−6 3.9629082 × 10−8

NE 1.3053417 × 10−5 6.2295139 × 10−6 3.9629082 × 10−8

2 GE 0 6.2295139 × 10−6 3.9629082 × 10−8

NE 0 6.2295139 × 10−6 3.9629082 × 10−8

3 GE 0 6.2295139 × 10−6 3.9629082 × 10−8

NE 0 6.2295139 × 10−6 3.9629082 × 10−8

GE, Gaussian elimination; NE, Neville elimination.

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International Journal of Computer Mathematics 351

In this table it can be also seen how the relative error for Neville elimination decreases whereasthe backward error ω is similar to the unit roundoff of IEEE single precision.

In the case where n = 14 we have that

9

4γ15‖A‖∞‖A−1‖∞ = 0.3326375 <

1

2.

We can observe in Table 2 that, for both elimination procedures ‖x(3) − x(2)‖∞ = 0 whereas therelative error decreases negligibly for Neville elimination and increases negligibly for Gaussianelimination in the first iteration, and it keeps constant in the following iterations.

Finally, when n = 29

9

4γ30‖A‖∞‖A−1‖∞ = 9137.289 >

1

2,

and hence, the sufficient condition in Theorem 3.7 does not hold. So we cannot assure theconvergence of iterative refinement. In fact, in Table 3 we can check as in this case the iterativerefinement clearly diverges for both elimination procedures.

We can also see in the previous tables that iterative refinement through Gaussian elimination hasa very similar behaviour to the iterative refinement through Neville elimination. We could haveexpected this behaviour since the backward error bound presented in Theorem 3.6, from whichwe derived the sufficient condition for convergence, is only slightly better than the one obtainedfor the Gaussian elimination by the Boor and Pinkus in ref. [4] (see Remark 3). Nevertheless,let us remind ourselves that Neville elimination is more useful for TP matrices than Gaussianelimination due to many reasons. Let us summarize some of these reasons.

Table 2. Iterative refinement through NE and GE for the Bernstein matrix of order 15.

Iteration Method ‖x(i) − x(i−1)‖∞ ‖x(i) − x‖∞/‖x‖∞ ω

0 GE 3.9707919 × 10−4 8.0003232 × 10−8

NE 2.1378661 × 10−3 6.6124521 × 10−8

1 GE 1.1821985 × 10−3 1.0698345 × 10−3 1.0051913 × 10−7

NE 9.9265575 × 10−4 1.0695126 × 10−3 6.7012749 × 10−8

2 GE 4.7683716 × 10−7 1.0693516 × 10−3 1.0051913 × 10−7

NE 2.3841858 × 10−7 1.0693516 × 10−3 1.0051913 × 10−7

3 GE 0 1.0693516 × 10−3 1.0051913 × 10−7

NE 0 1.0693516 × 10−3 1.0051913 × 10−7

GE, Gaussian elimination; NE, Neville elimination.

Table 3. Iterative refinement through NE and GE for the Bernstein matrix of order 30.

Iteration Method ‖x(i) − x(i−1)‖∞ ‖x(i) − x‖∞/‖x‖∞ ω

0 GE 8.601958 1.6308462 × 10−7

NE 5.817883 2.4425256 × 10−7

1 GE 10.44222 19.88214 4.5845301 × 10−8

NE 16.87044 21.01000 1.4842574 × 10−7

2 GE 21.15958 42.58482 3.8880344 × 10−8

NE 95.28648 123.9429 1.6128924 × 10−7

3 GE 41.72396 86.63822 3.6512297 × 10−8

NE 499.4199 662.6565 1.9544008 × 10−7

GE, Gaussian elimination; NE, Neville elimination.

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352 P. Alonso et al.

Table 4. Iterative refinement through NE for the TP matrix (min(i, j) max(i, j)1/4)1≤i,j≤10.

Iteration ‖x(i) − x(i−1)‖∞ ‖x(i) − x‖∞/‖x‖∞ ω

0 3.271085 4.8415866 × 10−2

1 7.392051 4.714156 6.6341490 × 10−2

2 4.301459 6.7517467 × 10−2 1.4447814 × 10−3

3 6.2507734 × 10−2 6.3260954 × 10−6 1.4706372 × 10−7

4 4.8577785 × 10−6 4.6037235 × 10−6 9.4124736 × 10−8

NE, Neville elimination.

1. Neville elimination provides a test to check the total positivity of a matrix with much lowercomputational cost than tests using Gaussian elimination [9].

2. Neville elimination provides natural parameters for the factorization of TP matrices which arevery useful for many applications, including the obtention of algorithms with high-relativeaccuracy (see [6]).

3. In [10] it was shown for a family of TP matrices that Neville elimination has lowercomputational cost than Gaussian elimination.

4. In the context of CAGD, Neville elimination is closely related with corner-cutting algorithmassociated to TP matrices, which are the fundamental algorithms in this field.

Finally, taking into account the remarks in the previous paragraphs and that in the previousexamples iterative refinement only improves slightly the solution got by Neville elimination, wepresent another example where this technique is crucial to get an accurate solution with Nevilleelimination.

Let A = (aij )1≤i,j≤10 be the matrix whose entries are given by aij = min(i, j) max(i, j)1/4 forall i, j ∈ {1, . . . , 10}, and let us consider x and b generated as in the previous examples. The matrixA is TP since it is the particular case of Section 7(f) in ref. [3] with f (t) = t and g(t) = t1/4.Then we have solved the system Ax = b with iterative refinement through Neville eliminationwithout pivoting. For the matrix A, the condition for the convergence presented previously holdsand observing Table 4 we can check that the iterative refinement process is crucial to get a accurateenough solution.

Let us summarize the main conclusions of this paper. We show that the iterative refinementcan be used with Neville elimination, an elimination procedure alternative to Gauss elimination.We provide a sufficient condition in order to guarantee that the iterative refinement converges.In addition, we improve this sufficient condition for the particular class of TP matrices, which isvery important in many applications and for which Neville elimination presents some advantagesover Gauss elimination.

Acknowledgements

This work has been partially supported by the Spanish Research Grant MTM2006-03388 and under MEC and FEDERGrant TIN2004-05920. The authors thank to an anonymous referee for helping to improve the paper.

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