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Computing Derivatives of JacobiPolynomials Using BernsteinTransformation and DifferentiationMatrixAbedallah Rababah a , Mohammed Al-Refai a & Radwan Al-Jarrah ba Department of Mathematics, Jordan University of Science andTechnology, Irbid, Jordanb College of Arts and Sciences, Southwestern Oklahoma StateUniversity, Weatherford, Oklahama, USAPublished online: 04 Jun 2008.

To cite this article: Abedallah Rababah , Mohammed Al-Refai & Radwan Al-Jarrah (2008) ComputingDerivatives of Jacobi Polynomials Using Bernstein Transformation and Differentiation Matrix,Numerical Functional Analysis and Optimization, 29:5-6, 660-673, DOI: 10.1080/01630560802099761

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Numerical Functional Analysis and Optimization, 29(5–6):660–673, 2008Copyright © Taylor & Francis Group, LLCISSN: 0163-0563 print/1532-2467 onlineDOI: 10.1080/01630560802099761

COMPUTING DERIVATIVES OF JACOBI POLYNOMIALSUSING BERNSTEIN TRANSFORMATIONAND DIFFERENTIATION MATRIX

Abedallah Rababah,1 Mohammed Al-Refai,1 and Radwan Al-Jarrah2

1Department of Mathematics, Jordan University of Science and Technology,Irbid, Jordan2College of Arts and Sciences, Southwestern Oklahoma State University,Weatherford, Oklahama, USA

� In this paper, we give a new, simple, and efficient method for evaluating the pth derivativeof the Jacobi polynomial of degree n. The Jacobi polynomial is written in terms of the Bernsteinbasis, and then the pth derivative is obtained. The results are given in terms of both Bernsteinbasis of degree n − p and Jacobi basis form of degree n − p and presented in a matrix form.Numerical examples and comparisons with other well-known methods are presented.

Keywords Basis transformation; Bernstein polynomials; Derivative evaluation; Jacobipolynomials.

AMS Subject Classification 33C45; 41A58; 41A10.

1. INTRODUCTION AND MOTIVATION

We are interested in evaluating the pth derivative of Qn(x), where

Qn(x) =n∑

�=0

c�P (�,�)� (x), n = 0, 1, 2, � � � (1.1)

and P (�,�)� (x) are the Jacobi polynomials.

Usually, the method of differentiation matrix is used, see for exampleBayliss, Class, and Matkowsky [5], Baltensperger [2], Barrio [3], and

Address correspondence to Radwan Al-Jarrah, College of Arts and Sciences, SouthwesternOklahoma State University, 100 Campus Drive, Weatherford, OK 73096, USA; E-mail: radwan.aljarrah@swosu.edu

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Computing Derivatives of Jacobi Polynomials 661

Don and Solomonoff [8], where the approximate solution is written interms of the Lagrange interpolation form

Qn(x) =n∑

�=0

Qn(x�)��(x), (1.2)

where ��(x) is the Lagrange polynomial of degree � at the Chebyshev–Gauss–Lobatto nodes

x� := cos��

n, � = 0, 1, � � � ,n�

To approximate the pth derivative of (1.1), the relation (1.2) isdifferentiated and evaluated at the collocation points, i.e.,

Q (p)n (x�) =

n∑�=0

Qn(x�)�(p)� (x�), � = 0, 1, � � � ,n� (1.3)

Using u = [Qn(x0), � � � ,Qn(xn)]t , u(p) = [Q (p)n (x0), � � � ,Q

(p)n (xn)]t and the

(n + 1) × (n + 1) matrix D(p) with entries D(p)�� = �

(p)� (x�), �, � = 0, 1, � � � ,n,

then the system in (1.3) can be written in the matrix form

u(p) = D(p)u�

Differentiation matrices D(p) based on Lagrange interpolation on theChebyshev nodes are used to evaluate the pth derivative, see also forexample Baltensperger and Berrut [1] and Tang and Trummer [15]. Thisleads to growth n4 in the error of evaluating the first derivative and growthn6 in the error of evaluating the second derivative, see Breuer and Everson[7].

Recently in Barrio and Peña [4], the recurrence relation of the Jacobipolynomials is used to evaluate the pth derivative of (1.1).

In this paper, we give a new approach: basis transformation intoBernstein polynomials is carried out first, and the Bernstein polynomialis differentiated by defining a differentiation matrix Dn,p . This method isjustified because the Bernstein polynomials are stable.

In scientific computation, it is practical and convenient to use thematrix form to compute and evaluate derivatives of functions andpolynomials. The matrix form gives an easy and compact form forcomputing the derivative.

The differentiation matrix Dn,p can be computed by simplecalculations.

We end this section with the definitions of the Jacobi and Bernsteinpolynomials.

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662 A. Rababah et al.

The Jacobi polynomials P (�,�)n (x), are orthogonal polynomials on [−1, 1]

with respect to the weight function w(x) = (1 − x)�(1 + x)�, �, � > −1. Formore information on the Jacobi polynomials, see Szegö [14].

The Bernstein polynomials of degree n on [0, 1] are given by theformula

Bn� (u) =

(n�

)(1 − u)n−�u�, � = 0, 1, � � � ,n, (1.4)

where (n�

)= n!

�!(n − �)! , � = 0, 1, � � � ,n�

For more information on the Bernstein polynomials, see Farin [9] andHoschek and Lasser [11].

2. INTRODUCING THE METHOD

Traditionally, the Bernstein polynomials Bn� (u) are defined for

u ∈ [0, 1], and the classic Jacobi polynomials P (�,�)� (x) are usually defined

and orthogonal for x ∈ [−1, 1]. Throughout this paper, we shift the Jacobipolynomials into [0,1].

Consider the polynomial

Qn(u) =n∑

�=0

c�P (�,�)� (u) =: c tJn ,u ∈ [0, 1], (2.1)

where c t := [c0, c1, � � � , cn] and Jtn := [P (�,�)0 (u),P (�,�)

1 (u), � � � ,P (�,�)n (u)]. We

write Qn(u) in terms of the Bernstein basis

Qn(u) =n∑

�=0

d�Bn� (u) =: dtBn , (2.2)

where dt := [d0, d1, � � � , dn] and Btn := [Bn

0 (u),Bn1 (u), � � � ,B

nn (u)]. We find

the pth derivative of (2.2), and then express the results in terms of bothBernstein basis of degree n − p and Jacobi basis form of degree n − p.

The entries of the transformation matrix Mn that transforms the Jacobicoefficients �c�n�=0 of (2.1) into the Bernstein coefficients �d�n�=0 of (2.2)are defined by

d� =n∑

�=0

M��c��

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Computing Derivatives of Jacobi Polynomials 663

This can be written in matrix-form as:

d = Mnc �

The entries M��, �, � = 0, 1, � � � ,n of the matrix Mn of transformation ofJacobi polynomial basis into Bernstein polynomial basis of degree n aregiven by (see Rababah [13])

M�� = 1( n�

) min(�,�)∑i=max(0,�+�−n)

(−1)�−i

(n − �

� − i

) (� + �

i

) (� + �

� − i

)� (2.3)

The entries M−1�� , �, � = 0, 1, � � � ,n of the matrix M−1

n of transformation ofBernstein polynomial basis into Jacobi polynomial basis of degree n aregiven by (see Rababah [13])

M−1�� =

(�0

h(�,�)0

+ �0

h(�,�)�

) (n�

) �∑i=0

(−1)�−i

(� + �

i

) (� + �

� − i

)× B(� + � + i + 1,n + � + � − � − i + 1) (2.4)

where B(x , y) is the beta function, and

�0 ={1, if � = 00, if � �= 0

and �0 ={0, if � = 01, if � �= 0

�

It is shown in Farouki [10] and Rababah [12] that the Legendre–Bernsteinand Chebyshev–Bernstein basis transformations are well conditioned,respectively. In Rababah [13], it is pointed out that the Jacobi–Bernsteinbasis transformation can be used to find the derivative of Jacobipolynomials, which is fulfilled in this paper.

3. DIFFERENTIATION MATRIX Dn,p

We first introduce the operator � as follows

�0d� = d�, �d� = d�+1 − d�, �pd� = �(�p−1d�

)�

The operator � can be written in the form (see Farin [9] and Hoschekand Lasser [11])

�pd� =p∑

j=0

(pj

)(−1)p−j d�+j �

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664 A. Rababah et al.

Thus, the pth derivative of the polynomial Qn(u) in terms of the Bernsteinbasis is given by

dp

dupQn(u) = n!

(n − p)!n−p∑�=0

�pd�Bn−p� (u), p = 1, 2, � � � ,n� (3.1)

The pth derivative of the Bézier curve Qn(u) can be computed andexpressed in terms of the intermediate points generated by the deCasteljau algorithm

dp

dupQn(u) = n!

(n − p)!�pdn−p

0 (u), p = 1, 2, � � � ,n�

In this paper, we follow a matrix form approach.The operator � on the vector d is defined by

�d = (�d0,�d1, � � � ,�dn−1)t , �pd = (�pd0, � � � ,�pdn−p)

t �

Thus, the operator � has the matrix form

�d = Dnd �

The matrix Dn with dimension n × (n + 1) is defined by

Dn =

−1 1 0 0 · · · 0 0 0

0 −1 1 0 · · · 0 0 0

0 0 −1 1 · · · 0 0 0���

������

���� � �

������

���

0 0 0 0 · · · −1 1 0

0 0 0 0 · · · 0 −1 1

�

In general, �pd can be written in the matrix form

�pd = Dn,pd �

The differentiation matrix Dn,p has dimension (n − p + 1) × (n + 1),

Dn,p = Dn−p+1Dn−p+2 � � �Dn−1Dn �

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Computing Derivatives of Jacobi Polynomials 665

4. PROPERTIES OF DIFFERENTIATION MATRIX Dn,p

We note that for any (n + 1) × m matrix B, the (i , j)th element of DnBis given by

(DnB)i ,j =n+1∑k=1

dikbkj = diibij + di ,i+1bi+1,j = −bij + bi+1,j �

Theorem 4.1. The elements of the kth row of the matrix Dn,p are given by

(0, 0, � � � , 0︸ ︷︷ ︸k−1 fold

, d11, � � � , d1 p+1︸ ︷︷ ︸p+1 fold

, 0, 0, � � � , 0︸ ︷︷ ︸n−p−k+1 fold

)�

Proof. We prove this theorem by induction on p. It is clear that the resultis true for p = 1� Assume the result is true for m = p − 1, then Dn,p =Dn−p+1Dn,p−1. Using the note preceding this theorem, the (i , j)th elementof Dn,p is given by (Dn,p)i ,j = −dij + di+1,j , and hence the elements of thekth row of Dn,p have the form

(0, 0, � � � , 0︸ ︷︷ ︸k−1 fold

,−d11,−d12 + d11, � � � ,−d1p + d1 p−1, d1p︸ ︷︷ ︸p+1 fold

, 0, 0, � � � , 0︸ ︷︷ ︸n−p−k+1 fold

)�

This completes the proof. �

Thus the matrix Dn,p has p + 1 nonzero entries in each row, andthese nonzero coefficients are shifted one column to the right in any twoconsecutive rows. That is, the nonzero entries of Dn,p satisfy

dij = di−1j−1 = d1,j−i+1 for j ≥ i ≥ 2�

Therefore, we can write the matrix Dn,p by knowing its first row only. Inthe following, we concentrate on studying the first row of Dn,p , and wedenote the nonzero entries in the first row of Dn,p by �cpj : j = 1, 2, � � � , p +1, p ≥ 1. We refer to the matrix C = (cpj), p ≥ 1 and j = 1, � � � ,n + 1 as thedifferentiation matrix. Knowing that Dn,p = Dn−p+1Dn,p−1 we have

cp+1,j = −cpj + cp,j−1� (4.1)

Theorem 4.2. For p ≥ 1, the entries cpj of the differentiation matrix satisfy

(1) cpj = 0, j ≥ p + 2(2) cp,p+1 = 1,

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666 A. Rababah et al.

(3) cp1 = (−1)p

(4) cpp = −p�

The proof is straightforward by induction on p and using (4.1).

Theorem 4.3. The following statements hold for cpj :

(1) cpj = (−1)pcp,p+2−j for j = 1, 2, � � � , p + 1, p ≥ 1(2) cpj = (−1)p�(−1)j+1 + ∑p−1

n=j−1(−1)n+1cn,j−1 for p ≥ j ≥ 2�

Proof. (1) For p = 1, c1j = −c1,3−j is true for j = 1 and 2. Suppose theresult is true for p, that is, cpj = (−1)pcp,p+2−j for j = 1, 2, � � � , p + 1� Then,

cp+1,j = −cpj + cp,j−1 = (−1)p+1cp,p+2−j + (−1)pcp,p+3−j

= (−1)p+1(cp,p+2−j − cp,p+3−j

) = (−1)p+1cp+1,p+3−j � (4.2)

Equation (4.2) is valid for j = 1, 2, � � � , p + 1, and from Theorem 4.2(2) theequation is also true for j = p + 2, which proves the result.

(2) For p = 2, we have j = 2 and c22 = (−1)2�(−1)3 + ∑1n=1

(−1)n+1cn1 = −1 + c11 = −2, and so the result is true. Suppose the resultis true for p = m, that is

cmj = (−1)m((−1)j+1 +

m−1∑n=j−1

(−1)n+1cn,j−1

)for m ≥ j ≥ 2�

Now, for m ≥ j ≥ 2, we have

cm+1,j = −cmj + cm,j−1

= (−1)m+1

((−1)j+1 +

m−1∑n=j−1

(−1)n+1cn,j−1

)+ cm,j−1

= (−1)m+1

((−1)j+1 +

m∑n=j−1

(−1)n+1cn,j−1

)�

This proves the result for m ≥ j . For j = m + 1 the result is also true, ascm+1,m+1 = (−1)m+1

((−1)m+2 + (−1)m+1cmm

) = −1 + cmm = −cm,m+1 + cmm �

This completes the proof. �

From Theorem 4.3(1), it is enough to find cpj for j = 1, 2, � � � , [p/2] +1, and use the symmetry cpj = (−1)pcp,p+2−j to find the rest of entries.Theorem 4.3(2) along with Theorem 4.2 help us in finding thedifferentiation matrix.

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Computing Derivatives of Jacobi Polynomials 667

Theorem 4.4. The first five columns of the differentiation matrix Dn,p aredetermined by

(1) cp1 = (−1)p

(2) cp2 = p(−1)p+1

(3) cp3 = (−1)p(1 + (p+1)(p−2)

2

)(4) cp4 = (−1)p+1(p − 2)

(1 + (p−3)(p+2)

6

)(5) cp5 = (−1)p�1 + (p − 4)

( p−32 + 1 + p−3

6 �(p−4)(p−3)

4 + 2p − 92

)

Proof. (1) Has been done in Theorem 4.2(3).

(2) From Theorem 4.3(2), we have

cp2 = (−1)p(

− 1 +p−1∑n=1

(−1)n+1cn1

)= (−1)p

(− 1 +

p−1∑n=1

(−1)n+1(−1)n)

= (−1)p(

− 1 +p−1∑n=1

−1)

= p(−1)p+1�

(3) Again using Theorem 4.3(2), we have

cp3 = (−1)p(1 +

p−1∑n=2

(−1)n+1cn2

)= (−1)p

(1 +

p−1∑n=2

(−1)n+1n(−1)n+1

)

= (−1)p(1 +

p−1∑n=2

n)

= (−1)p(1 + (p + 1)(p − 2)

2

)�

The proofs of parts (4) and (5) are similar to parts (2) and (3). �

5. DERIVATIVE OF JACOBI POLYNOMIALS

Equation (3.1) can be written in the form

dp

dupQn(u) = n!

(n − p)!(�pd)tBn−p , p = 1, 2, � � � ,n�

= n!(n − p)!(Dn,pd)tBn−p

= n!(n − p)!(Dn,pMnc)tBn−p � (5.1)

Thus, the pth derivative of the Jacobi polynomial is the Bézier curveof degree n − p with Bézier coefficients n!

(n−p)! multiplied by the transpose

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668 A. Rababah et al.

of the differentiation matrix Dn,p , the transformation matrix Mn , and thevector c . We summarize this in the following theorem.

Theorem 5.1. The pth derivative of the Jacobi polynomial Qn(u) = c tJn isgiven by

dp

dupQn(u) = btBn−p , p = 1, 2, � � � ,n,

where the vector bt = (b0, � � � , bn−p)t := n!

(n−p)!(Dn,pMnc)t .

In particular,

dp

dupQn(0) = b0,

and

dp

dupQn(1) = bn−p �

Applying the de Casteljau algorithm on the vector b, we can see thatbn−p0 (u) is the pth derivative of Qn(u) at the parameter value u.

By using the Bernstein to Jacobi basis transformation and the previousresult, the pth derivative of Qn(u) can also be written in terms of Jacobipolynomials of degree ≤ n − p. This is summarized in the followingtheorem.

Theorem 5.2. The pth derivative of the Jacobi series Qn(u) = c tJn is given by

dp

dupQn(u) = g t Jn−p , p = 1, 2, � � � ,n,

where the vector g t = (g0, � � � , gn−p)t := n!

(n−p)!(M−1n−pDn,pMnc)t .

6. NUMERICAL EXAMPLES AND COMPARISONS

In the calculations to follow, we have used the L∞-norm

E∞ = maxu∈[0,1]

|Qn(u) − f (u)|,

and a machine precision 10−16. We compute the pth derivative for differentkinds of functions using our method, and then the results are comparedwith the ones obtained by well-known methods, such as Clenshaw–Smithderivative algorithm (CSD) (Barrio and Peñ [4]), the classic differentiation

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Computing Derivatives of Jacobi Polynomials 669

matrices (CM) (Boyd [6]), the Baltensperger and Berrut method(BB) (Baltensperger and Berrut [1]), the Tang and Trummer method(TT) (Tang and Trummer [15]), and the modified Clenshaw–Smithderivative algorithm (MCSD) (Barrio and Peñ [4]). Figures 1–3, graph

FIGURE 1 The errors in evaluating the first, second, third and fourth derivatives of f (x) = sin(x)using polynomials of degrees 16 (dashed), 32 (solid), and 64 (dotted).

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670 A. Rababah et al.

FIGURE 2 The errors in evaluating the first, second, third and fourth derivatives of f (x) =sin(8x)

(x+1�1)3/2using polynomials of degrees 16 (dashed), 32 (solid), and 64 (dotted).

the errors in evaluating the pth derivative of different functions, forp = 1, 2, 3, 4. We see that the error increases with p, and it depends onthe function and the degree m of polynomial used. Table 1 presentsthe maximum error in evaluating the pth derivative of f (x) = 1

1+x2 , for

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Computing Derivatives of Jacobi Polynomials 671

FIGURE 3 The errors in evaluating the first, second, third and fourth derivatives of f (x) = 11+x2

using polynomials of degrees 16 (dashed), 32 (solid), and 64 (dotted).

p = 1, 2, 3, 4. From the table, one can see that the MCSD and BB methodsgive approximate solutions that are about 3 significant digits better thanthe ones obtained by our method, whereas the results of our method arebetter than the results obtained by the CSD, CM, and TT methods. For

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TABLE 1 The maximum errors in evaluating the pth derivativesof f (x) = sin(x) for p = 1, 2, 3, 4, and m = 8, 16, 32, 64

m = 8 16 32 64

p = 1 3�0 × 10−9 3�0 × 10−14 1�6 × 10−13 4�2 × 10−13

2 1�6 × 10−7 4�3 × 10−12 7�3 × 10−11 5�5 × 10−10

3 4�9 × 10−6 5�0 × 10−10 2�5 × 10−8 1�1 × 10−6

4 1�0 × 10−4 3�4 × 10−8 6�5 × 10−6 1�4 × 10−3

TABLE 2 The maximum errors in evaluating the pth derivatives off (x) = sin(x) for p = 1, 2, 3, 4 using different methods with m = 64

CSD CM BB TT MCSD

p = 1 1�9 × 10−13 2�8 × 10−11 6�4 × 10−14 1�8 × 10−13 8�8 × 10−14

2 2�4 × 10−10 4�8 × 10−8 4�8 × 10−10 1�1 × 10−9 1�5 × 10−11

3 1�7 × 10−7 4�3 × 10−5 4�0 × 10−7 2�7 × 10−6 2�6 × 10−8

4 8�4 × 10−5 2�6 × 10−2 8�0 × 10−4 1�9 × 10−3 1�4 × 10−7

f (x) = sin(8x)(x+1�1)3/2 , the results of our method are about 5 significant digits

better than the ones obtained by the MCSD and BB methods, see Table 2.Noting that these methods give the best approximations among otherslisted above, our method has approximately the same accuracy as the bestmethods in evaluating the pth derivative of f (x) = sin(x), see Tables 3,and 4.

TABLE 3 The maximum errors in evaluating the pth derivativesof f (x) = 1

1+x2for p = 1, 2, 3, 4 and m = 8, 16, 32, 64

m = 8 16 32 64

p = 1 2�4 × 10−4 4�9 × 10−9 4�9 × 10−11 3�8 × 10−10

2 1�2 × 10−2 9�3 × 10−7 2�0 × 10−8 6�4 × 10−7

3 3�8 × 10−1 1�0 × 10−4 6�2 × 10−6 7�6 × 10−4

4 7.4 8�2 × 10−3 1�4 × 10−3 7�0 × 10−1

TABLE 4 The maximum errors in evaluating the pth derivativesof f (x) = sin(8x)

(x+1�1)3/2for p = 1, 2, 3, 4 and m = 8, 16, 32, 64

m = 8 16 32 64

1 5�1 × 10−2 5�8 × 10−8 1�6 × 10−13 4�7 × 10−13

2 2.9 1�1 × 10−5 2�5 × 10−11 7�6 × 10−10

3 1�0 × 102 1�3 × 10−3 1�2 × 10−8 1�0 × 10−6

4 2�2 × 103 1�0 × 10−1 3�4 × 10−6 1�1 × 10−3

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Computing Derivatives of Jacobi Polynomials 673

From the previous discussion, one can see that our method is efficientin evaluating the pth derivative of Jacobi polynomial, and, in some cases,it gives better approximations than some well-known methods in theliterature. Also, the matrix form of the method is fairly simple and userfriendly.

REFERENCES

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2. R. Baltensperger (2000). Improving the accuracy of the matrix differentiation method forarbitrary collocation points. Appl. Numer. Math. 33:143–149.

3. R. Barrio (2002). Rounding error for the Clenshaw and Forsythe algorithms for the evaluationof orthogonal polynomial series. J. Comput. Appl. Math. 138:185–204.

4. R. Barrio and J.M. Peña (2002). Numerical evaluation of the pth derivative of Jacobi series.Appl. Numer. Math. 43:335–357.

5. A. Bayliss, A. Class, and B. Matkowsky (1994). Roundoff error in computing derivative usingChebyshev differentiation matrix. J. Comput. Phys. 116:380–383.

6. J. Boyd (1989). Chebyshev and Fourier Spectral Methods. Lecture Notes in Engineering, Vol. 49.Springer Verlag, New York.

7. K. Breuer and R. Everson (1992). On the errors incurred calculating derivative using Chebyshevpolynomials J. Comput. Phys. 99:56–57.

8. W. Don and A. Solomonoff (1995). Accuracy and speed in computing the Chebyshevcollocation derivatives. SIAM J. Sci. Comput. 16:1253–1268.

9. G. Farin (1996). Curves and Surfaces for Computer Aided Geometric Design. Academic Press, Boston.10. R.T. Farouki (2000). Legendre-Bernstein basis transformations. J. Comput. Appl. Math. 119:

145–160.11. J. Hoschek and D. Lasser (1993). Fundamentals of Computer Aided Geometric Design. A.K Peters,

Wellesley, MA.12. A. Rababah (2003). Transformation of Chebyshev–Bernstein polynomial basis. Comput. Meth.

Appl. Math. 3(4):608–622.13. A. Rababah (2004). Jacobi-Bernstein basis transfoprmation. Comput. Meth. Appl. Math. 4(2):

206–214.14. G. Szegö (1975). Orthogonal Polynomials. 4th ed. American Mathematical Society, Providence, RI.15. T. Tang and M. Trummer (1996). Boundary layer resolving pseudospectral methods for singular

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