Differentiation matrices in polynomial bases

Transcription

1 Math Sci () 5 DOI /s9---x ORIGINAL RESEARCH Differentiation matrices in polynomial bases A Amiraslani Received January 5 / Accepted April / Published online April The Author(s) This article is published with open access at Springerlinkcom Abstract Explicit differentiation matrices in various polynomial bases are presented in this work The idea is to avoid any change of basis in the process of polynomial differentiation This article concerns both degree-graded polynomial bases such as orthogonal bases, and non-degree-graded polynomial bases including the Lagrange and Bernstein bases Keywords Polynomial interpolation Polynomial bases Differentiation Mathematics Subject Classification Introduction A5 A Consider a scalar (complex) polynomial P(x) of degree n and a basis given by fb ðxþ B ðxþ B n ðxþ B n ðxþg This basis determines the representation PðxÞ ¼ P n j¼ a jb j ðxþ, where a j C and, regardless of the basis, the coefficient of x n in the expression is nonzero Alternatively, this polynomial can be written as & A Amiraslani aamirasl@hawaiiedu STEM Department, University of Hawaii-Maui College, W Kaahumanu Ave, Kahului, HI 9, USA PðxÞ ¼½a a a n a n B ðxþ B ðxþ I ðþ B n ðxþ 5 B n ðxþ where I is the unit matrix of size n þ We want to find a matrix D, the differentiation matrix, of size n þ such that the kth order derivative of P(x), shown by dk PðxÞ or P ðkþ ðxþ, can be written as dx k B ðxþ B ðxþ P ðkþ ðxþ ¼½a a a n a n D k B n ðxþ 5 B n ðxþ ðþ For a polynomial of degree n, D has to be a nilpotent matrix of degree n þ D has a well-known structure and can be easily found for the monomial basis For convenience, let us assume n ¼ 5 and the generalizations for all positive n will be clear If x x PðxÞ ¼½a a a a a a 5 I x ðþ 5 then x x 5

2 Math Sci () 5 ðþ 5 5 Differentiation in bases other than the monomial basis has been occasionally studied One of the most important applications of polynomial differentiation in other bases is in spectral methods like collocation method [] Differentiation matrices for Chebyshev and Jacobi polynomials were computed [] The differentiation of Jacobi polynomials through Bernstein basis was studied [] Chirikalov [] computed the differentiation matrix for the Hermite basis In this paper, we present explicit formulas for D in different polynomial bases Constructing D is fairly straightforward and having D, we can easily find derivatives of higher order by raising D to higher powers accordingly Another important advantage of having a formula for D in a basis is that we do not need to change the basis often to the monomial basis to differentiate P(x) Conversion between bases has been exhaustively studied in [], but it can be unstable [] Section Degree-graded bases of this paper considers degree-graded bases and finds D in general for them Orthogonal bases are all among degree-graded bases Section Degree-graded bases then discusses other important special cases such as the monomial and Newton bases as well as the Hermite basis Section Bernstein basis and Lagrange basis concern the Bernstein and Lagrange bases, respectively, and find D for them The Bernstein (Bézier) basis and the Lagrange basis are most useful in computer-aided geometric design (see [5], for example) For some problems in partial differential equations with symmetries in the boundary conditions Legendre polynomials can be successfully used the most natural Finally, in approximation theory, Chebyshev polynomials have a special place due to their minimumnorm property (see eg, []) Degree-graded bases Real polynomials f/ n ðxþg n¼ with / nðxþ of degree n which are orthonormal on an interval of the real line (with respect to some nonnegative weight function) necessarily satisfy a three-term recurrence relation (see Chapter of [], for example) These relations can be written in the form x/ j ðxþ ¼a j / jþ ðxþþb j / j ðxþþc j / j ðxþ j ¼ ðþ where the a j b j c j are real, a j ¼, / ðxþ, / ðxþ The choices of coefficients a j b j c j defining three wellknown sets of orthogonal polynomials (associated with the names of Chebyshev and Legendre) are summarized in Table Orthogonal polynomials have well-established significance in mathematical physics and numerical analysis (see eg, []) More generally, any sequence of polynomials f/ j ðxþg j¼ with / jðxþ of degree j is said to be degreegraded and obviously forms a linearly independent set but is not necessarily orthogonal A scalar polynomial of degree n can now be written in terms of a set of degree-graded polynomials PðxÞ ¼ P n j¼ a j/ j ðxþ, where a j C and a n ¼ We can then write / ðxþ / ðxþ PðxÞ ¼½a a a n a n I ðþ / n ðxþ 5 / n ðxþ Table Three well-known orthogonal polynomials Polynomial T n ðxþ P n ðxþ U n ðxþ Name of polynomial Chebyshev (st kind) Legendre (Spherical) Chebyshev (nd kind) Weight function ð x Þ ð x Þ Orthogonality interval ½ ½ ½ Leading coefficient k n n ðnþ! n n ðn!þ a n for n ¼ otherwise nþ nþ b n c n n nþ

3 Math Sci () 5 9 Lemma If P(x) is given by (), then / ðxþ / ðxþ P ðkþ ðxþ ¼½a a a n a n D k / n ðxþ 5 / n ðxþ ðþ where Q 5 ðþ Q is a size n lower triangular matrix that has the following structure for i ¼ n Given that / ðxþ ¼ and / ðxþ does not appear in the system, we can eliminate them and rewrite the above system as / ðxþ / ðxþ / ðxþ / ðxþ 5 ¼ H / ðxþ / ðxþ 5 ðþ / ðxþ / ðxþ where a b x a H ¼ c b x a 5 ð9þ c b x a Since a i ¼, H exists and is a lower triangular matrix that has the following row structure in general for i ¼ n i >< a q ij ¼ i > ððb a j b i Þq ij þ a j q ij þ c j q ijþ c i q ij Þ i i ¼ j i [ j ð5þ Any entry, q, with a negative or zero index is set to in the above formula Proof We start by differentiating () to get / j ðxþ ¼a j / jþ ðxþþðb j xþ/ j ðxþþc j/ jðxþ j ¼ ðþ We can write this equation in a matrix-vector form Without loss of generality, we assume n ¼ / ðxþ / ðxþ / ðxþ 5 / ðxþ 5 / ðxþ a / ðxþ c b x a / ðxþ ¼ c b x a / ðxþ c b x a 5 / ðxþ 5 / ðxþ ðþ h ðþ ij >< ¼ > a i j ¼ i x b j ðh ðþ ijþ a Þc jþ ðh ðþ ijþþ j ¼ðiÞ j a j ðþ Now / ðxþ / ðxþ / ðxþ / ðxþ 5 ¼ / ðxþ H / ðxþ 5 ðþ / ðxþ / ðxþ It is obvious from () that the variable, x, appears in entries of H Through a fairly straightforward, but rather tedious process and using () to eliminate x from H, we get Q as given by (5), and we have / ðxþ / ðxþ / ðxþ / ðxþ 5 ¼ Q / ðxþ / ðxþ 5 ðþ / ðxþ / ðxþ

4 5 Math Sci () 5 From there, adding a row and a column, we find D and have / ðxþ / / ðxþ ðxþ / ðxþ / ðxþ ¼ / ðxþ ðþ / ðxþ Q 5 5 / ðxþ 5 / ðxþ / ðxþ Using these results, we can write the first derivative of a generic first kind Chebyshev polynomial of degree (see Table ) as P ðxþ ¼½a a a a a T ðxþ T ðxþ T ðxþ 5 T ðxþ 5 T ðxþ ðþ Similarly, for a second kind Chebyshev polynomial of degree, we can write the first derivative as P ðxþ ¼½a a a a a P ðxþ ¼½a a a a a U ðxþ U ðxþ U ðxþ 5 U ðxþ 5 U ðxþ ð5þ P ðxþ P ðxþ P ðxþ 5 5 P ðxþ 5 P ðxþ ðþ Special degree-graded bases As mentioned above, the family of degree-graded polynomials with recurrence relations of the form () include all the orthogonal bases, but are not limited to them Here, we discuss some of the famous non-orthogonal bases of this kind and, consequently, for which we find the differentiation matrix, D, formulas In particular, if in (), we let a j ¼ and b j ¼ c j ¼, it will become the monomial basis Using () and (5), we can easily verify that in this case, D has a form like () Another important basis of this kind is the Newton basis Let a polynomial P(x) be specified by the data f z j P j g n j¼ where the z j s are distinct If the Newton polynomials are defined by setting N ðxþ ¼ and, for k ¼ n N k ðxþ ¼ Yk ðx z j Þ then j¼ PðxÞ ¼½a a a n a n N ðxþ N ðxþ I N n ðxþ 5 N n ðxþ ðþ ðþ For j ¼ n, the a j s can be found by divided differences as follows a j ¼½P P P j where we have ½P j ¼P j, and ð9þ ½P i P iþj ¼ ½P iþ P iþj ½P i P iþj ðþ z iþj z i If in (), we let a j ¼, b j ¼ z j and c j ¼, it will become the Newton basis For n ¼, D, as given by (), has the following form z z ðz z Þðz z Þ z þ z þ z 5 ðz z Þðz z Þðz z Þ ðz z Þðz z þ z Þþðz z Þðz z Þ z þ z þ z þ z ðþ

5 Math Sci () 5 5 The confluent case Suppose that a polynomial P(x) ofdegreen as well as its derivatives are sampled at k nodes, ie, distinct (finite) points z z z k WewriteP j ¼ Pðz j Þ P j ¼ P ðz j Þ P ðs jþ j ¼ P ðsjþ ðz j Þ j ¼ k Here s ¼ðs s s k Þ shows the confluencies (ie, the orders of the derivatives) associated with the nodes and we have P k i¼ s i ¼ n þ k If for j ¼ k, all s j ¼, then k ¼ n þ and we have the Lagrange interpolation This is an interesting polynomial interpolation that deserves a better consideration the Hermite interpolation (See eg, [, ]) It is basically similar to the Lagrange interpolation, but at each node, we have the value of P(x) as well as its derivatives up to a certain order Now, we assume that at each node, z j, we have the value and the derivatives of P(x) up to the s j th order The nodes at which the derivatives are given are treated as extra nodes In fact we pretend that we have s j þ nodes, z j, at which the value is P j and remember that P k i¼ s i ¼ n þ k In fact, the first s þ nodes are z, the next s þ nodes are z and so on Using the divided differences technique, as given by (), to find a j s, whenever we get ½P j P j P j where P j is repeated m times, we have ½P j P j P j ¼ PðmÞ j ðm Þ! ðþ and all the values P j to Pðs jþ j for j ¼ k are given For more details see eg, [] The bottom line is that the Hermite basis can be seen as a special case of the Newton basis, thus a degree-graded basis For the Hermite basis, like the Newton basis, a j ¼, b j ¼ z i, and c j ¼, but some of the b j s are repeated Other than that, the differentiation matrix, D, can be similarly found for the Hermite basis For a data set like fðz Pðz ÞÞ ðz P ðz ÞÞ ðz Pðz ÞÞ ðz Pðz ÞÞ ðz Pðz Þg, b ¼ z, we have b ¼ z, b ¼ z, b ¼ z, and b ¼ z In this case, () becomes z þ z 5 ðz z Þðz z Þ z þ z þ z ðþ Bernstein basis A Bernstein polynomial (also called Bézier polynomial) defined over the interval [a, b] has the form b j ðxþ ¼ n ðx aþ j ðb xþ nj j ðb aþ n j ¼ n ðþ This is not a typical scaling of the Bernstein polynomials however, this scaling makes matrix notations related to this basis slightly easier to write The Bernstein polynomials are nonnegative in [a, b], ie, b j ðxþ for all x ½a b (j ¼ n) Bernstein polynomials are widely used in computer-aided geometric design (eg, see []) A polynomial P(x) written in the Bernstein basis is of the form b ðxþ b ðxþ PðxÞ ¼½a a a n a n I ðþ b n ðxþ 5 b n ðxþ where the a j s are sometimes called the Bézier coefficients (j ¼ n) Lemma If P(x) is given by (), then b ðxþ P ðkþ ðxþ ¼½a a a n b ðxþ a n D k b n ðxþ 5 b n ðxþ ðþ where D is a size n þ tridiagonal matrix that has the following structure for i ¼ n þ n ði Þ i ¼ j >< i d ij ¼ i ¼ j ðþ > ðn i þ Þ i ¼ j þ Proof A little computation using () shows that b kðxþ ¼k b n k kðxþþ b kðxþþ n k b kþðxþ ð5þ

6 5 Math Sci () 5 for k ¼ n Any b i ðxþ with either a negative or larger than n index is set to in (5) This is why D is tridiagonal and form here, it is easy to derive () for D For n ¼, the differentiation matrix is as follows Lagrange basis 5 ðþ Lagrange polynomial interpolation is traditionally viewed as a tool for theoretical analysis however, recent work reveals several advantages to computation using new polynomial interpolation techniques in the Lagrange basis (see eg, [, 9]) Suppose that a polynomial P(x) of degree n is sampled at n þ distinct points z z z n, and write p j ¼ Pðz j Þ Lagrange polynomials are defined by L j ðxþ ¼ ðxþw j j ¼ n ðþ x z j where the weights w j are w j ¼ and Yn m¼ m¼j z j z m h ðþ ðxþ ¼ Yn m¼ ðx z m Þ ðþ Then P(x) can be expressed in terms of its samples in the following form L ðxþ PðxÞ ¼½p p p n L ðxþ p n I L n ðxþ 5 ðþ L n ðxþ Lemma If P(x) is given by (), then L ðxþ P ðkþ ðxþ ¼½p p p n L ðxþ p n D k L n ðxþ 5 L n ðxþ ð5þ where D is a size n þ matrix that has the following structure P n >< k¼k¼i i ¼ j z d ij ¼ i z k w ðþ i > w j ðz j z i Þ i ¼ j Proof A little computation using () shows that L Xn w i iðxþ ¼ w j ðz j z i Þ L jðxþþ Xn L i ðxþ z i z k j¼j¼i k¼k¼i ðþ for i ¼ n Form here, it is easy to derive () for D For n ¼, the differentiation matrix is as follows þ þ w z z z z z z w ðz z Þ w w ðz z Þ w w w ðz z Þ w ðz z Þ w w w ðz z Þ w ðz z Þ w w ðz z Þ þ þ w z z z z z z w w ðz z Þ w w ðz z Þ w ðz z Þ þ þ w z z z z z z w w ðz z Þ w ðz z Þ þ þ z z z z z z 5 ðþ h

7 Math Sci () 5 5 Concluding remarks A differentiation matrix, D, of size n þ in a certain basis is a nilpotent matrix of degree n þ As such, all of its eigenvalues are zero Assuming n ¼ 5 and the generalizations for all positive n will be clear we can fairly easily verify that the Jordan form for D in any basis is J ¼ ð5þ 5 We can write J ¼ T DT The jth column of the transformation matrix, T, is the first column of D nþj for j ¼ n þ For example, for the D given by (), we have T ¼ ð5þ 5 Now, if we consider two different polynomial bases for which the differentiation matrices are D and D, then we can write J ¼ T D T and J ¼ T D T From here it is easy to check that D ¼ðT T Þ D ðt T Þ ð5þ which shows, as expected, the differentiation matrices in different bases are similar In this paper, we have found explicit formulas for the differentiation matrix, D, in various polynomial bases The most important advantage of having D explicitly is that there is no need to go from one basis to another (normally monomial) to differentiate a polynomial in a given basis Moreover, having D, we can easily find higher order derivatives of any polynomial in its original basis One may hope that new and more efficient polynomial-related algorithms, such as root-finding methods, can be developed using D These results can be easily extended to matrix polynomials in different bases Open Access This article is distributed under the terms of the Creative Commons Attribution International License (http//crea tivecommonsorg/licenses/by//), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made References Berrut, J, Trefethen, L Barycentric Lagrange interpolation SIAM Rev (), 5 5 () Chirikalov, VI Computation of the differentiation matrix for the Hermite interpolating polynomials J Math Sci (), (99) Davis, PJ Interpolation and Approximation Blaisdell, New York (9) Farin, G Curves and Surfaces for Computer-Aided Geometric Design Acadmeic Press, San Diego (99) 5 Farouki, RT, Goodman, TNT, Sauer, T Construction of orthogonal bases for polynomials in Bernstein form on triangular simplex domains Comput Aided Geom Design, 9 () Gander, W Change of basis in polynomial interpolation Numer Linear Algebra Appl (), 9 () Gautschi, W Orthogonal Polynomials Computation and Approximation Clarendon, Oxford () Hermann, T On the stability of polynomial transformations between Taylor, Bézier Hermite forms Numer Algo, (99) 9 Higham, N The numerical stability of barycentric Lagrange interpolation IMA J Numer Anal, 5 55 () Lorentz, G, Jetter, K, Riemenschneider, SD Birkhoff Interpolation Addison Wesley Publishing Company, Boston (9) Ravlin, T Chebyshev Polynomials Wiley, New York (99) Rabbah, A, Al-Refai, M, Al-Jarrah, R Computing derivatives of Jacobi polynomials using Bernstein transformation and differentiation matirx Numer Funct Anal Optim 9(5 ), () Shen, J, Tang, T, Wang, L Spectral Methods Algorithms, Analysis and Applications Springer Series in Computational Mathematics, vol Springer, New York () Wang, L, Samson, D, Zhao, X A well-conditioned collocation method using a pseudospectral integration matrix SIAM J Sci Comput (), 9 99 ()