Applied Mathematical Sciences, Vol. 6, 2012, no. 48, 2357 - 2368
On Powers of Tridiagonal Matrices with Nonnegative Entries Qassem M. Al-Hassan Department of Mathematics University of Sharjah Sharjah, United Arab Emirates
[email protected] Abstract In this paper, a method for calculating powers of tridiagonal matrices with nonnegative entries is introduced. This method employs the close relationship among tridiagonal matrices, second-order linear homogeneous difference equations, and orthogonal polynomials. Some examples are included to demonstrate the implementation of the method.
Mathematics Subject Classification: 65Q05; 39A05 Keywords: difference equations; tridiagonal matrices; orthogonal polynomials; powers and inverses
1
Introduction
Tridiagonal matrices have recently attracted the attention of many researchers. This was because such matrices arise in many applications, such as boundary value problems, parallel computing, telecommunication system analysis, interpolation with splines, solution of differential equations when the method of finite differences is used. Therefore, the computation of inverses and powers of tridiagonal matrices is needed for the solution of problems that arise in these applications. A quite large number of publications that address this subject have appeared recently. Some of these papers discuss inversion of these matrices [1,2,4,5,6,10,11], or inversion of certain specific types of them [7]. Some other papers discuss powers of certain specific types of tridiagonal matrices [8,9,12,13,14,15], and some discuss both inversion and powers of these matrices [3].
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Qassem M. Al-Hassan
This present work starts with converting the general tridiagonal matrix into a symmetric matrix, and then operates on this symmetric matrix to obtain inverse and/or powers of the general tridiagonal matrix. This paper uses a second-order linear homogeneous difference equation in order to generate a set of orthogonal polynomials up to degree n, when the size of the matrix is n × n. As a matter of fact, for i = 1, ..., n, each polynomial of degree i is the determinant of the i × i principal submatrix of the n × n matrix given in (2). The set of polynomials p0 , p1 , ..., pn along with the roots of pn play an important rule in constructing the set of eigenvalues and the eigenvectors of the tridiagonal matrix T in (6). This matrix is then converted into a similar symmetric matrix J in theorem 1, this matrix has the same eigenvalues as T. The matrix that contains the eigenvectors of J is then constructed using theorem 2. The powers of T are then computed using the eigendecomposition of the symmetric matrix J and the conversion mentioned above using equation (8). This author is aware that reference [3] is performing the same as this paper, but it is worth to mention that [3] works on a general tridiagonal matrix with distinct eigenvalues, and then derives formulas for the entries of the inverse or powers of the matrix, these formulas involve derivatives of pn , and pn−1 of the orthogonal polynomials, using the Christoffel-Darboux identity. This paper is organized as follows: Section 2 contains preliminaries of a mathematical introduction of the method. Section 3 presents the mathematical derivation of the algorithm of this method. Section 4 introduces the algorithm and its practical implementation. Section 5 contains examples that employ the algorithm in calculating powers (positive and/or negative) of some tridiagonal matrices with nonnegative entries.
2
Preliminaries
It is known that tridiagonal matrices, orthogonal polynomials, and second-order linear homogeneous difference equations, are very much closely related. this relation is a consequence of the fact that the following second-order linear homogeneous difference equation x−an pn−1 (x) − cbnn pn−2(x), n ≥ 2 bn 1 , where conditions: p0 (x) = 1, p1 (x) = x−a b1
pn (x) =
(1)
an ≥ 0, bn > 0, cn > 0, f or all with initial n ≥ 0,and c1 = bn = 1, defines the recursion relation for the set of orthogonal polynomials {pn (x)}n≥0 on an open interval (a, b) with respect to a nonnegative weight function w(t ). This is also linked with tridiagonal matrices by the fact that pn (x) = det(C), where C is the n × n tridiagonal matrix:
2359
On powers of tridiagonal matrices
⎡
x−a1 b1 c2 b2
⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎣ ...
1 x−a2 b2 c3 b3
...
0 1 x−a3 b3
... 0... 1... ...
... cn 0 bn Equation (1) can be rewritten as:
⎤
0 0 0 ...
⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(2)
x−an bn x p (x) bn n−1
which ⎡ 1 translates in 0 b1 ⎢ 0 1 0 ⎢ b2 ⎢ x. ⎢ 0 0 b13 0 ⎢ ⎣ ... ... ... 0 (3)
matrix ⎤ ⎡form to: 0 p0 (x) ⎥ ⎢ 0 ⎥ ⎢ p1 (x) ⎥⎢ . ⎥⎢ ⎥⎢ ⎦ ⎣ . ... 1 pn−1 (x) bn
⎤
⎡
⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎦ ⎣
a1 b1 c2 b2
=
an p (x) bn n−1
... 0 a2 0... 0 b2 c3 a3 0 b3 b3 1... 0 ... ... ... ... ... 0 cbnn abnn 1
0 1
+
cn p (x) bn n−2
⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣
p0 (x) p1 (x) . . pn−1 (x)
+ pn (x) ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+⎢ ⎥ ⎢ ⎦ ⎣
0 0 . . pn (x)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
or x.D.p = A.p + eTn .pn (x)
⎡
⎢ ⎢ ⎢ where D = Diag( b11 , b12 , ..., b1n ), and p = [p0 (x), p1 (x), ..., pn−1 (x)]T , A = ⎢ ⎢ ⎣
(4) 1 0 ... 0 a2 1 0... 0 b2 c3 a3 0 b3 b3 1... 0 ... ... ... ... ... 0 cbnn abnn a1 b1 c2 b2
and en = [0, 0, ..., 1]T ∈ Rn . It is known that when {pn (x)}n≥0 is a set of orthogonal polynomials on an interval [a, b] with respect to a positive weight function w(x), each polynomial pj (x), 0 ≤ j ≤ n,has j distinct real roots, all contained in (a, b). So, when x = xj in (4), where xj is a root of pn (x), equation (4) becomes: xj .D.p(xj )= A.p(xj ) (5) Therefore, xj , j = 1, ..., n,is the solution of the generalized eigenvalue problem: λ.Du = Au which is equivalent to the eigenvalue problem: (D−1 A)u = λu matrix, where D−1 A is the tridiagonal ⎡ ⎤ a1 b1 0 ... 0 ⎢ c a b 0 ⎥ ⎢ 2 2 2 ⎥ ⎢ ⎥ −1 (6) D A = T = ⎢ 0 c3 a3 b3 ⎥ ⎢ ⎥ ⎣ ... ... ... ... ... ⎦ cn an
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦
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Thus, xj , j = 1, ..., n, are the eigenvalues of the tridiagonal matrix T, and the vector [p0 (xj ), p1 (xj ), ..., pn−1 (xj )]T is the corresponding eigenvector. Furthermore, ⎡ the n × n matrix ⎤ p0 (x2 ) . . p0 (xn ) p0 (x1 ) ⎢ p (x ) p1 (x2 ) . . p1 (xn ) ⎥ ⎢ 1 1 ⎥ ⎢ ⎥ B =⎢ . . . . . ⎥ ⎢ ⎥ ⎣ ⎦ . . . . . pn−1 (x1 ) pn−1 (x2 ) . . pn−1 (xn ) whose columns are the eigenvectors of the matrix T, is nonsingular, because the matrix T has n distinct real eigenvalues, namely x1 , x2 , ..., xn . Thus the eigenvectors are linearly independent.
3
Mathematical Derivation
Given the tridiagonal matrix T, this matrix can be transformed to a symmetric tridiagonal matrix. This is done in the following theorem: Theorem 1 Given the tridiagonal matrix T, and let D1 = Diag(γ 1 , γ 2 , ..., γ n ) be a diagn onal matrix , where the sequence {γ i }i=1 is generated using the recursion relation: ci+2 ...cn .i = n − 1, n − 2, ..., 1. Then the matrix J = D1 .T.D1−1 γ n = 1,and γ i = cbii+1 bi+1 ...bn−1 is the symmetric tridiagonal matrix: ⎡
⎤ √ a1 b1 c2 0 ... 0 √ ⎢ √b c ⎥ a2 b2 c3 0 0 1 2 ⎢ ⎥ √ ⎢ ⎥ J =⎢ 0 b2 c3 a3 ... ⎥ ⎢ ⎥ ⎣ bn−1 cn ⎦ ... 0 bn−1 cn an Proof. The proof is omitted, because it is straight forward calculation. Remark 2 Since matrices T and J are similar, x1 , x2 , ..., xn are the eigenvalues of both matrices T and J .
Theorem 3 If pj = [p0 (xj ), p1 (xj ), ..., pn−1 (xj )]T is an eigenvector of the matrix T that corresponds to the eigenvalue xj , then D1 .pj is an eigenvector of the matrix J that corresponds to the same eigenvalue xj .
On powers of tridiagonal matrices
2361
Proof. We have T.pj = xj .pj , and this yields xj .pj = (D−1 1 .J.D1 ).pj , which implies that J.[D1 .pj ] = xj .[D1 .pj ]. It is known that any n×n symmetric matrix is diagonalizable, and has real eigenvalues. Furthermore, since the symmetric matrix J has n distinct eigenvalues, J is orthogonally diagonalizable, in the sense that there exists an orthogonal matrix U- That is UT = U−1 such that J = U.D2 .UT , where U is an orthogonal matrix, and D2 is a diagonal matrix with the distinct eigenvalues of J on its main diagonal. The columns of the matrix U are the normalized eigenvectors of J. Therefore, the columns of U are the vectors D1 .pj , j = 1, ..., n after being normalized.
4
The Algorithm
Given an n × n tridiagonal matrix T with nonnegative entries as in (6), the first step is to use the recursion relation in (1) to generate the polynomials p0 (x), p1 (x), ..., pn−1 (x), pn (x). Then we use a rootfinding technique to compute the n distinct real roots of pn (x), such as the bisection method or Newton’s method. Suppose that the roots are x1 , x2 , ..., xn . These are the eigenvalues of T. For each xj , j = 1, ..., n, the corresponding eigenvector is the vector pj = [p0 (xj ), p1 (xj ), ..., pn−1 (xj )]T . The next step is to construct the orthogonal matrix U, and this is done by assigning the vector D1 . pj = D1 .[p0 (xj ), p1 (xj ), ..., pn−1 (xj )]T = [γ 1 p0 (xj ), γ 2 p1 (xj ), ..., γ n pn−1 (xj )]T to the j th column of U, j = 1, 2, ..., n. Then each column is normalized. So, the matrix U will be as follows: ⎡ γ 1 p0,1 ⎤ γ 1 p0,2 γ 1 p0,n ... ... N1 N2 Nn γ 2 p1,2 γ 2 p1,n ⎢ γ 2 p1,1 ⎥ ... ... ⎢ ⎥ N1 N2 Nn ⎢ ⎥ U =⎢ ... ... ... ... ... ⎥ ⎢ ⎥ ⎣ ⎦ ... ... ... ... ... γ n pn−1,1 γ n pn−1,2 γ n pn−1,n ... ... N1 N2 Nn Where pi,j = pi (xj ), i = 0, 1, ..., n − 1, j = 1, 2, ..., n,and
n γ 2i p2i−1,j , j = 1, ..., n. Nj = i=1
T Now, employing the two equations: T = D−1 1 .J.D1 and J = U.D2 .U , we get −1 T = (D1 .U).D2 .(UT .D1 ). (7) To calculate positive powers of the matrix T, equation (7) is used as follows: 2 −1 k T k T (8) T2 = (D−1 1 .U).(D2 ).(U .D1 ),and T = (D1 .U).D2 .(U .D1 ), k ≥ 2 When the matrix T is nonsingular, equation(8) can be used for negative values of k as well.
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Whence the matrix U is constructed, the following steps are executed to compute T , k ≥ 2. ⎡ p0,1 p0,2 p0,n ⎤ ... ... N1 N2 Nn p1,2 p1,n ⎥ ⎢ p1,1 ... ... ⎥ ⎢ N1 N2 Nn ⎥ ⎢ pi−1,j −1 1. D1 .U = ⎢ ... ... ... ... ... ⎥ , thus D−1 1 .U = (uij ) = ( Nj ), i, j = ⎥ ⎢ ⎣ ... ... ... ... ... ⎦ pn−1,1 pn−1,2 p ... ... n−1,n N1 N2 Nn 1, 2, ..., n ⎤ ⎡ k
⎢ ⎢ ⎢ T 2. U .D1 = ⎢ ⎢ ⎢ ⎣ 1, ..., n
γ 21 p0,1 N1 γ 21 p0,2 N2
γ 22 p1,1 N1 γ 22 p1,2 N2
γ 21 p0,n Nn
γ 22 p1,n Nn
... ...
... ...
γ 2n pn−1,1 N1 γ 2n pn−1,2 N2
... ... ... ... ... ...
... ... ... ...
... ...
γ 2n pn−1,n Nn
⎥ ⎥ ⎥ ⎥ , thus UT .D1 = (vij ) = (uji .γ 2j ), i, j = ⎥ ⎥ ⎦
3. D2 = Diagonal(x1 , x2 , ..., xn ) n 4. T = (tij ) = uik .xk .vkj , i, j = 1, ..., n k=1
=
n
=
xk .(
k=1 n k=1
5. For m ≥ 2, Tm ==
pi−1,k p ).( j−1,k γ 2j ), i, j Nk Nk
xk .( n k=1
= 1, ..., n
(pi−1,k ).(pj−1,k ) )γ 2j , i, j Nk2
(xk )m .(
= 1, ..., n
(pi−1,k ).(pj−1,k ) )γ 2j , i, j Nk2
= 1, ..., n
6. When x1 , x2 , ..., xn are all nonzero, then the matrix T is nonsingular, and in this case we have, for m ≥ 1, n (p ).(p ) −m T == (xk )−m .( i−1,kN 2 j−1,k )γ 2j , i, j = 1, ..., n k=1
5
k
Examples 1. Legendre polynomials
(i) These are orthogonal on [-1,1] with respect to the weight function w(x) = 1, and their recurrence relation is: pn (x) = 2n−1 xpn−1 (x) − n−1 pn−2 (x), n ≥ 2, p0 (x) = 1, p1 (x) = x. n n
On powers of tridiagonal matrices
2363
⎡
⎤ 0 1 0 0 0 ⎢ 1 0 2 0 0 ⎥ ⎢ 3 ⎥ 3 ⎢ ⎥ (ii) T = ⎢ 0 25 0 35 0 ⎥ , ⎢ ⎥ ⎣ 0 0 37 0 47 ⎦ 0 0 0 49 0 p2 (x) = 12 (3x2 − 1), p3 (x) = 12 (5x3 − 3x), p4 (x) = 18 (35x4 − 30x2 + 3), p5 (x) = 1 (63x5 − 70x3 + 15x) 8 (iii) x1 = −0.90618, x2 = −0.538469, x3 = 0, x4 = 0.53847, x5 = 0.90618, −0.538469, 0, 0.538469, 0.90618) D2 = Diagonal(−0.90618, √ √ 1 √1 5 (iv) D1 = Diagonal( 3 , 3 , 3 , 37 , 1) ⎡ ⎤ 1.032556 1.467593 1.6 1.467593 1.032556 ⎢ −0.935681 −0.790254 0 0.790254 0.935681 ⎥ ⎢ ⎥ ⎢ ⎥ U = (v) D−1 0.755565 −0.095505 −0.8 −0.095505 0.755565 ⎢ ⎥ 1 ⎢ ⎥ ⎣ −0.517343 0.612547 0 −0.612547 0.517343 ⎦ 0.253736 −0.505587 0.6 −0.505587 0.253736 ⎤ ⎡ 0.114728 −0.311894 0.419658 −0.402378 0.253736 ⎢ 0.163066 −0.263418 −0.053059 0.476425 −0.505587 ⎥ ⎥ ⎢ ⎥ ⎢ T (vi) U .D1 = ⎢ 0.177778 0 −0.444444 0 0.6 ⎥ ⎥ ⎢ ⎣ 0.163066 0.263418 −0.053059 −0.476425 −0.505587 ⎦ 0.114728 0.311894 0.419658 0.402378 0.253736 ⎡ ⎤ 0.333333 0 0.66667 0 0 ⎢ ⎥ 0 0.6 0 0.399999 0 ⎢ ⎥ ⎢ ⎥ 2 (vii) T = ⎢ 0.133333 0 0.5238095 0 0.342857 ⎥ , ⎢ ⎥ ⎣ ⎦ 0 0.171429 0 0.511111 0 0 0 0.190476 0 0.253968 ⎡ ⎤ 0 0.6 0 0.4 0 ⎢ 0.2 0 0.57143 0 0.22857 ⎥ ⎢ ⎥ ⎢ ⎥ 3 T =⎢ 0 0.34286 0 0.4667 0 ⎥, ⎢ ⎥ ⎣ 0.057143 0 0.3333 0 0.29206 ⎦ 0 0.07619 0 0.22716 0 ⎡ ⎤ 0.2 0 0.571429 0 0.228571 ⎢ ⎥ 0 0.428571 0 0.444444 0 ⎢ ⎥ ⎢ ⎥ 4 T = ⎢ 0.114286 0 0.428571 0 0.2666667 ⎥ ⎢ ⎥ ⎣ ⎦ 0 0.190476 0 0.329806 0 0.025397 0 0.148148 0 0.129806 2. Chebyshev polynomials of the first kind (i) These are orthogonal on (-1,1) with respect to the weight function w(x) = and their recurrence relation is:
√ 1 , 1−x2
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Qassem M. Al-Hassan
pn (x) = 2xpn−1 (x) − pn−2 (x), n ≥ 2, p0(x) = 1, p1 (x) = x. ⎤ ⎡ 0 1 0 0 0 ⎢ 1 0 1 0 0 ⎥ ⎥ ⎢ 2 2 ⎥ ⎢ (ii) T = ⎢ 0 12 0 12 0 ⎥ , ⎥ ⎢ ⎣ 0 0 12 0 12 ⎦ 0 0 0 12 0 p2 (x) = 2x2 −1, p3 (x) = 4x3 −3x, p4 (x) = 8x4 −8x2 +1, p5(x) = 16x5 −20x3 +5x (iii) x1 = −0.951057, x2 = −0.587785, x3 = 0, x4 = 0.587785, x5 = 0.951057 D2 = Diagonal(−0.951057, −0.587785, 0, 0.587785, 0.951057) (iv) D1 = Diagonal( √12 , 1, 1, 1, 1) ⎤ ⎡ 0.632456 0.632456 0.632456 0.632456 0.632456 ⎢ −0.601501 −0.371748 0 0.371748 0.601501 ⎥ ⎥ ⎢ ⎥ ⎢ U = (v) D−1 0.511667 −0.19544 −0.632456 −0.19544 0.511667 ⎥ ⎢ 1 ⎥ ⎢ ⎣ −0.371748 0.601501 0 −0.601501 0.371748 ⎦ 0.19544 −0.511667 0.632456 −0.511667 0.19544 ⎤ ⎡ 0.316228 −0.601501 0.511667 −0.371748 0.19544 ⎢ 0.316228 −0.371748 −0.19544 0.601501 −0.511667 ⎥ ⎥ ⎢ ⎥ ⎢ T (vi) U .D1 = ⎢ 0.316228 0 −0.632456 0 0.632456 ⎥ ⎥ ⎢ ⎣ 0.316228 0.371748 −0.19544 −0.601501 −0.511667 ⎦ 0.316228 0.601501 0.511667 0.371748 0.19544 ⎡ ⎤ 0.5 0 0.5 0 0 ⎢ 0 0.75 0 0.25 0 ⎥ ⎢ ⎥ ⎢ ⎥ 2 (vii) T = ⎢ 0.25 0 0.5 0 0.25 ⎥ , ⎢ ⎥ ⎣ 0 0.25 0 0.25 0 ⎦ 0 0 0.25 0 0.25 ⎤ ⎡ 0 0.75 0 0.25 0 ⎢ 0.375 0 0.5 0 0.125 ⎥ ⎥ ⎢ ⎥ ⎢ 3 T =⎢ 0 0.5 0 0.0.375 0 ⎥, ⎥ ⎢ ⎣ 0.125 0 0.375 0 0.25 ⎦ 0 0.125 0 0.25 0 ⎤ ⎡ 0.375 0 0.5 0 0.125 ⎥ ⎢ 0 0.625 0 0.3125 0 ⎥ ⎢ ⎥ ⎢ 4 T = ⎢ 0.25 0 0.4375 0 0.1875 ⎥ ⎥ ⎢ ⎦ ⎣ 0 0.3125 0 0.3125 0 0.0625 0 0.1875 0 0.125 3. This is the matrix in [14] (i) The matrix represents the orthogonal polynomials:
On powers of tridiagonal matrices
2365
pn (x) = (x − an )pn−1 (x) − pn−2 (x), n ≥ 2, a1 = an = 1, aj = 0, 1 < j < n, p0 (x) = 1, p1 (x) = x − 1. ⎤ 1 1 0 0 0 ⎢ 1 0 1 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ (ii) T = ⎢ 0 1 0 1 0 ⎥ , ⎥ ⎢ ⎣ 0 0 1 0 1 ⎦ 0 0 0 1 1 2 p2 (x) = x −x−1, p3 (x) = x3 −x2 −2x+1, p4 (x) = x4 −x3 −3x2 +2x+1, p5 (x) = x5 − 2x4 − 3x3 + 6x2 + x − 2 (iii) x1 = −1.618034, x2 = −0.618034, x3 = 0.618034, x4 = 1.618034, x5 = 2 D2 = Diagonal(−1.618034, −0.618034, 0.618034, 1.618034, 2) (iv) D1 = Diagonal(1, ⎡ 1, 1, 1, 1) ⎤ 0.19544 0.371748 0.511667 0.601501 0.447214 ⎢ −0.511667 −0.601501 −0.19544 0.371748 0.447214 ⎥ ⎢ ⎥ ⎢ ⎥ U = (v) D−1 0.632456 0 −0.632456 0 0.447214 ⎢ ⎥ 1 ⎢ ⎥ ⎣ −0.511667 0.601501 −0.19544 −0.371748 0.447214 ⎦ 0.19544 −0.371748 0.511667 −0.601501 0.447214 ⎡ ⎤ 0.19544 −0.511667 0.632456 −0.511667 0.19544 ⎢ 0.371748 −0.601501 0 0.601501 −0.371748 ⎥ ⎢ ⎥ ⎢ ⎥ (vi) UT .D1 = ⎢ 0.511667 −0.19544 −0.632456 −0.19544 0.511667 ⎥ ⎢ ⎥ ⎣ 0.0.601501 0.371748 0 −0.371748 −0.601501 ⎦ 0.447214 0.447214 0.447214 0.447214 0.447214 ⎡ ⎤ ⎡ ⎤ 2 1 1 0 0 6 4 4 1 1 ⎢ 1 2 0 1 0 ⎥ ⎢ 4 6 1 4 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 4 ⎢ ⎥ 2 (vii) T = ⎢ 1 0 2 0 1 ⎥ , T = ⎢ 4 1 6 1 4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0 1 0 2 1 ⎦ ⎣ 1 4 1 6 4 ⎦ 0 0 1 1 2 1 1 4 4 6 ⎤ ⎤ ⎡ 1 ⎡ 5 1 1 1 1 1 −2 −2 2 − 34 − 34 14 2 2 4 4 ⎢ 1 −1 1 ⎢ −3 5 1 1 − 12 ⎥ − 34 − 14 ⎥ ⎥ ⎥ ⎢ 2 ⎢ 4 4 2 2 2 4 ⎥ ⎢ ⎢ 1 1 1 5 1 1 3 3 ⎥ −2 = T−1 = ⎢ − 12 21 , T − − − ⎥ ⎢ 2 2 2 4 4 ⎥ ⎥ ⎢ 1 1 ⎢ 14 4 3 41 1 1 1 ⎥ 5 ⎣ −2 2 ⎣ 4 −4 4 −2 2 ⎦ − 34 ⎦ 2 4 1 1 1 1 − 12 − 12 12 − 34 − 34 54 2 2 4 4 ⎡
4. Hermite polynomials x2
(i) These are orthogonal on [-∞,∞] with respect to the weight function w(x) = e− 2 , and their recurrence relation is: pn (x) = 2x.pn−1 (x) − 2(n − 1).pn−2 (x), n ≥ 2, p0 (x) = 1, p1(x) = 2x.
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Qassem M. Al-Hassan
⎤ 0 12 0 0 ⎥ ⎢ ⎢ 1 0 12 0 ⎥ (ii) T = ⎢ ⎥, ⎣ 0 2 0 12 ⎦ 0 0 3 0 p2 (x) = 4x2 − 2, p3 (x) = 8x3 − 12x, p4 (x) = 16x4 − 48x2 + 12 (iii) x1 = −1.650680124, x2 = −0.524647623, x3 = 0.0.524647623, x4 = 1.650680124 D2 = Diagonal(−1.650680124, −0.524647623, 0.0.524647623, 1.650680124) √ √ √ 3, 2 6, 6, 1) (iv) D1 = Diagonal(4 ⎤ ⎡ 0.030915 0.097267 0.097267 0.030915 ⎥ ⎢ ⎢ −0.102062 −0.102062 0.102062 0.102062 ⎥ U = (v) D−1 ⎥ ⎢ 1 ⎣ 0.275113 −0.087441 −0.087441 0.275113 ⎦ −0.5 0.5 −0.5 0.5 ⎡ ⎤ 1.483928 −2.44949 1.65068 −0.5 ⎢ ⎥ ⎢ 4.668828 −2.44949 −0.524648 0.5 ⎥ T (vi) U .D1 = ⎢ ⎥ ⎣ 4.668828 2.44949 −0.524648 −0.5 ⎦ 1.483928 2.44949 1.65068 0.5 ⎤ ⎤ ⎡ ⎡ 1 1 3 0 4 0 18 0 4 0 2 ⎥ ⎥ ⎢ ⎢ ⎢ 0 32 0 14 ⎥ 3 ⎢ 23 0 32 0 ⎥ , T = (vii) T2 = ⎢ ⎥ ⎥ ⎢ ⎣ 2 0 52 0 ⎦ ⎣ 0 6 0 54 ⎦ 6 0 15 0 0 6 0 32 2 ⎡ ⎤ 3 0 3 0 ⎢ 4 15 4 3 ⎥ ⎢ 0 4 0 4 ⎥ T4 = ⎢ ⎥ 0 ⎦ ⎣ 6 0 27 4 0 18 0 15 4 ⎡
5.This is an example of the matrix in [9] (i) The matrix represents the orthogonal polynomials: pn−1 (x) − 32 pn−2 (x), n ≥ 2, p0 (x) = 1, p1(x) = pn (x) = (x−1) 2 ⎡ ⎢ ⎢ 1. (ii) T = ⎢ ⎣
1 3 0 0
2 1 3 0
0 2 1 3
0 0 2 1
x−1 2
⎤ ⎥ ⎥ ⎥, ⎦
2
3
2
4
3
2
+32x+19 , p3 (x) = x −3x 8−9x+11 , p4 (x) = x −4x −12x p2 (x) = x −2x−5 4 16 (iii) x1 = −2.963358, x2 = −0.513868, x3 = 2.513868, x4 = 4.963358 −0.513868, 2.513868, 4.963358) D2 = Diagonal(−2.963358, √
(iv) D1 = Diagonal( 3 4 6 , 32 ,
3 , 1) 2
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On powers of tridiagonal matrices
⎤ 0.202354 0.327416 0.327416 0.202354 ⎥ ⎢ ⎢ −0.401001 −0.247832 0.247832 0.401001 ⎥ U = (v) D−1 ⎥ ⎢ 1 ⎣ 0.491123 −0.303531 −0.303531 0.491123 ⎦ −0.371748 0.601501 −0.601501 0.371748 ⎤ ⎡ 0.682945 −0.902251 0.736685 −0.371748 ⎥ ⎢ ⎢ 1.105028 −0.557622 −0.455296 0.601501 ⎥ T (vi) U .D1 = ⎢ ⎥ ⎣ 1.105028 0.557622 −0.455296 −0.601501 ⎦ 0.682945 0.902251 0.736685 0.371748 ⎡ ⎤ ⎡ 7 4 4 0 109 104 96 32 ⎢ ⎥ ⎢ ⎢ 6 13 4 4 ⎥ 4 ⎢ 156 253 152 96 (vii) T2 = ⎢ ⎥,T = ⎢ ⎣ 9 6 13 4 ⎦ ⎣ 216 228 253 104 0 9 6 7 108 216 156 109 ⎤ ⎡ −0.578947 0.526316 0.210526 −0.421053 ⎥ ⎢ ⎢ 0.789474 −0.263158 −0.105263 0.210526 ⎥ −1 T =⎢ ⎥, ⎣ 0.473684 −0.157895 −0.263158 0.526316 ⎦ −1.421053 0.473684 0.789474 −0.578947 ⎤ ⎡ 1.448753 −0.6759 −0.5651 0.709141 ⎥ ⎢ ⎢ −1.01385 0.601108 0.387812 −0.565097 ⎥ T−2 = ⎢ ⎥ ⎣ −1.271468 0.581717 0.601108 −0.6759 ⎦ 2.393352 −1.271468 −1.01385 1.448753 ⎡
⎤ ⎥ ⎥ ⎥ ⎦
References 1. Aiat Hadj, A. D., and Elouafi, M., A fast numerical algorithm for the inverse of a tridiagonal and pentadiagonal matrix, Applied Mathematics and Computation, vol. 202(2), 2008, pp 441-445. 2. Al-Hassan, Q., An algorithm for computing inverses of tridiagonal matrices with applications, Soochow Journal of Mathematics, vo. 31, no. 3, 2005, pp 449-466. 3. Elouafi, M., and Aiat Hadj, A. D., On the powers and the inverse of a tridiagonal matrix, Applied Mathematics and Computation, vol. 211, 2009, pp 137-141. 4. Elouafi, M., and Aiat Hadj, A. D., A new recursive algorithm for inverting Hessenberg matrices, Applied Mathematics and Computation, 214(2009), pp497-499.
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5. El- Mikkkawy, M., On the inverse of a general tridiagonal matrix, Applied Mathematics and Computation,150(2004), pp 669-679. 6. El- Mikkkawy, M., and Karawia, A., Inversion of general tridiagonal matrices, Applied Mathematics Letters, 19(2006), pp712-720. 7. El-Shehawey, M. A., El-Shreef, Gh. A., and Al-Henawy, A. Sh., Analytical inversion of general periodic tridiagonal matrices, Journal of Mathematical Analysis and Applications, 345(2008), pp123-134. 8. Gutierrez-Gutierrez, J., Positive integer powers of certain tridiagonal matrices, Applied Mathematics and Computation, 202(2008), pp133-140. 9. Gutierrez-Gutierrez, J., Powers of tridiagonal matrices with constant diagonals, Applied Mathematics and Computation, 206(2008), pp885-891. 10. Hou-Biao Li, Ting-Zhu Huang, Xing-Ping Liu, and Hong Li, On the inverses of general tridiagonal matrices, Linear Algebra and Its Applications, 433(2010), pp 965-983. 11. Kilic, Emrah, Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions, Applied Mathematics and Computation, 197(2008), pp 345-357. 12. Rimas, Jonas, On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order-I, Applied Mathematics and Computation, 168(2005), pp 783-787. 13. Rimas, Jonas, On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of odd order-I, Applied Mathematics and Computation, 171(2005), pp 1214-1217. 14. Rimas, Jonas, On computing of arbitrary positive integer powers for tridiagonal matrices with elements 1, 0,0,...,0,1 in principal and 1,1,...1 in neighbouring diagonalsII, Applied Mathematics and Computation, 187(2007), pp 1472-1475. 15. Rimas, Jonas, On computing of positive integer powers for tridiagonal matrices with elements -1, 0,0,...,0,1 in principal and 1,1,...1 in neighbouring diagonals-II, Applied Mathematics and Computation, 188(2007), pp 2020-2024. Received: November, 2011
