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Banded Matrices and Discrete SturmLiouville Eigenvalue Problems
Advances in Difference Equations volumeÂ 2009, ArticleÂ number:Â 362627 (2010)
Abstract
We consider eigenvalue problems for selfadjoint SturmLiouville difference equations of any even order. It is well known that such problems with Dirichlet boundary conditions can be transformed into an algebraic eigenvalue problem for a banded, realsymmetric matrix, and vice versa. In this article it is shown that such a transform exists for general separated, selfadjoint boundary conditions also. But the main result is an explicit procedure (algorithm) for the numerical computation of this banded, realsymmetric matrix. This construction can be used for numerical purposes, since in the recent paper by Kratz and Tentler (2008) there is given a stable and superfast algorithm to compute the eigenvalues of banded, realsymmetric matrices. Hence, the SturmLiouville problems considered here may now be treated by this algorithm.
1. Introduction
In [1] it was shown that every discrete SturmLiouville eigenvalue problem (where )
with Dirichlet boundary conditions is equivalent with an algebraic eigenvalue problem [2] for a symmetric, banded matrix with bandwidth , where and are fixed integers with (see [1, Theoremâ€‰â€‰1, Remarkâ€‰â€‰1(i)]). Note that (SL) is irrelevant for in the case of Dirichlet boundary conditions, so that in [1] equation (SL) is considered only for .
In this article we treat the SturmLiouville difference equations (SL) with general separated, selfadjoint boundary conditions. These boundary conditions include the socalled natural boundary conditions, when no or "not enough'' boundary conditions are imposed [3, page 51, equation (2.3.9)]. More precisely, given (SL) and the (imposed and natural) boundary conditions, then we show that this eigenvalue problem is equivalent with an algebraic eigenvalue problem for a realsymmetric, banded matrix with bandwidth , and we will construct this matrix explicity. For our general boundary conditions we must assume that the coefficients are unequal to zero at "the beginning and at the end'' (see (4.7) below). This leads via [4] or [5] to a numerical algorithm to compute the eigenvalues of these SturmLiouville eigenvalue problems.
Therefore the present paper is to some extent a continuation of the articles [1, 4]. The paper [4] presents superfast (i.e., with numerical operations) and stable algorithms for the computation of some of the eigenvalues for a realsymmetric and banded matrix with bandwidth , where or for the most interesting divisionfree algorithms. These algorithms are based on the bisection method, and they generalize the wellknown procedure for realsymmetric and tridiagonal matrices. As is shown in [1] and used in [4] the algebraic eigenvalue problems for realsymmetric, banded matrices with bandwidth are equivalent to eigenvalue problems for selfadjoint SturmLiouville difference equations of order with Dirichlet boundary conditions. Hence, these discrete SturmLiouville eigenvalue problems can be treated by those algorithms.
Summarizing, the main goal of this article is to provide an algorithm to calculate some of the eigenvalues of eigenvalue problems for selfadjoint SturmLiouville difference equations with general separated, selfadjoint boundary conditions (and not only for Dirichlet conditions as in [4, 5]). To be more precise, we provide a construction to transform these eigenvalue problems into such eigenvalue problems with Dirichlet boundary conditions. This construction incorporates "somehow'' the general boundary conditions into the first and last equations of the SturmLiouville difference equations. These transformations are stable (using essentially orthogonal transforms) and the required computational work depends on only (most interesting ) but not on , so that the overall combined (with [4]) algorithm remains superfast (i.e., O() numerical operations) and stable. Hence, the results of this paper are of interest mainly for numerical applications.
Because of our intention above this article, more precisely Sections 2 to 5 of it, consists essentially of the following central parts:

(i)
deriving the transform to an explicit algebraic algebraic eigenvalue problem for a symmetric, banded matrix with bandwidth ,

(ii)
providing the required formulas, so that an implementation of the construction is easily "accessible'' for the reader,

(iii)
proving that the construction is always successful under the conditions (4.4) and (4.7) (which is the contents of Theorem 6.1).
This means that Sections 2 to 5 have to be quite technical. Our proceeding in these sections provides simultaneously the construction, the derivation, and the proof that the construction always works.
As already said the asserted equivalence in Theorem 6.1 is not the crucial result of this article. Actually, under the assumptions (4.4) and (4.7) this equivalence may be shown quite easily using Lemma 2.3 and Proposition 3.2. Moreover, assumption (4.4) is necessary and sufficient for selfadjointness (see, e.g., [3, Propositionâ€‰â€‰2.1.1] or elsewhere), if (4.7) holds. The assumption (4.7) is also necessary for all considered boundary conditions simultaneously, but for a particular boundary condition it may be weakened. But this can be seen during the procedure. The equivalence via Lemma 2.3 and Proposition 3.2 directly leads in general to a banded symmetric matrix with larger bandwidth. On the other hand the "minimal bandwidth'' tridiagonal (i.e., ) can always be achieved in a stable way by the wellknown methods of Givens and Householder, but these algorithms require numerical operations as discussed in [4], which would make our whole approach obsolete.
Note that our main goal is equivalence to an "ordinary'' algebraic eigenvalue problem for a realsymmetric and banded matric rather than to some matrix pencil or generalized eigenvalue problem of the form . This is so, because the algorithm via bisection works only for these algebraic eigenvalue problems, which are well posed, while those general problems are in general not well posed. Note also that the equivalence to "some matrix pencil'' can be seen immediately from (SL) or (4.1) and also (4.2). Moreover, the reduction of separated boundary conditions to Dirichlet boundary conditions via an extension of the system to a larger interval is also well known for discrete Hamiltonian or symplectic systems (see, e.g., [6] or [7]). Hence, this reduction combined with the transformation of the SturmLiouville equations (SL) to a linear Hamiltonian system by Lemma 2.2 would also lead to a problem with Dirichlet boundary conditions but for a larger matrix, more precisely, it would lead to some matrix pencil (possible also with larger bandwidth), which cannot be treated by the algorithms of [4, 5].
The discussion of assumption (4.7) in the Concluding Remarks of Section 6 does not focus on the necessity or sufficiency of it. It is quite "natural'' to assume that the leading coefficient never vanishes, because it is the case in most applications. But the main point is that the incorporation of our general boundary conditions into the the difference equations by our construction leads in general to problems with Dirichlet boundary conditions, but where the leading coefficient may vanish for some 's at the beginning and the end. Therefore, algorithms via corresponding Hamiltonian or Riccati equations cannot be used anymore, so that the divisionfree algorithms (i.e., no divisions by ) are needed as remarked in Concluding Remarks (i) of Section 6.
Let us shortly motivate why to consider the discrete SturmLiouville eigenvalue problems of this article, particularly for and , and for general boundary conditions.

(i)
The discretization of a second order SturmLiouville equation
(1.1)of higher order leads to a banded matrix with bandwidth with , and then even Dirichlet boundary conditions lead for the discrete problem to the boundary conditions , which have to be complemented by additional "natural boundary conditions'' in the usual way. Therefore, such problems of second order with Dirichlet boundary conditions cannot be treated (at least for higherorder discretization) directly by the algorithm of [4]. By using the construction of this article we obtain faster algorithms than the known ones.

(ii)
Linear discretization of th and thorder SturmLiouville difference equations leads to bandwidths and , and the numerical treatment of Dirichlet boundary conditions via [4] requires also the construction of this article.
Let us shortly discuss the setup of this paper. In the next section we provide the formulae, which transform the difference equations (SL) into the corresponding matrix equation (see Lemma 2.1 below) based on [1]. In Section 3 we derive via partial summation the socalled Dirichlet's formula, which yields the crucial identities (3.3). In Section 4 we formulate our discrete SturmLiouville eigenvalue problems. In particular, we introduce (based on [8]) and discuss shortly the corresponding general separated, selfadjoint boundary conditions. In Section 5 we carry out our construction of the symmetric, banded matrix, so that the corresponding algebraic eigenvalue problem is equivalent with our SturmLiouville problem. Hence, our proceeding in Sections 2 to 5 provides simultaneously the construction, the derivation, and the proof that the construction always works. This is formulated as our main result in the last Section 6 by adding some concluding remarks.
2. Discrete SturmLiouville Difference Equations, Banded Matrices, and Hamiltonian Systems
Let , and let reals for and be given. Then, for , we consider the SturmLiouville difference operator defined by
where is the forward difference operator, that is, , which will always operate with respect to the variable . Then, by [1, Theoremâ€‰â€‰1], we have that
where is a symmetric, banded matrix with bandwidth , given by
for and all .
This formula yields the following.
Lemma 2.1.
For and define vectors
Then,
where is a symmetric matrix, and where
is lower triangular, and it is invertible, if for . Moreover, the equation
is equivalent with
Next, we have by [1, Lemmaâ€‰â€‰3] or [9, Remarkâ€‰â€‰2] the following.
Lemma 2.2.
For and define vectors
Then, for any , the equation
is equivalent with the Hamiltonian system
provided that , where one uses the following notation: , , , are matrices defined by
For the next lemma; see, for example, [10, formulae (â€‰6) andâ€‰â€‰(â€‰9)] for the case of constant coefficients. It follows easily from our formulae (2.4) and (2.9) by computing the finite differences via .
Lemma 2.3.
Let (correspondly ), , and be defined by (2.4) and (2.9). Then,
where is invertible with , and where , are matrices with
and is invertible, if for .
3. Dirichlet's Formula
The next lemma is a discrete version of the continuous Dirichlet's formula [3, Lemmaâ€‰â€‰8.4.3].
Lemma 3.1.
For and two sequences , let the operator , the vectors , , , , and the matrices and be defined as in the previous section by (2.1) and Lemma 2.2. Then, for any ,
Proof.
A straight forward calculation using (2.1) and (2.9) yields
which proves our assertion (3.1) using also the definition of and by Lemma 2.2.
Proposition 3.2.
The matrices , , , and from Lemmas 2.3 and 2.1 satisfy
Proof.
We consider the functional , with and defined by (2.4) of Lemma 2.1, where we put . Then, by (2.5) of Lemma 2.1 we have that
From the definition of by (2.4) and Dirichlet's formula (3.1) we obtain that
where is a symmetric matrix. Comparing this last formula (observe that , , are completely free) with (3.4) we can conclude that
because is symmetric by Lemma 2.1. This yields our assertion (3.3).
4. Discrete SturmLiouville Eigenvalue Problems with Separated, SelfAdjoint Boundary Conditions
Let integers with (see (5.4) below) and real coefficients be given. Then, we consider the following discrete eigenvalue problem, which we will denote by . It consists of the selfadjoint SturmLiouville difference equations of even order (see (SL), Lemmas 2.1 and 2.2 above):
and it consists of the linearly independent, separated, and selfadjoint boundary conditions
at the beginning, and
at the end, where , , , are defined by (2.9) of Lemma 2.2, and where the real matrices , , , satisfy the following conditions:
By Lemma 2.3, (4.2) and (4.3) lead to the following equivalent conditions on , , , , defined by (2.4), that is, on and on , , respectively:
where , , and where , , are defined by Lemma 2.3. Moreover, the equivalence of (4.2), (4.3) with (4.5), (4.6) requires the assumption that and are invertible, which means by Lemma 2.3 that
We assume this from now on.
The selfadjointness of follows from general theory of linear Hamiltonian difference systems [11], and from the equivalence of our difference equation (4.1) with such systems, which is stated in Lemma 2.2. In addition, the selfadjointness of the boundary conditions via the assumption (4.4) is stated or discussed, for example, in [3, Definitionâ€‰â€‰2.1.2], [8, Remarkâ€‰â€‰2(iii)], [12, Propositionâ€‰â€‰2], or [7, Definitionâ€‰â€‰1].
5. Construction of the Symmetric, Banded Matrix
First, by Lemma 2.1, the SturmLiouville difference equations (4.1) may be written in matrix notation, namely,
where the coefficient matrix is of the form
which is , and where
and where we have written the first and the last rows of in blocked form according to (2.4) and (2.5) of Lemma 2.1. It is the aim of this section to incorporate the boundary conditions (4.2) or (4.5) and (4.3) or (4.6) into the first and the last equations of (4.1), that is, the first and last block rows of , respectively. This requires in general that
As a result we will obtain an algebraic eigenvalue problem for a symmetric, banded matrix of size with integers depending on the boundary conditions. This algebraic eigenvalue problem will be equivalent with our given SturmLiouville eigenvalue problem from Section 4 under the assumptions (4.4) and (4.7).
5.1. Boundary Conditions at the Beginning
We consider the boundary conditions (4.2) or (4.5) at the beginning by assuming (4.4) and (4.7). Hence, and are invertible by Lemmas 2.1 and 2.3. Let , be real matrices such that (see (4.5))
where , The existence (including construction) follows from [3, Corollaryâ€‰â€‰3.3.9], because
holds. This follows from (4.4) and (4.5) by the calculations
is symmetric, where we used (3.3) of Proposition 3.2. Moreover, (C1) and (5.5) imply by [3, Propositionâ€‰â€‰1.1.5] that
We conclude from (C1) and (5.7) that our boundary conditions (4.2) or (4.5) are equivalent with
Since is realsymmetric by (5.7), there exists by the spectral theorem [13] an orthogonal matrix such that
where is diagonal and invertible, so that
We use this block structure from now on including the extreme cases and , where the zeromatrices or do not occur. By the GramSchmidt process (or QRfactorization [13]), there exists an orthogonal matrix such that is lower triangular, where with the blockstructure above, that is, . Then is orthogonal, and
where is symmetric and invertible. Let
By (5.5) and (C1), , and
is symmetric. Hence, , because is invertible, and , so that is invertible.
Altogether, we have constructed such that
with a symmetric and invertible matrix , where ,
with an invertible matrix and so that is symmetric, and
so that is lower triangular, and where is orthogonal.
It follows immediately from (C1) and (C2) that our boundary conditions (4.2) or (4.5) are equivalent with
and therefore with
where
with while and remain free.
We say that (5.18) is the boundary conditions in normalized form.
Now, we consider the firstequations or the first block row of our difference equations (4.1) or (5.1), that is, by Lemma 2.1 and (5.2),
We obtain from (5.7) and (5.8) that
Hence, under (5.8) (i.e., the boundary conditions), (5.19) is equivalent with (use also the notation of (5.18) and (C2))
where is symmetric by Lemma 2.1 and (5.7). Hence, by (C2), equation (5.19) is equivalent with (under the boundary conditions (5.18))
Now, (5.22) defines independently of , which was free by (5.18). Note that is symmetric, because and are symmetric by Lemma 2.1 and (5.7). Moreover, is lower triangular by (C2), so that (5.23) leads to bandwidth and symmetry. More precisely, we drop the first columns of , and the first rows are replaced by the following rows, which constitute the firstrows of the symmetric, banded matrix under construction:
and where .
The next equations of (4.1) are given by
where by (C2), because by (5.18). Hence, the nextrows from of our matrix under construction have to be defined by
This completes the construction concerning the boundary conditions at the beginning. Thus, possible eigenvectors have to be of the form , where the boundary conditions are satisfied by putting and defining and by (5.18) and (5.22), respectively.
5.2. Boundary Conditions at the End
We proceed similarly as in the previous subsection. Therefore, we can skip some details. We shall use for convenience the same notation for auxiliary matrices or vectors here, but of course, with a different meaning. Observe that the situation is nevertheless not symmetric (see the concluding remarks (ii) below).
We consider the boundary conditions (4.3) or (4.6) at the end by assuming (4.4) and (4.7), so that and are invertible. Let , be real matrices such that (see (4.6))
where , .
These matrices exist, because (5.5) holds by (4.4) and (4.6). Note that
by Proposition 3.2 and (4.6). Moreover, (5.7) holds as before. Hence, the boundary conditions (4.3) or (4.6) are equivalent with
By the spectral theorem there exists an orthogonal matrix such that
where is diagonal and invertible, so that
As before we use this block structure including the extreme cases and . By QRfactorization there exists an orthogonal matrix such that is upper triangular, where with the blockstructure above, that is, . Then is orthogonal, and
where is symmetric and invertible. Let
By (5.5) and (C5) we obtain that and that is invertible.
Altogether, we have constructed such that
with a symmetric and invertible matrix , where ,
with an invertible matrix and so that is symmetric, and
so that is upper triangular, and where is orthogonal.
It follows from (C5) and (C6) that our boundary conditions (4.3) or (4.6) or (5.26) are equivalent with
and therefore with
with , , , and remaining free. We say that (5.34) is the boundary conditions (at the end) in normalized form.
Next, we consider the lastequations of our difference equations (4.1) or (5.1), that is, by Lemma 2.1 and (5.2),
Then, under (5.34) (i.e., the boundary conditions) this is equivalent with (using the notation of (5.34) and (C6))
that is
where is symmetric by Lemma 2.1. Hence, by (C6), equation (5.35) is equivalent with (under the boundary conditions (5.34))
Now, (5.38) defines independently of , which was free by (5.34). Note that is symmetric, because is symmetric by Lemma 2.1 and because is symmetric by (C6). Moreover, is upper triangular by (C6), so that (5.39) leads to bandwidth and symmetry. More precisely, we drop the last columns of , and the last rows are replaced by the following rows, which constitute the lastrows of the symmetric, banded matrix under construction:
and where
The last but one equations of (4.1) are given by
Note that for this overlaps with (5.24) of the previous subsection, and may have been changed by the construction there. We use here this new from Section 5.1, but note that it is irrelevant here. By (C6) and (5.34) we have that
Hence, the rows before the rows of (C7) of our matrix under construction have to be defined by
This completes the construction. Thus, possible eigenvectors must be of the form
6. Main Result and Concluding Remarks
Altogether we have shown by the construction of Section 5 the following result.
Theorem 6.1.
Assume (4.4) and (4.7). Then the construction of Section 5 transforms the SturmLiouville eigenvalue problem (given by (4.1) and (4.2)) of Section 4 into an equivalent algebraic eigenvalue problem for a realsymmetric, banded matrix with bandwidth . This matrix is of size with given by (5.10) and (5.28), and it is constructed from (defined by (5.2)) by (C1)â€“(C8).
Concluding Remarks

(i)
By our theorem every discrete SturmLiouville eigenvalue problem is equivalent with an algebraic eigenvalue problem for a banded, symmetric matrix under the assumptions (4.4) and (4.7). On the other hand, by [1, Remarkâ€‰â€‰1(i)], such an algebraic eigenvalue problem is equivalent with a discrete SturmLiouville eigenvalue problem with Dirichlet boundary conditions. Moreover, if for all , then our eigenvalue problem can be written as an eigenvalue problem for a corresponding Hamiltonian system or symplectic system [14] according to Lemma 2.2. Note that it is quite natural to assume that the leading coefficient never vanishes, because it is the case in most applications. But the main point is that the incorporation of our general boundary conditions into the difference equations by our construction leads in general to a problem where the leading coefficient may vanish for some 's at the beginning and the end. To be more precise, our construction may cause that matrix elements become zero at the beginning and at the end, so that its equivalent SturmLiouville problem with Dirichlet boundary conditions will not satisfy for all anymore, because by (2.3). Hence, it cannot be written as an eigenvalue problem for a linear Hamiltonian difference system, so that the corresponding recursion formulae (based on the Hamiltonian or associated Riccati difference system [4, 11]) cannot be applied for numerical purposes as in [4, Theoremsâ€‰â€‰A and 2]. Therefore, divisions by must be avoided, which is done by the divisionfree algorithms presented in [4, 5]. Hence, these divisionfree algorithms are crucial for our purposes.

(ii)
Note that in contrast to the corresponding continuous SturmLiouville problems (or the corresponding Hamiltonian differential systems), there is no symmetry with respect to the endpoints in the discrete case. This is quite obvious by the difference equation (4.1) with forward differences. Therefore the treatment of the boundary conditions at the endpoints in the subsections above had to be done separately. Actually the results of this treatment are quite different as can be seen also from the next remark.

(iii)
We discuss the extreme cases or of our construction. It follows from (5.10) and (5.28) that if and only if is invertible, that is, by (4.5) (this includes Dirichlet conditions for ).
if and only if , that is, by (4.5) (so that or is free). if and only if or is invertible, that is, by (4.6) or by (4.3) (this includes natural boundary conditions for ). if and only if , that is, (Dirichlet conditions).
Hence, our construction leads to the maximal size (i.e., ) of the constructed matrix for Dirichlet conditions (more general for ) at the beginning and for natural boundary conditions (more general for ) at the end. The construction leads to the minimal size for (or for free ) at the beginning and for Dirichlet conditions at the end.
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This research is supported by Grant KR 1157/21 of DFG.
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Kratz, W. Banded Matrices and Discrete SturmLiouville Eigenvalue Problems. Adv Differ Equ 2009, 362627 (2010). https://doi.org/10.1155/2009/362627
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DOI: https://doi.org/10.1155/2009/362627