Decompositions of Some Special Block Tridiagonal Matrices

In this paper, we present a unified approach to decomposing a special class of block tridiagonal matrices ( ) , K α β into block diagonal matrices using similarity transformations. The matrices ( ) , pq pq K α β × ∈ are of the form ( ) [ ] block-tridiag , , , K B A B α β β α = for three special pairs of ( ) , α β : ( ) 1,1 K , ( ) 1,2 K and ( ) 2,2 K , where the matrices A and B, , p p A B × ∈ , are general square matrices. The decomposed block diagonal matrices ( ) , K α β  for the three cases are all of the form: ( ) ( ) ( ) ( ) 1 2 , , , , q K D D D α β α β α β α β = ⊕ ⊕ ⊕   , where ( ) ( ) ( ) , 2cos , k k D A B α β θ α β = + , in which ( ) , k θ α β , 1,2, , k q =  , depend on the values of α and β. Our decomposition method is closely related to the classical fast Poisson solver using Fourier analysis. Unlike the fast Poisson solver, our approach decomposes ( ) , K α β into q diagonal blocks, instead of p blocks. Furthermore, our proposed approach does not require matrices A and B to be symmetric and commute, and employs only the eigenvectors of the tridiagonal matrix ( ) [ ] tridiag , , , T b a b α β β α = in a block form, where a and b are scalars. The transformation matrices, their inverses, and the explicit form of the decomposed block diagonal matrices are derived in this paper. Numerical examples and experiments are also presented to demonstrate the validity and usefulness of the approach. Due to the decoupled nature of the decomposed matrices, this approach lends itself to parallel and distributed computations for solving both linear systems and eigenvalue problems using multiprocessors.


Introduction
In this paper, we present explicit similarity transformations to decompose block tridiagonal matrices ( ) , pq pq K α β × ∈  of the following form: for some special pairs of ( ) , α β , where , ( ) ( ) ( ) ( ) 1 2 , , , , where the operation symbol ⊕ denotes the matrix direct sum and the diagonal submatrices are explicitly known, although they depend on the values of α and β . Our decomposition method is closely related to the classical fast Poisson solver [1] [2] using Fourier analysis.
The block decomposition scheme to be addressed has been presented by the author in [3] and formal proof was given for ( ) , k D α β , each of size only p by p. This block decomposition scheme provides a much more efficient means for solving eigenvalue problems with this type of coefficient matrices. It can also be employed for solving linear systems with efficiency because the transformation matrices are explicitly known. In addition, the decoupled structure of the transformed matrix lends itself to parallel computation with coarse-grain parallelism.

Decompositions
In this following, we present our key observations that lead to the proposed block decomposition method for this class of matrices ( ) , K α β using transformation matrices whose entries are inherent in the special block tridiagonal form of ( ) , K α β . Whenever there is no confusion, we shall simply use K to denote ( ) , K α β . Throughout the paper, the operation symbols ⊕ and ⊗ are used to denote the matrix direct sum and the Kronecker product. , K α β is orthogonally similar to the block diagonal matrix This will be the case when 2 x k = π for any integer k. Now since ( ) ( ) ( ) Likewise, since ( ) The denominator of ( ) ij S u will never be equal to zero. Accordingly, Finally from (3) and (4), we obtain and, therefore, This completes the proof.
Theorem 2. When 2 α β = = , the block tridiagonal matrix ( ) , K α β is similar to the block diagonal matrix Proof. This block diagonalization was mentioned previously in Corollary 2 in [3] without a proof. Unfortunately, the eigenvectors T k v used to form the transformation matrix Q and the decomposed submatrices D k consist of errors, in which 1) the vector as stated in that paper should be replaced by In this paper, we give a formal proof with the correct eigenvectors and provide the explicit form of the inverse of the transformation matrix Q for . We now show that the si- It deserves mentioning that Q in this case is not orthogonal. 1 Q − , however, exists and is explicitly known. Therefore, It suffices to show that Equation (5) holds for all j, 1 j q ≤ ≤ . Accordingly, by arranging all j V together to form the matrix Q, we obtain KQ QD = . In other words, the trans- This is a similarity transformation and, therefore, all eigenvalues of ( )

2, 2 K
are preserved in the decomposed matrix D. It is worth mentioning that obtaining all the eigenvalues from D is far more efficient than from the original matrix K since D consists of only q diagonal blocks: When it comes to solving linear systems in the transformed space that involves Q, however, one needs to employ the LU decomposition of Q or to find 1 Q − . Normally, finding the LU decomposition is more efficient and preferred. However, it does not make sense to find the LU decomposition of Q if the inverse of Q is readily available. In the following, we show that 1 Q − can be obtained explicitly.
Let C be the matrix formed by k v : , whose explicit form is    where we have used the following two properties of Kronecker products [5]: Note that the matrix Q here is not orthogonal either. It can be shown that In the following, we show that 1 Q − is almost identical to T Q and, therefore, can be explicitly obtained from T Q without any difficulty. Again, let C be the matrix formed by k v : as was done in the previos section. We have in this case: , with R consisting of the first ( ) 1 q − rows and r being the last row of C, we clearly see that Following the same derivation as we have done in Theorem 2, we conclude that: Note that C in this case is neither symmetric nor orthogonal.

Numerical Experiments
To demonstrate the validity and advantage of this block decomposition ap-  Table 1 where all eigenvalues are listed in the order produced by Octave without reordering.
We then present in Table 2, Table 3, and Table 4 the eigenvalues obtained directly from the decomposed diagonal blocks D 1 through D 5 (Equation (8)) of ( ) , respectively, where each k D is a 4 by 4 matrix.
As can be seen from these tables, all eigenvalues are preserved after the block decomposition. For example, all the eigenvalues shown in Table 2 are identical to those of ( ) Table 1 , a significant saving in computation. The advantage of the decomposition is obvious, not to mention the additional advantage that can be exploited from the coarse-grain parallelism offered by the block decomposition when the problem is to be solved using multiple processors.

Conclusions
In this paper, we have presented a unified block decomposition scheme for three special cases of block tridiagonal matrices of the form ( ) , K α β , as shown in Equation (1). This class of block tridiagonal matrices arises frequently from the finite difference approximation to solving certain partial differential equations such as the Laplace's, Poisson's, or Helmholtz equations using five-or nine-point schemes, over a rectangular or cubic domain [7]. They can also arise from some finite-element discretization of the same equation [8] and from surface fitting with the B-spline functions [9]. The values of α and β typically depend on the boundary conditions of the physical problem:

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.