On the approximation of inverse of some band matrices and their applications in local splines

In this paper, we obtain approximate inverses of popular tri-diagonal and penta-diagonal matrices which are used to construct local (or a discrete quasi-interpolant) interpolatory and integro splines.


Introduction
The band matrices often arise in a range of science and engineering applications such as numerical solutions of ordinary and partial differential equations, spline approximation, image and signal processing, and parallel computing, see [1,5,6,11] and references therein.In many of these areas, inversion of the tri-diagonal matrix is required.In particular, in [11] Yamamoto obtained explicit formulas for the entries of the inverse of nonsingular tri-diagonal matrices.In [7], Jia and Li derived the numerical or symbolic algorithms for the inverses of k-diagonal matrices.Moreover, in [9] Smolarski discussed a particular type of banded matrix, namely a diagonally striped matrix, and the structure of its inverse.Bickel and Lindner in [3] proved that if an infinite matrix A, which is invertible as a bounded operator on l 2 , can be uniformly approximated by banded matrices then so can the inverse of A. Although there are explicit formulas for entries of the inverse of band matrices but most of the time, practically, they are not suitable for simple and hand calculations.In some cases, it suffices to find only approximate inverse of these matrices.On the other hand, for band matrices, it is well established that the entries of its inverse decay exponentially away from the main diagonal; see for example [4].Therefore, we only need to find approximate entries α i j of the main and its few adjacent diagonals i.e. we need For example, in constructing interpolatory splines and integro-splines with small degrees, it is often required to solve a system of linear equations where A is a band matrix.In particular, we consider the following cases: 4. A =Penta-diag{1, 26, 66, 26, 1},
If we have approximate inverse A −1 = (α i j ), |i − j| ≤ k then we obtain the approximate solution of (1.2) as follows: 3) The error of approximate solution given by (1.3) is estimated as From (1.4) it is clear that it is better to restrict k by small values, because of the exponential decay of entries of inverse of band matrices [4].To find approximate inverse A −1 = (α i j ) we use the approximate solution of system (1.2) known in some cases.First we consider the system (1.2) with matrix A =Tri-diag{1, 4, 1}.Such system arises in constructing interpolatory cubic spline on the uniform partition [a, b] with knots 2 Approximate inverse of special tri-diagonal matrices and its applications Let S 3 (x) be a cubic C 2 spline satisfying the interpolation conditions By using the B-spline representation of S 3 (x) ∈ C 2 , we have: where B i (x) are normalized cubic B-splines that constitute basis for S 3 ∈ C 2 [a, b] cubic splines space, see [12].Then the interpolatory conditions (2.5) implies Now we will find the entries of near of the main diagonal, using the approximate explicit formula given in [8,12] with accuracy O(h 4 ).Using (2.8), we write (2.7b) as If we use explicit formulas given in [11] for matrix A =Tri-diag{1, 4, 1} then it is easy to show that α i j = α ji , and α i,i+ j = α i,i− j for j = 1, 2, • • • . (2.9) Therefore, the last expression can be rewritten as If we take into account the formula which holds for f (x) ∈ C 4 , then the expression (2.10) becomes (2.11) where α i j = O(h 4 ), |i − j| ≥ 3. From (2.12) the unknowns α i,i−1 and α ii are expressed by α i,i−2 as We know that which leads to Hence, from (2.13) we find and Thus, the entries of i-th row of A −1 are given by explicit formula (2.15).Further, if we use the notation M i = S ′′ 3 (x i ) then we have the following system of equations [12] (2.16) The matrix of system (2.16) is, as preceding case, A =Tri-diag{1, 4, 1}.Then according to (2.13) we have The Taylor expansions of f (x i + kh) give us , from these we have Using the last formulas and (2.15a), we have Thus, we find the solution of (2.16) with accuracy O(h 2 ) without solving it.One can write system for m i = S ′ 3 (x i ) which has the same matrix A as (2.16).Consequently, using the same technique as above, we find (2.17) Note that the system (1.2) with matrix A =Tri-diag{1, 4, 1} arises also in constructing integro splines.

Application of approximate inverse matrices on constructing integro splines
In a uniform partition case the integro quadratic spline S 2 (x) satisfies relations [10] where where B i (x) are a normalized quadratic B-splines that forms a basis for C 1 quadratic splines space.For convenience, we present here B i as: The values of B i (x) and B ′ i (x) at the knots are given in Table 1. Table From (3.20) and using the properties of B-spline in Table 1, we obtain Taking into account (3.22), the relations (3.18) can be written in term of coefficients b i as: where From (3.23b) we deduce Analogously, using (3.22) and the relations one can obtain Thus, we have the systems (3.18), (3.25), and (3.27) with the same matrix but different right-hand sides.Since the matrix of these system is A=Tri-diag{1, 4, 1}, we can use the above computed approximate inverse of this matrix.Using (2.15), from (3.18) we find (3.28)For the values of I i and y ∈ C 4 , the following property holds We can simplify (3.28) by using (3.29).As a result we have Using the same technique, as preceding case, in (3.25) and (3.27) we obtain Thus, we first obtain approximate explicit formulas for S 2 (x i ), b i , and S ′ 2 (x i ).In [10], we have the following estimation but no estimation for the first derivative is given.Due to the explicit formula (3.32) one can obtain Another application of approximate inverse of matrix A=Tri-diag{1, 4, 1} is the well-known relations in [15]: where n i = S ′′′ 5 (x i ).Such system appears in constructing quintic integro spline.As above, from (3.34) it follows that Using (2.15) and into the last formula, we obtain and In [14], we obtained the approximate formula (3.39) As above, we denote the entries of inverse matrix A −1 by α i j .Then from (3.37) and (3.39) we get 1 144 (3.40) Using symmetry of A −1 and matching the coefficients of I i on both sides (3.40) we obtain Thus we find the entries of i-th row of A −1 by formulas (3.43).Now we can use (3.43) to determine T i from (3.38).
As above, we get Using (3.29) and (3.43) into the last formula, we obtain the well-known explicit formula that was derived first in [14] Now we consider the matrix A=Tri-diag{1, d, 1} with |d| > 2. Obviously, the above two cases are particular cases of this matrix with d = 4 and d = 10.From Theorem 2.1 in [7], we get the following explicit formula for where In [14] we obtained approximate and explicit formula It is easy to show that by using expression (3.29), the c i can be rewritten in more symmetric form As before, we denote the entries of inverse of matrix A =Penta-diag{1, 26, 66, 26, 1} of system (4.48) by α i j .Then, from (4.48) and (4.49) we get 1 28 800 (13 in which we have used symmetry of α i j and α i,i− j = α i,i+ j , j = 1, 2, 3. Using (3.35), from (4.50) we have 1 28 800 (13(I Equating the coefficients of I i− j + I i+ j for j = 0, 1, 2 in both sides of last expression we get where m i = S ′ 5 (x i ) and n i = S ′′′ 5 (x i ) and

.57)
These systems arise in constructing integro quintic spline [2].An explicit and approximate solution of system (4.55) can be found in [15] as In this case, using analogous technique, as above, we find that 81α i,i−3 + 16α i,i−2 + α i,i−1 = 1 2 700 ,   Note that, in [13] Z-folding algorithm was proposed for solving the penta-diagonal system of linear equations, which allows us to reduce the system by solving two tri-diagonal systems sequentially.We can find the approximate inverse of penta-diagonal matrices by using the Z-folding algorithm and (3.46), but this approach is not suitable to obtain explicit formulas as (3.36) and (4.58).

Conclusion
For some application cases it is not necessary to find all entries of the inverse matrices of band matrices.The main advantage of our approach are simple and explicit formulas for only main diagonal and its few adjacent diagonals entries of the inverse matrices.
i.e. the integral values I i of function y(x) are known on the subintervals [x i , x i+1 ], h = (b − a)/n.Obviously, one can use the B-spline representation of S 2 (x):