The LA = U Decomposition Method for Solving Systems of Linear Equations

A method for solving systems of linear equations is presented based on direct decomposition of the coefficient matrix using the form LAX Elements of the reducing lower triangular matrix L can be determined using either row wise or column wise operations and are demonstrated to be sums of permutation products of the Gauss pivot row multipliers. These sums of permutation products can be constructed using a tree structure that can be easily memorized or alternatively computed using matrix products. The method requires only storage of the L matrix which is half in size compared to storage of the elements in the LU decomposition. Equivalence of the proposed method with both the Gauss elimination and LU decomposition is also shown in this paper.


Introduction
Systems of linear equations or equations linearized for iterative solutions arise in many science and engineering problems [1]. Practical applications of systems of linear equations are many, examples of such application include applications in digital signal processing, linear programming problems, numerical analysis of non-linear problems and least square curve fitting [2]. Systems of equations are also historically reported to have provided a motivation for the development of digital computer as less cumbersome way of solving the equations [3].
Gaussian elimination is a systematic way of reducing systems of linear equations into a triangularised matrix through addition of the independent equations

Method Development
The method proposed in this paper is based on reducing the coefficient matrix A in the system of linear equations AX = B using a single lower triangular reducing matrix L. The original coefficient matrix A is transformed into an upper triangular matrix U that allows solution through back substitution as is usual with both LU decomposition as well as Gauss elimination methods. For the original system of n by n linear equations given as: The matrix representation of Equation (1) will be: where A is the coefficient matrix having the elements a ij of the original equations and B is the right hand side column vector containing the elements 1 2 , , , n b b b  . The proposed method establishes a solution that transforms both the coefficient matrix A and the right hand side column vector B as follows: In other words the coefficient matrix and the right hand side column vector B are transformed through the equations: The procedure, therefore, essentially centers on determining the lower triangular matrix L that reduces the coefficient matrix A to an upper triangular matrix U. Let this matrix L be given through its elements l ij so that: The operation LA = U will reduce the coefficient matrix A in to an upper triangular matrix U given by: However, this proposed method does not need storage of the U matrix as only the L matrix needs to be determined and used to reduce both the A matrix and the right hand side column vector B. This is easily seen through the matrix operation involving the reducing matrix L only, namely, In this method, the l ij elements will be written in terms of the Gauss pivot row multipliers m ij of the Gauss elimination, and, as will be shown shortly, the l ij elements are the sum of the permutation products of the m ij multipliers assembled into a tree like structure for easy memorization. The elements l ij will not remain constant during the reduction process as is normally the case with Gauss elimination or LU decomposition, but change as the reduction of A to U matrix progresses column wise or row wise as new members of the Gauss pivot row multipliers are added to the element l ij .
Unlike the Gauss method which is restricted to column wise operation, in this method it is also possible to proceed row wise. In fact the row wise procedure will be followed to derive the l ij elements.
Starting with row 2 of the lower triangular L matrix,, the only unknown is l 21 and in terms of the Gauss elimination pivot row multipliers m ij , the pivot operation to educe u 21 to zero is given as: 21  For a 4 × 4 L matrix, summarizing the l ij elements, expressed in terms of the Gauss pivot row multipliers m shown above, will give the L matrix shown in Equation (15).
It is easy to show that the m terms in the L matrix in Equation (15) form permutation products where by the number of terms correspond to coefficients of the binomial series expansion. For any element l ij of the L matrix, the number of m-product terms is given by: The power of binomial expansion ( ) , K i j is given by;

Tree-Like Structure of the m-Permutation Products
It is easy to enumerate the m-permutation products of l ij as these products can be arranged in a tree-like structure. Taking the example of elements of l 51 for example, the tree structure shown in Figure 1 is formed.

Formula for Calculation of the Sum of Permutation Products
For the element l ij of the lower triangular matrix L, with the number of m products N m corresponding to the binomial coefficients of power K(i,j), the binomial coefficients N m (r) for 0,1, 2, , r K =  is given by:  Similarly for l 51 with ( )  In general for any element l ij , the m ij sum of products can be calculated using the formula: ( ) Finally, the element l ij is computed by summing the M ij sum of products as follows:

Matrix Solution to the Computation the lij Elements of the Lower Triangular Matrix L
The computation of elements of the lower triangular matrix L can be easily carried out using matrix multiplication. For any element l ij the matrix multiplication takes the following form: A. T. Tiruneh et al.
Equation (26) shows the l ij can be determined from already determined previous values of l kj where 1 j k i + < < and the Gauss pivot row multipliers The negative of the corresponding Gauss pivot row multipliers m rs that are already determined at this stage are given by the matrix form L LU ; The matrix L LU is simply the L matrix of the LU decomposition method. This can be verified as follows: To avoid confusion, let traditional LU decomposition method have its L matrix relabelled L LU to make it different from the L matrix of the proposed direct decomposition procedure.
From the relationship A = L LU U as well as LA = U, it follows that: It follows then that: in which I is the identity matrix. Therefore the L matrix is simply the inverse of the L matrix L LU of the LU decomposition method. The L matrix elements as shown in Equation (15) This computation will be illustrated for the 4 by 4 matrix of L shown in Equation 5 and later for the example of the 4 by 4 system of linear equations solved in the section that follows. Starting with the element l 21 the matrix form of Equation (30) will take the form:

Number of Operations Required
The number of operations required N p are related to the determination of the elements of the L matrix only. It is apparent that similar to the LU decomposition, the order of operations is of power 2, i.e., for n by n matrix the number of operations required grows proportional to n 2 . This is clearly seen as the number For example for a 4 by 4 L matrix ( ) The l ij elements of the L matrix shown in Equation (5) show the six elements to be determined. Compared to the LU decomposition, the proposed method requires only half of the operations required for the LU decomposition. The reason is, unlike the LU method the LAX LB B′ = = method does not require storage of the U elements, i.e., only the L matrix is needed to solve the system of linear equations.

Procedure for Determining Elements of the L Matrix
The computation of the l ij elements of the lower triangular matrix L can be carried out either row wise or column wise using more or less the same procedure as outlined in the following step by step procedure.
Step 1: Initially set all the Gaussian pivot row multipliers m rs of the element l ij to zero values. During computation of a particular value of m rs the most recent values of the other pivot row multipliers will be used. In other words, the values of m rs will be updated once their values change because of successive row wise or column wise computation.
Step 2: Starting with the first column and second row and proceeding either row wise or column wise, calculate the m rs value for which r = i and s = j. For example for the element l 21 , the m value to be calculated is that of m 21 and at l 53 it would be m 53 that will be calculated. Step 4: After the computation of all the m values of the Gauss pivot multipliers is completed, form the L matrix elements l ij using the summation rules of the permutation products involving the m products as given by Equation (24) and Equation (25) or using the matrix product given in Equation (30).
Step 5: Once the L matrix is formed compute the solution X vector of the system of equations AX = B using the formula shown in Equation (3), namely, In other words, the product LA results in the upper triangular matrix U which will allow the computation of the solution vector elements of X using back subs-A. T. Tiruneh et al.

titution.
As in the Gauss method, it is possible to check if a zero appears on the diagonal of the U = LA matrix, i.e., to check if u ii = 0 for a given row i during the computation of the l ij elements. In other words, for a given row i, a check can be made for the value of u ii using the formula: If the condition u ii = 0 becomes true, row interchange can be made with rows from below in the equation. Figure 2 shows a flow chart of the steps outlined above in solving a system of linear equations using the LA = U method. The procedure stated above will be illustrated with an example given below which is a 4 × 4 system of linear equations. Two methods are given, Method 1 using column wise operations and Method 2 using row wise operations.

Application Examples
Example 1: The 4 × 4 system of linear equation shown below will be used to illustrate the x

Method 1 (Column wise operation) Column 1 operations:
Initially all the m values will be set to zero as outlined in the steps for solving the system of equations. Starting with column 1 and at row 2, the equation The reduced matrix LA becomes; Similarly, the operation LB B′ = becomes; Finally the reduced equation LAX LB B′ = = takes the form: The elements of the solution vector X can now be determined by back substitution. Starting from the fourth row, x 4 is determined; Finally, the solution vector X is computed from LAX LB B′ = = : The elements of the solution vector X can now be determined by back substitution. Starting from the fourth row, x 4 is determined;

Discussion
The proposed method, developed and demonstrated with examples so far, shows that solution to linear systems of equation can be obtained through direct decomposition of the A matrix using the operation LAX LB B′ = = . The method provides a clear procedure for direct computation of the L matrix, the only matrix that is needed to transform the original equation AX = B in to a reduced form, i.e., LAX = BX unlike for example the LU method which requires that both the L and U matrix be stored to find the solution through AX LUX B = = . The elements l ij of the lower triangular matrix L are shown to be sums of permutation products of the Gauss pivot row multipliers m rs . The relationship between l ij and m rs is clearly established through a formula and it is easy to visually construct this relationship using a tree diagram that will assist in easy memorisation of the relationship. In addition (and as an alternative procedure) the relationship so established between elements l ij of the lower triangular matrix L and the Gauss pivot row multipliers m rs enables construction of the L matrix directly from the Gauss elimination steps.
The characteristic of Gauss elimination method is that the reduction to an A. T. Tiruneh et al.
upper triangular matrix can only proceed column wise. It is not possible to proceed row wise in the Gauss method. On the other hand, the LU decomposition requires alternate transition between the L and U elements for determining the LU compact matrix. By contrast, the proposed LA = U reduction method can proceed either column wise or row wise essentially giving the same result. This flexibility is demonstrated in the example shown above where it is easily seen that the computation of the Gauss pivot row multipliers remains more or less the same for both the row wise and column wise operations.
The storage requirement during the reduction process is related to the generation of the L matrix. Unlike the LU method, storage is needed only for the L matrix since the solution directly proceeds from the reduction LAX LB B′ = = in which there is no need to store the U matrix. The number of elements that need change is of the order O(n 2 ) as shown in Equation 38 and is typically half the number of operations required for the LU decomposition because in the LU decomposition both the L and U elements need to be determined and stored.

Conclusions
A direct decomposition of the coefficient matrix forming part of a system of linear equations using a single lower triangular reducing matrix L has been demonstrated as shown in this paper. The method allows solution to the system of linear equations to proceed through storage of a single lower triangular matrix L only, through which both the coefficient matrix A and the right hand side column vector B are transformed. Elements of the reducing matrix L are shown to be sums of permutation products of the pivot row multipliers of the Gauss elimination technique. These sums of permutation products, for any element of the reducing matrix L, can be easily constructed using a tree diagram that is relatively easy to memorize besides using the formula developed for the purpose. These L matrix elements can also be alternatively computed using matrix products. In the process of determining the elements of the L matrix, either row wise or column wise procedure can be followed essentially giving the same result which provides added flexibility to the proposed method. Equivalence of this newly proposed method with both the Gauss elimination and LU decomposition techniques has been established. In the case of the equivalence with Gauss elimination technique, elements of the L matrix are specified as functions of the Gauss pivot row multipliers. This also implies that it is possible to construct the reducing L matrix of the proposed direct decomposition method using the Gauss pivot row multipliers. As has been demonstrated, the L matrix can be directed constructed from the Gauss pivot row multipliers using the matrix product L Lu L = I. For the LU decomposition, the L matrix of the proposed method is simply the inverse of the L matrix of the LU decomposition. In terms of storage of computed values, it can be seen that the proposed method of direct decomposition using the transformation LAX LB B′ = = needs only storage of the L matrix elements which is half in size compared with storage of all the L and U elements in the LU de- Apart from providing added flexibility and simplicity, the proposed method would be of good educational value providing an alternative procedure for solving systems of linear equations.