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"Linear Algebraic Equations"
LU Decomposition Method
Solution from the LU Decomposition Method

Matrix A: = { , , }, { , , }, { , , }

Vector b: = { , , }

[ Initial MatrixValues A: { 2, 1, 1 }, { 3, 1, 2 }, { 2, 1, 2 } ]
[ Initial VectorValues b: { 8, 11, 3 } ]

LU Decomposition

The LU Decomposition is a matrix decomposition which represents a matrix as the product of a lower and upper triangular matrix. This decomposition is used in numerical analysis to solve systems of linear equations and to find the inverse of a matrix.


If a Matrix A can be written as the product of two matrices

A = L . U

where L is the lower triangular and U is the upper triangular. It is possible then to use this decomposition to solve the linear equations,

A . x = (L .U) . x = L . (U . x) = b

by first solving for the vecor y such that

L . y = b

and then solving

U . x = y

The advantage of decomposing one linear system into two succesive ones is that the solution of a triangular set of equations can be easily obtained just by a forward or backward substitution.

Matrix Inverse

Using the above LU decomposition and substitution methods, computing the inverse of a matrix column by column is straightforward.

Testing LU Decomposition Method

In order to test the method as defined above, a new TestLUDecomposition() static method has been added and executed. Supporting code and methods are not shown.

A set of initial equations for the test has been defined for Matrix A and Vector b.

           static void TestLUDecomposition();
                 LinearSystem ls = new LinearSystem();
                 MatrixR A = new MatrixR(new double[3, 3] { { t1, t2, t3 }, { t4, t5, t6 }, { t7, t8, t9 } });
                 VectorR b = new VectorR(new double[3] { t10, t11, t12 });
                 MatrixR AA = A.Clone();
                 MatrixR BB = A.Clone();
                 double d = ls.LUCrout(A, b);
                 MatrixR inv = ls.LUInverse(AA);
                 ListBox1.Items.Add("Inverse of A = " + inv.ToString());
                 ListBox1.Items.Add(" " + " ");
                 ListBox1.Items.Add("Solution of the equations = " + b.ToString());
                 ListBox2.Items.Add("Determinant of A = -1");
                 ListBox2.Items.Add(" " + " ");
                 ListBox2.Items.Add("Test Inverse: BB*Inverse = " + BB * inv);

An initial set of data points was defined for matrix A and vector b. The user can manipulate all values and try variations on the matrix and vector themselves.

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