solution of systems of linear equations

Algebraic, transcendental (i.e., involving trigonometric and exponential functions), ordinary

differential equations, or partial differential equations...

Solution of Systems of Linear Equations

ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology


Basic Terminology1. Analytical Method – is one that produces either exact or approximate solutions in closed form2. Components – the elements of a vector

3. Conformable – matrices with identical dimensions4. Accuracy – is a measure of the nearness of a value for the true value5. Precision – is a measure of the clustering of values near each other


Basic Terminology6. Triangular matrix – a square matrix in which all the elements on one side of the diagonal are zero 7. Gauss elimination – methods for solving a system; reducing the matrix to the upper triangular form, and then back to substitution8. Double sequence – is a function of domain of ordered pairs (i, j)of integer and with range consisting of a portion of the real number system9. Non- singular matrix – a square matrix with a non-zero determinant


In general, a system of linear algebraic equations may be of the form

: : : : :

: : : : :

Where xj (j=1,2,…m) denotes the unknown variable

aij (i=1,2,…n; j=1,2,…m) denotes the coefficients of the unknown variable

bi (i=1,2,…n) denotes the non-homogeneous terms


Analytical method

One that produces either exact or approximate solutions in closed form.


Four possible solutions to a system of linear algebraic equations:

1. A unique solution – a consistent set of solutions2. No solution – an inconsistent set of equations3. An infinite number of solutions – a redundant set of equations

4. The trival solution xj = 0 – a set of homogenous equations


Two fundamental approaches for solving systems of linear algebraic equations:

1. Direct methods – are systematic procedures, based on algebraic elimination, that obtain the solution in a fixed number of operations.2. Iterative methods – obtain the solution asymptotically by an iterative procedure. A trial solution is assumed, the trial solution is substituted into the system of equations to determine the mismatch in the trial solution, and an improved solution is obtained from the mismatch data.


Properties of Matrices and Determinants:

Definition 1An (m x n) or (m, n) matrix is a rectangular array of quantities arranged in m rows and n columns.



Definition 2A matrix with only one row is a special kind of matrix known as a row vector.



Definition 3A matrix with only one column is a special kind of matrix known as a column vector.



Definition 4The (n x m) or (n, m) matrix obtained from a given(m x n) or (m, n) A by interchanging its rows and columns is called the transpose of A denoted by the symbol AT.



Definition 5A square matrix is a matrix where the dimensions m is equal to n.



Definition 6A symmetric matrix is one where aij = aji for all i’s and j’s.



Definition 7A square matrix in which each element not on the principal diagonal is zero is called a diagonal matrix.



Definition 8A square matrix in which every element below the principal diagonal is zero is said to be upper triangular matrix.



Definition 9A square matrix in which every element above the principal diagonal is zero is said to be lower triangular matrix.



Definition 10A square matrix in which all elements equal to zero, with the exception of a band centered on the main diagonal is called a bonded matrix (e.g. tridiagonal matrix).



Definition 11A diagonal matrix in which each diagonal element is 1 is called a unit matrix or identity matrix.



Definition 12A matrix in which every element is zero is called a null matrix or zero matrix.



Definition 13The determinant of an (n, n) square matrix A is written as lAl and is defined by either of

or

in which cij is known as the cofactor of the element aij.



Definition 14The cofactor cij of an (n, n) square matrix A is obtained by first removing row i and column j to form an (n-1, n-1) matrix and then performing the operation



Definition 15The augmented matrix is obtained by adjoining the column vector b to the coefficient matrix A.



Definition 16A coefficient matrix with a zero determinant is singular, a unique solution for x requires a non-singular coefficient matrix.


Direct Methods of Linear Systems:

A. Methods for Triangular MatricesIt involves reduction of matrix equation into

one of the forms:

, L = lower triangular matrix

, U = upper triangular matrix



B. Cramer’s Rule Gives the components xi of x in terms of determinants according to:



B. Cramer’s Rule Example Use the Cramer’s rule to solve:

0.3x1 + 0.52x2 + x3 = -0.010.5x1 + x2 + 1.9x3 = 0.670.1x1 + 0.3x2 + 0.5x3 = -0.44



C. Gaussian Elimination a method for solving a system of the type (A• x = b) wherein the goal is to reduce it to the upper triangular form and then use the back substitution scheme to obtain the components from each of the remaining equations.



C. Gaussian Elimination ExampleUse Gaussian elimination to solve:

3x1 - 0.1x2 - 0.2x3 = 7.850.1x1 + 7x2 - 0.3x3 = -19.30.3x1 – 0.2x2 + 10x3 = 71.4



D. Gauss-Jordan Method a variation of Gauss Elimination wherein the goal is to reduce the original matrix to a diagonal form. not popular since there is neither a reduction in programming complexity nor increased efficiency.



D. Gauss-Jordan ExampleUse Gauss-Jordan to solve the previous problem:

3x1 - 0.1x2 - 0.2x3 = 7.850.1x1 + 7x2 - 0.3x3 = -19.30.3x1 – 0.2x2 + 10x3 = 71.4



E. LU Decomposition Method is another elimination method of solving general systems of linear algebraic equations wherein the objective is to find a lower triangular factor L and an upper triangular factor U such that the system of equations can be transformed according to

Where A* = matrix after row exchange have been made to allow the factors L and U to be computed accurately; b* = vector b after an identical set of row exchanges.



E. LU Decomposition ExampleSolve the previous problem:

3x1 - 0.1x2 - 0.2x3 = 7.850.1x1 + 7x2 - 0.3x3 = -19.30.3x1 – 0.2x2 + 10x3 = 71.4


Pitfalls of Elimination Methods:

1. Division by Zero2. Round-off Errors3. Ill-Condition Systems

is one where a small changes in one or more of the coefficients results in large changes in the solution. 4. Singular Systems is worse than ill-conditioned because two equations in the system are identical.


Techniques for Improving Solutions:

1. Pivoting2. Use of more significant figures 3. Scaling


Examples:

1. Solve the following systems:x1 + 2x2 = 101.1x1 + 2x2 = 10.4Then solve it again, but with the coefficient of x1 in the second equation modified slightly to 1.05.


Examples:

2. Evaluate the determinant of the following systems:3x1 + 2x2 = 18-x1 + 2x2 = 2And solve also the determinant in prob. 1.


Iterative Methods of Linear Systems:

A. Gauss-Seidel Method Iterative or approximate methods . Start the process by assigning initial values (guessing a value) and then use a systematic method to obtain a refined estimate of the root. Then solve for the subsequent values of x1 , x2 , x3 , etc. , using the following equations:


Iterative Methods of Linear Systems:

A. Gauss-Seidel Method



A. Gauss-Seidel ExampleSolve the previous problem:

3x1 - 0.1x2 - 0.2x3 = 7.850.1x1 + 7x2 - 0.3x3 = -19.30.3x1 – 0.2x2 + 10x3 = 71.4



Convergence criterion for the Gauss-Seidel

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solution of systems of linear equations

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