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Solution of Systems of Linear Equations

Solution of Systems of Linear Equations. Algebraic, transcendental (i.e., involving trigonometric and exponential functions), ordinary differential equations, or partial differential equations. Basic Terminology.

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Solution of Systems of Linear Equations

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  1. Solution of Systems of Linear Equations Algebraic, transcendental (i.e., involving trigonometric and exponential functions), ordinary differential equations, or partial differential equations... ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  2. Basic Terminology 1. Analytical Method – is one that produces either exact or approximate solutions in closed form 2. Components – the elements of a vector 3. Conformable – matrices with identical dimensions 4. Accuracy – is a measure of the nearness of a value for the true value 5. Precision – is a measure of the clustering of values near each other ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  3. Basic Terminology 6. 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 substitution 8. 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 system 9. Non- singular matrix – a square matrix with a non-zero determinant ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

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

  5. Analytical method • One that produces either exact or approximate solutions in closed form. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  6. Four possible solutions to a system of linear algebraic equations: 1. A unique solution – a consistent set of solutions 2. No solution – an inconsistent set of equations 3. An infinite number of solutions – a redundant set of equations 4. The trival solution xj = 0 – a set of homogenous equations ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

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

  8. Properties of Matrices and Determinants: Definition 1 An (m x n) or (m, n) matrix is a rectangular array of quantities arranged in m rows and n columns. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  9. Properties of Matrices and Determinants: Definition 2 A matrix with only one row is a special kind of matrix known as a row vector. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  10. Properties of Matrices and Determinants: Definition 3 A matrix with only one column is a special kind of matrix known as a column vector. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  11. Properties of Matrices and Determinants: Definition 4 The (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. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  12. Properties of Matrices and Determinants: Definition 5 A square matrix is a matrix where the dimensions m is equal to n. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  13. Properties of Matrices and Determinants: Definition 6 A symmetric matrix is one where aij = ajifor all i’s and j’s. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  14. Properties of Matrices and Determinants: Definition 7 A square matrix in which each element not on the principal diagonal is zero is called a diagonal matrix. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  15. Properties of Matrices and Determinants: Definition 8 A square matrix in which every element below the principal diagonal is zero is said to be upper triangular matrix. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  16. Properties of Matrices and Determinants: Definition 9 A square matrix in which every element above the principal diagonal is zero is said to be lower triangular matrix. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  17. Properties of Matrices and Determinants: Definition 10 A 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). ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  18. Properties of Matrices and Determinants: Definition 11 A diagonal matrix in which each diagonal element is 1 is called a unit matrix or identity matrix. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  19. Properties of Matrices and Determinants: Definition 12 A matrix in which every element is zero is called a null matrix or zero matrix. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  20. Properties of Matrices and Determinants: Definition 13 The 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. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  21. Properties of Matrices and Determinants: Definition 14 The 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 ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  22. Properties of Matrices and Determinants: Definition 15 The augmented matrix is obtained by adjoining the column vector b to the coefficient matrix A. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  23. Properties of Matrices and Determinants: Definition 16 A coefficient matrix with a zero determinant is singular, a unique solution for x requires a non-singular coefficient matrix. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  24. Direct Methods of Linear Systems: A. Methods for Triangular Matrices It involves reduction of matrix equation into one of the forms: , L = lower triangular matrix , U = upper triangular matrix ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  25. Direct Methods of Linear Systems: B. Cramer’s Rule • Gives the components xi of x in terms of determinants according to: ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  26. Direct Methods of Linear Systems: B. Cramer’s Rule Example • Use the Cramer’s rule to solve: • 0.3x1 + 0.52x2 + x3 = -0.01 • 0.5x1 + x2 + 1.9x3 = 0.67 • 0.1x1 + 0.3x2 + 0.5x3 = -0.44 ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  27. Direct Methods of Linear Systems: 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. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  28. Direct Methods of Linear Systems: C. Gaussian Elimination Example Use Gaussian elimination to solve: 3x1 - 0.1x2 - 0.2x3 = 7.85 0.1x1 + 7x2 - 0.3x3 = -19.3 0.3x1 – 0.2x2 + 10x3 = 71.4 ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  29. Direct Methods of Linear Systems: 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. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  30. Direct Methods of Linear Systems: D. Gauss-Jordan Example Use Gauss-Jordan to solve the previous problem: 3x1 - 0.1x2 - 0.2x3 = 7.85 0.1x1 + 7x2 - 0.3x3 = -19.3 0.3x1 – 0.2x2 + 10x3 = 71.4 ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  31. Direct Methods of Linear Systems: 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. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  32. Direct Methods of Linear Systems: E. LU Decomposition Example Solve the previous problem: 3x1 - 0.1x2 - 0.2x3 = 7.85 0.1x1 + 7x2 - 0.3x3 = -19.3 0.3x1 – 0.2x2 + 10x3 = 71.4 ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  33. Pitfalls of Elimination Methods: 1. Division by Zero 2. Round-off Errors 3. 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. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  34. Techniques for Improving Solutions: 1. Pivoting 2. Use of more significant figures 3. Scaling ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  35. Examples: Solve the following systems: x1 + 2x2 = 10 1.1x1 + 2x2 = 10.4 Then solve it again, but with the coefficient of x1 in the second equation modified slightly to 1.05. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  36. Examples: 2. Evaluate the determinant of the following systems: 3x1 + 2x2 = 18 -x1 + 2x2 = 2 And solve also the determinant in prob. 1. ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

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

  38. Iterative Methods of Linear Systems: A. Gauss-Seidel Method ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  39. Direct Methods of Linear Systems: A. Gauss-SeidelExample Solve the previous problem: 3x1 - 0.1x2 - 0.2x3 = 7.85 0.1x1 + 7x2 - 0.3x3 = -19.3 0.3x1 – 0.2x2 + 10x3 = 71.4 ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

  40. Direct Methods of Linear Systems: Convergence criterion for the Gauss-Seidel and ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

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