A LINEAR SYSTEMS OF EQUATIONS. By : Dewi Rachmatin

Transcription

1 A LINEAR SYSTEMS OF EQUATIONS By : Dewi Rachmatin

2 Back Substitution We will now develop the backsubstitution algorithm, which is useful for solving a linear system of equations that has an upper-triangular coefficient matrix. Definition (Upper-Triangular Matrix). An nxn matrix is called uppertriangular provided that the elements satisfy whenever i > j.

3 If A is an upper-triangular matrix, then AX = B is said to be an upper-triangular system of linear equations. (1)

4 Theorem (Back Substitution). Suppose that AX=B is an upper-triangular system with the form given above in (1). If for unique solution. then there exists a

5 The back substitution algorithm To solve the upper-triangular system AX=B by the method of back-substitution. Proceed with the method only if all the diagonal elements are nonzero. First compute and then use the rule for

6 Example 1 (a). Use the back-substitution method to solve the upper-triangular linear system. Example 1 (b). Use the back-substitution method to solve the upper-triangular linear system.

7 Definition (Lower-Triangular Matrix). An nxn matrix is called lower-triangular provided that the elements satisfy whenever i < j. If A is an lower-triangular matrix, then AX=B is said to be a lower-triangular system of linear equations. (2)

8 Theorem (Forward Substitution) Suppose that AX=B is an lower-triangular system with the form given above in (2). If for then there exists a unique solution.

9 The forward substitution algorithm To solve the lower-triangular system AX=B by the method of forward-substitution. Proceed with the method only if all the diagonal elements are nonzero. First compute and then use the rule for

10 Example 1. Use the forward-substitution method to solve the lower-triangular linear system

11 Cholesky, Doolittle and Crout Factorization Definition (LU-Factorization). The nonsingular matrix A has an LUfactorization if it can be expressed as the product of a lower-triangular matrix L and an upper triangular matrix U: A = LU

12 When this is possible we say that A has an LU-decomposition. It turns out that this factorization (when it exists) is not unique. If L has 1's on it's diagonal, then it is called a Doolittle factorization. If U has 1's on its diagonal, then it is called a Crout factorization. When U=L T (or L=U T ), it is called a Cholesky decomposition.

13 Theorem (A = LU; Factorization with NO Pivoting) Assume that A has a Doolittle, Crout or Cholesky factorization. The solution X to the linear system AX=B, is found in three steps: 1. Construct the matrices L dan U, if possible. 2. Solve LY=B for Y using forward substitution. 3. Solve UX=Y for X using back substitution.

14 Doolittle factorization A = LU. =

15 Crout factorization A = LU. =

16 Mathematica Subroutine (Doolittle)

17 Mathematica Subroutine (Crout)

18 Theorem (A = LU; Cholesky Factorization) Assume that A has a Cholesky factorization A = U T U, where L = U T. =

19 Or if you prefer to write the Cholesky factorization as A = L L T, where U = L T. =

20 Mathematica Subroutine (Cholesky factorization)

21 Example 1 (a). Find the A = LU factorization for the matrix Use the Doolittle method, Crout method and Cholesky method.

22 Jacobi and Gauss-Seidel Iteration Iterative schemes require time to achieve sufficient accuracy and are reserved for large systems of equations where there are a majority of zero elements in the matrix. Often times the algorithms are taylor-made to take advantage of the special structure such as band matrices. Practical uses include applications in circuit analysis, boundary value problems and partial differential equations.

23 Iteration is a popular technique finding roots of equations. Generalization of fixed point iteration can be applied to systems of linear equations to produce accurate results. The method Jacobi iteration is attributed to Carl Jacobi ( ) and Gauss-Seidel iteration is attributed to Johann Carl Friedrich Gauss ( ) and Philipp Ludwig von Seidel ( ).

24 Consider that the n n square matrix A is split into three parts, the main diagonal D, below diagonal L and above diagonal U. We have A = D - L - U.

25 = - -

26 Definition (Diagonally Dominant) The matrix A is strictly diagonally dominant if for.

27 Theorem (Jacobi Iteration) The solution to the linear system AX=B can be obtained starting with P 0, and using iteration scheme where and. If P 0 is carefully chosen a sequence is generated which converges to the solution P, i.e. AP=B. A sufficient condition for the method to be applicable is that A is strictly diagonally dominant or diagonally dominant and irreducible.

28 Mathematica Subroutine (Jacobi Iteration)

29 Theorem (Gauss-Seidel Iteration) The solution to the linear system AX=B can be obtained starting with P 0, and using iteration scheme where and. If P 0 is carefully chosen a sequence is generated which converges to the solution P, i.e. AP=B. A sufficient condition for the method to be applicable is that A is strictly diagonally dominant or diagonally dominant and irreducible.

30 Mathematica Subroutine (Gauss-Seidel Iteration)

31 Example 1. Use Jacobi iteration to solve the linear system Try 10, 20 and 30 iterations. Example 2. Use Jacobi iteration to attempt solving the linear system Try 10 iterations. Observe that something is not working. In example 5 we will check to see if this matrix is diagonally dominant.