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Numerical Analysis – Linear Equations(I)

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Numerical Analysis – Linear Equations(I)

Hanyang University

Jong-Il Park

- N unknowns, M equations

coefficient matrix

where

- Direct methods
- Gauss elimination
- Gauss-Jordan elimination
- LU decomposition
- Singular value decomposition
- …

- Iterative methods
- Jacobi iteration
- Gauss-Seidel iteration
- …

- Definition
- element
- row
- column
- row matrix, column matrix
- square matrix
- order= MxN (M rows, N columns)
- diagonal matrix
- identity matrix : I
- upper/lower triangular matrix
- tri-diagonal matrix
- transposed matrix: At
- symmetric matrix: A=At
- orthogonal matrix: At A= I

- Diagonal dominance
- Transpose facts

- Matrix multiplication

C

- Over-determined problem (m>n)
- least-square estimation,
- robust estimation etc.

- Under-determined problem (n<m)
- singular value decomposition

Upper triangular matrix

Lower triangular matrix

- Step 1: Gauss reduction
- =Forward elimination
- Coefficient matrix upper triangular matrix

- Step 2: Backward substitution

Gauss

reduction

- Harmful effect of round-off error in pivot coefficient

Pivoting strategy

- To determine the smallest such that
and perform

Partial pivoting

dramatic enhancement!

- Scaling is to ensure that the largest element in each row has a relative magnitude of 1 before the comparison for row interchange is performed.

Too much!

1) Augmented matrix의 행을 최대값이 1이 되도록 scaling(=normalization)

2) 첫 번째 열에 가장 큰 원소가 오도록 partial pivoting

3) 둘째 행 이하의 첫 열을 모두 0이 되도록 eliminating

4) 2행에서 n행까지 1)- 3)을 반복

5) backward substitution으로 해를 구함

0

0

0

Backward substitution

For each column

- Principle: Solving a set of linear equations based on decomposing the given coefficient matrix into a product of lower and upper triangular matrix.

A=LU

L-1

Ax = b LUx = b L-1 LUx = L-1 b

Upper triangular

L-1 b=c

U x = c

(1)

L

Lower triangular

L L-1 b = Lc

L c = b

(2)

By solving the equations (2) and (1) successively, we get the solution x.

- Doolittle decomposition
- L의 diagonal element 를 모두 1로 만들어줌

- Crout decomposition
- U의 diagonal element 를 모두 1로 만들어줌

- Cholesky decomposition
- L과 U의 diagonal element 를 같게 만들어줌
- symmetric, positive-definite matrix에 적합

- Solving a set of linear equations

- Obtaining inverse matrix

- Calculating the determinant of a matrix

[Due: 10/22]