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Numerical Analysis – Linear Equations(I). Hanyang University Jong-Il Park. Linear equations. N unknowns, M equations. coefficient matrix. where. Solving methods. Direct methods Gauss elimination Gauss-Jordan elimination LU decomposition Singular value decomposition …

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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

Over-determined/Under-determined problem

• 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]