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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

Linear equations
• N unknowns, M equations

coefficient matrix

where

Solving methods
• Direct methods
• Gauss elimination
• Gauss-Jordan elimination
• LU decomposition
• Singular value decomposition
• Iterative methods
• Jacobi iteration
• Gauss-Seidel iteration
Basic properties of matrices(I)
• 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
Basic properties of matrices(II)
• Diagonal dominance
• Transpose facts
Over-determined/Under-determined problem
• Over-determined problem (m>n)
• least-square estimation,
• robust estimation etc.
• Under-determined problem (n<m)
• singular value decomposition
Substitution

Upper triangular matrix

Lower triangular matrix

Gauss elimination
• Step 1: Gauss reduction
• =Forward elimination
• Coefficient matrix  upper triangular matrix
• Step 2: Backward substitution
Gauss reduction

Gauss

reduction

Troubles in Gauss elimination
• Harmful effect of round-off error in pivot coefficient

Pivoting strategy

Pivoting strategy
• To determine the smallest such that

and perform

 Partial pivoting

dramatic enhancement!

Scaled partial pivoting
• 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.
Summary: Gauss elimination

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

Eg. Obtaining inverse matrix(II)

Backward substitution

For each column

LU decomposition
• 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.

Various LU decompositions
• 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
Programming using NR in C(I)
• Solving a set of linear equations
Programming using NR in C(II)
• Obtaining inverse matrix
Programming using NR in C(III)
• Calculating the determinant of a matrix