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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

Numerical Analysis – Linear Equations(I)

Hanyang University

Jong-Il Park

linear equations
Linear equations
  • N unknowns, M equations

coefficient matrix

where

solving methods
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
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
Basic properties of matrices(II)
  • Diagonal dominance
  • Transpose facts
over determined under determined problem
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
Substitution

Upper triangular matrix

Lower triangular matrix

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

Gauss

reduction

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

Pivoting strategy

pivoting strategy
Pivoting strategy
  • To determine the smallest such that

and perform

 Partial pivoting

dramatic enhancement!

scaled partial pivoting
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
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
Eg. Obtaining inverse matrix(II)

Backward substitution

For each column

lu decomposition
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
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
Programming using NR in C(I)
  • Solving a set of linear equations
programming using nr in c ii
Programming using NR in C(II)
  • Obtaining inverse matrix
programming using nr in c iii
Programming using NR in C(III)
  • Calculating the determinant of a matrix
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