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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)

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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


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

Basic properties of matrices iii

Basic properties of matrices(III)

  • Matrix multiplication




Determinant facts i

Determinant facts(I)

Determinant facts ii

Determinant facts(II)

Geometrical interpretation of determinant

Geometrical interpretation of determinant

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

Augmented matrix

Augmented matrix

Cramer s rule

Cramer’s rule

Triangular coefficient matrix

Triangular coefficient matrix



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



Eg gauss elimination i

Eg. Gauss elimination(I)

Eg gauss elimination ii

Eg. Gauss elimination(II)

Troubles in gauss elimination

Troubles in Gauss elimination

  • Harmful effect of round-off error in pivot coefficient

Pivoting strategy

Eg trouble i

Eg. Trouble(I)

Eg trouble ii

Eg. Trouble(II)

Pivoting strategy

Pivoting strategy

  • To determine the smallest such that

    and perform

     Partial pivoting

    dramatic enhancement!

Effect of partial pivoting

Effect of partial pivoting

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.

Eg effect of scaling

Eg. Effect of scaling

Complexity of gauss elimination

Complexity of Gauss elimination

 Too much!

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




Gauss jordan elimination

Gauss-Jordan elimination

Eg obtaining inverse matrix i

Eg. Obtaining inverse matrix(I)

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.



Ax = b  LUx = b  L-1 LUx = L-1 b

Upper triangular

 L-1 b=c 

U x = c



Lower triangular

L L-1 b = Lc 

L c = b


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

Crout decomposition

Crout decomposition

Implementation of crout method

Implementation of Crout method

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

Homework 5 cont

Homework #5 (Cont’)

[Due: 10/22]

Cont homework 5

(Cont’) Homework #5

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