Numerical Analysis – Linear Equations(I)
description
Transcript of Numerical Analysis – Linear Equations(I)
Numerical Analysis – Numerical Analysis – Linear Equations(I)Linear Equations(I)
Hanyang University
Jong-Il Park
Linear equationsLinear equations N unknowns, M equations
where
coefficient matrix
Solving methodsSolving 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
Diagonal dominance
Transpose facts
Basic properties of Basic properties of matrices(II)matrices(II)
Basic properties of matrices(III)Basic properties of matrices(III) Matrix multiplication
DeterminantDeterminant
C
Determinant facts(I)Determinant facts(I)
Determinant facts(II)Determinant facts(II)
Geometrical interpretation of Geometrical interpretation of determinantdeterminant
Over-determined/Over-determined/Under-determined problemUnder-determined problem
Over-determined problem (m>n) least-square estimation, robust estimation etc.
Under-determined problem (n<m) singular value decomposition
Augmented matrixAugmented matrix
Cramer’s ruleCramer’s rule
Triangular coefficient matrixTriangular coefficient matrix
SubstitutionSubstitution
Upper triangular matrix
Lower triangular matrix
Gauss eliminationGauss elimination
1. Step 1: Gauss reduction =Forward elimination Coefficient matrix upper triangular matrix
2. Step 2: Backward substitution
Gauss reductionGauss reduction
Gaussreduction
Eg. Gauss elimination(I)Eg. Gauss elimination(I)
Eg. Gauss elimination(II)Eg. Gauss elimination(II)
Troubles in Gauss eliminationTroubles 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 strategyPivoting strategy To determine the smallest such that
and perform
Partial pivotingdramatic enhancement!
Effect of partial pivotingEffect of partial pivoting
Scaled partial pivotingScaled 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 scalingEg. Effect of scaling
Complexity of Gauss eliminationComplexity of Gauss elimination
Too much!
Summary: Gauss eliminationSummary: Gauss elimination
1) Augmented matrix 의 행을 최대값이 1 이 되도록 scaling(=normalization)
2) 첫 번째 열에 가장 큰 원소가 오도록 partial pivoting
3) 둘째 행 이하의 첫 열을 모두 0 이 되도록 eliminating
4) 2 행에서 n 행까지 1)- 3) 을 반복5) backward substitution 으로 해를 구함
00
0
Gauss-Jordan eliminationGauss-Jordan elimination
Eg. Obtaining inverse matrix(IEg. Obtaining inverse matrix(I))
Eg. Obtaining inverse matrix(IIEg. Obtaining inverse matrix(II))
Backward substitution For each column
LU decompositionLU 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 bA=LU L-1
L-1 b=c U x = c
L
L L-1 b = Lc L c = b
(1)
(2) By solving the equations (2) and (1) successively, we get the solution x.
Upper triangular
Lower triangular
Various LU decompositionsVarious LU decompositions
Doolittle decompositionL의 diagonal element 를 모두 1 로 만들어줌
Crout decompositionU의 diagonal element 를 모두 1 로 만들어줌
Cholesky decomposition L 과 U 의 diagonal element 를 같게 만들어줌 symmetric, positive-definite matrix 에 적합
Crout decompositionCrout decomposition
Implementation of Crout methodImplementation 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