Numerical Analysis – Linear Equations(I) Hanyang University Jong-Il Park.
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Transcript of Numerical Analysis – Linear Equations(I) Hanyang University Jong-Il Park.
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Numerical Analysis – Numerical Analysis – Linear Equations(I)Linear Equations(I)
Hanyang University
Jong-Il Park
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Linear equationsLinear equations N unknowns, M equations
where
coefficient matrix
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Solving methodsSolving methods Direct methods
Gauss elimination Gauss-Jordan elimination LU decomposition Singular value decomposition …
Iterative methods Jacobi iteration Gauss-Seidel iteration …
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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
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Diagonal dominance
Transpose facts
Basic properties of Basic properties of matrices(II)matrices(II)
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Basic properties of matrices(III)Basic properties of matrices(III) Matrix multiplication
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DeterminantDeterminant
C
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Determinant facts(I)Determinant facts(I)
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Determinant facts(II)Determinant facts(II)
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Geometrical interpretation of Geometrical interpretation of determinantdeterminant
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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
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Augmented matrixAugmented matrix
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Cramer’s ruleCramer’s rule
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Triangular coefficient matrixTriangular coefficient matrix
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SubstitutionSubstitution
Upper triangular matrix
Lower triangular matrix
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Gauss eliminationGauss elimination
1. Step 1: Gauss reduction =Forward elimination Coefficient matrix upper triangular matrix
2. Step 2: Backward substitution
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Gauss reductionGauss reduction
Gaussreduction
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Eg. Gauss elimination(I)Eg. Gauss elimination(I)
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Eg. Gauss elimination(II)Eg. Gauss elimination(II)
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Troubles in Gauss eliminationTroubles in Gauss elimination Harmful effect of round-off error in pivot
coefficient
Pivoting strategy
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Eg. Trouble(I)Eg. Trouble(I)
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Eg. Trouble(II)Eg. Trouble(II)
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Pivoting strategyPivoting strategy To determine the smallest such that
and perform
Partial pivotingdramatic enhancement!
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Effect of partial pivotingEffect of partial pivoting
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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.
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Eg. Effect of scalingEg. Effect of scaling
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Complexity of Gauss eliminationComplexity of Gauss elimination
Too much!
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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
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Gauss-Jordan eliminationGauss-Jordan elimination
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Eg. Obtaining inverse matrix(IEg. Obtaining inverse matrix(I))
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Eg. Obtaining inverse matrix(IIEg. Obtaining inverse matrix(II))
Backward substitution For each column
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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
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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
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Crout decompositionCrout decomposition
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Implementation of Crout methodImplementation of Crout method
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Programming using NR in C(I)Programming using NR in C(I) Solving a set of linear equations
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Programming using NR in C(II)Programming using NR in C(II) Obtaining inverse matrix
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Programming using NR in C(III)Programming using NR in C(III) Calculating the determinant of a matrix
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Homework #5 (Cont’)Homework #5 (Cont’)[Due: 10/22]
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(Cont’) Homework #5(Cont’) Homework #5