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set1 - $ Set 1 Problems and Decompositions of Interest Kyle...

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a39 a38 a36 a37 Set 1: Problems and Decompositions of Interest Kyle A. Gallivan Department of Mathematics Florida State University Numerical Linear Algebra Fall 2010 1
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a39 a38 a36 a37 Problems of Interest Solving systems square – nonsingular and singular linear least squares – full rank and rank deficient dense, sparse, structured Eigenvalue problems Hermitian and symmetric non-Hermitian and non-symmetric generalized forms dense, sparse, structured Singular value problems standard and generalized forms dense, sparse, structured 2
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a39 a38 a36 a37 Algorithms of Interest Efficient acceptable computational complexity acceptable storage complexity Efficiently exploit algebraic structure of problem representation structure, e.g., free parameters for the matrices architectural structure of computing platform Robust numerically stable detect problems where the algorithm is not reliable 3
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a39 a38 a36 a37 Overall Strategy Our discussion is based on matrix decompositions matrix transformations iterations (finite and convergent) decompositions are used for analytical purposes, e.g., to characterize solutions and their properties decompositions are used for algorithmic purposes, e.g., they serve as goals for the matrix transformations and influence accuracy and complexity decompositions are often computed via repeated application of one or more types transformations 4
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a39 a38 a36 a37 Examples of Matrix Decompositions/Problems Cholesky decomposition: A = A H and x H Ax 0 A = LL H where L is lower triangular with positive diagonal elements and the same rank as A . LU decomposition: P L AP R = LDU where P L and P R are permutation matrices D is diagonal, L is unit lower trapezoidal and is U unit upper trapezoidal and D , L and U have the same rank as A . 5
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a39 a38 a36 a37 Examples of Matrix Decompositions/Problems QR decomposition: P L AP R = QR with Q unitary and R upper triangular with the same rank as A . polar decomposition: given m × n matrix A with m n A = ZP Z is m × n with orthonormal columns and P is symmetric positive semidefinite. 6
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a39 a38 a36 a37 Examples of Matrix Decompositions/Problems eigenvalue problem/decomposition: given n × n matrix A find scalar/vector pairs ( λ i , x i ) Ax i = x i λ i x i linearly independent from all other x j . λ i may or may not be distinct from all other λ j . The number of such pairs possible determines the form of the decomposition. singular value decomposition given m × n matrix A find r = rank ( A ) scalar/vector triples ( σ i , u i , v i ) Av i = u i σ i , σ i > 0 , bardbl u i bardbl 2 = bardbl v i bardbl 2 = 1 u H i u j = 0 and v H i v j = 0 , when i negationslash = j 7
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a39 a38 a36 a37 Examples of Matrix Decompositions/Problems CS decomposition: Given, k 1 and k 2 , defining an arbitrary partition of an n × n orthogonal matrix Q Q 11 Q 12 Q 21 Q 22 = U 1 0 U 2 Σ 1 Σ 2 Σ 2 Σ 3 V T 1 0 V T 2 Σ 1 = I 0 0 0 C 0 0 0 0
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