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Accurate Eigenvalue Decomposition of Arrowhead Matrices and Applications Accurate eigenvalue decomposition of arrowhead matrices and applications Accurate eigenvalue decomposition of arrowhead matrices and applications Nevena Jakovˇcevi´cStor FESB, University of Split joint work with Ivan Slapniˇcar and Jesse Barlow IWASEP9 June 4th, 2012. 1/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. 2/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. Under certain conditions, the aheig algorithm computes all eigenvalues and all components of corresponding eigenvectors with high rel. acc. in O(n2) operations. 2/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. Under certain conditions, the aheig algorithm computes all eigenvalues and all components of corresponding eigenvectors with high rel. acc. in O(n2) operations. 2/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. Under certain conditions, the aheig algorithm computes all eigenvalues and all components of corresponding eigenvectors with high rel. acc. in O(n2) operations. The algorithm is based on shift-and-invert technique and limited finite O(n) use of double precision arithmetic when necessary. 2/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. Under certain conditions, the aheig algorithm computes all eigenvalues and all components of corresponding eigenvectors with high rel. acc. in O(n2) operations. The algorithm is based on shift-and-invert technique and limited finite O(n) use of double precision arithmetic when necessary. Orthogonality of eigenvectors computed by aheig algorithm is consequence of their accuracy (no need for follow-up orthogonalization). 2/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. Under certain conditions, the aheig algorithm computes all eigenvalues and all components of corresponding eigenvectors with high rel. acc. in O(n2) operations. The algorithm is based on shift-and-invert technique and limited finite O(n) use of double precision arithmetic when necessary. Orthogonality of eigenvectors computed by aheig algorithm is consequence of their accuracy (no need for follow-up orthogonalization). Each eigenvalue and corresponding eigenvector can be computed independently, which makes the algorithm adaptable for parallel computing. 2/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. Under certain conditions, the aheig algorithm computes all eigenvalues and all components of corresponding eigenvectors with high rel. acc. in O(n2) operations. The algorithm is based on shift-and-invert technique and limited finite O(n) use of double precision arithmetic when necessary. Orthogonality of eigenvectors computed by aheig algorithm is consequence of their accuracy (no need for follow-up orthogonalization). Each eigenvalue and corresponding eigenvector can be computed independently, which makes the algorithm adaptable for parallel computing. We also present the applications of aheig algorithm to hermitian arrowhead, symmetric tridiagonal matrices and diagonal plus rank one matrices. 2/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction Outline 1 Introduction 2 The idea of the aheig algorithm 3 Example 4 Accuracy of the aheig algorithm 5 Application to hermitian arrowhead matrices 6 Application to tridiagonal symmetric matrices 7 Application to ”diagonal + rank-one” matrices (D + uuT ) 3/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction Let D z A = zT α be n n real symmetric arrowhead matrix, where × T D = diag(d1,d2,...,dn−1), z = ζ1 ζ2 ζn−1 , α R ··· ∈ and 4/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction Let D z A = zT α be n n real symmetric arrowhead matrix, where × T D = diag(d1,d2,...,dn−1), z = ζ1 ζ2 ζn−1 , α R ··· ∈ and A = V ΛV T eigenvalue decomposition of A, where Λ = diag(λ1,λ2,...,λn) and V = v1 vn . ··· 4/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We assume that A is irreducible: ζi = 0, i, di = dj, i = j. 6 ∀ 6 ∀ 6 Without loss of generality we can assume that ζi > 0, i and ∀ that diagonal elements of D are ordered, that is d1 >d2 > >dn−1. ··· 5/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We assume that A is irreducible: ζi = 0, i, di = dj, i = j. 6 ∀ 6 ∀ 6 Without loss of generality we can assume that ζi > 0, i and ∀ that diagonal elements of D are ordered, that is d1 >d2 > >dn−1. ··· The assumptions on D imply the interlacing property λ1 >d1 >λ2 >d2 > >dn−2 >λn−1 >dn−1 >λn ··· where λi,i = 1,...,n, are eigenvalues of matrix A. 5/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction The eigenvalues of A are the zeros of function: n−1 2 ζi T −1 ϕA (λ)= α λ = α λ z (D λI) z. − − di λ − − − Xi=1 − 6/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction The eigenvalues of A are the zeros of function: n−1 2 ζi T −1 ϕA (λ)= α λ = α λ z (D λI) z. − − di λ − − − Xi=1 − and the eigenvectors are given by: −1 xi (D λiI) z vi = , xi = − , i = 1,...,n. xi 1 k k2 − 6/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction The eigenvalues of A are the zeros of function: n−1 2 ζi T −1 ϕA (λ)= α λ = α λ z (D λI) z. − − di λ − − − Xi=1 − and the eigenvectors are given by: −1 xi (D λiI) z vi = , xi = − , i = 1,...,n. xi 1 k k2 − Problem: λi is not exact = vi may not be orthogonal. ⇒ 6/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction The existing algorithms obtain orthogonal eigenvectors with the following procedure. Compute eigenvalues of A. Construct new matrix A (inverse problem) with prescribed eigenvalues λ, and diagonal matrix D (that is compute new z b and α). b b Computeb eigenvectors of A with the previous formula. This way computed, eigenvectorsb are not the exact eigenvectors of starting matrix A, they are exact eigenvectors of matrix D z A = , zT α b b b b i λ − d n−1 λ − d v j i j i |zi| = u di − λn λ1 − di , u db− − d bd − d u jY=2 j 1 i j=Yi+1 j i b t b b n−1 α = λn + λj − dj . jX=1 b b b 7/31 Accurate eigenvalue decomposition of arrowhead matrices and applications The idea of the aheig algorithm Our algorithm has a different concept. Accuracy of the eigenvectors and their orthogonality follow from accuracy of computed eigenvalues. Thus, there is no need for follow-up orthogonalization. 8/31 Accurate eigenvalue decomposition of arrowhead matrices and applications The idea of the aheig algorithm Our algorithm has a different concept. Accuracy of the eigenvectors and their orthogonality follow from accuracy of computed eigenvalues. Thus, there is no need for follow-up orthogonalization. Let di be the diagonal element (pole) in A which is closest to λ. 8/31 Accurate eigenvalue decomposition of arrowhead matrices and applications The idea of the aheig algorithm Our algorithm has a different concept. Accuracy of the eigenvectors and their orthogonality follow from accuracy of computed eigenvalues. Thus, there is no need for follow-up orthogonalization. Let di be the diagonal element (pole) in A which is closest to λ. From the interlacing property it follows that either λ = λi or λ = λi+1. 8/31 Accurate eigenvalue decomposition of arrowhead matrices and applications The idea of the aheig algorithm Let Ai be the shifted matrix, D1 0 0 z1 0 0 0 ζi Ai = A diI = , − 0 0 D2 z2 zT ζ zT a 1 i 2 9/31 Accurate eigenvalue decomposition of arrowhead matrices and applications The idea of the aheig algorithm Let Ai be the shifted matrix, D1 0 0 z1 0 0 0 ζi Ai = A diI = , − 0 0 D2 z2 zT ζ zT a 1 i 2 where D1 = diag(d1 − di,...,di−1 − di) positive definite, D2 = diag(di+1 − di,...,dn−1 − di) negative definite, T z1 = ζ1 ζ2 ··· ζi−1 , T z2 = ζi+1 ζi+2 ··· ζn−1 , a = α − di. 9/31 Accurate eigenvalue decomposition of arrowhead matrices and applications The idea of the aheig algorithm Let Ai be the shifted matrix, D1 0 0 z1 0 0 0 ζi Ai = A diI = , − 0 0 D2 z2 zT ζ zT a 1 i 2 where D1 = diag(d1 − di,...,di−1 − di) positive definite, D2 = diag(di+1 − di,...,dn−1 − di) negative definite, T z1 = ζ1 ζ2 ··· ζi−1 , T z2 = ζi+1 ζi+2 ··· ζn−1 , a = α − di. Obviously µ = λ di − is an eigenvalue of Ai iff λ is an eigenvalue of A. 9/31 Accurate eigenvalue decomposition of arrowhead matrices and applications The idea of the aheig algorithm Now −1 D1 w1 0 0 T T −1 w1 b w2 1/ζi A = − , i 0 w D 1 0 2 2 0 1/ζ 0 0 i 10/31 Accurate eigenvalue decomposition of arrowhead matrices and applications The idea of the aheig algorithm Now −1 D1 w1 0 0 T T −1 w1 b w2 1/ζi A = − , i 0 w D 1 0 2 2 0 1/ζ 0 0 i where −1 1 w1 = −D1 z1 , ζi −1 1 w2 = −D2 z2 , ζi 1 T − T − b = −a + z D 1z + z D 1z .
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