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DEF: Characteristic Polynomial (of degree n) QR - Algorithm Note: 1) QR – Algorithm is different from QR-Decomposition 2) a procedure to calculate the eigenvalues of a matrix The eigenvalues of A are the n roots of its characteristic polynomial 1 2 3 The set of these roots is called the spectrum of A and denoted by for i=1:50 [Q,R]=qr(A); A=R*Q; end QR Algorithm:(Basic version) Find QR factorization Multiply in the reverse order

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DEF: Characteristic Polynomial (of degree n) Eigenvalues & Eigenvectors The eigenvalues of A are the n roots of its characteristic polynomial 1 2 3 The set of these roots is called the spectrum of A and denoted by 4 5 6 If A is symmetric then all eigenvalues are reals

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DEF: the nonzero vectors x that satisfy Eigenvalues & Eigenvectors are called eigenvectors. DEF: THM: same eigenvalues

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Special Matrices DiagonalTridiagonalUpper triangular Upper Hessenberg Upper bidiagonal

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Schur Factorization Schur Factorization: 1)A and T are similar 2)They have same eigenvalues 3)Eigenvalues of T are the diagonal entries

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Example: Compute all eigenvalues of A: Schur Factorization IDEA: We need to construct the Schur factorization for i=1:50 [Q,R]=qr(A); A=R*Q; end QR Algorithm:(Basic version) >> eig(A)‘ -100.0111 -10.0988 -0.8901 A5 = -100.0111 0.0000 0.0000 0.0000 -10.0988 0.0000 0 0.0000 -0.8901 A3 = -100.0111 0.0010 0.0000 0.0010 -10.0988 0.0007 0 0.0007 -0.8901

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All A’s are similar Compute all eigenvalues of A: Schur Factorization IDEA: We need to construct the Schur factorization for i=1:50 [Q,R]=qr(A); A=R*Q; end QR Algorithm:(Basic version) All A’s have same eigenvalues

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Compute all eigenvalues of A: Schur Factorization IDEA: We need to construct the Schur factorization QR-factorization

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Idea QR-factorization Bad Idea All zeros are destroyed

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Idea Bad Idea Good Idea zeros are not destroyed

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Idea Good Idea It is not upper triangular Hessenberg Givens

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Idea Good Idea PHASE-I PHASE-II QR-Algorithm for i=1:50 [Q,R]=qr(A); A=R*Q; end QR Algorithm: for i=1:50 [Q,R]=qr(H); H=R*Q; end QR Algorithm: General matrix A Hessenberg H O(n^3) All H’s are Hessenberg (why?) We can upper triangularize H with a sequence of Givens O(n^2) 6n^2

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for k=1:n-1 v=house(A(k+1:n,k); A(k+1:n,k:n)=(I-2*v*v’/v’*v)*A(k+1:n,k:n); A(1:n,k+1:n)=A(1:n,k+1:n)(I-2*v*v’/v*v); end Reduce H Idea PHASE-I PHASE-II for i=1:50 [Q,R]=qr(H); H=R*Q; end QR Algorithm: If A is symmetric matrix, then H is Hessenberg and symmetric A symmetric The work reduces to 50% or less

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Example Slow Convergence PHASE-I PHASE-II A = 1 2 3 4 4 5 6 7 2 1 5 0 4 2 1 0 H = 1.0000 -5.0000 -1.5395 -1.2767 -6.0000 8.5556 0.1085 3.6986 0 -5.9182 -2.1689 -1.1428 0 0 -0.1428 3.6133 >> eig(A )' 11.1055 -3.8556 3.5736 0.1765 H20 = 11.1055 -4.7403 3.9060 -4.0296 -0.0000 -3.8526 -0.6899 1.2559 0 -0.0322 3.5706 0.1571 0 0 -0.0000 0.1765 H5 = 11.1328 4.5792 3.9941 -4.0372 -0.0905 -3.8908 0.5327 -1.2342 0 -0.1006 3.5815 0.1342 0 0 -0.0000 0.1765 Any solution for low convergence

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Remark: Deflation Remark: We assume that H is unreduced. If not then we have Decouple and Deflation The problem decouples into two smaller problem

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QR Algorithm with shift PHASE-II QR Algorithm with shift: A20 = 11.1055 -4.7403 3.9060 -4.0296 -0.0000 -3.8526 -0.6899 1.2559 0 -0.0322 3.5706 0.1571 0 0 -0.0000 0.1765 >> eig(A )' 11.1055 -3.8556 3.5736 0.1765 The theorem says that if we shift an exact eigenvalue, then in exact arithmetic deflation occurs in one step.

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Example: QR Algorithm with shift PHASE-II QR Algorithm with shift: >> eig(A)' 11.1055 -3.8556 3.5736 0.1765 H = 1.0000 -5.0000 -1.5395 -1.2767 -6.0000 8.5556 0.1085 3.6986 0 -5.9182 -2.1689 -1.1428 0 0 -0.1428 3.6133 H1 = 3.3236 -5.5508 4.4353 2.6929 -8.0984 2.9128 -6.1528 -2.4688 0 -2.2250 1.1900 1.0296 0 0 -0.0017 3.5736 H2 = 2.4924 -9.7341 -2.5648 2.3834 -5.9560 5.0131 3.0854 -2.9507 0 -1.4702 -0.0791 0.0767 0 0 -0.0000 3.5736 deflation

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Example: QR Algorithm with shift H = 1.0000 -5.0000 -1.5395 -1.2767 -6.0000 8.5556 0.1085 3.6986 0 -5.9182 -2.1689 -1.1428 0 0 -0.1428 3.6133 H1 = 3.3236 -5.5508 4.4353 2.6929 -8.0984 2.9128 -6.1528 -2.4688 0 -2.2250 1.1900 1.0296 0 0 -0.0017 3.5736 H2 = 2.4924 -9.7341 -2.5648 2.3834 -5.9560 5.0131 3.0854 -2.9507 0 -1.4702 -0.0791 0.0767 0 0 -0.0000 3.5736 Theory: quadratic conv Symmetric case : cubic H5 = 11.0798 5.2490 -3.2292 -0.0099 0.1708 1.2672 0 -0.0000 -3.8640 H4 11.0926 5.2199 -3.2299 -0.0368 0.1589 1.2723 0 -0.0029 -3.8649 H3 = 11.1418 5.2542 2.9806 -0.1409 0.0453 -1.4768 0 -0.1735 -3.8005 deflation

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Eigenvalues of Symmetric Matrices PHASE-I PHASE-II for i=1:50 [Q,R]=qr(H); H=R*Q; end QR Algorithm: If A is symmetric matrix, then H is Hessenberg and symmetric A symmetric T is tridiagonal O(n) Schur Factorization:

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Wilkinson shift: QR Algorithm with wilkinson shift PHASE-II QR Algorithm with shift: A more effective choice is to shift by the eigenvalue of That is closer to Wilkinson has shown that: cubically convergent. Can we use this shift to non-symmetric matrix

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