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Computing the eigenvalues of a companion matrix Marc Van Barel Computing the eigenvalues of a Joint work with Raf Vandebril and Paul Van Dooren companion matrix Structured Linear Algebra Problems: Analysis, Algorithms, and Applications

  1. Computing the eigenvalues of a companion matrix Marc Van Barel Computing the eigenvalues of a Joint work with Raf Vandebril and Paul Van Dooren companion matrix Structured Linear Algebra Problems: Analysis, Algorithms, and Applications The problem Cortona, Italy Companion matrix Working with Givens September 15-19, 2008 transformations Representation Fusion and shift-through operations Unitary plus rank one matrices Structure under a QR -step A representation for the unitary matrix Representation of the unitary plus rank one matrix Implicit QR -algorithm with single shift Initialization Marc Van Barel The chasing Joint work with Raf Vandebril and Paul Van Dooren The last Givens transformation Dept. Computer Science, K.U.Leuven, Belgium Numerical Experiments Dept. of Math. Eng., Catholic University of Louvain, Belgium Scaling Comparison 1 / 64

  2. Contents Computing the eigenvalues of a companion matrix Marc Van Barel Joint work with Raf The problem Vandebril and Paul Van Dooren Companion matrix Working with Givens transformations Representation Fusion and shift-through operations The problem Companion matrix Unitary plus rank one matrices Working with Givens transformations Structure under a QR -step Representation A representation for the unitary matrix Fusion and shift-through operations Representation of the unitary plus rank one matrix Unitary plus rank one matrices Implicit QR -algorithm with single shift Structure under a QR -step A representation for the Initialization unitary matrix Representation of the The chasing unitary plus rank one matrix Implicit QR -algorithm The last Givens transformation with single shift Initialization Numerical Experiments The chasing The last Givens Scaling transformation Numerical Comparison Experiments Scaling Comparison 2 / 64

  3. Outline Computing the eigenvalues of a companion matrix Marc Van Barel Joint work with Raf The problem Vandebril and Paul Van Dooren Companion matrix Working with Givens transformations Representation Fusion and shift-through operations The problem Companion matrix Unitary plus rank one matrices Working with Givens transformations Structure under a QR -step Representation A representation for the unitary matrix Fusion and shift-through operations Representation of the unitary plus rank one matrix Unitary plus rank one matrices Implicit QR -algorithm with single shift Structure under a QR -step A representation for the Initialization unitary matrix Representation of the The chasing unitary plus rank one matrix Implicit QR -algorithm The last Givens transformation with single shift Initialization Numerical Experiments The chasing The last Givens Scaling transformation Numerical Comparison Experiments Scaling Comparison 3 / 64

  4. Outline Computing the eigenvalues of a companion matrix Marc Van Barel Joint work with Raf The problem Vandebril and Paul Van Dooren Companion matrix Working with Givens transformations Representation Fusion and shift-through operations The problem Companion matrix Unitary plus rank one matrices Working with Givens transformations Structure under a QR -step Representation A representation for the unitary matrix Fusion and shift-through operations Representation of the unitary plus rank one matrix Unitary plus rank one matrices Implicit QR -algorithm with single shift Structure under a QR -step A representation for the Initialization unitary matrix Representation of the The chasing unitary plus rank one matrix Implicit QR -algorithm The last Givens transformation with single shift Initialization Numerical Experiments The chasing The last Givens Scaling transformation Numerical Comparison Experiments Scaling Comparison 4 / 64

  5. Companion matrix Computing the eigenvalues of a companion matrix Marc Van Barel Definition Joint work with Raf Vandebril and Paul Van Given a monic polynomial Dooren p ( z ) = p 0 + p 1 z + p 2 z 2 + ... + p n − 1 z n − 1 + z n with the coefficients p i ∈ R or C The problem the associated companion matrix C p is defined as: Companion matrix Working with Givens transformations   0 − p 0 Representation Fusion and shift-through ... operations   1 − p 1   C p = . Unitary plus rank one  .  ... matrices .   . Structure under a QR -step   A representation for the 1 − p n − 1 unitary matrix Representation of the unitary plus rank one matrix Implicit QR -algorithm with single shift The eigenvalues of the companion matrix coincide with the Initialization The chasing zeros of the associated polynomial p ( z ) , because The last Givens transformation p ( z ) = det ( zI − C p ) . Numerical Experiments Scaling Comparison 5 / 64

  6. Unitary plus rank one matrix Computing the eigenvalues of a companion matrix Marc Van Barel Joint work with Raf Vandebril and Paul Van C p = H upper Hessenberg Dooren U + uv H = unitary plus rank one with   0 0 ··· 0 ± 1 The problem 1 0 ··· 0 0 Companion matrix   Working with Givens   0 1 ··· 0 0 U = transformations   . . . .  ...  Representation . . . .   . . . . Fusion and shift-through   operations 0 0 ··· 1 0 Unitary plus rank one matrices v T Structure under a QR -step = ( 0 , 0 ,..., 0 , 1 ) A representation for the unitary matrix u T = ( − p 0 ∓ 1 , − p 1 ,..., − p n − 1 ) Representation of the unitary plus rank one matrix Hence, computing the zeros of the polynomial p ( z ) is Implicit QR -algorithm with single shift equivalent to computing the eigenvalues of the upper Initialization The chasing Hessenberg, unitary plus rank one matrix The last Givens transformation C p = H = U + uv H . Numerical Experiments Scaling Comparison 6 / 64

  7. Outline Computing the eigenvalues of a companion matrix Marc Van Barel Joint work with Raf The problem Vandebril and Paul Van Dooren Companion matrix Working with Givens transformations Representation Fusion and shift-through operations The problem Companion matrix Unitary plus rank one matrices Working with Givens transformations Structure under a QR -step Representation A representation for the unitary matrix Fusion and shift-through operations Representation of the unitary plus rank one matrix Unitary plus rank one matrices Implicit QR -algorithm with single shift Structure under a QR -step A representation for the Initialization unitary matrix Representation of the The chasing unitary plus rank one matrix Implicit QR -algorithm The last Givens transformation with single shift Initialization Numerical Experiments The chasing The last Givens Scaling transformation Numerical Comparison Experiments Scaling Comparison 7 / 64

  8. Outline Computing the eigenvalues of a companion matrix Marc Van Barel Joint work with Raf The problem Vandebril and Paul Van Dooren Companion matrix Working with Givens transformations Representation Fusion and shift-through operations The problem Companion matrix Unitary plus rank one matrices Working with Givens transformations Structure under a QR -step Representation A representation for the unitary matrix Fusion and shift-through operations Representation of the unitary plus rank one matrix Unitary plus rank one matrices Implicit QR -algorithm with single shift Structure under a QR -step A representation for the Initialization unitary matrix Representation of the The chasing unitary plus rank one matrix Implicit QR -algorithm The last Givens transformation with single shift Initialization Numerical Experiments The chasing The last Givens Scaling transformation Numerical Comparison Experiments Scaling Comparison 8 / 64

  9. Representation Computing the eigenvalues of a companion matrix ◮ Givens transformations are a powerful tool for working with Marc Van Barel Joint work with Raf structured matrices. Vandebril and Paul Van Dooren ◮ representation of sequences of Givens transformations by a graphical scheme. ◮ example: the QR -factorization of a 6 × 6 Hessenberg matrix The problem ➊ × × × × × × Companion matrix � � ➋ × × × × × Working with Givens � � transformations ➌ × × × × � Representation � ➍ × × × � Fusion and shift-through � × × operations ➎ � � × ➏ Unitary plus rank one matrices 5 4 3 2 1 Structure under a QR -step A representation for the unitary matrix The figure corresponds to G 5 G 4 ... G 1 R . Representation of the Rewriting the formula we have G H 1 ... G H unitary plus rank one matrix 5 H = R . Implicit QR -algorithm Hence G H 5 annihilates the first subdiagonal element, G H with single shift 4 Initialization the second and so forth. The chasing The last Givens transformation Numerical Experiments Scaling Comparison 9 / 64

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