Algorithms for hyperbolic quadratic eigenvalue problems
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Abstract
We consider the quadratic eigenvalue problem (or the QEP) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis lamda squared upper A plus lamda upper B plus upper C right-parenthesis x equals 0"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi> λ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>A</mml:mi> <mml:mo>+</mml:mo> <mml:mi> λ </mml:mi> <mml:mi>B</mml:mi> <mml:mo>+</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">(\lambda ^2 A+\lambda B + C)x=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A comma upper B comma"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A, B,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are Hermitian with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> positive definite. The QEP is called hyperbolic if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Superscript asterisk Baseline upper B x right-parenthesis squared greater-than 4 left-parenthesis x Superscript asterisk Baseline upper A x right-parenthesis left-parenthesis x Superscript asterisk Baseline upper C x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mi>B</mml:mi> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mspace width="negativethinmathspace"/> <mml:mo>></mml:mo> <mml:mspace width="negativethinmathspace"/> <mml:mn>4</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mi>C</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x^*Bx)^2\!>\!4(x^*Ax)(x^*Cx)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all nonzero <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of double-struck upper C Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">x\in {\mathbb C}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that a relatively efficient test for hyperbolicity can be obtained by computing the eigenvalues of the QEP. A hyperbolic QEP is overdamped if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is positive definite and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is positive semidefinite. We show that a hyperbolic QEP (whose eigenvalues are necessarily real) is overdamped if and only if its largest eigenvalue is nonpositive. For overdamped QEPs, we show that all eigenpairs can be found efficiently by finding two solutions of the corresponding quadratic matrix equation using a method based on cyclic reduction. We also present a new measure for the degree of hyperbolicity of a hyperbolic QEP.
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