calculus approach to matrix eigenvalue algorithms - hueper

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1、ACalculusApproachtoMatrixEigenvalueAlgorithmsHabilitationsschriftderFakultÄatfurÄMathematikundInformatikderBayerischenJulius-Maximilians-UniversitÄatWurzburgÄfurÄdasFachMathematikvorgelegtvonKnutHupÄerWurzburgÄimJuli20022MeinerFrauBarbaraundunserenKindernLea,JuvalundNoagewidmetContents1Introdu

2、ction52Jacobi-typeAlgorithmsandCyclicCoordinateDescent82.1Algorithms.............................82.1.1JacobiandCyclicCoordinateDescent.........92.1.2BlockJacobiandGroupedVariableCyclicCoordinateDescent..........................102.1.3ApplicationsandExamplesfor1-dimensionalOptimiza-tion.......

3、.....................122.1.4ApplicationsandExamplesforBlockJacobi......222.2LocalConvergenceAnalysis...................232.3Discussion.............................313Re¯ningEstimatesofInvariantSubspaces323.1LowerUnipotentBlockTriangularTransformations......333.2Algorithms......................

4、.......373.2.1MainIdeas........................373.2.2FormulationoftheAlgorithm..............403.2.3LocalConvergenceAnalysis...............443.2.4FurtherInsighttoOrderings...............483.3OrthogonalTransformations...................523.3.1TheAlgorithm......................573.3.2LocalConver

5、genceAnalysis...............593.3.3DiscussionandOutlook.................624RayleighQuotientIteration,QR-Algorithm,andSomeGen-eralizations634.1LocalCubicConvergenceofRQI................64CONTENTS44.2ParallelRayleighQuotientIterationorMatrix-valuedShiftedQR-Algorithms..........................69

6、4.2.1Discussion.........................724.3LocalConvergencePropertiesoftheShiftedQR-Algorithm..73Chapter1IntroductionTheinteractionbetweennumericallinearalgebraandcontroltheoryhascru-ciallyin°uencedthedevelopmentofnumericalalgorithmsforlinearsystemsinthepast.Sincetheperformanceofacontrolsyst

7、emcanoftenbemea-suredintermsofeigenvaluesorsingularvalues,matrixeigenvaluemethodshavebecomeanimportanttoolfortheimplementationofcontrolalgorithms.StandardnumericalmethodsforeigenvalueorsingularvaluecomputationsarebasedontheQR-algorithm.