An extension of the QZ algorithm for solving the generalized matrix eigenvalue problem
National Aeronautics and Space Administration, 1973 - Mathematics - 52 pages
This algorithm is an extension of Moler and Stewart's QZ algorithm with some added features for saving time and operations. Also, some additional properties of the QR algorithm which were not practical to implement in the QZ algorithm can be generalized with the combination shift QZ algorithm. Numerous test cases are presented to give practical application tests for algorithm. Based on results, this algorithm should be preferred over existing algorithms which attempt to solve the class of generalized eigenproblems where both matrices are singular or nearly singular.
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COMBINATION SHIFT QZ ALGORITHM
THEORETICAL COMPARISON OF THE COMBINATION SHIFT
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60 percent real annihilate the element Average percentage average Row standard Column average column of AB-1 Column standard combination shift QZ complex conjugate complex eigenvalues computed consecutive small subdiagonals determine diag double shift iteration double shift QZ eigen eigenproblems eigenvectors equal to zero Hessenberg matrix Householder reflection Householder transformation implicit QZ iteration itera ITERATION TIME COMPARISON Langley Research Center MATRIX EIGENVALUE PROBLEM matrix which annihilates nearly singular negligible operation count orthogonal matrix Percent of QZ percent real eigenvalues percentage of real Postmultiplying QZ for matrix real eigenvalues Category REAL EIGENVALUES Test reduce Row average Row row of Q Row standard deviation saved shift estimate shift implicit QZ shift QZ algorithm shift QZ iteration shifts are real single shift implicit single shift iteration singular or nearly subdiagonal elements symmetric symmetric matrix test case II-5 Test case Percent test the algorithms unitary matrix unitary transformations unstable eigenvalues upper triangular form Wilkinson ref