Numerical Computation of the Characteristic Polynomial of aComplex Matrix

eigenvalues;singular values;characteristic polynomials

Rehman, Rizwana ; Stephen L. Campbell, Committee Member,Carl D. Meyer, Committee Member,Dean J. Lee, Committee Co-Chair,Ilse C. F. Ipsen, Committee Chair,Rehman, Rizwana ; Stephen L. Campbell ; Committee Member ; Carl D. Meyer ; Committee Member ; Dean J. Lee ; Committee Co-Chair ; Ilse C. F. Ipsen ; Committee Chair

University:North Carolina State University

关键词: eigenvalues;    singular values;    characteristic polynomials;   

Others  :  https://repository.lib.ncsu.edu/bitstream/handle/1840.16/6262/etd.pdf?sequence=1&isAllowed=y

美国|英语

来源: null

【 摘 要 】

REHMAN, RIZWANA. Numerical Computation of the Characteristic Polynomial ofa Complex Matrix. (Under the direction of Ilse C.F. Ipsen.)In this dissertation we present algorithms, and sensitivity and stability analysesfor the numerical computation of characteristic polynomials of complex matrices. InQuantum Physics, for instance, characteristic polynomials are required to calculatethermodynamic properties of systems of fermions.The general consensus seems to be that numerical methods for computing characteristicpolynomials are numerically inaccurate and unstable. However, in order tojudge the numerical accuracy of a method, one needs to investigate the sensitivity ofthe coeffcients of the characteristic polynomial to perturbations in the matrix. Wederive forward error bounds for the coeffcients of the characteristic polynomial of ann x n complex matrix. These bounds consist of elementary symmetric functions ofsingular values. Furthermore, we investigate the numerical stability of two methodsfor the computation of characteristic polynomials. The frst method determines thecoeffcients of the characteristic polynomial of a matrix from its eigenvalues. The secondmethod requires a preliminary reduction of a complex matrix A to its Hessenbergform H. The characteristic polynomial of H is obtained from successive computationsof characteristic polynomials of leading principal submatrices of H. Our numericalexperiments suggest that the second method is more accurate than the determinationof the characteristic polynomial from eigenvalues.