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| 研究生： |
范慶辰 Fan, Ching chen |
|---|---|
| 論文名稱： |
計算一個逆特徵值問題 Computing an Inverse Eigenvalue Problem |
| 指導教授： | 王太林 |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 1996 |
| 畢業學年度： | 84 |
| 語文別： | 英文 |
| 論文頁數： | 24 |
| 中文關鍵詞： | 逆特徵值問題 、蘭克澤斯演算法 、QR 演算法 |
| 外文關鍵詞： | Inverse eigenvalue problem, Lanczos algorithm, QR algorithm |
| 相關次數： | 點閱：111 下載：0 |
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In this thesis three methods LMGS, TQR and GR are applied to
solve an inverseeigenvalue problem. We list the numerical
results and compare the accuracy of the computed Jacobi matrix $T$ and the associated orthogonal matrix $Q$, wherethe columns of $Q^T$ are the eigenvectors of $T$. In the application of this inverse eigenvalue problem, the Fourier coefficients of $h(x)=e^x$ relative to the orthonormal polynomials associatedwith $T$ are evaluated, and these values are used to compute the least squarescoefficients of $h$ relative to the Chebyshev polynomials. We list thesenumerical results and compare them as our conclusion.
1 Introduction……1
1.1 An Inverse Eigenvalue Problem……1
1.2 Lanczos Process……2
1.3 Orthogonal Polynomials……4
1.4 TQR Method……5
1.5 GR Method……7
2 Example and Numerical Results……10
2.1 Examples……10
2.2 Difference between L and LMGS……10
2.3 Comparison of LMGS, TQR and GR……13
3 Application to the Least Squares Problem……16
3.1 Fourier Coefficients……16
3.2 Polynomial Least Squares Approximation……20
4 Conclusion……23
References……23
〔1〕N. Barkakati, Turbo C++ Bible, Howard W. Sams 1991.
〔2〕C. de Boor, G. H. Golub, The Numerically Stable Reconstruction of A Jacobi Matris from Spectral Data, Linear Algebra and Its Applications 21(1978), pp.245-260.
〔3〕J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, LINPACK User’s Guide, STAM 1979.
〔4〕W. Gautschi, Is the Recurrence Relation for Orthogonal Polynomials Always Stable?, BIT 33(1993), pp.277-284.
〔5〕W. B. Gragg, W. J. Harrod, The Numerically Stable Reconstruction of Jacobi Matrices from Spectral Data, Numer. Math. 44(1984), pp.317-335.
〔6〕B. N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall 1980.
〔7〕L. Reichel, Fast QR decomposition of Vandermonde-Like Matrices and Polynomial Least Squares Approximation, SIAM J. Matrix Anal. Appl., 12(1991), pp.552-564.
〔8〕T. L. Wang, The QR Transformation for Normal Hessenberg Matrices, unpublished manuscript (1998).
〔9〕D. S. Watkins, Fundamentals of Matrix Computations, John Wiley 1991.
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