On the Approximate Computation of Extreme Eigenvalues and the Condition Number of Nonsingular Matrices

Lei Quang-yao

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Article Navigation > Applied Mathematics and Mechanics > 1992 > 13(2): 181-186

Lei Quang-yao. On the Approximate Computation of Extreme Eigenvalues and the Condition Number of Nonsingular Matrices[J]. Applied Mathematics and Mechanics, 1992, 13(2): 181-186.

Citation:

Lei Quang-yao. On the Approximate Computation of Extreme Eigenvalues and the Condition Number of Nonsingular Matrices[J]. Applied Mathematics and Mechanics, 1992, 13(2): 181-186.

Citation:

Lei Quang-yao. On the Approximate Computation of Extreme Eigenvalues and the Condition Number of Nonsingular Matrices[J]. Applied Mathematics and Mechanics, 1992, 13(2): 181-186.

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On the Approximate Computation of Extreme Eigenvalues and the Condition Number of Nonsingular Matrices

Lei Quang-yao

Institute of Applied Mathematics, Academia Sinica, Beijing

Received Date: 1990-05-03
Publish Date: 1992-02-15

Abstract

Abstract

From the formulas of the conjugate gradient, a similarity between a symmetric positive definite(SPD) matrix A and a tridiagonal matrix B is obtained.The elements of the matrix B are determined by the parameters of the conjugate gradient.The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B.The approximation of extreme eigenvalues of A can be obtained as a ‘by-product' in the computation of the conjugate gradient if a computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient.If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on A^TA.Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.
- symmetric positive definite matria,
- conjugate gradient,
- eigenvalues,
- condition number

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References(6)

References

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