Summary
We consider the following problem: Compute a vectorx such that ∥Ax−b∥2=min, subject to the constraint ∥x∥2=α. A new approach to this problem based on Gauss quadrature is given. The method is especially well suited when the dimensions ofA are large and the matrix is sparse.
It is also possible to extend this technique to a constrained quadratic form: For a symmetric matrixA we consider the minimization ofx T A x−2b T x subject to the constraint ∥x∥2=α.
Some numerical examples are given.
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Dahlquist, G., Golub, G.H., Nash, St.G. (1979): Bounds for the Error in Linear Systems. In: Proc. of the Workshop on Semi-Infinite Programming, ed. R. Hettich. Springer, Berlin Heidelberg New York, pp. 154–172
Davis, P.J., Rabinowitz, P. (1984): Methods of Numerical Integration. Academic Press, Orlando
Gander, W. (1981): Least Squares with a Quadratic Constraint. Numer. Math.36, 291–307
Gander, W., Golub, G.H., von Matt, U. (1989): A Constrained Eigenvalue Problem. Linear Algebra Appl.114/115, 815–839
Golub, G.H. (1973): Some Modified Matrix Eigenvalue Problems. SIAM Review15, 318–334
Golub, G.H. (1974): Bounds for Matrix Moments, Rocky Mountain. J. Math.,4, 207–211
Golub, G.H., Kahan, W. (1965): Calculating the singular values and pseudo-inverse of a matrix. SIAM J. Numer. Anal.2, 205–224
Golub, G.H., O'Leary, D.P. (1989): Some History of the Conjugate Gradient and Lanczos Algorithms: 1948–1976. SIAM Review31, 50–102
Golub, G.H., Van Loan, C.F. (1989): Matrix Computations, 2nd Ed. The Johns Hopkins University Press, Baltimore
Golub, G.H., Wwlsch, J.H. (1969): Calculation of Gauss Quadrature Rules. Math. Comp.23, 221–230
Householder, A.S. (1975): The Theory of Matrices in Numerical Analysis. Dover Publications, New York
Karlin, S., Studden, W.J. (1966): Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience Publishers, New York
Krylov, V.I. (1962): Approximate Calculation of Integrals. translated by A.H. Stroud. Macmillan, New York
Lawson, C., Hanson, R. (1974): Solving Least Squares Problems. Prentice-Hall, Englewood Cliffs, New Jersey
Paige, C.C. (1972): Computational Variants of the Lanczos Method for the Eigenproblem. J. Inst. Math. Appl.10, 373–381
Paige, C.C., Saunders, M.A. (1982): LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares. ACM Trans. Math. Softw.8, 43–71
Paige, C.C., Saunders, M.A. (1982): LSQR: Sparse Linear Equations and Least Squares Problems. ACM Trans. Math. Softw.8, 195–209
Parlett, B.N. (1980): The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs
Reinsch, C.H. (1971): Smoothing by Spline Functions. II. Numer. Math.16, 451–454
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This work was in part supported by the National Science Foundation under Grant DCR-8412314 and by the National Institute of Standards and Technology under Grant 60NANB9D0908.
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Golub, G.H., von Matt, U. Quadratically constrained least squares and quadratic problems. Numer. Math. 59, 561–580 (1991). https://doi.org/10.1007/BF01385796
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DOI: https://doi.org/10.1007/BF01385796