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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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