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Multifrontal Computation with the Orthogonal Factors of Sparse Matrices
 SIAM Journal on Matrix Analysis and Applications
, 1994
"... . This paper studies the solution of the linear least squares problem for a large and sparse m by n matrix A with m n by QR factorization of A and transformation of the righthand side vector b to Q T b. A multifrontalbased method for computing Q T b using Householder factorization is presented ..."
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. This paper studies the solution of the linear least squares problem for a large and sparse m by n matrix A with m n by QR factorization of A and transformation of the righthand side vector b to Q T b. A multifrontalbased method for computing Q T b using Householder factorization is presented. A theoretical operation count for the K by K unbordered grid model problem and problems defined on graphs with p nseparators shows that the proposed method requires O(NR ) storage and multiplications to compute Q T b, where NR = O(n log n) is the number of nonzeros of the upper triangular factor R of A. In order to introduce BLAS2 operations, Schreiber and Van Loan's StorageEfficientWY Representation [SIAM J. Sci. Stat. Computing, 10(1989),pp. 5557] is applied for the orthogonal factor Q i of each frontal matrix F i . If this technique is used, the bound on storage increases to O(n(logn) 2 ). Some numerical results for the grid model problems as well as HarwellBoeing problems...
Structured low rank approximations of the Sylvester resultant matrix for approximate GCDs of Bernstein polynomials, 2006. Submitted to Computer Aided Geometric Design
"... Abstract. A structured low rank approximation of the Sylvester resultant matrix S(f, g) of the Bernstein basis polynomials f = f(y) and g = g(y), for the determination of their approximate greatest common divisors (GCDs), is computed using the method of structured total least norm. Since the GCD of ..."
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Abstract. A structured low rank approximation of the Sylvester resultant matrix S(f, g) of the Bernstein basis polynomials f = f(y) and g = g(y), for the determination of their approximate greatest common divisors (GCDs), is computed using the method of structured total least norm. Since the GCD of f(y) and g(y) is equal to the GCD of f(y) and αg(y), where α is an arbitrary nonzero constant, it is more appropriate to consider a structured low rank approximation S ( ˜ f, ˜g) of S(f, αg), where the polynomials ˜ f = ˜ f(y) and ˜g = ˜g(y) approximate the polynomials f(y) and αg(y), respectively. Different values of α yield different structured low rank approximations S ( ˜ f, ˜g), and therefore different approximate GCDs. It is shown that the inclusion of α allows to obtain considerably improved approximations, as measured by the decrease of the singular values σi of S ( ˜ f, ˜g), with respect to the approximation obtained when the default value α = 1 is used. An example that illustrates the theory is presented and future work is discussed. Key words. Bernstein polynomials, structured low rank approximation, Sylvester resultant matrix.
RowWise Backward Stable Elimination Methods for the Equality Constrained Least Squares Problem
 Manchester Centre for Computational Mathematics
, 1999
"... . It is well known that the solution of the equality constrained least squares (LSE) problem minBx=d #b  Ax# 2 is the limit of the solution of the unconstrained weighted least squares problem min x h d b i  h B A i x 2 as the weight tends to infinity, assuming that [ B T A ..."
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. It is well known that the solution of the equality constrained least squares (LSE) problem minBx=d #b  Ax# 2 is the limit of the solution of the unconstrained weighted least squares problem min x h d b i  h B A i x 2 as the weight tends to infinity, assuming that [ B T A T ] T has full rank. We derive a method for the LSE problem by applying Householder QR factorization with column pivoting to this weighted problem and taking the limit analytically, with an appropriate rescaling of rows. The method obtained is a type of direct elimination method. We adapt existing error analysis for the unconstrained problem to obtain a rowwise backward error bound for the method. The bound shows that, provided row pivoting or row sorting is used, the method is wellsuited to problems in which the rows of A and B vary widely in norm. As a byproduct of our analysis, we derive a rowwise backward error bound of precisely the same form for the standard elimination m...