G. H. Golub. Numerical methods for solving linear least squares problems. Numer. Math., 7:206--216, 1965.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Van Loan [16] shows. Let 6 6 6 6 6 6 6 6 6 1 3 1 1 Gamma1 1 Gamma 7 7 7 7 7 7 7 7 7 : 1. 4) We applied Householder QR factorization in Matlab with = 10 , using no pivoting, and combinations of row pivoting (defined later) and the standard column pivoting strategy introduced by Golub [7]. The unit roundoff u 1:1 Theta 10 . Table 1.1 reports the backward errors kA Pi Gamma b Q b Rk 2 kAk 2 ; j R = max k(A Pi Gamma b Q b R) i; k 2 In these expressions, b Q denotes the computed product of the Householder transformations and Pi is the permutation matrix produced by ....

G. H. Golub. Numerical methods for solving linear least squares problems. Numer. Math., 7:206--216, 1965.

.... m i=1 of S, i.e. a generating set for S which satisfies ## i , # j # = # ij , i, j = 1, m. For then, 2.10) P S u = m # i=1 #u, # i ## i . Given a basis # i m 1 for S, an orthonormal basis # i for S may be constructed from it by a variety of techniques (e.g. [3], 6] The best known of these is the Gram Schmidt (G. S. orthonormalization procedure, in which each # i is computed as the normalized error of the best approximation to # i by elements in the span of # j i 1 j=1 , i.e. by successfully solving a least squares approximation problem m 1 ....

G. Golub, Numerical methods for solving linear least-squares problems, Numer. Math. 7 (1965), 206--216. 15

....A. Thus, the algorithm is stable in the sense of backward error analysis. Note that kA T A Gamma R T Rk=kA T Ak is small, but k Q Gamma Qk and k R Gamma Rk=kRk are not necessarily small. Bounds on k Q Gamma Qk and k R Gamma Rk=kRk depend on , and are discussed in [36, 66, 81]. A different algorithm is to compute (the upper triangular part of) A T A, and then compute the Cholesky factorization of A T A by the usual (stable) algorithm. The computed result R is such that R T R is close to A T A. However, this does not imply the existence of b A and b ....

....examples show that they are unstable. It may be surprising that fast algorithms for computing an orthogonal factorization (6.1) are unstable. The classical O(n 3 ) algorithms are stable because they form Q as a product of elementary orthogonal matrices (usually Givens or Householder matrices [36, 38, 81]) Unlike the classical algorithms, the O(n 2 ) algorithms do not form Q in a numerically stable manner as a product of matrices which are (close to) orthogonal. This observation explains both their speed and their instability For example, the algorithms of BBH [6] and Chun et al. [23] depend ....

G. H. Golub, Numerical methods for solving linear least squares problems, Numer. Math. 7 (1965), 206--216.

....form of the algorithm is the triangularization by a J Gammaunitary matrix (and for a recursive solution, a sequence of elementary circular and hyperbolic rotations reflections) of a suitably defined prearray. Such algorithms were first proposed, for numerical reasons, in least squares theory [93, 146] and were introduced for Levinson type algorithms in [122] They now are the main focus of many computationally effective and recursive filtering algorithms (see, e.g. 179] This focus was given further impetus by the rederivation [135] using state space concepts and some embedding results ....

G. H. Golub, Numerical methods for solving linear least squares problems, Numer. Math., 7 (1965), pp. 206--216.

....algorithm can be implemented by an array of elementary cells composed of elementary rotations and time variant tapped delay filters. We then consider the special case of the recursive least squares problem and show that the derived algorithm collapses to the now widely studied QR algorithm (see [3,4,12 14] and the references therein) with the extra ingredient of allowing for a parallel extraction of the weight vector. To conclude this introduction, let us state a readily established matrix result that since [15] has played an important role in the derivation of all so called square root ....

....L Phi (t) 0 ; 9) where Gamma(t) is any unitary transformation ( Gamma(t) Gamma (t) I n 1 ) that produces the block zero in the postarray. Expression (9) is the so called QR recursion that updates the Cholesky factor of the autocorrelation matrix through a sequence of unitary rotations [3,4,12]. Once L Phi (t) is determined, then one way to obtain the weight vector w(t) is to solve the following triangular system (via backsubstitution) L Phi (t)w(t) L Gamma1 Phi (t) t) This however, does not yield a fully parallelizable algorithm. We now extend an embedding ....

G. H. Golub, "Numerical methods for solving linear least squares problems," Numer. Math., vol. 7, pp. 206--216, 1965.

....v xk 2 2 kLD 1=2 w xk 2 2 jo , in which D v and Dw are diagonal weight matrices corresponding to v and w, and L = bidiag( Gamma1; 1) Via the regularized normal equations he then derived the expression x = i A T A 2 (D v L T Dw L) j Gamma1 A T b. In 1965 Gene H. Golub [9] was the first to propose a modern approach to solving (1) via the least squares formulation (3) and QR factorization of the associated coefficient matrix. Golub proposed this approach in connection with Riley s iterative scheme, which includes the computation of x as the first step. G. Ribiere ....

G. H. Golub, Numerical methods for solving linear least squares problems, Numer. Math., 7 (1965), pp. 206--216.

....only working with matrix R. The major advantage of the orthogonal factorization is its stability. Pivoting for stability is as a rule not used in the codes based on orthogonal decomposition techniques, although column interchanges are recommended when the matrix is very ill conditioned, cf. Golub [17] and Powell Reid [26] The performance on parallel computers is normally more efficient when the orthogonal factorization is carried out without pivoting for numerical stability. Column interchanges are also (explicitly or implicitly) needed if one wants to compute a rank revealing ....

G. H. Golub, Numerical methods for solving linear least squares problems, Numer. Math., Vol. 7, 1965, pp. 206-216.

....contains the column indices of all nonzero elements of A(L, and that J# J=#. We reduce J to the set of the most profitable indices j with smallest # j and add it to J . We note that equation (9) was also used in [4] to estimate the reduction in the residual and can already be found in [8]. Using the augmented set of indices J , we solve the sparse least squares problem (4) again. This yields a better approximation m k of the kth column of A 1 . We repeat this process for each k =1, nuntil the residual satisfies a prescribed tolerance or a maximum amount of fill in has been ....

G. GOLUB, Numerical methods for solving linear least squares problems, Numer. Math., 7 (1965), pp. 206--216.

....These matrices are obtained by a recursive differentiation algorithm which appends new rows to the previous matrices. The process of incorporating a new row or column in a matrix is called updating. Other applications are the solution of underdetermined rank deficient least squares problems [12, 14, 20], subset selection problems [13, 14] and information retrieval [2] The singular value factorization (SV D) 14, p. 246] is known to be an extremely reliable tool for computing the numerical rank and bases for the null spaces of a matrix. However, the SV D is too expensive when it comes to ....

G. H. Golub, Numerical methods for solving linear least squares problems, Numer. Math. 7, (1965), 206-216.

....xk 2 2 kLD 1=2 w xk 2 2 jo , in which D v and Dw are diagonal weight matrices corresponding to v and w, and L = bidiag( Gamma1; 1) Via the regularized normal equations he then derived the expression x = i A T A 2 (D v L T Dw L) j Gamma1 A T b. In 1965 Gene H. Golub [8] was the first to propose a modern approach to solving (1) via the least squares formulation (3) and QR factorization of the associated coefficient matrix. Golub proposed this approach in connection with Riley s iterative scheme, which includes the computation of x as the first step. G. Ribiere ....

G. H. Golub, Numerical methods for solving linear least squares problems, Numer. Math., 7 (1965), pp. 206--216.

....while its inverse is called the covariance matrix, as it measures the expected error in x. An accurate and stable way of computing the LS solution consists in first applying a sequence of orthogonal transformations, such that A is reduced to an upper triangular matrix (QR factorization) [3, 4] Q T 1 A z R 0 # 1x = Q T 1 b z y z # where Q is an m2m orthogonal matrix QQ T = I m2m , and R is n2n upper triangular. The least squares solution can then be computed by solving the triangular system R 1 x LS = y Computing the QR factorization takes O(mn 2 ) ....

G.H. Golub. Numerical methods for solving linear least squares problems. Numer. Math. 7, 1965, pp 206-216.

....6 6 6 6 6 6 6 6 4 1 1 1 1 3 1 1 Gamma1 1 1 1 1 Gamma 3 7 7 7 7 7 7 7 7 7 5 : 1. 4) We applied Householder QR factorization in Matlab with = 10 12 , using no pivoting, and combinations of row pivoting (defined later) and the standard column pivoting strategy introduced by Golub [7]. The unit roundoff u 1:1 Theta 10 Gamma16 . Table 1.1 reports the backward errors j = kA Pi Gamma b Q b Rk 2 kAk 2 ; j R = max i k(A Pi Gamma b Q b R) i; k 2 kA(i; k 2 : In these expressions, b Q denotes the computed product of the Householder transformations and Pi is the ....

G. H. Golub. Numerical methods for solving linear least squares problems. Numer. Math., 7:206--216, 1965.

....kAx Gamma bk 2 (1) lies at the heart of many challenging computational problems frequently arising in geodetic survey, photogrammmetry, tomography, structural analysis, surface fitting, numerical optimization, etc. Let A be an M Theta N matrix with full column rank. The QR factorization method [7] for solving (1) first computes the QR factorization A = Q R 0 : 2) Then the solution to (1) is obtained by solving the triangular system Rx = c, where c is the first N components of Q t b. The matrix R is referred to as the upper triangular factor of A. This method is numerically ....

.... in a multifrontal manner is also considered in the context of solving systems of sparse linear equations [18] The parallel sparse QR factorization algorithm [19] and the parallel sparse back substitution algorithm described in section 3 can also be used to implement the QR factorization method [7] for solving sparse linear least squares problems as discussed in [20] Acknowledgements. I would like to thank Professor Thomas F. Coleman for many discussions relating to this work and for his helpful comments on the manuscript. ....

G. Golub, Numerical methods for solving linear least squares problems, Numer. Math., 7 (1965), pp. 206--216.

....triangular matrix. Step 3: Solve the following system, via back substitution, for y: U 1 y = Z T 1 PD Gamma1=2 b: 5) Step 4: To get y, multiply the result of Step 3 by Q: y = Q y: 6) Solution of least squares problems via QR factorization with column pivoting was introduced by Golub [3]. Note that the QR factorization for the least squares problem occurs in Step 2. The QR factorization in Step 1 is to make the algorithm stable. First, we compare this algorithm to Vavasis s NSH method [12] The NSH method is the only algorithm in the literature known to stably (in the sense of ....

....level) with respect to the original norm of that column. If so, we change those entries to zeros. Notice that this test requires very little additional work because the usual QR factorization algorithm with column pivoting already monitors the norms of the residual portions of the columns [3]. In this way, if there are exact dependences among the rows of A, our algorithm does not miss them. If this numerical test fails to detect exact dependence, then our stability analysis no longer holds. It can be shown that the test we have proposed will fail only in the case that there is ....

G. Golub. Numerical methods for solving linear least squares problems. Numer. Math., 7:206--216, 1965.

....the heart of many challenging computational problems frequently arising in geodetic survey, photogrammmetry, tomography, structural analysis, surface fitting, numerical optimization, etc. Let A be an M Theta N matrix with full column rank. The QR factorization method for solving (1) due to Golub [9] is to reduce A to an upper triangular matrix R by an orthogonal matrix Q Q t A = R 0 ; 2) and transform b by forming Q t b. The solution to (1) is then obtained by solving the triangular system Rx = c, where c is the first N components of Q t b. This method is numerically backward ....

G. Golub, Numerical methods for solving linear least squares problems, Numer. Math., 7 (1965), pp. 206--216.

....n are the number of rows and columns of the problem respectively. Here, we use m n as an approximation of the average value of s in section 4.2, since P n i s i = m, and s = max i s i and is presumed to be constant. We also solve the linear least squares problems given in (1) by QR method by [18] using our method for computing Q T b. We compare the QR method with the method of correct semi normal equations method (CSNE) by [6] The CSNE method 20 S. M. LU AND J. L. BARLOW Table 3 Residuals for the model problem. K QR CSNE 10 4.08 4.08 20 9.21 9.21 30 13.99 13.99 40 19.54 19.19 50 ....

G. H. Golub, Numerical methods for solving linear least squares problems, Numer.Math., 7(1965), pp. 206--216.

....y, multiply the result of Step 3 by Q: y = Q y: 6) Note that the QR factorization for the least squares problem occurs in Step 2. The QR factorization in Step 1 is to make the algorithm stable. Solution of leastsquares problems via QR factorization with column pivoting was introduced by Golub [5]. The term complete orthogonal decomposition refers to a factorization of the form QRZ in which Q and Z are orthogonal and R is triangular [6] Therefore we have chosen this name for the above algorithm, which computes a particular kind of complete orthogonal decomposition. In exact arithmetic, ....

....level) with respect to the original norm of that column. If so, those entries are changed to zeros. Notice that this test requires very little additional work because the usual QR factorization algorithm with column pivoting already monitors the norms of the residual portions of the columns [5]. In this way, if there are exact dependences among the rows of A, the algorithm does not miss them. If this numerical test fails to detect exact dependence, then the following stability analysis no longer holds. It can be shown that the test we have proposed will fail only in the case that there ....

G. Golub. Numerical methods for solving linear least squares problems. Numer. Math., 7:206--216, 1965.

....to A. Thus, the algorithm is stable in the sense of backward error analysis. Note that kA T A Gamma R T Rk=kA T Ak is small, but k Q Gamma Qk and k R Gamma Rk=kRk are not necessarily small. Bounds on k Q Gamma Qk and k R Gamma Rk=kRk depend on , and are discussed in [35, 71, 84]. A different algorithm is to compute (the upper triangular part of) A T A, and then compute the Cholesky factorization of A T A by the usual (stable) algorithm. The computed result R is such that R T R is close to A T A. However, this does not imply the existence of b A and b Q ....

G. H. Golub, "Numerical methods for solving linear least squares problems", Numer. Math. 7 (1965), 206--216.

....the column indices of all nonzero elements of A(L; and that J J = We reduce J to the set of the most profitable indices j with smallest ae j and add it to J . We note that equation (9) was also used in [4] to estimate the reduction in the residual, and can already be found in [8]. Using the augmented set of indices J , we solve the sparse least squares problem (4) again. This yields a better approximation m k of the k th column of A Gamma1 . We repeat this process for each k = 1; n until the residual satisfies a prescribed tolerance or a maximum amount of ....

G. Golub, Numerical methods for solving linear least squares problems, Num. Mathematik, 7, 1965, pp. 206--216.

....8.3.3] we have oe min (R(1 : k; 1 : k) oe k (A) and oe max (R(k 1 : n; k 1 : n) oe k 1 (A) 4) Hence, to satisfy condition (3) we need to pursue two tasks: Task 1. Find a permutation P that maximizes oe min (R 11 ) Task 2. Find a permutation P that minimizes oe max (R 22 ) Golub [1965] suggested what is commonly called the QR factorization with column pivoting. Given a set of already selected columns, this algorithm chooses as the next pivot column the one that is farthest away in the Euclidean norm from the subspace spanned by the columns already chosen [Golub and Loan ....

....2 (see, for example, Golub and Loan 1989, p. 196] The application of a Householder matrix B : H(u)A involves a matrix vector product z : A T u and a rank one update B : A Gamma 2uz T . Figure 1 describes the Golub Householder QR factorization algorithm with traditional column pivoting [Golub 1965] for computing the QR decomposition of an m Theta n matrix A. The primitive operation [u; y] genhh(x) computes u such that y = H(u)x is a multiple of e 1 , while the primitive operation B : apphh(u; A) overwrites B with H(u)A. After step i is completed, the values res j ; j = i 1; ....

Golub, G. H. 1965 . Numerical methods for solving linear least squares problems. Numerische Mathematik 7, 206--216.

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G. H. Golub. Numerical methods for solving linear least squares problems. Numer. Math., 7:206--216, 1965.

No context found.

G. H. GOLUB, Numerical methods for solving linear least squares problems, Numerische Mathematik, 7, pp. 206-216, 1965.

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G. H. Golub. Numerical methods for solving linear least squares problems. Numerical Mathematics, 7:206#216, 1965.

No context found.

G. H. Golub, "Numerical methods for solving linear least squares problems," Numer. Math., vol. 7, pp. 206--216, 1965.

No context found.

G. H. Golub, Numerical methods for solving linear least squares problems, Numerische Mathematik, 7, pp. 206--216, 1965.

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