LAPACK Users' Guide by E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J.

By E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen

LAPACK is a library of numerical linear algebra subroutines designed for top functionality on workstations, vector desktops, and shared reminiscence multiprocessors. liberate 3.0 of LAPACK introduces new exercises and extends the performance of present workouts. the main major new exercises and capabilities comprise: 1) a swifter singular price decomposition computed via divide-and-conquer 2) speedier exercises for fixing rank-deficient least squares difficulties: utilizing QR with column pivoting utilizing the SVD in response to divide-and-conquer three) new workouts for the generalized symmetric eigenproblem: speedier workouts in line with divide-and-conquer workouts in accordance with bisection/inverse generation, for computing a part of the spectrum four) quicker regimen for the symmetric eigenproblem utilizing ''relatively powerful eigenvector algorithm'' five) new basic and professional drivers for the generalized nonsymmetric eigenproblem, together with blunders bounds 6) solver for generalized Sylvester equation, utilized in five) 7) computational exercises utilized in five)

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5: • xGEES: a simple driver that computes all or part of the Schur factorization of A, with optional ordering of the eigenvalues; • xGEESX: an expert driver that can additionally compute condition numbers for the average of a selected subset of the eigenvalues, and for the corresponding right invariant subspace; • xGEEV: a simple driver that computes all the eigenvalues of A, and (optionally) the right or left eigenvectors (or both); • xGEEVX: an expert driver that can additionally balance the matrix to improve the conditioning of the eigenvalues and eigenvectors, and compute condition numbers for the eigenvalues or right eigenvectors (or both).

3 for details. 1). xSTEGR This routine uses the relatively robust representation (RRR) algorithm to find eigenvalues and eigenvectors. This routine uses an LDLT factorization of a number of translates T — sI of T, for one shift s near each cluster of eigenvalues. For each translate the algorithm computes very accurate eigenpairs for the tiny eigenvalues. xSTEGR is faster than all the other routines except in a few cases, and uses the least workspace. 3 for details. xPTEQR This routine applies to symmetric positive definite tridiagonal matrices only.

3) and [81][55, page 252]). 3) as QTd = QTAx + QTBy = Rx + TZy. We partition this as where and can be computed by xORMQR (or xUNMQR). 4. COMPUTATIONAL ROUTINES from which we obtain the desired solutions and which can be computed by xTRSV, xGEMV and xORMRQ (or xUNMRQ). 2 Generalized RQ Factorization The generalized RQ (GRQ) factorization of an ra-by-n matrix A and a p-by-n matrix B is given by the pair of factorizations A = RQ and B = ZTQ where Q and Z are respectively n-by-n and p-by-p orthogonal matrices (or unitary matrices if A and B are complex).

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