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  • 1
    Monograph available for loan
    Monograph available for loan
    MATLAB guide (2005)
    Philadelphia : SIAM
    Details Availability
    Call number: 19/M 07.0053
    Description / Table of Contents: Contents: 1. A brief tutorial; 2. Basics; 3. Distinctive features ofMATLAB; 4 Arithmetic; 5. Matrices; 6. Operators and flow control; 7.M-files; 8. Graphics; 9. Linear algebra; 10. More on functions; 11.Numerical methods Part I; 12. Numerical methods Part II; 13. Input and output; 14. Troubleshooting; 15. Sparse matrices; 16. Further M-files;17. Handle graphics; 18. Other data types and multidimensional arrays;19. The symbolic math toolbox; 20. Optimizing M-files; 21. Tricks and tips; 22. Case studies
    Type of Medium: Monograph available for loan
    Pages: XXIII, 382 S. , graph. Darst.
    Edition: 2nd ed.
    ISBN: 0898715784
    Classification:
    Mathematics
    Location: Reading room
    Branch Library: GFZ Library
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  • 2
    Monograph available for loan
    Monograph available for loan
    Accuracy and stability of numerical algorithms (1996)
    Philadelphia : Society for Industrial and Applied Mathematics
    Details Availability
    Call number: 19/M 97.0141
    Type of Medium: Monograph available for loan
    Pages: xxviii, 688 S.
    ISBN: 0898713552
    Classification:
    C.1.8.
    Language: English
    Location: Reading room
    Branch Library: GFZ Library
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  • 3
    Electronic Resource
    Electronic Resource
    Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems (1987)
    Springer
    Numerische Mathematik 50 (1987), S. 613-632 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65G05, 65F05 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A forward error analysis is presented for the Björck-Pereyra algorithms used for solving Vandermonde systems of equations. This analysis applies to the case where the points defining the Vandermonde matrix are nonnegative and are arranged in increasing order. It is shown that for a particular class of Vandermonde problems the error bound obtained depends on the dimensionn and on the machine precision only, being independent of the condition number of the coefficient matrix. By comparing appropriate condition numbers for the Vandermonde problem with the forward error bounds it is shown that the Björck-Pereyra algorithms introduce no more uncertainty into the numerical solution than is caused simply by storing the right-hand side vector on the computer. A technique for computing “running” a posteriori error bounds is derived. Several numerical experiments are presented, and it is observed that the ordering of the points can greatly affect the solution accuracy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01408579
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  • 4
    Electronic Resource
    Electronic Resource
    Estimating the matrixp-norm (1992)
    Springer
    Numerische Mathematik 62 (1992), S. 539-555 
    Details
    ISSN: 0945-3245
    Keywords: 65F35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The Hölderp-norm of anm×n matrix has no explicit representation unlessp=1,2 or ∞. It is shown here that thep-norm can be estimated reliably inO(mn) operations. A generalization of the power method is used, with a starting vector determined by a technique with a condition estimation flavour. The algorithm nearly always computes ap-norm estimate correct to the specified accuracy, and the estimate is always within a factorn 1−1/p of ‖A‖ p . As a by-product, a new way is obtained to estimate the 2-norm of a rectangular matrix; this method is more general and produces better estimates in practice than a similar technique of Cline, Conn and Van Loan.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396242
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  • 5
    Electronic Resource
    Electronic Resource
    Computing the field of values and pseudospectra using the Lanczos method with continuation (1996)
    Springer
    BIT 36 (1996), S. 422-440 
    Details
    ISSN: 1572-9125
    Keywords: Field of values ; pseudospectra ; Lanczos method ; Chebyshev acceleration ; continuation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The field of values and pseudospectra are useful tools for understanding the behaviour of various matrix processes. To compute these subsets of the complex plane it is necessary to estimate one or two eigenvalues of a large number of parametrized Hermitian matrices; these computations are prohibitively expensive for large, possibly sparse, matrices, if done by use of the QR algorithm. We describe an approach based on the Lanczos method with selective reorthogonalization and Chebyshev acceleration that, when combined with continuation and a shift and invert technique, enables efficient and reliable computation of the field of values and pseudospectra for large matrices. The idea of using the Lanczos method with continuation to compute pseudospectra is not new, but in experiments reported here our algorithm is faster and more accurate than existing algorithms of this type.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01731925
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  • 6
    Electronic Resource
    Electronic Resource
    Accuracy and Stability of the Null Space Method for Solving the Equality Constrained Least Squares Problem (1999)
    Springer
    BIT 39 (1999), S. 34-50 
    Details
    ISSN: 1572-9125
    Keywords: Constrained least squares problem ; null space method ; rounding error analysis ; condition number ; generalized QR factorization ; LAPACK
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The null space method is a standard method for solving the linear least squares problem subject to equality constraints (the LSE problem). We show that three variants of the method, including one used in LAPACK that is based on the generalized QR factorization, are numerically stable. We derive two perturbation bounds for the LSE problem: one of standard form that is not attainable, and a bound that yields the condition number of the LSE problem to within a small constant factor. By combining the backward error analysis and perturbation bounds we derive an approximate forward error bound suitable for practical computation. Numerical experiments are given to illustrate the sharpness of this bound.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022365107361
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  • 7
    Electronic Resource
    Electronic Resource
    Numerically Stable Generation of Correlation Matrices and Their Factors (2000)
    Springer
    BIT 40 (2000), S. 640-651 
    Details
    ISSN: 1572-9125
    Keywords: Random correlation matrix ; Bendel-Mickey algorithm ; eigenvalues ; singular value decomposition ; test matrices ; forward error bounds ; relative error bounds ; IMSL ; NAG Library ; Jacobi method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Correlation matrices—symmetric positive semidefinite matrices with unit diagonal—are important in statistics and in numerical linear algebra. For simulation and testing it is desirable to be able to generate random correlation matrices with specified eigenvalues (which must be nonnegative and sum to the dimension of the matrix). A popular algorithm of Bendel and Mickey takes a matrix having the specified eigenvalues and uses a finite sequence of Givens rotations to introduce 1s on the diagonal. We give improved formulae for computing the rotations and prove that the resulting algorithm is numerically stable. We show by example that the formulae originally proposed, which are used in certain existing Fortran implementations, can lead to serious instability. We also show how to modify the algorithm to generate a rectangular matrix with columns of unit 2-norm. Such a matrix represents a correlation matrix in factored form, which can be preferable to representing the matrix itself, for example when the correlation matrix is nearly singular to working precision.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022384216930
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  • 8
    Electronic Resource
    Electronic Resource
    Iterative refinement enhances the stability ofQR factorization methods for solving linear equations (1991)
    Springer
    BIT 31 (1991), S. 447-468 
    Details
    ISSN: 1572-9125
    Keywords: Primary ; 65F05 ; 65G05 ; Iterative refinement ; linear system ; least squares problem ; QR factorization ; Gaussian elimination ; partial pivoting ; rounding error analysis ; backward error ; componentwise error bounds ; Householder transformations ; Givens transformations ; confluent Vandermonde-like matrices
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Iterative refinement is a well-known technique for improving the quality of an approximate solution to a linear system. In the traditional usage residuals are computed in extended precision, but more recent work has shown that fixed precision is sufficient to yield benefits for stability. We extend existing results to show that fixed precision iterative refinement renders anarbitrary linear equations solver backward stable in a strong, componentwise sense, under suitable assumptions. Two particular applications involving theQR factorization are discussed in detail: solution of square linear systems and solution of least squares problems. In the former case we show that one step of iterative refinement suffices to produce a small componentwise relative backward error. Our results are weaker for the least squares problem, but again we find that iterative refinement improves a componentwise measure of backward stability. In particular, iterative refinement mitigates the effect of poor row scaling of the coefficient matrix, and so provides an alternative to the use of row interchanges in the HouseholderQR factorization. A further application of the results is described to fast methods for solving Vandermonde-like systems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01933262
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  • 9
    Electronic Resource
    Electronic Resource
    Backward Error Bounds for Constrained Least Squares Problems (1999)
    Springer
    BIT 39 (1999), S. 210-227 
    Details
    ISSN: 1572-9125
    Keywords: Equality constrained least squares problem ; least squares minimization over a sphere ; null space method ; elimination method ; method of weighting ; backward error ; backward stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We derive an upper bound on the normwise backward error of an approximate solution to the equality constrained least squares problem min Bx=d ‖b − Ax‖2. Instead of minimizing over the four perturbations to A, b, B and d, we fix those to B and d and minimize over the remaining two; we obtain an explicit solution of this simplified minimization problem. Our experiments show that backward error bounds of practical use are obtained when B and d are chosen as the optimal normwise relative backward perturbations to the constraint system, and we find that when the bounds are weak they can be improved by direct search optimization. We also derive upper and lower backward error bounds for the problem of least squares minimization over a sphere: $$\min _{\left\| x \right\|_2 \leqslant \alpha } \left\| {b - Ax} \right\|_2 $$ .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022385611904
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  • 10
    Electronic Resource
    Electronic Resource
    The symmetric procrustes problem (1988)
    Springer
    BIT 28 (1988), S. 133-143 
    Details
    ISSN: 1572-9125
    Keywords: Primary 65F20 ; Least squares ; symmetric Procrustes problem ; singular value decomposition ; normal equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The following “symmetric Procrustes” problem arises in the determination of the strain matrix of an elastic structure: find the symmetric matrixX which minimises the Frobenius (or Euclidean) norm ofAX — B, whereA andB are given rectangular matrices. We use the singular value decomposition to analyse the problem and to derive a stable method for its solution. A perturbation result is derived and used to assess the stability of methods based on solving normal equations. Some comparisons with the standard, unconstrained least squares problem are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01934701
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