! IR using function values only. Many questions about the stability and accuracy of algorithms in matrix computations can be expressed in terms of the maximum value of some easily computable function f. For a variety of algorithms it is shown that direct search is capable of revealing instability or poor performance, even when such failure is difficult to discover using theoretical analysis or numerical tests with random or non-random data. Informative numerical examples generated by direct search provide the impetus for further analysis and improvement of an algorithm. The direct search methods used are the method of alternating directions and the multi-directional search method of Dennis and Torczon. The problems examined include the reliability of matrix condition number estimators and the stability of Strassen's fast matrix inversion method.
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607
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A simplex method for function minimization
– Nelder, Mead
- 1965
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230
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Gaussian elimination is not optimal
– STRASSEN
- 1969
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95
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Direct search methods on parallel machines
– Dennis, Torczon
- 1991
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53
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Error analysis of direct methods of matrix inversion
– Wilkinson
- 1961
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50
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Multi-Directional Search: A Direct Search Algorithm for Parallel Machines
– Torczon
- 1989
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49
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Condition estimates
– Hager
- 1984
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38
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Stability of Block Algorithms with Fast Level 3
– Demmel, Higham
- 1990
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33
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On the compatibility of a given solution with the data of a linear system
– Rigal, Gaches
- 1967
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32
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An estimate for the condition number of a matrix
– Cline, Moler, et al.
- 1979
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29
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Incremental condition estimation
– Bischof
- 1990
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18
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Stability of Methods for Matrix Inversion
– Croz, Higham
- 1992
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17
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On growth in Gaussian elimination with complete pivoting
– Gould
- 1991
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16
|
Software for Roundoff Analysis of Matrix Algorithms
– Miller, Wrathall
- 1980
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|
14
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Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems
– Higham
- 1987
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|
13
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Optimization on microcomputers: the NelderMead simplex algorithm
– Woods
- 1987
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|
13
|
Direct search methods, in Numerical Methods for Unconstrained Optimization
– Swann
- 1972
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|
12
|
Large growth factors in Gaussian elimination with pivoting
– Higham, Higham
- 1989
|
|
12
|
Functional Stability Analysis of Numerical Algorithms
– Rowan
- 1990
|
|
12
|
Reliable solution of special event location problems for ODEs
– SHAMPINE, GLADWELL, et al.
- 1991
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|
10
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Generalizing the LINPACK Condition Estimator
– Cline, Conn, et al.
- 1982
|
|
10
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Dongarra and Eric Grosse. Distribution of mathematical software via electronic mail
– Jack
- 1987
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|
9
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A set of counter-examples to three condition number estimators
– CLINE, REW
- 1983
|
|
8
|
Software for roundoff analysis
– MILLER, SPOONER
- 1978
|
|
8
|
The block decomposition of a Vandermonde matrix and its applications
– Tang, Golub
- 1981
|
|
7
|
PC-Ma&b User’s Guide. The MathWorks
– MOLER, LITTLE, et al.
- 1987
|
|
7
|
On estimating the condition of eigenvalues and eigenvectors
– Loan
- 1987
|
|
6
|
A Strassen--Newton algorithm for high-speed parallelizable matrix inversion
– Bailey, Ferguson
- 1988
|
|
6
|
A stable variant of the secant method for solving nonlinear equations
– Gragg, Stewart
- 1976
|
|
5
|
A Probabilistic Round-off Error Propagation Model. Application to the Eigenvalue Problem
– Chatelin, Brunet
- 1990
|
|
5
|
Fast adaptive condition estimation
– Pierce, Plemmons
- 1990
|
|
4
|
Discrete Chebyshev approximation by interpolating rationals
– Almacany, Dunham, et al.
- 1984
|
|
4
|
codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674
– FORTRAN
- 1988
|
|
4
|
A survey of numerical methods for unconstrained optimization
– Powell
- 1970
|
|
4
|
Performance of the super modified simplex
– Routh, Swartz, et al.
- 1977
|
|
3
|
Algorithm 594: Software for relative error analysis
– Larson, Pasternak, et al.
- 1983
|
|
3
|
Adaptive condition estimation for rank-one updates of QR factorizations
– Shroff, Bischof
- 1990
|
|
2
|
Direct search methods in the optimisation of cancer chemotherapy
– Berenbaum
- 1991
|
|
2
|
Automatic program instrumentation with applications in performance and error analysis
– Bliss, Brunet, et al.
- 1992
|
|
2
|
Contribution `a la Fiabilit'e de Logiciels Num'eriques et `a L'analyse de Leur Comportement: Une Approche Statistique
– Brunet
- 1989
|
|
2
|
Algorithms for roundoff error analysis---A relative error approach
– Larson, Sameh
- 1980
|
|
2
|
Solving nonlinear Vandermonde systems
– Trapp, Squire
- 1975
|
|
2
|
Errors and Perturbations in Vandermonde Systems
– Varah
- 1993
|
|
1
|
Bj orck and Tommy Elfving, Algorithms for confluent Vandermonde systems
– Ake
- 1973
|
|
1
|
Bj orck and Victor Pereyra, Solution of Vandermonde systems of equations
– Ake
- 1970
|
|
1
|
The probability of large diagonal elements in the QR factorization
– Foster
- 1990
|
|
1
|
solution of Vandermonde-like systems involving orthogonal polynomials
– Fast
- 1988
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