# Bibliography

For more information about the BLAS, Sparse BLAS, LAPACK, ScaLAPACK, Sparse Solver, Interval Solver, VML, VSL, and FFT functionality, refer to the following publications:

- BLAS Level 1

  C. Lawson, R. Hanson, D. Kincaid, and F. Krough. Basic Linear Algebra Subprograms for Fortran Usage, ACM Transactions on Mathematical Software, Vol.5, No.3 (September 1979) 308-325.

- BLAS Level 2

  J. Dongarra, J. Du Croz, S. Hammarling, and R. Hanson. An Extended Set of Fortran Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software, Vol.14, No.1 (March 1988) 1-32.

- BLAS Level 3

  J. Dongarra, J. DuCroz, I. Duff, and S. Hammarling. A Set of Level 3 Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software (December 1989).

- Sparse BLAS

  D. Dodson, R. Grimes, and J. Lewis. Sparse Extensions to the FORTRAN Basic Linear Algebra Subprograms, ACM Transactions on Math Software, Vol.17, No.2 (June 1991).

  D. Dodson, R. Grimes, and J. Lewis. Algorithm 692: Model Implementation and Test Package for the Sparse Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software, Vol.17, No.2 (June 1991).

  [Duff86]

    I.S.Duff, A.M.Erisman, and J.K.Reid. Direct Methods for Sparse Matrices. Clarendon Press, Oxford, UK, 1986.

  [CXML01]

    Compaq Extended Math Library. Reference Guide, Oct.2001.

  [Rem05]

    K.Remington. A NIST FORTRAN Sparse Blas User's Guide. (available on http://math.nist.gov/~KRemington/fspblas/)

  [Saad94]

    Y.Saad. SPARSKIT: A Basic Tool-kit for Sparse Matrix Computation. Version 2, 1994.(http://www.cs.umn.edu/~saad)

  [Saad96]

    Y.Saad. Iterative Methods for Linear Systems. PWS Publishing, Boston, 1996.

- LAPACK

  [LUG]

    E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK Users' Guide, Third Edition, Society for Industrial and Applied Mathematics (SIAM), 1999.

  [Golub96]

    G. Golub and C. Van Loan. Matrix Computations, Johns Hopkins University Press, Baltimore, third edition,1996.

  [Bischof92]

    http://citeseer.ist.psu.edu/bischof92framework.html

  [Marques06]

    O.Marques, E.J.Riedy, and Ch.Voemel. Benefits of IEEE-754 Features in Modern Symmetric Tridiagonal Eigensolvers, SIAM Journal on Scientific Computing, Vol.28, No.5, 2006. (Tech report version in LAPACK Working Note 172 with the same title.)

  [Kahan66]

    W. Kahan. Accurate Eigenvalues of a Symmetric Tridiagonal Matrix, Report CS41, Computer Science Dept., Stanford University, July 21, 1966.

  [Dhillon04]

    I. Dhillon, B. Parlett. Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices, Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.

  [Dhillon04-02]

    I. Dhillon, B. Parlett. Orthogonal Eigenvectors and * Relative Gaps, SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. (Also LAPACK Working Note 154.)

  [Dhillon97]

    I. Dhillon. A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem, Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

- ScaLAPACK

  [SLUG]

    L. Blackford, J. Choi, A.Cleary, E. D'Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K.Stanley, D. Walker, and R. Whaley. ScaLAPACK Users' Guide, Society for Industrial and Applied Mathematics (SIAM), 1997.

- Sparse Solver

  [Duff99]

    I. S. Duff and J. Koster. The Design and Use of Algorithms for Permuting Large Entries to the Diagonal of Sparse Matrices. SIAM J. Matrix Analysis and Applications, 20(4):889-901, 1999.

  [Dong95]

    J. Dongarra, V.Eijkhout, A.Kalhan. Reverse Communication Interface for Linear Algebra Templates for Iterative Methods. UT-CS-95-291, May 1995. http://www.netlib.org/lapack/lawnspdf/lawn99.pdf

  [Karypis98]

    G. Karypis and V. Kumar. A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs. SIAM Journal on Scientific Computing, 20(1):359-392, 1998.

  [Li99]

    X.S. Li and J.W. Demmel. A Scalable Sparse Direct Solver Using Static Pivoting. In Proceeding of the 9th SIAM conference on Parallel Processing for Scientific Computing, San Antonio, Texas, March 22-34,1999.

  [Liu85]

    J.W.H. Liu. Modification of the Minimum-Degree algorithm by multiple elimination. ACM Transactions on Mathematical Software, 11(2):141-153, 1985.

  [Menon98]

    R. Menon L. Dagnum. OpenMP: An Industry-Standard API for Shared-Memory Programming. IEEE Computational Science & Engineering, 1:46-55, 1998. http://www.openmp.org.

  [Saad03]

    Y. Saad. Iterative Methods for Sparse Linear Systems. 2nd edition, SIAM, Philadelphia, PA, 2003.

  [Schenk00]

    O. Schenk. Scalable Parallel Sparse LU Factorization Methods on Shared Memory Multiprocessors. PhD thesis, ETH Zurich, 2000.

  [Schenk00-2]

    O. Schenk, K. Gartner, and W. Fichtner. Efficient Sparse LU Factorization with Left-right Looking Strategy on Shared Memory Multiprocessors. BIT, 40(1):158-176, 2000.

  [Schenk01]

    O. Schenk and K. Gartner. Sparse Factorization with Two-Level Scheduling in PARDISO. In Proceeding of the 10th SIAM conference on Parallel Processing for Scientific Computing, Portsmouth, Virginia, March 12-14, 2001.

  [Schenk02]

    O. Schenk and K. Gartner. Two-level scheduling in PARDISO: Improved Scalability on Shared Memory Multiprocessing Systems. Parallel Computing, 28:187-197, 2002.

  [Schenk03]

    O. Schenk and K. Gartner. Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO. Journal of Future Generation Computer Systems, 20(3):475-487, 2004.

  [Schenk04]

    O. Schenk and K. Gartner. On Fast Factorization Pivoting Methods for Sparse Symmetric Indefinite Systems. Technical Report, Department of Computer Science, University of Basel, 2004, submitted.

  [Sonn89]

    P. Sonneveld. CGS, a Fast Lanczos-Type Solver for Nonsymmetric Linear Systems. SIAM Journal on Scientific and Statistical Computing, 10:36-52, 1989.

  [Young71]

    D.M.Young. Iterative Solution of Large Linear Systems. New York, Academic Press, Inc., 1971.

- VSL

  [VSL Notes]

    Document included with Intel® MKL product (file name vslnotes.pdf).

  [Bratley87]

    Bratley P., Fox B.L., and Schrage L.E. A Guide to Simulation. 2nd edition. Springer-Verlag, New York, 1987.

  [Bratley88]

    Bratley P. and Fox B.L. Implementing Sobol`s Quasirandom Sequence Generator, ACM Transactions on Mathematical Software, Vol. 14, No. 1, Pages 88-100, March 1988.

  [Bratley92]

    Bratley P., Fox B.L., and Niederreiter H. Implementation and Tests of Low-Discrepancy Sequences, ACM Transactions on Modeling and Computer Simulation, Vol. 2, No. 3, Pages 195-213, July 1992.

  [Coddington94]

    Coddington, P. D. Analysis of Random Number Generators Using Monte Carlo Simulation. Int. J. Mod. Phys. C-5, 547, 1994.

  [Gentle98]

    Gentle, James E. Random Number Generation and Monte Carlo Methods, Springer-Verlag New York, Inc., 1998.

  [L'Ecuyer94]

    L'Ecuyer, Pierre. Uniform Random Number Generation. Annals of Operations Research, 53, 77-120, 1994.

  [L'Ecuyer99]

    L'Ecuyer, Pierre. Tables of Linear Congruential Generators of Different Sizes and Good Lattice Structure. Mathematics of Computation, 68, 225, 249-260, 1999.

  [L'Ecuyer99a]

    L'Ecuyer, Pierre. Good Parameter Sets for Combined Multiple Recursive Random Number Generators. Operations Research, 47, 1, 159-164, 1999.

  [L'Ecuyer01]

    L'Ecuyer, Pierre. Software for Uniform Random Number Generation: Distinguishing the Good and the Bad. Proceedings of the 2001 Winter Simulation Conference, IEEE Press, 95-105, Dec. 2001.

  [Kirkpatrick81]

    Kirkpatrick, S., and Stoll, E. A Very Fast Shift-Register Sequence Random Number Generator. Journal of Computational Physics, V. 40. 517-526, 1981.

  [Knuth81]

    Knuth, Donald E. The Art of Computer Programming, Volume 2, Seminumerical Algorithms. 2nd edition, Addison-Wesley Publishing Company, Reading, Massachusetts, 1981.

  [Matsumoto98]

    Matsumoto, M., and Nishimura, T. Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator, ACM Transactions on Modeling and Computer Simulation, Vol. 8, No. 1, Pages 3-30, January 1998.

  [Matsumoto00]

    Matsumoto, M., and Nishimura, T. Dynamic Creation of Pseudorandom Number Generators, 56-69, in: Monte Carlo and Quasi-Monte Carlo Methods 1998, Ed. Niederreiter, H. and Spanier, J., Springer 2000, http://www.math.sci.hiroshima-u.ac.jp/%7Em-mat/MT/DC/dc.html.

  [NAG]

    NAG Numerical Libraries. http://www.nag.co.uk/numeric/numerical_libraries.asp

  [Sobol76]

    Sobol, I.M., and Levitan, Yu.L. The production of points uniformly distributed in a multidimensional cube. Preprint 40, Institute of Applied Mathematics, USSR Academy of Sciences, 1976 (In Russian).

- FFT

  [1]

    E. Oran Brigham, The Fast Fourier Transform and Its Applications, Prentice Hall, New Jersey, 1988.

  [2]

    Athanasios Papoulis, The Fourier Integral and its Applications, 2nd edition, McGraw-Hill, New York, 1984.

  [3]

    Ping Tak Peter Tang, DFTI, a New API for DFT: Motivation, Design, and Rationale, July 2002.

  [4]

    Charles Van Loan, Computational Frameworks for the Fast Fourier Transform, SIAM, Philadelphia, 1992

- VML

  J.M.Muller. Elementary functions: algorithms and implementation, Birkhauser Boston, 1997.

  IEEE Standard for Binary Floating-Point Arithmetic. ANSI/IEEE Std 754-1985.

- Interval Solver

  [Alefeld83]

    G. Alefeld and J. Herzberger, Introduction to Interval Computations. - Academic Press, New York, 1983.

  [Bentbib02]

    A.H.Bentbib, Solving the full rank interval least squares problem // Applied Numerical Mathematics. - 2002. - Vol. 41. - P. 283-294.

  [Bliek92]

    Ch. Bliek, Computer methods for design automation, Ph.D. Thesis. - Dept. of Ocean Engineering, Massachusetts Institute of Technology, 1992.

  [Hammer95]

    R. Hammer, M. Hocks, U. Kulisch, D. Ratz, C++ Toolbox for Verified Computing I. Basic Numerical Problems. - Berlin-Heidelberg: Springer, 1995.

  [Hansen92]

    E. Hansen, Bounding the solution of interval linear equations // SIAM Journal on Numerical Analysis. - 1992. - Vol. 29, No. 5. - P. 1493-1503.

  [Herzberger94]

    J. Herzberger, Iterative methods for the inclusion of the inverse of a matrix // Topics in Validated Computations, J. Herzberger, ed. - Amsterdam: Elsevier, 1994. -

  [Jansson91]

    Ch.Jansson, Interval linear systems with symmetric matrices, skew-symmetric matrices, and dependencies in the right hand side // Computing. - 1991. - Vol. 46. - P. 265 - 274.

  [Kearfott96]

    R.B. Kearfott, Rigorous Global Search: Continuous Problems. - Dordrecht, Kluwer, 1996.

  [Kearfott]

    R.B. Kearfott, M.T. Nakao, A. Neumaier, S.M. Rump, S.P. Shary, P. van Hentenryck, Standardized notation in interval analysis. - An electronic version of the paper is accessible at http://www.mat.univie.ac.at/~neum/software/int/

  [Kreinovich97]

    V. Kreinovich, A. Lakeyev, J. Rohn, P. Kahl, Computational Complexity and Feasibility of Data Processing and Interval Computations. - Kluwer, Dordrecht, 1997.

  [Neumaier90]

    A. Neumaier, Interval Methods for Systems of Equations. - Cambridge, Cambridge University Press, 1990.

  [Neumaier99]

    A.Neumaier, A simple derivation of Hansen-Bliek-Rohn-Ning-Kearfott enclosure for linear interval equations // Reliable Computing. - 1999. - Vol. 5, No. 2. - P. 131-136.

  [Ning97]

    S. Ning, R.B. Kearfott, A comparison of some methods for solving linear interval equations // SIAM Journal on Numerical Analysis. - 1997. - Vol. 34, No. 4. - P. 1289-1305.

  [Rex99]

    G.Rex, J.Rohn, Sufficient conditions for regularity and singularity of interval matrices // SIAM Journal on Numerical Analysis. - 1999. - Vol. 20. - P. 437-445.

  [Rohn93]

    J. Rohn, Cheap and tight bounds: the recent result by E. Hansen can be made more efficient // Interval Computations. - 1993. - No. 4. - P. 13-21.

  [Rump83]

    S. M.Rump, Solving algebraic problems with high accuracy // A New Approach to Scientific Computation; Kulisch U. W. and Miranker W. L., eds. - New York: Academic Press, 1983. - P. 51-120.

  [Rump84]

    S. M.Rump, Solution of linear and nonlinear algebraic problems with sharp guaranteed bounds // Computing Supplement. - 1984. - Vol. 5. - P. 147-168.

  [Rump80]

    S. M.Rump, Kaucher E. Small bounds for the solution of systems of linear equations // Computing Supplement. - 1980. - Vol. 2. - P. 157-164.

  [Rump]

    S. Rump, INTLAB -- INTerval LABoratory. - 21 p. - An electronic version of the paper is accessible at http://www.ti3.tu-harburg.de/ rump/intlab/.

  [Shary92]

    S.P. Shary, A new class of algorithms for optimal solution of interval linear systems // Interval Computations. - 1992, No. 2(4). - P. 18-29.

  [Shary95]

    S.P. Shary, On optimal solution of interval linear equations // SIAM Journal on Numerical Analysis. - 1995. - Vol. 32, No. 2. - P. 610-630.

  [Shary02]

    S. P. Shary, A new technique in systems analysis under interval uncertainty and ambiguity // Reliable Computing. - 2002. - Vol. 8. - 321-419.

  [Shary]

    S.P. Shary, A new class of methods for optimal enclosing solution sets to interval linear systems // Journal of Computational Mathematics. - to be published.

- Optimization Solvers

  [Conn00]

    A. R. Conn, N. I.M. Gould, P. L. Toint.Trust-region Methods.SIAM Society for Industrial & Applied Mathematics, Englewood Cliffs, New Jersey, MPS-SIAM Series on Optimization edition, 2000.

  [Dong95]

    J. Dongara, V. Eijkhout, A. Kalhan.Reverse communication interface for linear algebra templates for iterative methods.1995.

For a reference implementation of BLAS, sparse BLAS, LAPACK, and ScaLAPACK packages (without platform-specific optimizations) visit www.netlib.org