Constructing a Generator of Matrices with Pattern

Halkos, George and Tsilika, Kyriaki (2012): Constructing a Generator of Matrices with Pattern. Published in: International Journal of Information Science and Computer Mathematics , Vol. 4, No. 2 (2011): pp. 101-117.

Abstract

Computations with large matrices work out faster with computer software, even faster creating automatically the matrix of the size and pattern needed. In this paper we propose free computer algebra system Xcas resources to display particular matrices that can be called up directly. Our computer codes provide shortcuts for entering random block diagonal matrices, random triangular matrices, random and specialized band matrices, elementary matrices Eij, Fourier matrices. As for matrices needed in the study of mathematical issues concerning the properties of the roots of a polynomial, we create features with polynomial coefficients. We also propose codes for immediate construction of functional matrices such as Jacobian, bordered Hessian and Wronskian. The computer codes proposed provide visual representation of the matrix pattern (which is traditionally explained using indices and numerals), infinite number of examples using random numbers and immediate construction of large matrices of various forms.

• Item Type: MPRA Paper
• Language: English
• Keywords: Matrices with pattern; functional programming; computer software
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling; C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C8 - Data Collection and Data Estimation Methodology ; Computer Programs > C88 - Other Computer Software; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62 - Existence and Stability Conditions of Equilibrium

References

Anton H. (2000). Elementary Linear Algebra. John Wiley & Sons, Inc., New York.

Anton H., Busby R.C., Knoll C. and Martinez-Garza C. (2003). Contemporary Linear Algebra, J. Willey, Hoboken NJ.

Batten, D and Martellato, D. (1985). Classical versus Modern Approaches to Interregional Input-Output Analysis. The Annals of Regional Science, 19(3): 1-15.

Brauer F., Nohel J.A. (1989). Qualitative Theory of Ordinary Differential Equations. Reprint, Dover.

Cassetti M. (1995). A new method for the identification of patterns in input-output matrices. Economic Systems Research, 7: 363-381.

Chiang A. (1984). Fundamental Methods of Mathematical Economics. 3rd edition, McGraw-Hill Book, Singapore.

Cushing J.M. (2004). Differential Equations: An Applied Approach. Prentice Hall.

Elsner L. and Johnson C.R. (1989). Nonnegative matrices, zero patterns and spectral inequalities. Linear Algebra Applications, 120: 225-236.

Goldberg J. L. (1991). Matrix theory with applications. McGraw-Hill, Inc., Columbus.

Halkos G.E. and Tsilika K.D. (2012a). Stability Analysis in Economic Dynamics: A Computational Approach. MPRA paper 41371, University Library of Munich, Germany.

Halkos G.E. and Tsilika K.D. (2012b). Programming identication criteria in simultaneous equation models. MPRA paper, 43467, University Library of Munich, Germany.

Hall F.J., Li Z. and Wang D. (2001). Symmetric sign pattern matrices that require unique inertia. Linear Algebra Applications, 338(1-3): 153-169.

Heppell S.S., Caswell H. and Crowder L.B. (2000). Life history and elasticity patterns, perturbation analysis for species with minimal demographic data. Ecology, 81(3): 654-665.

Hertz G.Z., Hartzell G.W. III and Stormo G.D. (1990). Identification of consensus patterns in unaligned DNA sequences known to be functionally related. Computer Applications in the Biosciences 6(2): 81-92.

Johnson C.R. (1983). Sign patterns of inverse nonnegative matrices. Linear Algebra Applications, 55: 69-80.

Kaven A. and Sayarinejad M.A. (2004). Eigensolutions for factorable matrices of special patterns. Communications in Numerical Methods in Engineering, 20: 133-146.

Lipschutz S. (1987). Theory and problems of linear algebra. McGraw-Hill, Inc., Columbus.

Lloyd W.P. and Lee C.F. (1976). Block Recursive Systems in Asset Pricing Models, The Journal of Finance, 31(4): 1101-1113.

McDonald J.J., Olesky D.D., Tsatsomeros M.J. and van den Driessche P. (1997). Ray patterns of matrices and nonsingularity. Linear Algebra Applications, 267: 359-373.

Moler C., Van Loan C. (1978). Nineteen dubious ways to compute the exponential of a matrix. SIAM Review, 20: 801-836.

Moler C., Van Loan C. (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review, 45: 3-49.

Noble B. (1969). Applied Linear Algebra. Prentice Hall.

Parisse B. An Introduction to the Xcas Interface, available at http://www-fourier.ujf-grenoble.fr/~parisse/giac/tutoriel_en.pdf

Quarteroni A. and Saleri F. (2006). Scientific Computing with MATLAB and Octave, 2nd edition, Springer-Verlag, Heidelberg, 2006.

Samuelson P.A. (1947). Foundations of Economic Analysis. Harvard Univrersity Press.

Stadelmaier M.W., Rose N.J., Poole G.D. and Meyer C.D. (1982). Nonnegative matrices with power invariant zero patterns. Linear Algebra Applications, 42: 23-29.

Strang G. (1988). Linear Algebra and its Applications. 3rd edition, Harcount Brace Jovanovich College, Philadelphia, New York.

Tardos G. (2005). On 0-1 matrices and small excluded submatrices, Journal of Combinatorial Theory Series A, 111(2): 266-288.

Tarr D.G. (1976). Distributed Lags, Morishima Matrices, and the Stability of Economic Models. Econometrica, 44(3): 597-600.

Veinott A.F. (1969). Minimum concave-cost solution of Leontief substitution models of multi-facility inventory systems, Operational Research, 17(2): 262-291.

Wermuth N. (1992). On block-recursive linear regression equations. Revista Brasileira de Probabilidade e Estatistica, 6, 1-56.

Wold H. (1954). Causality and Econometrics. Econometrica, 22: 162-177.