Inverses of Multivariable Polynomial Matrices by Discrete Fourier Transforms

The problem of the fast computation of the Moore–Penrose and Drazin inverse of a multi-variable polynomial matrix is addressed. The algorithms proposed, use evaluation-interpolation techniques and the Fast Fourier transform. They proved to be faster than other known algorithms. The efficiency of the algorithms is illustrated via randomly generated examples.

[1]  Predrag S. Stanimirovic,et al.  A problem in computation of pseudoinverses , 2003, Appl. Math. Comput..

[2]  Bede Liu,et al.  Accumulation of Round-Off Error in Fast Fourier Transforms , 1970, JACM.

[3]  Tolga Güyer,et al.  A new method for computing the solutions of differential equation systems using generalized inverse via Maple , 2001, Appl. Math. Comput..

[4]  R. Penrose A Generalized inverse for matrices , 1955 .

[5]  Steven G. Johnson,et al.  FFTW: an adaptive software architecture for the FFT , 1998, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181).

[6]  V. Pan,et al.  Polynomial and Matrix Computations , 1994, Progress in Theoretical Computer Science.

[7]  Nicholas P. Karampetakis,et al.  Generalized inverses of two-variable polynomial matrices and applications , 1997 .

[8]  George E. Antoniou Transfer function computation for generalized n-dimensional systems , 2001, J. Frankl. Inst..

[9]  Michael Sebek,et al.  Fast Fourier Transform and Linear Polynomial Matrix Equations , 2001 .

[10]  John D. Lipson The fast Fourier transform its role as an algebraic algorithm , 1976, ACM '76.

[11]  Dimitrios Alexios Karras,et al.  Transfer function determination of singular systems using the DFT , 1989 .

[12]  Henry P. Decell An application of the Cayley-Hamilton theorem to generalized matrix inversion. , 1965 .

[13]  P. Tzekis,et al.  On the computation of the generalized inverse of a polynomial matrix , 2001 .

[14]  Nicholas P. Karampetakis,et al.  Computation of the Generalized Inverse of a Polynomial Matrix and Applications , 1997 .

[15]  M. Drazin Pseudo-Inverses in Associative Rings and Semigroups , 1958 .

[16]  Nicholas P. Karampetakis,et al.  On the Computation of the Drazin Inverse of a Polynomial Matrix , 2001 .

[17]  Gene H. Golub,et al.  Matrix computations , 1983 .

[18]  P. Hippe,et al.  Inversion of polynomial matrices by interpolation , 1992 .

[19]  A finite algorithm for computing the weighted Moore-Penrose inverse A +MN , 1987 .

[20]  G. Pierobon,et al.  FFT calculation of a determinantal polynomial , 1976 .

[21]  T. Greville,et al.  The Souriau-Frame algorithm and the Drazin pseudoinverse , 1973 .

[22]  Predrag S. Stanimirovic,et al.  A finite algorithm for generalized inverses of polynomial and rational matrices , 2003, Appl. Math. Comput..

[23]  Nicholas P. Karampetakis,et al.  The Computation and Application of the Generalized Inverse via Maple , 1998, J. Symb. Comput..

[24]  Predrag S. Stanimirovic,et al.  SYMBOLIC IMPLEMENTATION OF LEVERRIER-FADDEEV ALGORITHM AND APPLICATIONS , .

[25]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[26]  Dan E. Dudgeon,et al.  Multidimensional Digital Signal Processing , 1983 .