of a hierarchy of (r × r) tri-diagonal matrices of increasing size. ANITI through the French “Investing for the Future PIA3” program. Automatic Performance Tuning of Sparse Matrix Kernels; Updating LAPACK and ScaLAPACK Ideal: program in your favorite high level language (Matlab, PETSc). Use matlab to simulate the power control algorithm (1), starting (a) A matrix A ∈ Rn×n is tridiagonal if Aij = 0 for |i − j| > 1. FOREX SINHALA BLOGS

Moreover, this representation reveals already the QR factorization of the semisepar- able matrix S: the Givens transformations appearing in the representation of the matrix are exactly the same as the Givens transformations appearing in the Q factor of the QR factorization. More information can be found in [42, 34]. In [42, 44] other algorithms, based on the Givens-vector representation can be found. The algorithms for retrieving the representation and for constructing the full semiseparable matrix, given the Givens-vector representation, can be deduced easily with the inform- ation in the previous section.

Here an O n implementation of a matrix vector multiplication will be given. The formulas will be given for nonsymmetric semiseparable matrices, who have two se- quences of Givens transformations and two vectors. To deduce the algorithm, we have to decompose the matrix into a strict uppertriangular, and a lower triangular part of S.

Rewriting the formulas reveals the order O n algorithm for the multiplication. Combining the last 2 equalities of all the yi one can derive an order n algorithm to perform the multiplication of S1 and v. The multiplication of S2 and v can be derived in a completely analogous way. We have given an altern- ative definition and a corresponding representation which keeps the interesting proper- ties of the standard representation but does not exhibit the disadvantages.

Appendix A Our main result here will be to prove, as in [21, Chapter II] , that the inverse of a one-pair matrix is an unreducible tridiagonal matrix and vice versa. First some notation has to be introduced. Proposition 7. This proves This matrix will then appear to be a one-pair matrix.

This means that S i1 ,. Theorem 7. Scand, —60, Barrett and P. Inverses of banded matrices. Linear Algebra and Its Applications, —, Chandrasekaran, P. Dewilde, M. Gu, T. Pals, X. Sun, A. Fast stable solvers for sequentially semi—separable linear sys- tems of equations and least squares problems. Pals, and A. Fast stable solver for sequentially seni-separable linear systems of equations.

Lecture Notes in Computer Science, —, Chandrasekaran and M. Fast and stable eigendecomposition of symmet- ric banded plus semi-separable matrices. A fast and stable solver for recursively semi- separable systems of linear equations. Contemporary Mathematics, —53, A divide and conquer algorithm for the eigen- decomposition of symmetric block-diagonal plus semi-separable matrices. Nu- merische Mathematik, Fast and stable algorithms for banded plus semiseparable systems of linear equations.

SIAM J. Matrix Anal. Colton and R. Inverse acoustic and electromagnetic scattering theory. Springer-Verlag, second edition, Dewilde and A. Time-varying systems and computations. Kluwer academic publishers, Boston, June Eidelman and I. Inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices. A look ahead block schur algorithm for diag- onal plus semiseparable matrices. Some observations on inverses of band matrices and low rank perturb- ations of triangular matrices.

Fallat, M. Fiedler, and T. Generalized oscillatory matrices. Linear Algebra and Its Applications, —90, Fasino and L. Direct and inverse eigenvalue problems, for diagonal-plus-semiseparable matrices. Submitted for publication, Structural and computational properties of possibly singular semiseparable matrices.

Linear Algebra and its Applications, — , Fasino, N. Mastronardi, and M. Van Barel. Fast and stable algorithms for re- ducing diagonal plus semi separable matrices to tridiagonal and bidiagonal form. Contemporary Mathematics, —, Structure ranks of matrices. Basic matrices. Fiedler and Z. Generalized Hessenberg matrices. Submitted for pub- lication. Gantmacher and M. Oscillation matrices and kernels and small vibrations of mechanical systems.

AMS Chelsea publishing, Gemignani and D. Fast and stable solution of banded-plus- semiseparable linear systems. Calcolo, 39 4 —, Gohberg and M. Time varying linear systems with boundary con- ditions and integral operators, I. Integral Equations and Operator Theory, —, Gonzales, J. Eisert, I. Koltracht, M.

Neumann, and G. Integ- ral equation method for the continuous spectrum radial schrodinger equation. Journal of computational physics, —, Matrices with applications in statistics. Wadsworth international group, Belmont, California, Greengard and V. On the numerical solution of two-point boundary value problems. Communications on Pure and Applied Mathematics, — , Groenewald, M.

Petersen, and A. Factoriza- tion of integral operators with semiseparable kernel and symmetries. Kang, I. Koltracht, and G. High accuracy method for integ- ral equations with discontinuous kernels. Kavcic and M. Matrices with banded inverses: inversion algorithms and factorization of gauss-markov processes. Lee and L. A fast adaptive numerical method for stiff two-point boundary value problems. Mastronardi, S. Chandrasekaran, and S. Van Huffel.

Fast and stable two-way algorithm for diagonal plus semi-separable systems of linear equations. Numer- ical Linear Algebra with Applications, 8 1 :7—12, Fast and stable algorithms for reducing diagonal plus semiseparable matrices to tridiagonal and bidiagonal form. BIT, 41 1 —, Mastronardi, M. Van Barel, R. Vandebril, and L. A Lanczos subspace iteration method for the symmetric rank revealing factorization.

In preparation, Mastronardi, E. Van Camp, and M. Divide and conquer type al- gorithms for computing the eigendecomposition of diagonal plus semiseparable matrices. Band matrices and semi-separable matrices. Colloquia Mathematica Societatis Janos Bolyai, —, On the inverse of band matrices.

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Van Barel, and M. An implicit QR algorithm for semiseparable matrices to compute the eigendecomposition of symmetric matrices. Leuven, Leuven, Belgium, August A QR method for computing the singular values via semiseparable matrices.

Jun 14, at 1 Thank you so much. And there was a point where he said: When implementing the normal equation in octave we want to use the 'pinv' function rather than 'inv. Jun 15, at 1 RishabhKumarSingh indeed!

But this is a mathematical property of of the pinv, it has nothing to do with precision. If inv exists, then pinv will necessarily be equal to inv.

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