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### Algorithm for generating orthogonal matrices with rational elements [article]

Ruslan Sharipov
2009 arXiv   pre-print
Special orthogonal matrices with rational elements form the group SO(n,Q), where Q is the field of rational numbers.  ...  This theorem yields an algorithm for generating such matrices by means of random number routines.  ...  I have encountered the problem of generating orthogonal matrices with rational elements in designing computer programs for testing students (see [5] ).  ...

### Rational orthogonal approximations to orthogonal matrices

Victor J. Milenkovic, Veljko Milenkovic
1997 Computational geometry
Several algorithms are presented for approximating an orthogonal rotation matrix M in three dimensions by an orthogonal matrix with rational entries.  ...  In practice, the second algorithm generates an approximation with v ~ 1.5 and is much faster than the third algorithm.  ...  Acknowledgements The authors would like to thank Steve Fortune for suggesting Lemma 2.3.  ...

### Page 3412 of Mathematical Reviews Vol. , Issue 99e [page]

1999 Mathematical Reviews
Again, the algorithm of Euclid can serve as an in- troduction to the rapidly growing literature on fast algorithms for matrices with special structure, of which Toeplitz and Hankel matrices are only the  ...  This connects the Euclidean algorithm with (rational) approximation theory.  ...

### A fast algorithm for rational interpolation via orthogonal polynomials

{Ömer Eğecioğlu, Çetin K. Koç
1989 Mathematics of Computation
A new algorithm for rational interpolation is proposed.  ...  Given the data set, the algorithm generates a set of orthogonal polynomials by the classical threeterm recurrence relation and then uses Newton interpolation to find the numerator and the denominator polynomials  ...  We have described two algorithms for rational interpolation. Given that the Hankel matrices Ho, Hi,...  ...

### Computation of coprime factorizations of rational matrices

A. Varga
1998 Linear Algebra and its Applications
We propose numerically reliable state-space algorithms for computing several coprime factorizations of rational matrices: (1) factorizations with factors having poles in a given stability domain; (2) factorizations  ...  The new algorithms are based on a recursive generalized Schur algorithm for pole dislocation.  ...  All computational algorithms use the equivalent descriptor-system (or generalized state-space) representations of rational matrices.  ...

### Page 1014 of Mathematical Reviews Vol. , Issue 94b [page]

1994 Mathematical Reviews
Summary: “A new algorithm for computing the generalized sin- gular value decomposition that diagonalizes two matrices is intro- duced.  ...  Then, the number of outliers depends on the or- der of the rational generating function, and the clustering radius is proportional to the magnitude of the last elements in the gen- erating sequence used  ...

### The cone of 5× 5 completely positive matrices [article]

Max Pfeffer, Jose Alejandro Samper
2021 arXiv   pre-print
A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary.  ...  We study the cone of completely positive (cp) matrices for the first interesting case n = 5.  ...  We also thank Sascha Timme and Paul Breiding for help with the computation of the degree of V Hi using HomotopyContinuation.  ...

### Orthogonal matrices with rational components in composing tests for High School students [article]

Ruslan A. Sharipov
2000 arXiv   pre-print
Is it a reason for pessimism? No, since arithmetics if entire numbers contains broad variety of problems with a simple statement, which might be not less intricate.  ...  Formula (3.4) gives an algorithm for constructing orthogonal matrices with rational components in dimension 3. We shall call it a regular algorithm. Some generalizations and open questions.  ...  Constructing orthogonal matrices with rational components. Constructing orthogonal matrices with rational components is a separate problem.  ...

### Quasi-orthogonal matrices with level based on ratio of Fibonacci numbers

Nikolay Balonin, Mikhail Sergeev
2015 Applied Mathematical Sciences
This paper discusses quasi-orthogonal matrices which were first highlighted by J. J. Sylvester and J. Hadamard, who showed that two level matrices exist for even orders 4t, t integer.  ...  The example of continuous matrices with varying levels is used to show, that the golden section matrices branch is closely associated with Hadamard and conference matrices.  ...  The authors wish to sincerely thank Tamara Balonina for converting this paper into printing format.  ...

### Application of subspace projection approaches for reduced-order modeling of electromagnetic systems

Tingdong Zhou, S.L. Dvorak, J.L. Prince
2003 IEEE Transactions on Advanced Packaging
Subspace projection approaches, including the Padé via Lanczos (PVL), Krylov, and rational Krylov algorithms, are used for reduced-order modeling of wide-band electromagnetic systems.  ...  A frequency segmentation technique has also been used with the Lanczos algorithm to obtain benchmark data of electromagnetic fields and for scattering parameter extraction from the calculated electromagnetic  ...  The two matrices and are generated to be bi-orthogonal, orthogonal, or neither depending on the orthogonalization choices in the rational Krylov algorithm [7] .  ...

### Inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matrices

Thomas Mach, Marc Van Barel, Raf Vandebril
2014 Journal of Computational and Applied Mathematics
Again there is a close link with orthogonal function theory, the extended tridiagonal matrix captures the recurrence coefficients of bi-orthogonal rational functions.  ...  Moreover, it is again sort of inverse of the nonsymmetric Lanczos algorithm: given spectral properties, we reconstruct the two basis Krylov matrices linked to the nonsymmetric Lanczos algorithm Article  ...  For completeness also the orthogonal basis vectors for the rational Krylov space, stored in V , as well as their indices are given.  ...

### Page 5179 of Mathematical Reviews Vol. , Issue 95i [page]

1995 Mathematical Reviews
Three algorithms are proposed for inversion of regular polynomial and rational square matrices D(A).  ...  Algorithms for computing the pseudoinverses of singular polynomial and rational matrices are also considered.  ...

### Algebraic methods for modified orthogonal polynomials

David Galant
1992 Mathematics of Computation
The stability of the algorithms is discussed.  ...  Write The proof follows from examining the matrices generated and comparing them with the matrices in Theorem 1, using Pit) = oit-v). They are identical.  ...  Associated with each weight function is a sequence of monic orthogonal polynomials {tp¡), j = 1,2,..., deg(6?  ...

### Algebraic Methods for Modified Orthogonal Polynomials

David Galant
1992 Mathematics of Computation
The stability of the algorithms is discussed.  ...  Write The proof follows from examining the matrices generated and comparing them with the matrices in Theorem 1, using Pit) = oit-v). They are identical.  ...  Associated with each weight function is a sequence of monic orthogonal polynomials {tp¡), j = 1,2,..., deg(6?  ...

### Page 9045 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews
The second algorithm (a generalization of the Schur method) is associated with the reduction of the matrix 0 -I W = [cr to Schur form using an orthogonal matrix U.  ...  Hence it suffices to compute the ranks (and co-ranks) of the boundary matrices of Q (along with the basis transformations to compute the generators), which is what the algorithm suggested actually does  ...
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