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Algorithm for generating orthogonal matrices with rational elements
[article]

2009
*
arXiv
*
pre-print

Special

arXiv:cs/0201007v2
fatcat:tkmk6jjcmzguta5m4tosaizns4
*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] ). ...##
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Rational orthogonal approximations to orthogonal matrices

1997
*
Computational geometry
*

Several

doi:10.1016/0925-7721(95)00048-8
fatcat:psi35x7ccvdizfdsyqy6fex2h4
*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. ...##
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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. ...##
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A fast algorithm for rational interpolation via orthogonal polynomials

1989
*
Mathematics of Computation
*

A new

doi:10.1090/s0025-5718-1989-0972369-4
fatcat:idcmbjchvjb3fkmuwelypz5olq
*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,... ...##
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Computation of coprime factorizations of rational matrices

1998
*
Linear Algebra and its Applications
*

We propose numerically reliable state-space

doi:10.1016/s0024-3795(97)00256-5
fatcat:a6dmwdzvjvhfzn426wfq6mefa4
*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*. ...##
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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]

2021
*
arXiv
*
pre-print

A numerical

arXiv:2108.11928v2
fatcat:6v6qv4c4h5hhnb6l6udb2zrxka
*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. ...##
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Orthogonal matrices with rational components in composing tests for High School students
[article]

2000
*
arXiv
*
pre-print

Is it a reason

arXiv:math/0006230v1
fatcat:nifhq23zbbb4xizqusd6hwsdqe
*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. ...##
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Quasi-orthogonal matrices with level based on ratio of Fibonacci numbers

2015
*
Applied Mathematical Sciences
*

This paper discusses quasi-

doi:10.12988/ams.2015.53263
fatcat:erutj4tu65ay5imyggnrj67udq
*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. ...##
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Application of subspace projection approaches for reduced-order modeling of electromagnetic systems

2003
*
IEEE Transactions on Advanced Packaging
*

Subspace projection approaches, including the Padé via Lanczos (PVL), Krylov, and

doi:10.1109/tadvp.2003.821072
fatcat:2yv44zy7vzbzpb4vm2hfre7lj4
*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] . ...##
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Inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matrices

2014
*
Journal of Computational and Applied Mathematics
*

Again there is a close link

doi:10.1016/j.cam.2014.03.015
fatcat:hklm6tp7cjdmxet57zt2ec2sb4
*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. ...##
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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. ...##
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Algebraic methods for modified orthogonal polynomials

1992
*
Mathematics of Computation
*

The stability of the

doi:10.1090/s0025-5718-1992-1140648-6
fatcat:tmwj5ttlfbhm3jqwjejbiwtnrm
*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? ...##
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Algebraic Methods for Modified Orthogonal Polynomials

1992
*
Mathematics of Computation
*

The stability of the

doi:10.2307/2153072
fatcat:b6f4iv4rafaitoykfw45dr3ari
*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? ...##
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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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