Topic
Squaring the square
About: Squaring the square is a research topic. Over the lifetime, 25 publications have been published within this topic receiving 341 citations.
Papers
TL;DR: Improvements to the inverse scaling and squaring method for the matrix logarithm are made, including backward error analysis to replace the previous forward error analysis; obtain backward error bounds in terms of the quantities $p$ instead of $\|A\|$; and use special techniques to compute the argument of the Pad\'e approximant more accurately.
Abstract: A popular method for computing the matrix logarithm is the
inverse scaling and squaring method,
which essentially carries out the steps of the scaling and squaring
method for the matrix exponential in reverse order.
Here we make several improvements to the method,
putting its development on a par with our recent version
[\emph{SIAM J. Matrix Anal.\ Appl.}, 31 (2009), pp.\ 970--989]
of the scaling and squaring method for the exponential.
In particular,
we introduce backward error analysis to replace the previous forward
error analysis;
obtain backward error bounds in terms of the quantities
$\|A^p\|^{1/p}$, for several small integer $p$, instead of $\|A\|$;
and use special techniques to compute the argument of the
Pad\'e approximant more accurately.
We derive one algorithm
that employs a Schur decomposition,
and thereby works with triangular matrices,
and another that requires only matrix multiplications and the solution
of multiple right-hand side linear systems.
Numerical experiments show the new algorithms to be generally faster and more
accurate than their existing counterparts
and suggest that the Schur-based method is the method of choice for computing
the matrix logarithm.
102 citations
Book•
1 Jan 1970TL;DR: In this paper, Van Schouten's problem of solutions to exercises has been studied in the context of Pythagorean arithmetic, including the Farey series and the isoperimetric problem.
Abstract: 1. Probability and p 2. Odd and even numbers 3. Sylvester's problem of collinear triads 4. The algebra of statements 5. The Farey series 6. A property of an 7. Squaring the square 8. Writing a number as sum of two squares 9. The isoperimetric problem 10. Five curiosities from arithmetic 11. A problem of Regiomontanus 12. Complementary sequences 13. Pythagorean arithmetic 14. Abundant numbers 15. Macheroni and Steiner 16. A property of some repeating decimals 17. The theorem of Barbier 18. The series of reciprocals of primes 19. Van Schouten's problem Solutions to exercises Bibliography.
66 citations
[...]
TL;DR: In this paper, the general methods by which a square may be dissected into smaller unequal non-overlapping squares are described in detail and examples of such dissections are given.
Abstract: 1. Introduction. It is the object of this paper to describe in more detail than has hitherto been done the general methods by which a square may be
dissected into smaller unequal non-overlapping squares. Examples of such
dissections are given
56 citations
25 Jun 2007
TL;DR: This work presents efficient squaring formulae based on the Toom-Cook multiplication algorithm which are much superior to other known squaring algorithms for small input size and under some reasonable assumptions is faster than the recently proposed Montgomery's 5-way Karatsuba-likeformulae.
Abstract: We present efficient squaring formulae based on the Toom-Cook multiplication algorithm. The latter always requires at least one non-trivial constant division in the interpolation step. We show such non-trivial divisions are not needed in the case two operands are equal for three, four and five-way squarings. Our analysis shows that our 3-way squaring algorithms have much less overhead than the best known 3-way Toom-Cook algorithm. Our experimental results show that one of our new 3-way squaring methods performs faster than mpz_mul ( ) in GNU multiple precision library (GMP) for squaring integers of approximately 2400-6700 bits on Pentium IV Prescott 3.2 GHz. For squaring in Z[x], our 3-way squaring algorithms are much superior to other known squaring algorithms for small input size. In addition, we present 4-way and 5-way squaring formulae which do not require any constant divisions by integers other than a power of 2. Under some reasonable assumptions, our 5-way squaring formula is faster than the recently proposed Montgomery's 5-way Karatsuba-like formulae.
56 citations
TL;DR: In this article, the authors prove that a compound perfect squared square must contain at least 22 subsquares, using elementary combinatoric and graph theoretic arguments and an extensive computer search.
21 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2019 | 2 |
| 2017 | 1 |
| 2015 | 1 |
| 2014 | 3 |
| 2012 | 2 |
| 2011 | 1 |