Theory of Equations. By J. V. Uspensky. Pp. vii, 353. 27s. 1948. (McGraw-Hill)

The Geometry of the Zeros of a Polynomial in a complex variable. By M. Marden. Pp. ix, 183. $5. 1949. Mathematical Surveys, 3. (American Mathematical Society)

This paper studies the secure multiple-antenna transmission in slow fading channels for the cognitive radio network, where a multiple-input, single-output, multieavesdropper (MISOME) primary network coexisting with a multiple-input single-output secondary user (SU) pair. The SU can get the transmission opportunity to achieve its own data traffic by providing the secrecy guarantee for the PU with artificial noise. Different from the existing works, which adopt the instantaneous secrecy rate as the performance metric, with only the statistical channel state information (CSI) of the eavesdroppers, we maximize the secrecy throughput of the PU by designing and optimizing the beamforming, rate parameters of the wiretap code adopted by the PU, and power allocation between the information signal and the artificial noise of the SU, subjected to the secrecy outage constraint at the PU and a throughput constraint at the SU. We propose two design strategies: 1) nonadaptive secure transmission strategy (NASTS) and 2) adaptive secure transmission strategy, which are based on the statistical and instantaneous CSIs of the primary and secondary links, respectively. For both strategies, the exact rate parameters can be optimized through numerical methods. Moreover, we derive an explicit approximation for the optimal rate parameters of the NASTS at high SNR regime. Numerical results are illustrated to show the efficiency of the proposed schemes.

On the Secrecy Throughput Maximization for MISO Cognitive Radio Network in Slow Fading Channels

https://pure.mpg.de/pubman/item/item_2076012_3/component/file_2214467/1308.4088v2

Computing real roots of real polynomials

https://faculty.nps.edu/bneta/papers/AMCReviewPaper.pdf

Multipoint methods for solving nonlinear equations: A survey

Numerical Methods for Roots of Polynomials - Part II along with Part I (9780444527295) covers most of the traditional methods for polynomial root-finding such as interpolation and methods due to Graeffe, Laguerre, and Jenkins and Traub. It includes many other methods and topics as well and has a chapter devoted to certain modern virtually optimal methods. Additionally, there are pointers to robust and efficient programs. This book is invaluable to anyone doing research in polynomial roots, or teaching a graduate course on that topic. It is the first comprehensive treatment of Root-Finding in several decades with a description of high-grade software and where it can be downloaded. It offers a long chapter on matrix methods and includes Parallel methods and errors where appropriate. It proves invaluable for research or graduate course.

Numerical methods for roots of polynomials

We discuss the secant method: x i + 1 = x i - f i x i - x i - 1 f i - f i - 1 ( i = 1 , 2 , 3 , … ) where ( x 0 , f 0 ) , ( x 1 , f 1 ) are initial guesses. In the Regula Falsi variation we start with initial guesses ( a , f ( a ) ) and ( b , f ( b ) ) such that f ( a ) f ( b ) 0 ; after an iteration similar to the above we replace either a or b by the new value x i + 1 depending on which of f ( a ) or f ( b ) has the same sign as f ( x i + 1 ) . Often one of the points gets “stuck,” and several variants such as the Illinois or Pegasus methods and variations are used to “unstick” it. We discuss convergence and efficiency of most of the methods considered. We treat methods involving quadratic of higher order interpolation and rational approximation. We also discuss the bisection method where again f ( a ) f ( b ) 0 and we set c = a + b 2 . We replace a or b by c according to the sign of f ( c ) as in the Regula Falsi method. Various generalizations are described, including some for complex roots. Finally we consider hybrid methods involving two or more of the previously described methods.

Bisection and Interpolation Methods

First we consider the Jenkins–Traub 3-stage algorithm. In stage 1 we define H ( 0 ) ( z ) = P ′ ( z ) ; H ( k + 1 ) ( z ) = 1 z H ( k ) ( z ) - H ( k ) ( 0 ) P ( 0 ) P ( z ) ( k = 0 , 1 , 2 , … , M ) In the second stage the factor 1 z is replaced by 1 z - s for fixed s , and in the third stage by 1 z - s k where s k is re-computed at each iteration. Then s k → a root. A slightly different algorithm is given for real polynomials. Another class of methods uses minimization, i.e. we try to find z ∗ such that V = R 2 + J 2 is a minimum, where f ( z ) = f ( x + iy ) = R ( x , y ) + iJ ( x , y ) . At this minimum we must have R = J = 0 , i.e. f ( z ∗ ) = 0 . Several authors search along the coordinate axes or at various angles with them, while others move along the negative gradient, which is probably more efficient. Some use a hybrid of Newton and minimization. Finally we come to Lin and Bairstow’s methods, which divide the polynomial by a quadratic and iteratively reduce the remainder to 0. This enables us to find pairs of complex roots using only real arithmetic.

Chapter 11 - Jenkins–Traub, Minimization, and Bairstow Methods

This chapter treats several topics, starting with Bernoulli’s method. This method iteratively solves a linear difference equation whose coefficients are the same as those of the polynomial. The ratios of successive iterates tends to the root of largest magnitude. Special versions are used for complex and/or multiple roots. The iteration may be accelerated, and Aitken’s variation finds all the roots simultaneously.

The Quotient-Difference algorithm uses two sequences
qk(m+1)=qk(m)+ek(m)-ek-1(m+1)
(with a similar one for ek(m)). Then, if the roots are well separated, qk(m)→1kthroot. Special techniques are used for roots of equal modulus.

The Lehmer–Schur method uses a test to determine whether a given circle contains a root or not. Using this test we find an annulus R<|z|<2R which contains a root, whereas the circle |z|<R does not. We cover the annulus with 8 smaller circles and test which one contains the roots. We repeat the process until a sufficiently small circle is known to contain the root.

We also consider methods using integration, such as by Delves–Lyness and Kravanja et al.

Chapter 10 - Bernoulli, Quotient-Difference, and Integral Methods

Whereas Newton’s method involves only the first derivative, methods discussed in this chapter involve the second or higher. The “classical” methods of this type (such as Halley’s, Euler’s, Hansen and Patrick’s, Ostrowski’s, Cauchy’s and Chebyshev’s) are all third order with three evaluations, so are slightly more efficient than Newton’s method. Convergence of some of these methods is discussed, as well as composite variations (some of which have fairly high efficiency). We describe special methods for multiple roots, simultaneous or interval methods, and acceleration techniques. We treat Laguerre’s method, which is known to be globally convergent for all-real-roots. The Cluster-Adapted Method is useful for multiple or near-multiple roots. Several composite methods are discussed, as well as methods using determinants or various types of interpolation, and Schroeder’s method.

J.M. McNamee

Papers

Numerical methods for roots of polynomials

Bisection and Interpolation Methods

Chapter 11 - Jenkins–Traub, Minimization, and Bairstow Methods

Chapter 10 - Bernoulli, Quotient-Difference, and Integral Methods

Methods Involving Second or Higher Derivatives