Multiple context-free grammar (MCFG) is an extension of context-free grammar (CFG), which generates tuples of words. The expressive power of MCFG is between CFG and context-sensitive grammar while MCFG inherits good properties of CFG. In this paper, we introduce weighted multiple context-free grammar (WMCFG) as a quantitative extension of MCFG. Then we investigate properties of WMCFG such as polynomial-time computability of basic problems, its closure property and expressive power.

Let M(y) be a matrix whose entries are polynomial in y, λ(y) and v(y) be a set of eigenvalue and eigenvector of M(y). Then, λ(y) and v(y) are algebraic functions of y, and λ(y) and v(y) have their power series expansionsλ(y) = β0 + β1 y + + βk yk + (βj C),(1) v(y) = γ0 + γ1 y + + γk yk + (γj Cn), (2)provided that y=0 is not a singular point of λ(y) or v(y). Several algorithms are already proposed to compute the above power series expansions using Newton's method (the algorithm in [4]) or the Hensel construction (the algorithm in[5],[12]). The algorithms proposed so far compute high degree coefficients βk and γk, using lower degree coefficients βj and γj (j=0,1,,k-1). Thus with floating point arithmetic, the numerical errors in the coefficients can accumulate as index k increases. This can cause serious deterioration of the numerical accuracy of high degree coefficients βk and γk, and we need to check the accuracy. In this paper, we assume that given matrix M(y) does not have multiple eigenvalues at y=0 (this implies that y=0 is not singular point of λ(y) or v(y)), and presents an algorithm to estimate the accuracy of the computed power series βi,γj in (1) and (2). The estimation process employs the idea in [9] which computes a coefficient of a power series with Cauchy's integral formula and numerical integrations. We present an efficient implementation of the algorithm that utilizes Newton's method. We also present a modification of Newton's method to speed up the procedure, introducing tuning parameter p. Numerical experiments of the paper indicates that we can enhance the performance of the algorithm by 1216%, choosing the optimal tuning parameter p.

Given the bivariate polynomial f(x,y), let φ(y) be a root of f(x,y) = 0 with respect to x, i.e. φ(y) is a function of y such that f(φ(y),y) = 0. If φ(y) is analytic at y = 0, then we have its power series expansion φ(y) = α0 + α1y + α2y2 + + αpyp + .(1)Let φ(p)(y) denote φ(y) truncated at yp, i.e. φ(p)(y) = α0 + α1y + α2y2 + + αpyp.(2) It is well-known that we can compute power series roots φ(p)(y) by Newton's method. In fact, given the initial value φ(0)(y) = α0 C, the following Newton's methodφ(k)(y) φ(k-1)(y) - (mod yk+1) (3) computes φ(k)(y) (1 k) in expression (2) efficiently (applying the above formula for k = 1,2,, we can compute the power series root φ(p)(y) of any degree p). The above formula (3) is referred to as "symbolic Newton's method" in this paper. From this formula (3), we can see that the numerical errors in the coefficients αs (s = 0,1,...,k - 1) directly affect the numerical error in the coefficient αk. This implies that the symbolic Newton's method is numerically unstable, i.e., a numerical error in the coefficient αk accumulates as the index k increases. Moreover, with the symbolic Newton's method, even if we need only one coefficient αk, we must compute all coefficients αs (s = 0,1,,k - 1). Thus, when we require only one high degree coefficient αk, the symbolic Newton's method is numerically unstable and inefficient. In this paper, given the integer k (> 0), we present a new algorithm to compute the coefficient αk of (1). The new algorithm is numerically stable and requires no computation of the coefficients other than αk itself.

Suppose that we need to design a controller for the system x(t) = A x(t) + B u, u = -K x(t), y(t) = C x(t), where matrices A, B and C are given and K is the matrix to to determine. It is required to determine K so that y(t) should not exceed prescribed value (i.e., the peak of output y(t) is limited). This kind of specification, in general, difficult to satisfy, since the peak ymax of y(t) (we define ymax to be max0 t |y(t)|) is a non-trivial function of design parameter K, which can not be expressed explicitly generally. Therefore, a controller design with such specifications often requires try and error process. In this paper, we approximate ymax in the form of formal power series and give an efficient algorithm to compute the series. We also give a design example of a control system as an application of the algorithm.

In Ref.[5], the author defines "approximate eigenvalues" and "approximate eigenvectors," which are, in short, Taylor series expansions of eigenvalues and eigenvectors of a polynomial matrix. In this paper, an efficient algorithm to compute the approximate eigenvalues and eigenvectors is presented. The algorithm performs the computations with an arbitrary degree of convergence.

In this paper, we present a numerical method with guaranteed accuracy to solve initial value problems (IVPs) of normal form simultaneous first order ordinary differential equations (ODEs) which have wide domain. Our method is based on the algorithm proposed by Kashiwagi, by which we can obtain inclusions of exact values at several discrete points of the solution curve of ODEs. The method can be regarded as an extension of the Lohner's method. But the algorithm is not efficient for equations which have wide domain, because the error bounds become too wide from a practical point of view. Our purpose is to produce tight bounds even for such equations. We realize it by combining Kashiwagi's algorithm with the mean value form. We also consider the wrapping effects to obtain tighter bounds.

This paper proposes a new distortion analysis method for frequency-dependent FET amplifiers, which uses a novel Frequency-Dependent Complex Power Series (FDCPS) model. This model consists of a frequency-independent nonlinear amplifier represented by an odd-order complex power series and frequency-dependent input and output filters. The in-band frequency characteristics of the saturation region are represented by the frequency-dependent output filter, while the in-band frequency characteristics of the linear region are represented by the frequency-dependent input and output filters. In this method, the time-domain analysis is carried out to calculate the frequency-independent nonlinear amplifier characteristics, and the frequency-domain analysis is applied to calculate the frequency-dependent input and output filter characteristics. The third-order intermodulation (IM3) calculated by this method for a GaAs MESFET amplifier is in good agreement with the numerical results obtained by the Harmonic Balance (HB) method. Moreover, the IM3 calculated by this method also agrees well with the measured data for an L-band 3-stage GaAs MESFET amplifier. It is shown that this method is effective for calculating frequency-dependent distortion of a nonlinear amplifier with broadband modulation signals.