Convolution number

Abstract

Convolution number is a new proposed name for the sequence of numbers that constitute the coefficients of polynomials and truncated Taylor expansions of functions. The arithmetic of the convolution number is the well-known arithmetic of sequences, where multiplication is the convolution of sequences, or the arithmetic of polynomials or formal power series where the multiplication is the Cauchy product. To separate the coefficients from the Taylor series formally, a Taylor transform is defined. Considering the sequence of coefficients as a single number is a new perspective which emphasizes the purely computational application, making a (digital) computer method out of a symbolic (computer) method. By this means, the whole well-known field of solution by Taylor series is cast in a simple numerical application oriented algebraic method. Convolution number analysis is an alternative computational tool to obtain numerical solutions for certain problems that would otherwise be determined by analytical analysis, or Finite Difference or Finite Element methods when analytical solution is too difficult. Taylor expansion of functions of a single variable, i.e., univariate polynomials, real or complex, are considered to develop the theory and the methods. All the methods and results are adapted to functions of two variables, i.e., bivariate polynomials, from which the extension to multivariate polynomials is then obvious. Simple programming of the four basic arithmetic operations on the convolution number is reviewed (and the square root operation), to be used as a set of subroutines, so that problems are formulated and programmed directly in terms of convolution numbers. It is shown how matrix algebra using convolution numbers as elements can be applied to vibration problems and illustrated with an example, resulting in a dynamic system matrix, although limited to small degree of freedom systems because of the large storage required. From the arithmetic, an algebra of convolution numbers is developed, considering the convolution number as a variable. An example of conformal mapping by convolution number algebra is given. A convolution function of a convolution variable is defined, with a compatibility condition analog to the Cauchy-Riemann equations of a complex function of a complex variable. The compatibility equations serve as a tool to derive some basic theorems of convolution variables which are necessary for the development of programming with convolution variables. Generally, the convolution number represents a truncated Taylor series of a function. The arithmetic is such that each coefficient has original machine accuracy and, therefore, the corresponding function could be evaluated to machine accuracy, although no theoretical general rule for the a priori required length of the convolution number and the radius of convergence is given. The well-known problems of polynomial composition and reversion of series are stated in terms of convolution number algebra. It is shown, by an example, how such problems are solved on a digital computer once the basic arithmetic routines are programmed. The well-known method to solve nonlinear ordinary differential equations by Taylor series is generalized to a simple computational solution of the integration process with the aid of a pointer in the computer storage of the convolution number. The solution process is posed in terms of a flow diagram, which is an exact copy of the analog computer diagram of a differential equation, from which the sequence of convolution number equations are programmed. Relation to the z-transform and digital filters is shown. The same example of conformal mapping is solved as a nonlinear equation, and the solution of a nonlinear ordinary differential equation by convolution number analysis is presented. In view of the applications, the radius of convergence of the function in the complex plane must be considered at all times, for which, unfortunately, no general theoretical determination can be given, and which may severely limit the advantage of a Taylor series solution. In the examples, it is shown how a practical estimate can be made. A final solution consists of the well-known method of patching up a sequence of Taylor series, similar to a sequence of high order Finite Difference solution values; the difference, however, being that the series accuracy can be tested analytically. The stability of the recursive solution routines is investigated. The convolution number and its algebra is defined for a bivariate polynomial and an example of the solution of a nonlinear partial differential equation given. Again, the solution is seriously limited by the region of convergence. Throughout, a distinctive and precise symbolic notation for convolution algebra has been attempted.