A conditionality criterion for systems of linear algebraic equations

The condition number of the double-period method

“…Subsystem (3) is complete on [−π/2, π/2]: any piecewise continuous u(x) is decomposable with respect to it into a Fourier series, converging in the L 2 norm. Hence, adding subsystem (4) makes the approxima tion problem overdetermined, if ; for finite N it is not so. Let u(x) be sufficiently smooth but nonperiodic.…”

“…In [4], a new quantitative criterion of conditionality, an angular one, was proposed. There, the condi tion number of a matrix is understood as such a number that quantity is the number of decimal digits being lost in the process of the numerical solution of a linear system with this matrix.…”

The method of odd continuation for Fourier approximations of nonperiodic functions

“…Then, for fixed , we subtract a linear function of from function (4) to obtain (5) This function vanishes on all four sides of the original square. After continuing it oddly to the domain , the resulting function can be regarded as periodic and continuous together with its first derivatives.…”

“…The off diagonal elements of the Gram matrix com prise . It was shown in [5] that, in this case, a sys tem of linear equations remains well conditioned. However, the computing complexity increases consid erably, since a linear system with a dense matrix has to be solved.…”

“…In [3,4] the double period method was proposed, which uses the entire smoothness of u(x) in an original manner. Approximations produced by this method are differentiable and equally well approximate u(x) inside the interval and on its boundaries.…”