Riemann Integral Reiteration 1.3 2005/11/03 16:31:33 US/Central 2005/11/30 11:12:32.396 US/Central Leif Faure Anderson lfanders@mail.uh.edu Leif Faure Anderson lfanders@mail.uh.edu Eigenfunction Eigenvalue Eigenvector Riemann Integral The notes contained herein outline the delta-x of the Riemann sum equation transformation into a function used to find the area spectrum of a data set. The transformation uses an eigenfunction by expanding the data set arrays into eigenvecotrs.

Delta-x Transformation through an Eigenfunction The formatting for the expressions in this module are currently being revised.Transformation from a Riemann sum equation to the function for area spectrum (A) is the eigenvalue: λ n 2 1 -1 The area spectrum can be stretched through array duplication across multipule dimensions forming a matrix. The matrices complimenting the eigenvalue have an equal number of values. For this instance of λ , a matrix formulation is necessary for the Riemann ∆x and (A) arrays by duplicating the vertical elements so that the horizontal size of each matrix equals the number of inputs of the original array: Δx Eigenvectors Subscript x nn  Subscript x 11 Subscript x 12 ⋯ Subscript x 1 n Subscript x 21 Subscript x 22 ⋯ Subscript x 2 n ⋮ ⋮ ⋱ ⋮ Subscript x n1 Subscript x n2 ⋯ Subscript x nn A Eigenvectors Subscript x nn  Subscript x 11 Subscript x 12 ⋯ Subscript x 1 n Subscript x 21 Subscript x 22 ⋯ Subscript x 2 n ⋮ ⋮ ⋱ ⋮ Subscript x n1 Subscript x n2 ⋯ Subscript x nn The eigenvalue representing a squared value of the n number of observations is congruent to the input in either matrix. Thereby, the eigenfunction for the vectors and data set independent variable is possible. ErrorBox TextData Ax=λΔx When ErrorBox TextData λ is applied to the Riemann ∆x, find the matrix transformation by the function of x. Set Ax n 2 1 -1 b -1 a n -1 Set A n b -1 a x -1 The area spectrum is calculated from a data set using the same variables as the Riemann sum equation and solved for A with f(x): x 1 n n b -1 a x -1 The resulting equation (5) represents all possible areas of a data set calculated by the Riemann sum equation.