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Riemann Integral Reiteration

Module by: Leif Anderson

Summary: 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.

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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=n21-1 λ n 2 1 -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 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=EigenvectorsSubscriptxnn Δx Eigenvectors Subscript x nn (2)
=Subscriptx11Subscriptx12⋯Subscriptx1nSubscriptx21Subscriptx22⋯Subscriptx2n⋮⋮⋱⋮Subscriptxn1Subscriptxn2⋯Subscriptxnn 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 (3)
A=EigenvectorsSubscript∂xnn A Eigenvectors Subscript x nn (4)
=Subscript∂x11Subscript∂x12⋯Subscript∂x1nSubscript∂x21Subscript∂x22⋯Subscript∂x2n⋮⋮⋱⋮Subscript∂xn1Subscript∂xn2⋯Subscript∂xnn 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 (5)

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.

ErrorBoxTextDataAx=λΔx ErrorBox TextData Ax=λΔx (6)

When ErrorBoxTextDataλ ErrorBox TextData λ is applied to the Riemann ∆x, find the matrix transformation by the function of x.

SetAxn21-1b+-1an-1 Set Ax n 2 1 -1 b -1 a n -1 (7)
SetAnb+-1ax-1 Set A n b -1 a x -1 (8)

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=1nnb+-1ax-1 x 1 n n b -1 a x -1 (9)

The resulting equation (5) represents all possible areas of a data set calculated by the Riemann sum equation.

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