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Riemann Integral Reiteration

Module by: Leif Anderson. E-mail the author

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.

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)
=Subscriptx11Subscriptx12Subscriptx1nSubscriptx21Subscriptx22Subscriptx2nSubscriptxn1Subscriptxn2Subscriptxnn 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=EigenvectorsSubscript0xnn A Eigenvectors Subscript x nn
(4)
=Subscript0x11Subscript0x12Subscript0x1nSubscript0x21Subscript0x22Subscript0x2nSubscript0xn1Subscript0xn2Subscript0xnn 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.

SetAx(n21-1)((b1a)n-1) Set Ax n 2 1 -1 b -1 a n -1
(7)
SetAn(b1a)x-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 =1nn(b1a)x-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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