Values of Trigonometric Functions of Angles (n*pi)/65537. Part XII.

				Values of Trigonometric Functions of Angles (n*pi)/65537. Part XII. by 'Literka'.

In the following text we'll be using the notations pi=3.14159... and w=(2*pi)/65537.

We'll be using all notations from the previous pages of Literka Values of trigonometric functions. Part 65537-1, Part 65537-2, Part 65537-3, Part 65537-4, Part 65537-5, Part 65537-6, Part 65537-7, Part 65537-8, Part 65537-9, Part 65537-10, Part 65537-11.

We'll proceed a similar way as on the page Angle257_5.
Let us introduce 2048 new numbers

to be the summations of terms cos(i*w), where i is and element of Ai (Sets A1, A2,...A2048 were defined previously). For example
Rozmiar: 2726 bajtów

The definitions of

(we already know the values of these numbers) show that these numbers are linear combinations of

with coefficients 0 or 1.
Hence, regarding

as unknown variables, we have 2047 linear equations with 2048 unknowns. We add one more equation (as the first one):

We obtain a system of 2048 equations. Let M be the matrix of this system.
By M[i][j] we denote the entry of this matrix corresponding to i-th row and j-th column. We assume that columns and rows are enumerated from 0 to 2047.
The matrix M can be easily constructed the following way:
We assume M[0][j]=1 (hence first column enumerated with number 0 consists of "1" only).
Let k be an integer less than 11 and let us assume that we have constructed columns enumerated from 0 to m-1,where

Let i be an integer from {0,1,2... 2048} and let j be an integer from {0,1...m}.
Let d=M[i][j] (we have it already defined) and e=(i div m) (integer division of i by m, for example (100 div 7) = 14).
We define M[m+i][j] to be

M[m+i][j]=1 if d=1 and e is even,
M[m+i][j]=1 if d=0 and e is odd,
M[m+i][j]=0 if d=0 and e is even,
M[m+i][j]=0 if d=1 and e is odd.

Matrix M is a matrix of our system of equations. It appears that M is a symmetric matrix i.e. M[i][j]=M[j][i]. M possesses many properties reminding orthogonality (of rows and columns). That is why it is easy to find the inverse matrix to M. Let us take a matrix K of size 2048 x 2048 such that
K[0][0]=-1023
K[i][j]=2*M[i][j]-1 if i or j is not equal 0.

This means that
K is equal 1, when M is equal 1,
K is equal -1, when M is equal 0,
except K[0][0], where it is -1023.

Then the inverse matrix to M is equal Rozmiar: 2592 bajtów

This matrix allows us to write ui as a linear combinations of ti (entries of rows are coefficients of these combinations). That is why further we'll assume that ui are known values.

See the next parts of this page Part 65537-13, Part 65537-14.

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