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 thread  Author  Topic: [RC] Chebyshev coefficients  (Read 269 times)
tenochtitlanuk
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xx [RC] Chebyshev coefficients
« Thread started on: Dec 31st, 2016, 05:19am »

Could do with some help here from mathematicians! If you can help me I'll wish you a VERY good New Year!
Sorry about mixed spelling- I can't type the more economical Cyrillic, and at least two spellings are regularly used...
Code:
Chebyshev coefficients is a draft programming task.
Chebyshev coefficients are the basis of polynomial approximations of functions.
Write a program to generate Chebyshev coefficients.
Calculate coefficients: cosine function, 10 coefficients, interval 0 1 

Rosetta Code
I'm familiar with Fourier- you can break any waveform down into a series of sin/cos terms and reassemble.
Chebyshev is similar- you can do the same with the (orthogonal) Tchebyshev functions. It gives better approximation near the end points, and can generate an equivalent series of power terms which will be fast to compute.
I've written code that does the analysis/synthesis across the range of -1 <x <1.
But I don't UNDERSTAND some of it!
Anyone able to comment or provide a GOOD reference? RC comment page points to confusion about the task. Googling has given me a lot, but tends to go straight to coding in Mathematica...
My code works but is in wrong range, and the constants differ from RC!!

Code:
    nomainwin

    '   use only up to first 8 Tchebyshev terms ( 0 to 7)

    n       =200

    global pi: pi =4 *atn( 1)

    for c =0 to 7
        c( c) =0
        for k =0 to n
            xk      =cos( ( ( 2 *k +1) /(2 *( n +1))) *pi)

            if c =0 then
                c( c)   =c( c) +( 1 /( n +1)) *funcn( xk)
            else
                c( c)   =c( c) +( 2 /( n +1)) *funcn( xk) *Tchebyshev( c, xk)
            end if

        next k
        'print using( "##.####", c( c))
    next c

    print

    WindowWidth  =840
    WindowHeight =460

    open "Tchebyshev" for graphics_nsb as #wg

    #wg "trapclose quit"

    #wg "down ; fill darkblue"
    #wg "color white"
    for h =0 to 9
        if h =4 then #wg "size 2" else #wg "size 1"
        #wg "up ; goto  0 "; 10 +50 *h; " ; down ; goto 420 "; 10 +50 *h
    next h

    for v =0 to 2
        if v =1 then #wg "size 2" else #wg "size 1"
        #wg "up ; goto "; 10 +200 *v; " 0 ; down ; goto "; 10 +200 *v; " 420"
    next v

    for x =-1 to 1 step 0.005
        print using( "##.##", x), using( "##.####",funcn( x)),
        #wg "color 255 255 0 ; set "; int( ( x +1) *200 +10); " "; int( 210 -funcn( x) *50)
        t =0
        for c =0 to 7
            t =t +c( c) *Tchebyshev( c, x)
        next c
        print using( "##.####", t)
        #wg "color 255 0 255 ; set "; int( ( x +1) *200 +10); " "; int( 210 -t *50)
    next x

    #wg "color white ; backcolor darkblue"

    for i =0 to 7
        #wg "up ; goto 450 "; 50 +i *20
        #wg "down"
        #wg "\"; i; "  "; c( i)
    next i

    wait'_____________________________________________________________________

    function Tchebyshev( n, x)
        select case
            case x >1
                Tchebyshev =cosh( n *arcCosh( x))
            case x <-1
                Tchebyshev =cos(  n *arcCosh( x))
            case ( ( x >=-1) and ( x <=1))
                Tchebyshev =cos(  n *acs(     x))
        end select
    end function

    function cosh( x)
        cosh    =0.5 *( exp( x) +exp( 0 -x))
    end function

    function arcCosh( x)
        if x >1 then arcCosh =log( x +( x^2 -1)^0.5)
    end function

    sub quit h$
        close #wg
        end
    end sub

    sub delay t
        timer t *1000, [o]
        wait
        [o]
        timer 0
    end sub

    function funcn( x)
        funcn =cos( x)
        'funcn4 *sin( x)^2
        'funcn =exp( x)  '   1.2661  1.1302  0.2715  0.0443 is expected
        'funcn =4 *x^2
        'funcn =4 *sin( pi *x)
    end function
 
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xx Re: [RC] Chebyshev coefficients
« Reply #1 on: Jan 5th, 2017, 1:09pm »

Further work on my site on the following pages
Fourier - - Tchebyshev - - Moused Fourier
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NB The getPixel method allows me to mouse-select-drag to alter amplitudes and phases. So related to the thread on selecting and altering drawn lines, circles, etc.)
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