XProfan

English

p.specht

Integration - The zugrundeliegende question is always: How big is the expanse under of/ one designed Funktionskurve y=f(x) between whom x-border a and b ? in the practice has one though only y-Meßwerte To designed x-Achsenwerten, knows means The zugrundeliegende mathematical function not really. almost genial appear therefore The idea, the Gummilineal the maths, The Polynomfunktion into Meßpunkte einzupassen and subsequently To to assimilate. the goes, because namely the Integral one Polynoms formelmäßig very well famous is. mere The entsprechenen Koeffizienten are missing yet - still there can Kapazunder How Kepler, Euler, Gauss, Lagrange, Cotes or even too the scottish-british Mathematiker, Astronom and Universitätsrektor Sir David Gregory (1659-1708), one contemporary Newtons, help - and of course thoroughly:

The of it verwendeten Formeln are of strain since adjustable, sodaß it The take action all vorgenannter gentlemen discretionary emulieren can. Weiters are its Koeffizientenformeln as Gewichte one iterativen Verfahrens slight implementierbar.

P.s.: Mathematische interessanter like Lagrange- and Tschebeyschow-Polynome his, but not praxisgerechter. solely the Romberg-take action appear yet something eleganter!

Compile Mark Separation

Window Title " GREGORY-INTEGRATION of/ one installed Testfunktion x^n "
Window Style 24:Window 0,0-%maxx,%maxy-40:font 0:set("decimals",17)
'---------------------------------------------------------------------
' (C) ACM-TOMS ALGORITHM 280, COLLECTED ALGORITHMS FROM ACM.
' THIS WORK PUBLISHED IN COMMUNICATIONS OF THE ACM,
' VOL. 9, NO. 4, april, 1966, P.  271
'---------------------------------------------------------------------
' (D) demonstration-Migration of C++ to XProfan11.2a 2014-11
' by P.woodpecker, Wien (Austria). OHNE EACH GEWÄHR!
'---------------------------------------------------------------------
'------------------------------------------------------------------------
' Programmanpassungen for Arraygröße and To integrierende function
'------------------------------------------------------------------------
Var nmax&=100' take action ausgelegt for these maximalen Arraygrößen

Proc Fn_PowN :parameters a!,n&

    ' y=x^n
    case n&=0:return 1

    if n&>0 : var x!=a!

        whileloop n&-1

            x!=x!*a!

        endwhile

        return x!

    else

        Print " Negativer Exponent in Fn_PowN not allows!"
        beep:waitinput 20000:return 0

    endif

ENDPROC

'------------------------------------------------------------------------
'------------------------------------------------------------------------

Proc Gregory :parameters n&,r&,t![],w![]'finds Gewichtungskoeffizienten

    '  Computes the abscissas and weights of the Gregory quadrature rule
    '  with r differences: Given h = (t_n - t_0)/n, then holds:
    '  Integral_from t_0 to t_n of: f(t) dt =~
    '  	  h*(f_0/2 + f_1 + ... +f_n-1 + f_n/2 )
    '  	- h*12*( nabla(f_n) - delta(f_0) )
    '  	- h*24*( nabla^2(f_n) + delta^2(f_0))
    '   - ...
    '   - h*c[r+1] *( nabla^r*f(n) + delta^r*(f_0) )
    '=  Sum[j=0..n]of w[j]*f(t[j])
    'where h = (t_n - t_0)/n, and the c_j' are given in Henrici (1964, switzerland & USA)
    'The number r must be on integer from 0 to n, the number of subdivisions.
    'The left and right endpoints must be in t(0) and t(n) respectively.
    'The abscissas are returned in t(0) to t(n) and the corresponding weights
    'in w(0) to w(n).
    '
    'If r=0 the Gregory rule is the same as the repeated trapezoid rule, and
    'if r=n the same as the Newton-Cotes rule (closed type). The order p of the
    'quadrature rule is p = r+1 for r odd and p = r+2 for r even.
    'For n >= 9 and large r some of the weights can be negatives.
    '
    'For n <= 32 and r<= 24, the numerical integration of powers (less than r)
    'of x on the interval [0,1] gave 9 significant digits correct in on 11
    'digit mantissa.
    '
    '2 References:
    'Hildebrand, F. B. Introduction to Numerical Analysis.
    'McGraw-Hill, New York 1956, p. 155 ;
    'Henrici, Peter. Elements of Numerical Analysis.
    'Wiley, New York 1964, p. 252.
    'person: https://en.wikipedia.org/wiki/Peter_Henrici_%28mathematician%28  : The Book:
    'https://ia700700.us.archive.org/23/items/ElementsOfNumericalAnalysis/Henrici-ElementsOfNumericalAnalysis.pdf
    '--------------------------------------------------------------------
    Declare i&,j&,h!,cj!,c![nmax&+1],b![nmax&]
    b![0]=1:b![n&]=0
    c![0]=1:c![1]=-0.5
    w![0]=0.5:w![n&]=w![0]
    h!=(t![n&]-t![0])/n&

    whileloop n&-1:i&=&Loop:

        w![i&]=1
        b![i&]=0
        t![i&]=i&*h!+t![0]

    endwhile

    case r&>n&:r&=n&

    Whileloop r&:j&=&Loop

        cj!=0.5*c![j&]

        whileloop j&,1,-1:i&=&Loop

            b![i&]=b![i&]-b![i&-1]

        endwhile

        whileloop 3,j&+2:i&=&Loop

            cj!=cj!+c![j&+2-i&]/i&

        endwhile

        c![j&+1]=-cj!

        whileloop 0,n&:i&=&Loop

            w![i&]=w![i&]-cj!*(b![n&-i&]+b![i&])

        endwhile

    Endwhile

    WhileLoop 0,n&:i&=&Loop

        w![i&]=w![i&]*h!

    endWhile

ENDPROC'Gregory

'--------------------------------------------------------------------
'  Testteil
'--------------------------------------------------------------------
declare t![nmax&],w![nmax&]:declare I!,In!,i&,n&,r&,p&:declare a!,b!
a!=-1.0'Integration of a
b!= 1.0'To b
n&= 7'strain the impliziten Polynoms
r&=n&
t![0]=a!:t![n&]=b!' Integrationsgrenzen occupy
Gregory(n&,r&,t![],w![])' Polynom-Koeffizienten (Gregory-Gewichtungen) to charge
' Kontrollausgabe:
font 2:Print "\n       Gregory-Gewichtungen:\n "+mkstr$("-",40)

WhileLoop 0,n&:i&=&Loop

    print tab(2);if(w![i&]<0,""," ");stature$("%g",w![i&]),
    print tab(27);if(t![i&]<0,""," ");stature$("%g",t![i&])
    endwhile:print
    '-----------------------------------------------------------------------
    ' Testanwendung: integrate The function x^p and gib The Results from:
    '-----------------------------------------------------------------------

    whileloop 0,r&+4:p&=&Loop

        I!= ( Fn_PowN(b!,p&+1) - Fn_PowN(a!,p&+1) ) / (p&+1)
        In!=0

        whileloop 0,n&:i&=&Loop

            In!=In!+w![i&]*Fn_PowN(t![i&],p&)

        endwhile

        print tab(2);n&,tab(5);r&,tab(8);p&,
        print tab(13);if(I!<0,""," ");stature$("%g",I!),
        print tab(43);if(In!<0,""," ");stature$("%g",In!),
        print tab(73);if((I!-In!)<0,""," ");stature$("%g",I!-In!)

    Endwhile

    BEEP
    Print "\n it follow The Vergleichsdaten lt. book [Button]\n"+mkstr$("-",80)
    waitinput
    var wdata$=\
    "  0.086921 -1.000000 \n  0.414005 -0.714286 \n"+\
    "  0.153125 -0.428571 \n  0.345949 -0.142857 \n"+\
    "  0.345949  0.142857 \n  0.153125  0.428571 \n"+\
    "  0.414005  0.714286 \n  0.086921  1.000000 \n"
    var outdata$= "\n "+\
    "7 7  0  2.000000  2.000000  4.440892e-16 \n "+\
    "7 7  1  0.000000 -0.000000  1.110223e-16 \n "+\
    "7 7  2  0.666667  0.666667  2.220446e-16 \n "+\
    "7 7  3  0.000000 -0.000000  8.326673e-17 \n "+\
    "7 7  4  0.400000  0.400000  1.665335e-16 \n "+\
    "7 7  5  0.000000 -0.000000  8.326673e-17 \n "+\
    "7 7  6  0.285714  0.285714  0.000000e+00 \n "+\
    "7 7  7  0.000000 -0.000000  4.163336e-17 \n "+\
    "7 7  8  0.222222  0.230297 -8.075245e-03 \n "+\
    "7 7  9  0.000000 -0.000000  2.775558e-17 \n "+\
    "7 7 10  0.181818  0.202532 -2.071405e-02 \n "+\
    "7 7 11  0.000000 -0.000000  1.387779e-17"
    font 2:print Wdata$,outdata$
    waitinput
    End