The Simpson Rule: Two numerical sweeties

Discussion in 'Scientific Statistics Math' started by Luis A. Afonso, Jul 7, 2009.

  1. The Simpson Rule: Two numerical sweeties


    _________Integral (from 0 to 1). [4/ (1+x^2)]
    ***************************************

    It provides, needing only 32 intervals, the value of pi with 10 decimal places exact, namely = 3.14159´26536.

    REM "pisimp"
    CLS
    DEFDBL A-Z
    DEF fnw (x) = 4 / (1 + x * x)
    nn = 32
    pi = 4 * ATN(1)
    h = 1 / nn
    FOR n = 0 TO nn
    IF INT(n / 2) = n / 2 THEN k = 2
    IF INT(n / 2) <> n / 2 THEN k = 4
    IF n = 0 OR n = nn THEN k = 1
    x = n * h
    s = s + h / 3 * k * fnw(x)
    NEXT n
    PRINT USING " #.########## "; s
    PRINT USING " #.########## "; pi
    END

    ___Normal Standard Dstribution Function
    ***************************************
    ___Phi ---> chose 6 or 10 exact decimal places.

    Example :
    ________ Phi(.5) = 0.691463
    ________ Phi(-1) = 0.15865´52539

    REM "simp"
    CLS
    DEFDBL A-Z
    INPUT " INPUT z ---> PHI(z) "; z
    PRINT " EXACT to 5 decimal places, input nn=200"
    PRINT " EXACT to 10, input nn=2000 "
    INPUT " nn= "; nn
    pi = 4 * ATN(1): c = 1 / SQR(2 * pi)
    h = (z + 20) / nn
    FOR n = 0 TO nn
    IF INT(n / 2) = n / 2 THEN k = 2
    IF INT(n / 2) <> n / 2 THEN k = 4
    IF n = 0 OR n = nn THEN k = 1
    x = -20 + n * h
    s = s + h / 3 * k * c * EXP(-x * x / 2)
    NEXT n
    IF nn = 200 THEN GOTO 200
    LOCATE 10, 50
    PRINT USING " #.########## "; s: GOTO 210
    200 LOCATE 10, 50
    PRINT USING " #.######"; s
    210 END



    Luis A. Afonso
     
    Luis A. Afonso, Jul 7, 2009
    #1
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