Formulas for numerical computation of derivatives f^(3) means third derivative f^(2)(x[1]) means second derivative computed at point x[1] f(x[2]) means function evaluated at point x[2] Points are x[0], x[1]=x[0]+h, x[2]=x0+2h, ... The error term gives the power of h and derivative order with z chosen for maximum value. The error terms, err=, are for test polynomials with h = 0.125 and order = 'pwr' computing order=1, npoints=5, at term=0 denom[0]=24 numer[0][0]=-1 numer[0][1]=-2 numer[0][2]=-3 numer[0][3]=-4 denom[1]=-6 numer[1][0]=0 numer[1][1]=-2 numer[1][2]=-3 numer[1][3]=-4 denom[2]=4 numer[2][0]=0 numer[2][1]=-1 numer[2][2]=-3 numer[2][3]=-4 denom[3]=-6 numer[3][0]=0 numer[3][1]=-1 numer[3][2]=-2 numer[3][3]=-4 denom[4]=24 numer[4][0]=0 numer[4][1]=-1 numer[4][2]=-2 numer[4][3]=-3 p(x)= 24 x^0 at term=0 p(x)= -50 x^1 at term=0 p(x)= 35 x^2 at term=0 p(x)= -10 x^3 at term=0 p(x)= 1 x^4 at term=0 p^(1)(x)= -50 x^0 at term=0 p^(1)(x)= 70 x^1 at term=0 p^(1)(x)= -30 x^2 at term=0 p^(1)(x)= 4 x^3 at term=0 p(x)= 0 x^0 at term=1 p(x)= -24 x^1 at term=1 p(x)= 26 x^2 at term=1 p(x)= -9 x^3 at term=1 p(x)= 1 x^4 at term=1 p^(1)(x)= -24 x^0 at term=1 p^(1)(x)= 52 x^1 at term=1 p^(1)(x)= -27 x^2 at term=1 p^(1)(x)= 4 x^3 at term=1 p(x)= 0 x^0 at term=2 p(x)= -12 x^1 at term=2 p(x)= 19 x^2 at term=2 p(x)= -8 x^3 at term=2 p(x)= 1 x^4 at term=2 p^(1)(x)= -12 x^0 at term=2 p^(1)(x)= 38 x^1 at term=2 p^(1)(x)= -24 x^2 at term=2 p^(1)(x)= 4 x^3 at term=2 p(x)= 0 x^0 at term=3 p(x)= -8 x^1 at term=3 p(x)= 14 x^2 at term=3 p(x)= -7 x^3 at term=3 p(x)= 1 x^4 at term=3 p^(1)(x)= -8 x^0 at term=3 p^(1)(x)= 28 x^1 at term=3 p^(1)(x)= -21 x^2 at term=3 p^(1)(x)= 4 x^3 at term=3 p(x)= 0 x^0 at term=4 p(x)= -6 x^1 at term=4 p(x)= 11 x^2 at term=4 p(x)= -6 x^3 at term=4 p(x)= 1 x^4 at term=4 p^(1)(x)= -6 x^0 at term=4 p^(1)(x)= 22 x^1 at term=4 p^(1)(x)= -18 x^2 at term=4 p^(1)(x)= 4 x^3 at term=4 f^(1)(x[0]) = (1/h^1) (-50/24 f(x[0]) + f^(1)(x[0]) = (1/h^1) (-24/-6 f(x[1]) + f^(1)(x[0]) = (1/h^1) (-12/4 f(x[2]) + f^(1)(x[0]) = (1/h^1) (-8/-6 f(x[3]) + f^(1)(x[0]) = (1/h^1) (-6/24 f(x[4]) + f^(1)(x[0])=(1/12h^1)(-25 f(x[0]) + f^(1)(x[0])=(1/12h^1)(48 f(x[1]) + f^(1)(x[0])=(1/12h^1)(-36 f(x[2]) + f^(1)(x[0])=(1/12h^1)(16 f(x[3]) + f^(1)(x[0])=(1/12h^1)(-3 f(x[4]) + f^(1)(x[0])=(1/12h^1)( -25 * f(x[0]) + 48 * f(x[1] + -36 * f(x[2] + 16 * f(x[3] + -3 * f(x[4] ) + O(1/h^5)f^(5)(z) err=0, sum=3, deriv=3, x=1, pwr=3 err=0, sum=4, deriv=4, x=1, pwr=4 err=0.00585938, sum=4.99414, deriv=5, x=1, pwr=5 err=0.0424805, sum=5.95752, deriv=6, x=1, pwr=6 computing order=1, npoints=5, at term=1 denom[0]=24 numer[0][0]=-1 numer[0][1]=-2 numer[0][2]=-3 numer[0][3]=-4 denom[1]=-6 numer[1][0]=0 numer[1][1]=-2 numer[1][2]=-3 numer[1][3]=-4 denom[2]=4 numer[2][0]=0 numer[2][1]=-1 numer[2][2]=-3 numer[2][3]=-4 denom[3]=-6 numer[3][0]=0 numer[3][1]=-1 numer[3][2]=-2 numer[3][3]=-4 denom[4]=24 numer[4][0]=0 numer[4][1]=-1 numer[4][2]=-2 numer[4][3]=-3 p(x)= 24 x^0 at term=0 p(x)= -50 x^1 at term=0 p(x)= 35 x^2 at term=0 p(x)= -10 x^3 at term=0 p(x)= 1 x^4 at term=0 p^(1)(x)= -50 x^0 at term=0 p^(1)(x)= 70 x^1 at term=0 p^(1)(x)= -30 x^2 at term=0 p^(1)(x)= 4 x^3 at term=0 p(x)= 0 x^0 at term=1 p(x)= -24 x^1 at term=1 p(x)= 26 x^2 at term=1 p(x)= -9 x^3 at term=1 p(x)= 1 x^4 at term=1 p^(1)(x)= -24 x^0 at term=1 p^(1)(x)= 52 x^1 at term=1 p^(1)(x)= -27 x^2 at term=1 p^(1)(x)= 4 x^3 at term=1 p(x)= 0 x^0 at term=2 p(x)= -12 x^1 at term=2 p(x)= 19 x^2 at term=2 p(x)= -8 x^3 at term=2 p(x)= 1 x^4 at term=2 p^(1)(x)= -12 x^0 at term=2 p^(1)(x)= 38 x^1 at term=2 p^(1)(x)= -24 x^2 at term=2 p^(1)(x)= 4 x^3 at term=2 p(x)= 0 x^0 at term=3 p(x)= -8 x^1 at term=3 p(x)= 14 x^2 at term=3 p(x)= -7 x^3 at term=3 p(x)= 1 x^4 at term=3 p^(1)(x)= -8 x^0 at term=3 p^(1)(x)= 28 x^1 at term=3 p^(1)(x)= -21 x^2 at term=3 p^(1)(x)= 4 x^3 at term=3 p(x)= 0 x^0 at term=4 p(x)= -6 x^1 at term=4 p(x)= 11 x^2 at term=4 p(x)= -6 x^3 at term=4 p(x)= 1 x^4 at term=4 p^(1)(x)= -6 x^0 at term=4 p^(1)(x)= 22 x^1 at term=4 p^(1)(x)= -18 x^2 at term=4 p^(1)(x)= 4 x^3 at term=4 f^(1)(x[1]) = (1/h^1) (-6/24 f(x[0]) + f^(1)(x[1]) = (1/h^1) (5/-6 f(x[1]) + f^(1)(x[1]) = (1/h^1) (6/4 f(x[2]) + f^(1)(x[1]) = (1/h^1) (3/-6 f(x[3]) + f^(1)(x[1]) = (1/h^1) (2/24 f(x[4]) + f^(1)(x[1])=(1/12h^1)(-3 f(x[0]) + f^(1)(x[1])=(1/12h^1)(-10 f(x[1]) + f^(1)(x[1])=(1/12h^1)(18 f(x[2]) + f^(1)(x[1])=(1/12h^1)(-6 f(x[3]) + f^(1)(x[1])=(1/12h^1)(1 f(x[4]) + f^(1)(x[1])=(1/12h^1)( -3 * f(x[0]) + -10 * f(x[1] + 18 * f(x[2] + -6 * f(x[3] + 1 * f(x[4] ) + O(1/h^5)f^(5)(z) err=0, sum=3.79688, deriv=3.79688, x=1.125, pwr=3 err=0, sum=5.69531, deriv=5.69531, x=1.125, pwr=4 err=-0.00146484, sum=8.0105, deriv=8.00903, x=1.125, pwr=5 err=-0.0108032, sum=10.823, deriv=10.8122, x=1.125, pwr=6 computing order=1, npoints=5, at term=2 denom[0]=24 numer[0][0]=-1 numer[0][1]=-2 numer[0][2]=-3 numer[0][3]=-4 denom[1]=-6 numer[1][0]=0 numer[1][1]=-2 numer[1][2]=-3 numer[1][3]=-4 denom[2]=4 numer[2][0]=0 numer[2][1]=-1 numer[2][2]=-3 numer[2][3]=-4 denom[3]=-6 numer[3][0]=0 numer[3][1]=-1 numer[3][2]=-2 numer[3][3]=-4 denom[4]=24 numer[4][0]=0 numer[4][1]=-1 numer[4][2]=-2 numer[4][3]=-3 p(x)= 24 x^0 at term=0 p(x)= -50 x^1 at term=0 p(x)= 35 x^2 at term=0 p(x)= -10 x^3 at term=0 p(x)= 1 x^4 at term=0 p^(1)(x)= -50 x^0 at term=0 p^(1)(x)= 70 x^1 at term=0 p^(1)(x)= -30 x^2 at term=0 p^(1)(x)= 4 x^3 at term=0 p(x)= 0 x^0 at term=1 p(x)= -24 x^1 at term=1 p(x)= 26 x^2 at term=1 p(x)= -9 x^3 at term=1 p(x)= 1 x^4 at term=1 p^(1)(x)= -24 x^0 at term=1 p^(1)(x)= 52 x^1 at term=1 p^(1)(x)= -27 x^2 at term=1 p^(1)(x)= 4 x^3 at term=1 p(x)= 0 x^0 at term=2 p(x)= -12 x^1 at term=2 p(x)= 19 x^2 at term=2 p(x)= -8 x^3 at term=2 p(x)= 1 x^4 at term=2 p^(1)(x)= -12 x^0 at term=2 p^(1)(x)= 38 x^1 at term=2 p^(1)(x)= -24 x^2 at term=2 p^(1)(x)= 4 x^3 at term=2 p(x)= 0 x^0 at term=3 p(x)= -8 x^1 at term=3 p(x)= 14 x^2 at term=3 p(x)= -7 x^3 at term=3 p(x)= 1 x^4 at term=3 p^(1)(x)= -8 x^0 at term=3 p^(1)(x)= 28 x^1 at term=3 p^(1)(x)= -21 x^2 at term=3 p^(1)(x)= 4 x^3 at term=3 p(x)= 0 x^0 at term=4 p(x)= -6 x^1 at term=4 p(x)= 11 x^2 at term=4 p(x)= -6 x^3 at term=4 p(x)= 1 x^4 at term=4 p^(1)(x)= -6 x^0 at term=4 p^(1)(x)= 22 x^1 at term=4 p^(1)(x)= -18 x^2 at term=4 p^(1)(x)= 4 x^3 at term=4 f^(1)(x[2]) = (1/h^1) (2/24 f(x[0]) + f^(1)(x[2]) = (1/h^1) (4/-6 f(x[1]) + f^(1)(x[2]) = (1/h^1) (0/4 f(x[2]) + f^(1)(x[2]) = (1/h^1) (-4/-6 f(x[3]) + f^(1)(x[2]) = (1/h^1) (-2/24 f(x[4]) + f^(1)(x[2])=(1/12h^1)(1 f(x[0]) + f^(1)(x[2])=(1/12h^1)(-8 f(x[1]) + f^(1)(x[2])=(1/12h^1)(0 f(x[2]) + f^(1)(x[2])=(1/12h^1)(8 f(x[3]) + f^(1)(x[2])=(1/12h^1)(-1 f(x[4]) + f^(1)(x[2])=(1/12h^1)( 1 * f(x[0]) + -8 * f(x[1] + 0 * f(x[2] + 8 * f(x[3] + -1 * f(x[4] ) + O(1/h^5)f^(5)(z) err=0, sum=4.6875, deriv=4.6875, x=1.25, pwr=3 err=0, sum=7.8125, deriv=7.8125, x=1.25, pwr=4 err=0.000976562, sum=12.2061, deriv=12.207, x=1.25, pwr=5 err=0.00732422, sum=18.3032, deriv=18.3105, x=1.25, pwr=6 computing order=1, npoints=5, at term=3 denom[0]=24 numer[0][0]=-1 numer[0][1]=-2 numer[0][2]=-3 numer[0][3]=-4 denom[1]=-6 numer[1][0]=0 numer[1][1]=-2 numer[1][2]=-3 numer[1][3]=-4 denom[2]=4 numer[2][0]=0 numer[2][1]=-1 numer[2][2]=-3 numer[2][3]=-4 denom[3]=-6 numer[3][0]=0 numer[3][1]=-1 numer[3][2]=-2 numer[3][3]=-4 denom[4]=24 numer[4][0]=0 numer[4][1]=-1 numer[4][2]=-2 numer[4][3]=-3 p(x)= 24 x^0 at term=0 p(x)= -50 x^1 at term=0 p(x)= 35 x^2 at term=0 p(x)= -10 x^3 at term=0 p(x)= 1 x^4 at term=0 p^(1)(x)= -50 x^0 at term=0 p^(1)(x)= 70 x^1 at term=0 p^(1)(x)= -30 x^2 at term=0 p^(1)(x)= 4 x^3 at term=0 p(x)= 0 x^0 at term=1 p(x)= -24 x^1 at term=1 p(x)= 26 x^2 at term=1 p(x)= -9 x^3 at term=1 p(x)= 1 x^4 at term=1 p^(1)(x)= -24 x^0 at term=1 p^(1)(x)= 52 x^1 at term=1 p^(1)(x)= -27 x^2 at term=1 p^(1)(x)= 4 x^3 at term=1 p(x)= 0 x^0 at term=2 p(x)= -12 x^1 at term=2 p(x)= 19 x^2 at term=2 p(x)= -8 x^3 at term=2 p(x)= 1 x^4 at term=2 p^(1)(x)= -12 x^0 at term=2 p^(1)(x)= 38 x^1 at term=2 p^(1)(x)= -24 x^2 at term=2 p^(1)(x)= 4 x^3 at term=2 p(x)= 0 x^0 at term=3 p(x)= -8 x^1 at term=3 p(x)= 14 x^2 at term=3 p(x)= -7 x^3 at term=3 p(x)= 1 x^4 at term=3 p^(1)(x)= -8 x^0 at term=3 p^(1)(x)= 28 x^1 at term=3 p^(1)(x)= -21 x^2 at term=3 p^(1)(x)= 4 x^3 at term=3 p(x)= 0 x^0 at term=4 p(x)= -6 x^1 at term=4 p(x)= 11 x^2 at term=4 p(x)= -6 x^3 at term=4 p(x)= 1 x^4 at term=4 p^(1)(x)= -6 x^0 at term=4 p^(1)(x)= 22 x^1 at term=4 p^(1)(x)= -18 x^2 at term=4 p^(1)(x)= 4 x^3 at term=4 f^(1)(x[3]) = (1/h^1) (-2/24 f(x[0]) + f^(1)(x[3]) = (1/h^1) (-3/-6 f(x[1]) + f^(1)(x[3]) = (1/h^1) (-6/4 f(x[2]) + f^(1)(x[3]) = (1/h^1) (-5/-6 f(x[3]) + f^(1)(x[3]) = (1/h^1) (6/24 f(x[4]) + f^(1)(x[3])=(1/12h^1)(-1 f(x[0]) + f^(1)(x[3])=(1/12h^1)(6 f(x[1]) + f^(1)(x[3])=(1/12h^1)(-18 f(x[2]) + f^(1)(x[3])=(1/12h^1)(10 f(x[3]) + f^(1)(x[3])=(1/12h^1)(3 f(x[4]) + f^(1)(x[3])=(1/12h^1)( -1 * f(x[0]) + 6 * f(x[1] + -18 * f(x[2] + 10 * f(x[3] + 3 * f(x[4] ) + O(1/h^5)f^(5)(z) err=0, sum=5.67188, deriv=5.67188, x=1.375, pwr=3 err=0, sum=10.3984, deriv=10.3984, x=1.375, pwr=4 err=-0.00146484, sum=17.8738, deriv=17.8723, x=1.375, pwr=5 err=-0.0111694, sum=29.5005, deriv=29.4893, x=1.375, pwr=6 computing order=1, npoints=5, at term=4 denom[0]=24 numer[0][0]=-1 numer[0][1]=-2 numer[0][2]=-3 numer[0][3]=-4 denom[1]=-6 numer[1][0]=0 numer[1][1]=-2 numer[1][2]=-3 numer[1][3]=-4 denom[2]=4 numer[2][0]=0 numer[2][1]=-1 numer[2][2]=-3 numer[2][3]=-4 denom[3]=-6 numer[3][0]=0 numer[3][1]=-1 numer[3][2]=-2 numer[3][3]=-4 denom[4]=24 numer[4][0]=0 numer[4][1]=-1 numer[4][2]=-2 numer[4][3]=-3 p(x)= 24 x^0 at term=0 p(x)= -50 x^1 at term=0 p(x)= 35 x^2 at term=0 p(x)= -10 x^3 at term=0 p(x)= 1 x^4 at term=0 p^(1)(x)= -50 x^0 at term=0 p^(1)(x)= 70 x^1 at term=0 p^(1)(x)= -30 x^2 at term=0 p^(1)(x)= 4 x^3 at term=0 p(x)= 0 x^0 at term=1 p(x)= -24 x^1 at term=1 p(x)= 26 x^2 at term=1 p(x)= -9 x^3 at term=1 p(x)= 1 x^4 at term=1 p^(1)(x)= -24 x^0 at term=1 p^(1)(x)= 52 x^1 at term=1 p^(1)(x)= -27 x^2 at term=1 p^(1)(x)= 4 x^3 at term=1 p(x)= 0 x^0 at term=2 p(x)= -12 x^1 at term=2 p(x)= 19 x^2 at term=2 p(x)= -8 x^3 at term=2 p(x)= 1 x^4 at term=2 p^(1)(x)= -12 x^0 at term=2 p^(1)(x)= 38 x^1 at term=2 p^(1)(x)= -24 x^2 at term=2 p^(1)(x)= 4 x^3 at term=2 p(x)= 0 x^0 at term=3 p(x)= -8 x^1 at term=3 p(x)= 14 x^2 at term=3 p(x)= -7 x^3 at term=3 p(x)= 1 x^4 at term=3 p^(1)(x)= -8 x^0 at term=3 p^(1)(x)= 28 x^1 at term=3 p^(1)(x)= -21 x^2 at term=3 p^(1)(x)= 4 x^3 at term=3 p(x)= 0 x^0 at term=4 p(x)= -6 x^1 at term=4 p(x)= 11 x^2 at term=4 p(x)= -6 x^3 at term=4 p(x)= 1 x^4 at term=4 p^(1)(x)= -6 x^0 at term=4 p^(1)(x)= 22 x^1 at term=4 p^(1)(x)= -18 x^2 at term=4 p^(1)(x)= 4 x^3 at term=4 f^(1)(x[4]) = (1/h^1) (6/24 f(x[0]) + f^(1)(x[4]) = (1/h^1) (8/-6 f(x[1]) + f^(1)(x[4]) = (1/h^1) (12/4 f(x[2]) + f^(1)(x[4]) = (1/h^1) (24/-6 f(x[3]) + f^(1)(x[4]) = (1/h^1) (50/24 f(x[4]) + f^(1)(x[4])=(1/12h^1)(3 f(x[0]) + f^(1)(x[4])=(1/12h^1)(-16 f(x[1]) + f^(1)(x[4])=(1/12h^1)(36 f(x[2]) + f^(1)(x[4])=(1/12h^1)(-48 f(x[3]) + f^(1)(x[4])=(1/12h^1)(25 f(x[4]) + f^(1)(x[4])=(1/12h^1)( 3 * f(x[0]) + -16 * f(x[1] + 36 * f(x[2] + -48 * f(x[3] + 25 * f(x[4] ) + O(1/h^5)f^(5)(z) err=0, sum=6.75, deriv=6.75, x=1.5, pwr=3 err=0, sum=13.5, deriv=13.5, x=1.5, pwr=4 err=0.00585938, sum=25.3066, deriv=25.3125, x=1.5, pwr=5 err=0.0454102, sum=45.5171, deriv=45.5625, x=1.5, pwr=6 computing order=2, npoints=5, at term=0 denom[0]=24 numer[0][0]=-1 numer[0][1]=-2 numer[0][2]=-3 numer[0][3]=-4 denom[1]=-6 numer[1][0]=0 numer[1][1]=-2 numer[1][2]=-3 numer[1][3]=-4 denom[2]=4 numer[2][0]=0 numer[2][1]=-1 numer[2][2]=-3 numer[2][3]=-4 denom[3]=-6 numer[3][0]=0 numer[3][1]=-1 numer[3][2]=-2 numer[3][3]=-4 denom[4]=24 numer[4][0]=0 numer[4][1]=-1 numer[4][2]=-2 numer[4][3]=-3 p(x)= 24 x^0 at term=0 p(x)= -50 x^1 at term=0 p(x)= 35 x^2 at term=0 p(x)= -10 x^3 at term=0 p(x)= 1 x^4 at term=0 p^(2)(x)= 70 x^0 at term=0 p^(2)(x)= -60 x^1 at term=0 p^(2)(x)= 12 x^2 at term=0 p(x)= 0 x^0 at term=1 p(x)= -24 x^1 at term=1 p(x)= 26 x^2 at term=1 p(x)= -9 x^3 at term=1 p(x)= 1 x^4 at term=1 p^(2)(x)= 52 x^0 at term=1 p^(2)(x)= -54 x^1 at term=1 p^(2)(x)= 12 x^2 at term=1 p(x)= 0 x^0 at term=2 p(x)= -12 x^1 at term=2 p(x)= 19 x^2 at term=2 p(x)= -8 x^3 at term=2 p(x)= 1 x^4 at term=2 p^(2)(x)= 38 x^0 at term=2 p^(2)(x)= -48 x^1 at term=2 p^(2)(x)= 12 x^2 at term=2 p(x)= 0 x^0 at term=3 p(x)= -8 x^1 at term=3 p(x)= 14 x^2 at term=3 p(x)= -7 x^3 at term=3 p(x)= 1 x^4 at term=3 p^(2)(x)= 28 x^0 at term=3 p^(2)(x)= -42 x^1 at term=3 p^(2)(x)= 12 x^2 at term=3 p(x)= 0 x^0 at term=4 p(x)= -6 x^1 at term=4 p(x)= 11 x^2 at term=4 p(x)= -6 x^3 at term=4 p(x)= 1 x^4 at term=4 p^(2)(x)= 22 x^0 at term=4 p^(2)(x)= -36 x^1 at term=4 p^(2)(x)= 12 x^2 at term=4 f^(2)(x[0]) = (1/h^2) (70/24 f(x[0]) + f^(2)(x[0]) = (1/h^2) (52/-6 f(x[1]) + f^(2)(x[0]) = (1/h^2) (38/4 f(x[2]) + f^(2)(x[0]) = (1/h^2) (28/-6 f(x[3]) + f^(2)(x[0]) = (1/h^2) (22/24 f(x[4]) + f^(2)(x[0])=(1/12h^2)(35 f(x[0]) + f^(2)(x[0])=(1/12h^2)(-104 f(x[1]) + f^(2)(x[0])=(1/12h^2)(114 f(x[2]) + f^(2)(x[0])=(1/12h^2)(-56 f(x[3]) + f^(2)(x[0])=(1/12h^2)(11 f(x[4]) + f^(2)(x[0])=(1/12h^2)( 35 * f(x[0]) + -104 * f(x[1] + 114 * f(x[2] + -56 * f(x[3] + 11 * f(x[4] ) + O(1/h^5)f^(5)(z) err=0, sum=6, deriv=6, x=1, pwr=3 err=0, sum=12, deriv=12, x=1, pwr=4 err=-0.195312, sum=20.1953, deriv=20, x=1, pwr=5 err=-1.4043, sum=31.4043, deriv=30, x=1, pwr=6 computing order=2, npoints=5, at term=1 denom[0]=24 numer[0][0]=-1 numer[0][1]=-2 numer[0][2]=-3 numer[0][3]=-4 denom[1]=-6 numer[1][0]=0 numer[1][1]=-2 numer[1][2]=-3 numer[1][3]=-4 denom[2]=4 numer[2][0]=0 numer[2][1]=-1 numer[2][2]=-3 numer[2][3]=-4 denom[3]=-6 numer[3][0]=0 numer[3][1]=-1 numer[3][2]=-2 numer[3][3]=-4 denom[4]=24 numer[4][0]=0 numer[4][1]=-1 numer[4][2]=-2 numer[4][3]=-3 p(x)= 24 x^0 at term=0 p(x)= -50 x^1 at term=0 p(x)= 35 x^2 at term=0 p(x)= -10 x^3 at term=0 p(x)= 1 x^4 at term=0 p^(2)(x)= 70 x^0 at term=0 p^(2)(x)= -60 x^1 at term=0 p^(2)(x)= 12 x^2 at term=0 p(x)= 0 x^0 at term=1 p(x)= -24 x^1 at term=1 p(x)= 26 x^2 at term=1 p(x)= -9 x^3 at term=1 p(x)= 1 x^4 at term=1 p^(2)(x)= 52 x^0 at term=1 p^(2)(x)= -54 x^1 at term=1 p^(2)(x)= 12 x^2 at term=1 p(x)= 0 x^0 at term=2 p(x)= -12 x^1 at term=2 p(x)= 19 x^2 at term=2 p(x)= -8 x^3 at term=2 p(x)= 1 x^4 at term=2 p^(2)(x)= 38 x^0 at term=2 p^(2)(x)= -48 x^1 at term=2 p^(2)(x)= 12 x^2 at term=2 p(x)= 0 x^0 at term=3 p(x)= -8 x^1 at term=3 p(x)= 14 x^2 at term=3 p(x)= -7 x^3 at term=3 p(x)= 1 x^4 at term=3 p^(2)(x)= 28 x^0 at term=3 p^(2)(x)= -42 x^1 at term=3 p^(2)(x)= 12 x^2 at term=3 p(x)= 0 x^0 at term=4 p(x)= -6 x^1 at term=4 p(x)= 11 x^2 at term=4 p(x)= -6 x^3 at term=4 p(x)= 1 x^4 at term=4 p^(2)(x)= 22 x^0 at term=4 p^(2)(x)= -36 x^1 at term=4 p^(2)(x)= 12 x^2 at term=4 f^(2)(x[1]) = (1/h^2) (22/24 f(x[0]) + f^(2)(x[1]) = (1/h^2) (10/-6 f(x[1]) + f^(2)(x[1]) = (1/h^2) (2/4 f(x[2]) + f^(2)(x[1]) = (1/h^2) (-2/-6 f(x[3]) + f^(2)(x[1]) = (1/h^2) (-2/24 f(x[4]) + f^(2)(x[1])=(1/12h^2)(11 f(x[0]) + f^(2)(x[1])=(1/12h^2)(-20 f(x[1]) + f^(2)(x[1])=(1/12h^2)(6 f(x[2]) + f^(2)(x[1])=(1/12h^2)(4 f(x[3]) + f^(2)(x[1])=(1/12h^2)(-1 f(x[4]) + f^(2)(x[1])=(1/12h^2)( 11 * f(x[0]) + -20 * f(x[1] + 6 * f(x[2] + 4 * f(x[3] + -1 * f(x[4] ) + O(1/h^5)f^(5)(z) err=0, sum=6.75, deriv=6.75, x=1.125, pwr=3 err=0, sum=15.1875, deriv=15.1875, x=1.125, pwr=4 err=0.0195312, sum=28.457, deriv=28.4766, x=1.125, pwr=5 err=0.141113, sum=47.9131, deriv=48.0542, x=1.125, pwr=6 computing order=2, npoints=5, at term=2 denom[0]=24 numer[0][0]=-1 numer[0][1]=-2 numer[0][2]=-3 numer[0][3]=-4 denom[1]=-6 numer[1][0]=0 numer[1][1]=-2 numer[1][2]=-3 numer[1][3]=-4 denom[2]=4 numer[2][0]=0 numer[2][1]=-1 numer[2][2]=-3 numer[2][3]=-4 denom[3]=-6 numer[3][0]=0 numer[3][1]=-1 numer[3][2]=-2 numer[3][3]=-4 denom[4]=24 numer[4][0]=0 numer[4][1]=-1 numer[4][2]=-2 numer[4][3]=-3 p(x)= 24 x^0 at term=0 p(x)= -50 x^1 at term=0 p(x)= 35 x^2 at term=0 p(x)= -10 x^3 at term=0 p(x)= 1 x^4 at term=0 p^(2)(x)= 70 x^0 at term=0 p^(2)(x)= -60 x^1 at term=0 p^(2)(x)= 12 x^2 at term=0 p(x)= 0 x^0 at term=1 p(x)= -24 x^1 at term=1 p(x)= 26 x^2 at term=1 p(x)= -9 x^3 at term=1 p(x)= 1 x^4 at term=1 p^(2)(x)= 52 x^0 at term=1 p^(2)(x)= -54 x^1 at term=1 p^(2)(x)= 12 x^2 at term=1 p(x)= 0 x^0 at term=2 p(x)= -12 x^1 at term=2 p(x)= 19 x^2 at term=2 p(x)= -8 x^3 at term=2 p(x)= 1 x^4 at term=2 p^(2)(x)= 38 x^0 at term=2 p^(2)(x)= -48 x^1 at term=2 p^(2)(x)= 12 x^2 at term=2 p(x)= 0 x^0 at term=3 p(x)= -8 x^1 at term=3 p(x)= 14 x^2 at term=3 p(x)= -7 x^3 at term=3 p(x)= 1 x^4 at term=3 p^(2)(x)= 28 x^0 at term=3 p^(2)(x)= -42 x^1 at term=3 p^(2)(x)= 12 x^2 at term=3 p(x)= 0 x^0 at term=4 p(x)= -6 x^1 at term=4 p(x)= 11 x^2 at term=4 p(x)= -6 x^3 at term=4 p(x)= 1 x^4 at term=4 p^(2)(x)= 22 x^0 at term=4 p^(2)(x)= -36 x^1 at term=4 p^(2)(x)= 12 x^2 at term=4 f^(2)(x[2]) = (1/h^2) (-2/24 f(x[0]) + f^(2)(x[2]) = (1/h^2) (-8/-6 f(x[1]) + f^(2)(x[2]) = (1/h^2) (-10/4 f(x[2]) + f^(2)(x[2]) = (1/h^2) (-8/-6 f(x[3]) + f^(2)(x[2]) = (1/h^2) (-2/24 f(x[4]) + f^(2)(x[2])=(1/12h^2)(-1 f(x[0]) + f^(2)(x[2])=(1/12h^2)(16 f(x[1]) + f^(2)(x[2])=(1/12h^2)(-30 f(x[2]) + f^(2)(x[2])=(1/12h^2)(16 f(x[3]) + f^(2)(x[2])=(1/12h^2)(-1 f(x[4]) + f^(2)(x[2])=(1/12h^2)( -1 * f(x[0]) + 16 * f(x[1] + -30 * f(x[2] + 16 * f(x[3] + -1 * f(x[4] ) + O(1/h^5)f^(5)(z) err=0, sum=7.5, deriv=7.5, x=1.25, pwr=3 err=0, sum=18.75, deriv=18.75, x=1.25, pwr=4 err=0, sum=39.0625, deriv=39.0625, x=1.25, pwr=5 err=0.00195312, sum=73.2402, deriv=73.2422, x=1.25, pwr=6 computing order=2, npoints=5, at term=3 denom[0]=24 numer[0][0]=-1 numer[0][1]=-2 numer[0][2]=-3 numer[0][3]=-4 denom[1]=-6 numer[1][0]=0 numer[1][1]=-2 numer[1][2]=-3 numer[1][3]=-4 denom[2]=4 numer[2][0]=0 numer[2][1]=-1 numer[2][2]=-3 numer[2][3]=-4 denom[3]=-6 numer[3][0]=0 numer[3][1]=-1 numer[3][2]=-2 numer[3][3]=-4 denom[4]=24 numer[4][0]=0 numer[4][1]=-1 numer[4][2]=-2 numer[4][3]=-3 p(x)= 24 x^0 at term=0 p(x)= -50 x^1 at term=0 p(x)= 35 x^2 at term=0 p(x)= -10 x^3 at term=0 p(x)= 1 x^4 at term=0 p^(2)(x)= 70 x^0 at term=0 p^(2)(x)= -60 x^1 at term=0 p^(2)(x)= 12 x^2 at term=0 p(x)= 0 x^0 at term=1 p(x)= -24 x^1 at term=1 p(x)= 26 x^2 at term=1 p(x)= -9 x^3 at term=1 p(x)= 1 x^4 at term=1 p^(2)(x)= 52 x^0 at term=1 p^(2)(x)= -54 x^1 at term=1 p^(2)(x)= 12 x^2 at term=1 p(x)= 0 x^0 at term=2 p(x)= -12 x^1 at term=2 p(x)= 19 x^2 at term=2 p(x)= -8 x^3 at term=2 p(x)= 1 x^4 at term=2 p^(2)(x)= 38 x^0 at term=2 p^(2)(x)= -48 x^1 at term=2 p^(2)(x)= 12 x^2 at term=2 p(x)= 0 x^0 at term=3 p(x)= -8 x^1 at term=3 p(x)= 14 x^2 at term=3 p(x)= -7 x^3 at term=3 p(x)= 1 x^4 at term=3 p^(2)(x)= 28 x^0 at term=3 p^(2)(x)= -42 x^1 at term=3 p^(2)(x)= 12 x^2 at term=3 p(x)= 0 x^0 at term=4 p(x)= -6 x^1 at term=4 p(x)= 11 x^2 at term=4 p(x)= -6 x^3 at term=4 p(x)= 1 x^4 at term=4 p^(2)(x)= 22 x^0 at term=4 p^(2)(x)= -36 x^1 at term=4 p^(2)(x)= 12 x^2 at term=4 f^(2)(x[3]) = (1/h^2) (-2/24 f(x[0]) + f^(2)(x[3]) = (1/h^2) (-2/-6 f(x[1]) + f^(2)(x[3]) = (1/h^2) (2/4 f(x[2]) + f^(2)(x[3]) = (1/h^2) (10/-6 f(x[3]) + f^(2)(x[3]) = (1/h^2) (22/24 f(x[4]) + f^(2)(x[3])=(1/12h^2)(-1 f(x[0]) + f^(2)(x[3])=(1/12h^2)(4 f(x[1]) + f^(2)(x[3])=(1/12h^2)(6 f(x[2]) + f^(2)(x[3])=(1/12h^2)(-20 f(x[3]) + f^(2)(x[3])=(1/12h^2)(11 f(x[4]) + f^(2)(x[3])=(1/12h^2)( -1 * f(x[0]) + 4 * f(x[1] + 6 * f(x[2] + -20 * f(x[3] + 11 * f(x[4] ) + O(1/h^5)f^(5)(z) err=0, sum=8.25, deriv=8.25, x=1.375, pwr=3 err=0, sum=22.6875, deriv=22.6875, x=1.375, pwr=4 err=-0.0195312, sum=52.0117, deriv=51.9922, x=1.375, pwr=5 err=-0.151855, sum=107.386, deriv=107.234, x=1.375, pwr=6 computing order=2, npoints=5, at term=4 denom[0]=24 numer[0][0]=-1 numer[0][1]=-2 numer[0][2]=-3 numer[0][3]=-4 denom[1]=-6 numer[1][0]=0 numer[1][1]=-2 numer[1][2]=-3 numer[1][3]=-4 denom[2]=4 numer[2][0]=0 numer[2][1]=-1 numer[2][2]=-3 numer[2][3]=-4 denom[3]=-6 numer[3][0]=0 numer[3][1]=-1 numer[3][2]=-2 numer[3][3]=-4 denom[4]=24 numer[4][0]=0 numer[4][1]=-1 numer[4][2]=-2 numer[4][3]=-3 p(x)= 24 x^0 at term=0 p(x)= -50 x^1 at term=0 p(x)= 35 x^2 at term=0 p(x)= -10 x^3 at term=0 p(x)= 1 x^4 at term=0 p^(2)(x)= 70 x^0 at term=0 p^(2)(x)= -60 x^1 at term=0 p^(2)(x)= 12 x^2 at term=0 p(x)= 0 x^0 at term=1 p(x)= -24 x^1 at term=1 p(x)= 26 x^2 at term=1 p(x)= -9 x^3 at term=1 p(x)= 1 x^4 at term=1 p^(2)(x)= 52 x^0 at term=1 p^(2)(x)= -54 x^1 at term=1 p^(2)(x)= 12 x^2 at term=1 p(x)= 0 x^0 at term=2 p(x)= -12 x^1 at term=2 p(x)= 19 x^2 at term=2 p(x)= -8 x^3 at term=2 p(x)= 1 x^4 at term=2 p^(2)(x)= 38 x^0 at term=2 p^(2)(x)= -48 x^1 at term=2 p^(2)(x)= 12 x^2 at term=2 p(x)= 0 x^0 at term=3 p(x)= -8 x^1 at term=3 p(x)= 14 x^2 at term=3 p(x)= -7 x^3 at term=3 p(x)= 1 x^4 at term=3 p^(2)(x)= 28 x^0 at term=3 p^(2)(x)= -42 x^1 at term=3 p^(2)(x)= 12 x^2 at term=3 p(x)= 0 x^0 at term=4 p(x)= -6 x^1 at term=4 p(x)= 11 x^2 at term=4 p(x)= -6 x^3 at term=4 p(x)= 1 x^4 at term=4 p^(2)(x)= 22 x^0 at term=4 p^(2)(x)= -36 x^1 at term=4 p^(2)(x)= 12 x^2 at term=4 f^(2)(x[4]) = (1/h^2) (22/24 f(x[0]) + f^(2)(x[4]) = (1/h^2) (28/-6 f(x[1]) + f^(2)(x[4]) = (1/h^2) (38/4 f(x[2]) + f^(2)(x[4]) = (1/h^2) (52/-6 f(x[3]) + f^(2)(x[4]) = (1/h^2) (70/24 f(x[4]) + f^(2)(x[4])=(1/12h^2)(11 f(x[0]) + f^(2)(x[4])=(1/12h^2)(-56 f(x[1]) + f^(2)(x[4])=(1/12h^2)(114 f(x[2]) + f^(2)(x[4])=(1/12h^2)(-104 f(x[3]) + f^(2)(x[4])=(1/12h^2)(35 f(x[4]) + f^(2)(x[4])=(1/12h^2)( 11 * f(x[0]) + -56 * f(x[1] + 114 * f(x[2] + -104 * f(x[3] + 35 * f(x[4] ) + O(1/h^5)f^(5)(z) err=0, sum=9, deriv=9, x=1.5, pwr=3 err=0, sum=27, deriv=27, x=1.5, pwr=4 err=0.195312, sum=67.3047, deriv=67.5, x=1.5, pwr=5 err=1.52539, sum=150.35, deriv=151.875, x=1.5, pwr=6 many formulas checked against Abramowitz and Stegun, Handbook of Mathematical Functions Table 25.2