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Newton-Cotes Formulas

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The Newton-Cotes formulas are numerical integration formulas obtained by integrating a Lagrange interpolating polynomial through equally spaced sampling points.

For a closed formula with n>=2 sampling points on [a,b], put x_i=a+(i-1)h, f_i=f(x_i), and h=(b-a)/(n-1). Integrating the fitting polynomial approximates the integral of f(x).

A closed formula includes both endpoints among its sampling points. An open formula uses n>=1 equally spaced sampling points strictly inside [x_0,x_(n+1)], omitting both endpoints. In formulas written as hsum_(i)w_if_i, the weights sum to n-1 for the closed rule and n+1 for the open rule. These sums follow by integrating the constant function 1.

When sampling points can be chosen freely, Gaussian quadrature is an alternative. An n-point Legendre-Gauss quadrature formula is exact for polynomials of polynomial degree at most 2n-1.

In the remainder formulas below, assume that the displayed derivatives are continuous on the interval containing the integration interval and all sampling points. Each intermediate argument xi, xi_1, or xi_2 lies in that interval and may differ between formulas.

TrapezoidalRule

The 2-point closed Newton-Cotes formula is called the trapezoidal rule because it approximates the area under a curve by a trapezoid with horizontal base and sloped top (connecting the endpoints x_1 and x_2). If the first point is x_1, then the other endpoint will be located at

x_2=x_1+h,
(1)

and the Lagrange interpolating polynomial through the points (x_1,f_1) and (x_2,f_2) is

P_2(x)=(x-x_2)/(x_1-x_2)f_1+(x-x_1)/(x_2-x_1)f_2
(2)
=(x-x_1-h)/(-h)f_1+(x-x_1)/hf_2
(3)
=x/h(f_2-f_1)+(f_1+(x_1)/hf_1-(x_1)/hf_2).
(4)

Integrating over the interval (i.e., finding the area of the trapezoid) then gives

int_(x_1)^(x_1+h)P_2(x)dx
(5)
=1/(2h)(f_2-f_1)[x^2]_(x_1)^(x_2)+(f_1+(x_1)/hf_1-(x_1)/hf_2)[x]_(x_1)^(x_2)
(6)
=1/(2h)(f_2-f_1)(x_2+x_1)(x_2-x_1)+(x_2-x_1)(f_1+(x_1)/hf_1-(x_1)/hf_2)
(7)
=1/2(f_2-f_1)(2x_1+h)+f_1h+x_1(f_1-f_2)
(8)
=x_1(f_2-f_1)+1/2h(f_2-f_1)+hf_1-x_1(f_2-f_1)
(9)
=1/2h(f_1+f_2).
(10)

The integral of f(x) is therefore given by the trapezoidal rule

int_(x_1)^(x_2)f(x)dx=1/2h(f_1+f_2)-1/(12)h^3f^('')(xi).
(11)

The absolute error is at most h^3max_(x in [x_1,x_2])|f^('')(x)|/12 (Ueberhuber 1997, p. 100).

The 3-point rule is known as Simpson's rule. The abscissas are

x_2=x_1+h
(12)
x_3=x_1+2h
(13)

and the Lagrange interpolating polynomial is

P_3(x)=((x-x_2)(x-x_3))/((x_1-x_2)(x_1-x_3))f_1+((x-x_1)(x-x_3))/((x_2-x_1)(x_2-x_3))f_2+((x-x_1)(x-x_2))/((x_3-x_1)(x_3-x_2))f_3
(14)
=(x^2-x(x_2+x_3)+x_2x_3)/(h(2h))f_1+(x^2-x(x_1+x_3)+x_1x_3)/(h(-h))f_2+(x^2-x(x_1+x_2)+x_1x_2)/(2h(h))f_3
(15)
=1/(h^2){x^2(1/2f_1-f_2+1/2f_3)+x[-1/2(2x_1+3h)f_1+(2x_1+2h)f_2-1/2(2x_1+h)f_3]+[1/2(x_1+h)(x_1+2h)f_1-x_1(x_1+2h)f_2+1/2x_1(x_1+h)f_3]}.
(16)

Integrating and simplifying gives

int_(x_1)^(x_1+2h)P_3(x)dx=1/3h(f_1+4f_2+f_3)
(17)

and Simpson's rule is

int_(x_1)^(x_3)f(x)dx=1/3h(f_1+4f_2+f_3)-1/(90)h^5f^((4))(xi)
(18)

(Ueberhuber 1997, p. 100).

The 4-point closed rule is Simpson's 3/8 rule,

int_(x_1)^(x_4)f(x)dx=3/8h(f_1+3f_2+3f_3+f_4)-3/(80)h^5f^((4))(xi)
(19)

(Ueberhuber 1997, p. 100). The 5-point closed rule is Boole's rule,

int_(x_1)^(x_5)f(x)dx=2/(45)h(7f_1+32f_2+12f_3+32f_4+7f_5)-8/(945)h^7f^((6))(xi)
(20)

(Abramowitz and Stegun 1972, p. 886). Higher order rules include the 6-point

int_(x_1)^(x_6)f(x)dx=5/(288)h(19f_1+75f_2+50f_3+50f_4+75f_5+19f_6)-(275)/(12096)h^7f^((6))(xi),
(21)

7-point

int_(x_1)^(x_7)f(x)dx=1/(140)h(41f_1+216f_2+27f_3+272f_4 +27f_5+216f_6+41f_7)-9/(1400)h^9f^((8))(xi),
(22)

8-point

int_(x_1)^(x_8)f(x)dx=7/(17280)h(751f_1+3577f_2+1323f_3+2989f_4+2989f_5+1323f_6+3577f_7+751f_8)-(8183)/(518400)h^9f^((8))(xi),
(23)

9-point

int_(x_1)^(x_9)f(x)dx=4/(14175)h(989f_1+5888f_2-928f_3+10496f_4-4540f_5+10496f_6-928f_7+5888f_8+989f_9)-(2368)/(467775)h^(11)f^((10))(xi)
(24)

(Ueberhuber 1997, p. 100), 10-point

int_(x_1)^(x_(10))f(x)dx=9/(89600)h[2857(f_1+f_(10))+15741(f_2+f_9)+1080(f_3+f_8)+19344(f_4+f_7)+5778(f_5+f_6)]-(4671)/(394240)h^(11)f^((10))(xi),
(25)

and 11-point

int_(x_1)^(x_(11))f(x)dx=5/(299376)h[16067(f_1+f_(11))+106300(f_2+f_(10))-48525(f_3+f_9)+272400(f_4+f_8)-260550(f_5+f_7)+427368f_6]-(1346350)/(326918592)h^(13)f^((12))(xi)
(26)

rules.

In general, the n-point closed rule approximates the integral by

int_(x_1)^(x_n)f(x)dx approx hsum_(i=1)^nH_(n-1,i)f_i,
(27)

where H_(n,r+1) denotes the weight at node r for n subintervals and n+1 sampling points:

H_(n,r+1)=((-1)^(n-r))/(r!(n-r)!)int_0^nproduct_(0<=j<=n; j!=r)(t-j)dt
(28)

for 0<=r<=n (Whittaker and Robinson 1967, p. 154). This gives the triangle of coefficients shown in the following table (OEIS A093735 and A093736).

n\r012345
11/21/2
21/34/31/3
33/89/89/83/8
4(14)/(45)(64)/(45)8/(15)(64)/(45)(14)/(45)
5(95)/(288)(125)/(96)(125)/(144)(125)/(144)(125)/(96)(95)/(288)

Note that

sum_(r=0)^nH_(n,r+1)=n.
(29)

Closed "extended" rules use multiple copies of lower order closed rules to build up higher order rules. By appropriately tailoring this process, rules with particularly nice properties can be constructed. For n>=5 tabulated points, using the trapezoidal rule (n-1) times and adding the results gives

int_(x_1)^(x_n)f(x)dx=(int_(x_1)^(x_2)+int_(x_2)^(x_3)+...+int_(x_(n-1))^(x_n))f(x)dx =1/2h[(f_1+f_2)+(f_2+f_3)+...+(f_(n-2)+f_(n-1))+(f_(n-1)+f_n)]-1/(12)(n-1)h^3f^('')(xi) =h(1/2f_1+f_2+sum_(i=3)^(n-2)f_i+f_(n-1)+1/2f_n)-1/(12)(n-1)h^3f^('')(xi)
(30)

(Ueberhuber 1997, p. 107). Using a series of refinements on the extended trapezoidal rule gives the method known as Romberg integration. A 3-point extended rule for odd n>=3 is

int_(x_1)^(x_n)f(x)dx=1/3hsum_(j=0)^((n-3)/2)(f_(2j+1)+4f_(2j+2)+f_(2j+3))-(n-1)/(180)h^5f^((4))(xi).
(31)

For n>=9, expanding the sum gives

int_(x_1)^(x_n)f(x)dx=h[(1/3f_1+4/3f_2+1/3f_3)+(1/3f_3+4/3f_4+1/3f_5)+...+(1/3f_(n-4)+4/3f_(n-3)+1/3f_(n-2))+(1/3f_(n-2)+4/3f_(n-1)+1/3f_n)]-(n-1)/(180)h^5f^((4))(xi) =1/3h(f_1+4f_2+2f_3+4f_4+2f_5+...+4f_(n-1)+f_n)-(n-1)/21/(90)h^5f^((4))(xi).
(32)

Applying Simpson's 3/8 rule, then Simpson's rule (3-point) twice, and adding gives

[int_(x_1)^(x_4)+int_(x_4)^(x_6)+int_(x_6)^(x_8)]f(x)dx approx h[(3/8f_1+9/8f_2+9/8f_3+3/8f_4)+(1/3f_4+4/3f_5+1/3f_6)+(1/3f_6+4/3f_7+1/3f_8)] =h[3/8f_1+9/8f_2+9/8f_3+(3/8+1/3)f_4+4/3f_5+(1/3+1/3)f_6+4/3f_7+1/3f_8] =h(3/8f_1+9/8f_2+9/8f_3+(17)/(24)f_4+4/3f_5+2/3f_6+4/3f_7+1/3f_8).
(33)

Taking the next Simpson's 3/8 rule step then gives

int_(x_8)^(x_(11))f(x)dx approx h(3/8f_8+9/8f_9+9/8f_(10)+3/8f_(11)).
(34)

Combining with the previous result gives

int_(x_1)^(x_(11))f(x)dx approx h[3/8f_1+9/8f_2+9/8f_3+(17)/(24)f_4+4/3f_5+2/3f_6+4/3f_7+(1/3+3/8)f_8+9/8f_9+9/8f_(10)+3/8f_(11)] =h(3/8f_1+9/8f_2+9/8f_3+(17)/(24)f_4+4/3f_5+2/3f_6+4/3f_7+(17)/(24)f_8+9/8f_9+9/8f_(10)+3/8f_(11)).
(35)

Continuing with Simpson's rule in the middle and Simpson's 3/8 rule at both ends gives, for odd n>=13,

int_(x_1)^(x_n)f(x)dx approx h(3/8f_1+9/8f_2+9/8f_3+(17)/(24)f_4+4/3f_5+2/3f_6+...+2/3f_(n-5)+4/3f_(n-4)+(17)/(24)f_(n-3)+9/8f_(n-2)+9/8f_(n-1)+3/8f_n).
(36)

For sufficiently smooth f on a fixed integration interval, averaging with the 3-point approximation

h(1/3f_1+4/3f_2+2/3f_3+4/3f_4+2/3f_5+...+4/3f_(n-1)+1/3f_n)
(37)

gives

int_(x_1)^(x_n)f(x)dx=h[(17)/(48)f_1+(59)/(48)f_2+(43)/(48)f_3+(49)/(48)f_4+(f_5+f_6+...+f_(n-5)+f_(n-4))+(49)/(48)f_(n-3)+(43)/(48)f_(n-2)+(59)/(48)f_(n-1)+(17)/(48)f_n]+O(n^(-4)).
(38)

Note that all the middle terms now have unity coefficients. Similarly, combining a 3-point with the (2+3)-point rule gives, for n>=4,

int_(x_1)^(x_n)f(x)dx=h(5/(12)f_1+(13)/(12)f_2+sum_(i=3)^(n-2)f_i+(13)/(12)f_(n-1)+5/(12)f_n)+O(n^(-3)).
(39)

Other Newton-Cotes rules occasionally encountered include Durand's rule (for n>=4)

int_(x_1)^(x_n)f(x)dx approx h(2/5f_1+(11)/(10)f_2+sum_(i=3)^(n-2)f_i+(11)/(10)f_(n-1)+2/5f_n)
(40)

(Beyer 1987), Hardy's rule

int_(x_0-3h)^(x_0+3h)f(x)dx=1/(100)h(28f_(-3)+162f_(-2)+220f_0+162f_2+28f_3) +9/(1400)h^7[2f^((6))(xi_2)-h^2f^((8))(xi_1)],
(41)

and Weddle's rule

int_(x_1)^(x_7)f(x)dx approx 3/(10)h(f_1+5f_2+f_3+6f_4+f_5+5f_6+f_7)
(42)

(Beyer 1987).

The open Newton-Cotes rules use sampling points strictly inside the integration interval, yielding the 1-point

int_(x_0)^(x_2)f(x)dx=2hf_1+1/3h^3f^('')(xi),
(43)

2-point

int_(x_1-h)^(x_1+2h)P_2(x)dx
(44)
=1/(2h)(f_2-f_1)[x^2]_(x_1-h)^(x_1+2h)+(f_1+(x_1)/hf_1-(x_1)/hf_2)[x]_(x_1-h)^(x_1+2h)
(45)
=3/2h(f_1+f_2),
(46)

and hence

int_(x_0)^(x_3)f(x)dx=3/2h(f_1+f_2)+3/4h^3f^('')(xi),
(47)

3-point

int_(x_0)^(x_4)f(x)dx=4/3h(2f_1-f_2+2f_3)+(28)/(90)h^5f^((4))(xi),
(48)

4-point

int_(x_0)^(x_5)f(x)dx=5/(24)h(11f_1+f_2+f_3+11f_4)+(95)/(144)h^5f^((4))(xi),
(49)

5-point

int_(x_0)^(x_6)f(x)dx=6/(20)h(11f_1-14f_2+26f_3-14f_4+11f_5)+(41)/(140)h^7f^((6))(xi),
(50)

6-point

int_(x_0)^(x_7)f(x)dx=7/(1440)h(611f_1-453f_2+562f_3+562f_4-453f_5+611f_6)+(5257)/(8640)h^7f^((6))(xi),
(51)

and 7-point

int_(x_0)^(x_8)f(x)dx=8/(945)h(460f_1-954f_2+2196f_3-2459f_4+2196f_5-954f_6+460f_7)+(3956)/(14175)h^9f^((8))(xi)
(52)

rules.

For n>=3, a 2-point extended formula using external sampling points x_0 and x_(n+1), in addition to x_1, ..., x_n, is

int_(x_1)^(x_n)f(x)dx=h[(1/2f_1+f_2+...+f_(n-1)+1/2f_n)+1/(24)(-f_0+f_2+f_(n-1)-f_(n+1))]+(11(n-1))/(720)h^5f^((4))(xi).
(53)

Single interval extrapolative rules estimate the integral over [x_0,x_1] from sampling points x_1, x_2, ... to its right. Examples are

int_(x_0)^(x_1)f(x)dx=hf_1+O(h^2f^')
(54)
int_(x_0)^(x_1)f(x)dx=1/2h(3f_1-f_2)+O(h^3f^(''))
(55)
int_(x_0)^(x_1)f(x)dx=1/(12)h(23f_1-16f_2+5f_3)+O(h^4f^((3)))
(56)
int_(x_0)^(x_1)f(x)dx=1/(24)h(55f_1-59f_2+37f_3-9f_4)+O(h^5f^((4))).
(57)

See also

Boole's Rule, Difference Equation, Durand's Rule, Finite Difference, Gaussian Quadrature, Hardy's Rule, Lagrange Interpolating Polynomial, Numerical Integration, Shovelton's Rule, Simpson's 3/8 Rule, Simpson's Rule, Trapezoidal Rule, Weddle's Rule, Woolhouse's Formulas

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Integration." §25.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 885-887, 1972.Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 127, 1987.Corbit, D. "Numerical Integration: From Trapezoids to RMS: Object-Oriented Numerical Integration." Dr. Dobb's J., No. 252, 117-120, Oct. 1996.Daniell, P. J. "Remainders in Interpolation and Quadrature Formulae." Math. Gaz. 24, 238, 1940.Fornberg, B. "Calculation of Weights in Finite Difference Formulas." SIAM Rev. 40, 685-691, 1998.Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 160-161, 1956.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Classical Formulas for Equally Spaced Abscissas." §4.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 124-130, 1992.Sloane, N. J. A. Sequences A093735 and A093736 in "The On-Line Encyclopedia of Integer Sequences."Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin, Germany: Springer-Verlag, 1997.Whittaker, E. T. and Robinson, G. "The Newton-Cotes Formulae of Integration." §76 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 152-156, 1967.

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Newton-Cotes Formulas

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Weisstein, Eric W. "Newton-Cotes Formulas." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Newton-CotesFormulas.html

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