# Newton-Cotes formulas

Newton-Cotes formula for n = 2

A Newton-Cotes formula (according to Isaac Newton and Roger Cotes ) is a numerical quadrature formula for the approximate calculation of integrals . These formulas is based on the idea that function to be integrated by a polynomial to interpolate and integrate this exactly as an approximation. The interpolation points are selected to be equidistant.

## Derivation

The interpolation points are for the interpolation polynomial of degree to be integrated${\ displaystyle p_ {n} (x)}$${\ displaystyle n}$

${\ displaystyle a \ leq x_ {0}

Equidistant with the constant distance chosen so that they are symmetrical to the interval center of the integration interval . Thus applies . ${\ displaystyle h = x_ {i + 1} -x_ {i}}$${\ displaystyle {\ tfrac {a + b} {2}}}$${\ displaystyle [a, b]}$${\ displaystyle x_ {ni} = a + b-x_ {i}}$

With (and thus ) one obtains intervals of length and thus and . These formulas are called closed Newton-Cotes formulas . ${\ displaystyle x_ {0} = a}$${\ displaystyle x_ {n} = b}$${\ displaystyle n}$${\ displaystyle h}$${\ displaystyle h = {\ tfrac {ba} {n}}}$${\ displaystyle x_ {i} = a + i \ cdot h}$

With (and thus ) one obtains open quadrature formulas:${\ displaystyle x_ {0} \ neq a}$${\ displaystyle x_ {n} \ neq b}$

• If you choose (and thus ), you get intervals of length and thus and . These formulas are called open Newton-Cotes formulas .${\ displaystyle x_ {0} = a + h}$${\ displaystyle x_ {n} = bh}$${\ displaystyle n + 2}$${\ displaystyle h}$${\ displaystyle h = {\ tfrac {ba} {n + 2}}}$${\ displaystyle x_ {i} = a + (1 + i) \ cdot h}$
• If you choose (and thus ), you get intervals of length and thus and . These formulas are called Maclaurin formulas .${\ displaystyle x_ {0} = a + {\ tfrac {h} {2}}}$${\ displaystyle x_ {n} = b - {\ tfrac {h} {2}}}$${\ displaystyle n + 1}$${\ displaystyle h}$${\ displaystyle h = {\ tfrac {ba} {n + 1}}}$${\ displaystyle x_ {i} = a + \ left ({\ tfrac {1} {2}} + i \ right) \ cdot h}$

For the numerical integration of , the interpolation polynomial of the function for the given support points is used. For this applies: ${\ displaystyle \ int \ limits _ {a} ^ {b} f (x) \, dx}$${\ displaystyle p_ {n} (x)}$${\ displaystyle f (x)}$

${\ displaystyle p_ {n} (x) = \ sum _ {i = 0} ^ {n} f (x_ {i}) l_ {i} (x)}$,

where are the Lagrange base polynomials . It follows: ${\ displaystyle l_ {i}}$

${\ displaystyle \ int \ limits _ {a} ^ {b} p_ {n} (x) \, dx = (ba) \ sum _ {i = 0} ^ {n} f (x_ {i}) {\ frac {1} {ba}} \ int \ limits _ {a} ^ {b} l_ {i} (x) \, dx}$.

## definition

For the Newton-Cotes formula it then follows:

${\ displaystyle \ int \ limits _ {a} ^ {b} f (x) \, dx \ approx \ int \ limits _ {a} ^ {b} p_ {n} (x) \, dx = (ba) \ sum _ {i = 0} ^ {n} w_ {i} f (x_ {i})}$

with the weights

${\ displaystyle w_ {i} = {\ frac {1} {ba}} \ int \ limits _ {a} ^ {b} l_ {i} (x) \, dx}$

The weights are symmetrical, that is . ${\ displaystyle w_ {ni} = w_ {i}}$

${\ displaystyle l_ {i} (x) = \ prod _ {\ begin {smallmatrix} 0 \ leq j \ leq n \\ j \ neq i \ end {smallmatrix}} {\ frac {x-x_ {j}} {x_ {i} -x_ {j}}} = {\ frac {(x-x_ {0}) \ dotsm (x-x_ {i-1}) (x-x_ {i + 1}) \ dotsm ( x-x_ {n})} {(x_ {i} -x_ {0}) \ dotsm (x_ {i} -x_ {i-1}) (x_ {i} -x_ {i + 1}) \ dotsm (x_ {i} -x_ {n})}}}$

Because of the particular choice of sampling points, the quadrature formulas for odd integrate polynomials up to degree , with straight even to the degree of precision. Quadrature formulas with an even number (i.e. an odd number of nodes) are therefore preferable to those with an odd number . This property is also called the degree of accuracy of the quadrature formula. ${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle n + 1}$${\ displaystyle n}$${\ displaystyle n}$

Especially applies to that and thus . ${\ displaystyle f (x) = 1}$${\ displaystyle \ int \ limits _ {a} ^ {b} f (x) \, dx = \ int \ limits _ {a} ^ {b} 1 \, dx = ba = (ba) \ sum _ {i = 0} ^ {n} w_ {i} \ cdot 1 = (ba) \ sum _ {i = 0} ^ {n} w_ {i}}$${\ displaystyle \ sum _ {i = 0} ^ {n} w_ {i} = 1}$

If what at weights with different signs is the case, there is a danger that the rounding errors get progressively or cancellation occurs. Therefore, for numerical reasons, quadrature formulas with positive weights are to be preferred. Since the interpolation polynomial is useless for large , quadrature formulas with large are also not recommended. If you want to achieve better approximations, the use of summed quadrature formulas is recommended. ${\ displaystyle \ sum _ {i = 0} ^ {n} | w_ {i} |> \ sum _ {i = 0} ^ {n} w_ {i} = 1}$${\ displaystyle n}$${\ displaystyle p_ {n} (x)}$${\ displaystyle n}$

${\ displaystyle E (f) = \ int \ limits _ {a} ^ {b} f (x) \, dx- \ int \ limits _ {a} ^ {b} p_ {n} (x) \, dx }$

is the error (procedural error) that is made when applying the quadrature formula. With the special choice of the support points for -mal on continuously differentiable real-valued functions, this always has the form ${\ displaystyle (p + 1)}$${\ displaystyle [a, b]}$${\ displaystyle f (x)}$

${\ displaystyle E (f) = K \ cdot f ^ {(p + 1)} (\ xi)}$,

where is a constant that is independent of and an intermediate value that is only known in exceptional cases. If it were generally known, one could calculate exactly, and thus also the integral, in contradiction to the fact that there are infinitely many integrals that cannot be calculated exactly. The error is zero for all functions whose -th derivative is zero, i.e. for all polynomials of degree less than / equal . Thus, the degree of accuracy is the quadrature formula. The value is also called the (polynomial) order of the quadrature formula. ${\ displaystyle K}$${\ displaystyle f (x)}$${\ displaystyle \ xi \ in [a, b]}$${\ displaystyle E (f)}$${\ displaystyle (p + 1)}$${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle p + 1}$

With the help of the procedural error, the error estimate is obtained:

${\ displaystyle | E (f) | \ leq | K | \ cdot \ max _ {a \ leq \ xi \ leq b} \ left | f ^ {(p + 1)} (\ xi) \ right |}$.

The exact error is always less than or equal to this error estimate, as the examples given below also show.

### Completed Newton-Cotes formulas

The sampling points specified apply to the integration interval : . The support points are for a general interval . ${\ displaystyle t_ {i}}$${\ displaystyle [0,1]}$${\ displaystyle t_ {0} = 0, t_ {i} = {\ frac {i} {n}}, t_ {n} = 1}$${\ displaystyle [a, b]}$${\ displaystyle x_ {i} = a + t_ {i} \ cdot (ba)}$

${\ displaystyle n}$ Surname Support points ${\ displaystyle t_ {i}}$ Weights ${\ displaystyle w_ {i}}$ ${\ displaystyle E (f)}$
1 Trapezoidal rule
Tendon trapezoidal rule
${\ displaystyle 0 \ quad 1}$ ${\ displaystyle {\ frac {1} {2}} \ quad {\ frac {1} {2}}}$ ${\ displaystyle - {\ frac {(ba) ^ {3}} {12}} f '' (\ xi)}$
2 Simpson's rule
Kepler's barrel rule
${\ displaystyle 0 \ quad {\ frac {1} {2}} \ quad 1}$ ${\ displaystyle {\ frac {1} {6}} \ quad {\ frac {4} {6}} \ quad {\ frac {1} {6}}}$ ${\ displaystyle - {\ frac {\ left ({\ frac {ba} {2}} \ right) ^ {5}} {90}} f ^ {(4)} (\ xi)}$
3 3/8 rule
Pulcherrima
${\ displaystyle 0 \ quad {\ frac {1} {3}} \ quad {\ frac {2} {3}} \ quad 1}$ ${\ displaystyle {\ frac {1} {8}} \ quad {\ frac {3} {8}} \ quad {\ frac {3} {8}} \ quad {\ frac {1} {8}}}$ ${\ displaystyle - {\ frac {3 \ left ({\ frac {ba} {3}} \ right) ^ {5}} {80}} f ^ {(4)} (\ xi)}$
4th Milne rule
Boole rule
${\ displaystyle 0 \ quad {\ frac {1} {4}} \ quad {\ frac {2} {4}} \ quad {\ frac {3} {4}} \ quad 1}$ ${\ displaystyle {\ frac {7} {90}} \ quad {\ frac {32} {90}} \ quad {\ frac {12} {90}} \ quad {\ frac {32} {90}} \ quad {\ frac {7} {90}}}$ ${\ displaystyle - {\ frac {8 \ left ({\ frac {ba} {4}} \ right) ^ {7}} {945}} f ^ {(6)} (\ xi)}$
5 6 point rule ${\ displaystyle 0 \ quad {\ frac {1} {5}} \ quad {\ frac {2} {5}} \ quad {\ frac {3} {5}} \ quad {\ frac {4} {5 }} \ quad 1}$ ${\ displaystyle {\ frac {19} {288}} \ quad {\ frac {75} {288}} \ quad {\ frac {50} {288}} \ quad {\ frac {50} {288}} \ quad {\ frac {75} {288}} \ quad {\ frac {19} {288}}}$ ${\ displaystyle - {\ frac {275 \ left ({\ frac {ba} {5}} \ right) ^ {7}} {12 \, 096}} f ^ {(6)} (\ xi)}$
6th Weddle rule (after Thomas Weddle , 1817-1853) ${\ displaystyle 0 \ quad {\ frac {1} {6}} \ quad {\ frac {2} {6}} \ quad {\ frac {3} {6}} \ quad {\ frac {4} {6 }} \ quad {\ frac {5} {6}} \ quad 1}$ ${\ displaystyle {\ frac {41} {840}} \ quad {\ frac {216} {840}} \ quad {\ frac {27} {840}} \ quad {\ frac {272} {840}} \ quad {\ frac {27} {840}} \ quad {\ frac {216} {840}} \ quad {\ frac {41} {840}}}$ ${\ displaystyle - {\ frac {9 \ left ({\ frac {ba} {6}} \ right) ^ {9}} {1400}} f ^ {(8)} (\ xi)}$

The abbreviated values ​​of all nodes to are: ${\ displaystyle n = 10}$

n = 1: {1/2, 1/2}

n = 2: {1/6, 2/3, 1/6}

n = 3: {1/8, 3/8, 3/8, 1/8}

n = 4: {7/90, 16/45, 2/15, 16/45, 7/90}

n = 5: {19/288, 25/96, 25/144, 25/144, 25/96, 19/288}

n = 6: {41/840, 9/35, 9/280, 34/105, 9/280, 9/35, 41/840}

n = 7: {751/17280, 3577/17280, 49/640, 2989/17280, 2989/17280, 49/640, 3577/17280, 751/17280}

n = 8: {989/28350, 2944/14175, -464/14175, 5248/14175, -454/2835, 5248/14175, -464/14175, 2944/14175, 989/28350}

n = 9: {2857/89600, 15741/89600, 27/2240, 1209/5600, 2889/44800, 2889/44800, 1209/5600, 27/2240, 15741/89600, 2857/89600}

n = 10: {16067/598752, 26575/149688, -16175/199584, 5675/12474, -4825/11088, 17807/24948, -4825/11088, 5675/12474, -16175/199584, 26575/149688, 16067 / 598752}

To applies to and To applies${\ displaystyle n = 8}$${\ displaystyle w_ {i} <0}$${\ displaystyle i = 2,4,6}$${\ displaystyle \ textstyle \ sum _ {i = 0} ^ {n} | w_ {i} | = 1 {,} 45 \ dotso}$${\ displaystyle n = 10}$${\ displaystyle \ sum _ {i = 0} ^ {n} | w_ {i} | = 3 {,} 06470 \, 9 \ dotso}$

Example: ${\ displaystyle \ int \ limits _ {1} ^ {3} {\ frac {1} {x}} \, dx = \ ln (3) - \ ln (1) = \ ln (3) = 1 {, } 09861 \, 23 \ dotso}$

Approximation with Simpson's rule ( ). It applies and . ${\ displaystyle n = 2}$${\ displaystyle h = {\ frac {ba} {n}} = {\ frac {2} {2}} = 1}$${\ displaystyle x_ {0} = a = 1}$

${\ displaystyle \ int \ limits _ {1} ^ {3} p_ {2} (x) \, dx = 2 \ cdot \ left ({\ frac {1} {6}} f (1) + {\ frac {4} {6}} f (2) + {\ frac {1} {6}} f (3) \ right) = 2 \ cdot \ left ({\ frac {1} {6}} \ cdot 1+ {\ frac {4} {6}} \ cdot {\ frac {1} {2}} + {\ frac {1} {6}} \ cdot {\ frac {1} {3}} \ right) = { \ frac {10} {9}} = 1 {,} {\ overline {1}}}$

Process error: With one receives with${\ displaystyle f ^ {(4)} (\ xi) = {\ frac {4!} {\ xi ^ {5}}}}$${\ displaystyle E (f) = - {\ frac {1} {90}} \ cdot \ left ({\ frac {2} {2}} \ right) ^ {5} \ cdot {\ frac {4!} {\ xi ^ {5}}} = - {\ frac {4} {15}} \ cdot {\ frac {1} {\ xi ^ {5}}}}$${\ displaystyle \ xi \ in [1,3]}$

Error estimation: ${\ displaystyle | E (f) | \ leq {\ frac {4} {15}} \ cdot \ max _ {1 \ leq \ xi \ leq 3} \ left | {\ frac {1} {\ xi ^ { 5}}} \ right | = {\ frac {4} {15}} \ cdot {\ frac {1} {1}} = 0 {,} 2 {\ overline {6}}}$

Exact error: ${\ displaystyle | E (f) | = \ left | \ int \ limits _ {1} ^ {3} {\ frac {1} {x}} \, dx- \ int \ limits _ {1} ^ {3 } p_ {2} (x) \, dx \ right | = \ left | 1 {,} 09861 \, 23 \ dotso -1 {,} {\ overline {1}} \ right | = 0 {,} 01249 \ .88 \ dotso <0 {,} 2 {\ overline {6}}}$

### Open Newton-Cotes formulas

The reference points are for the integration interval : . The support points are for a general interval . ${\ displaystyle t_ {i}}$${\ displaystyle [0,1]}$${\ displaystyle t_ {0} = {\ tfrac {1} {n + 2}}, t_ {i} = {\ tfrac {i + 1} {n + 2}}, t_ {n} = {\ tfrac { n + 1} {n + 2}}}$${\ displaystyle [a, b]}$${\ displaystyle x_ {i} = a + t_ {i} \ cdot (ba)}$

${\ displaystyle n}$ Surname Support points ${\ displaystyle t_ {i}}$ Weights ${\ displaystyle w_ {i}}$ ${\ displaystyle E (f)}$
0 Rectangle rule,
center
rule, Tangent trapezoid rule
${\ displaystyle {\ frac {1} {2}}}$ ${\ displaystyle 1 \ quad}$ ${\ displaystyle {\ frac {(ba) ^ {3}} {24}} f '' (\ xi)}$
1 ${\ displaystyle {\ frac {1} {3}} \ quad {\ frac {2} {3}}}$ ${\ displaystyle {\ frac {1} {2}} \ quad {\ frac {1} {2}}}$ ${\ displaystyle {\ frac {\ left ({\ frac {ba} {3}} \ right) ^ {3}} {4}} f '' (\ xi)}$
2 ${\ displaystyle {\ frac {1} {4}} \ quad {\ frac {2} {4}} \ quad {\ frac {3} {4}}}$ ${\ displaystyle {\ frac {2} {3}} \ quad - {\ frac {1} {3}} \ quad {\ frac {2} {3}}}$ ${\ displaystyle {\ frac {14 \ left ({\ frac {ba} {4}} \ right) ^ {5}} {45}} f ^ {(4)} (\ xi)}$
3 ${\ displaystyle {\ frac {1} {5}} \ quad {\ frac {2} {5}} \ quad {\ frac {3} {5}} \ quad {\ frac {4} {5}}}$ ${\ displaystyle {\ frac {11} {24}} \ quad {\ frac {1} {24}} \ quad {\ frac {1} {24}} \ quad {\ frac {11} {24}}}$ ${\ displaystyle {\ frac {95 \ left ({\ frac {ba} {5}} \ right) ^ {5}} {144}} f ^ {(4)} (\ xi)}$
4th ${\ displaystyle {\ frac {1} {6}} \ quad {\ frac {2} {6}} \ quad {\ frac {3} {6}} \ quad {\ frac {4} {6}} \ quad {\ frac {5} {6}}}$ ${\ displaystyle {\ frac {11} {20}} \ quad - {\ frac {14} {20}} \ quad {\ frac {26} {20}} \ quad - {\ frac {14} {20} } \ quad {\ frac {11} {20}}}$ ${\ displaystyle {\ frac {41 \ left ({\ frac {ba} {6}} \ right) ^ {7}} {140}} f ^ {(6)} (\ xi)}$
5 ${\ displaystyle {\ frac {1} {7}} \ quad {\ frac {2} {7}} \ quad {\ frac {3} {7}} \ quad {\ frac {4} {7}} \ quad {\ frac {5} {7}} \ quad {\ frac {6} {7}}}$ ${\ displaystyle {\ frac {611} {1440}} \ quad - {\ frac {453} {1440}} \ quad {\ frac {562} {1440}} \ quad {\ frac {562} {1440}} \ quad - {\ frac {453} {1440}} \ quad {\ frac {611} {1440}}}$ ${\ displaystyle {\ frac {5257 \ left ({\ frac {ba} {7}} \ right) ^ {7}} {8640}} f ^ {(6)} (\ xi)}$
6th ${\ displaystyle {\ frac {1} {8}} \ quad {\ frac {2} {8}} \ quad {\ frac {3} {8}} \ quad {\ frac {4} {8}} \ quad {\ frac {5} {8}} \ quad {\ frac {6} {8}} \ quad {\ frac {7} {8}}}$ ${\ displaystyle {\ frac {460} {945}} \ quad - {\ frac {954} {945}} \ quad {\ frac {2196} {945}} \ quad - {\ frac {2459} {945} } \ quad {\ frac {2196} {945}} \ quad - {\ frac {954} {945}} \ quad {\ frac {460} {945}}}$ ${\ displaystyle {\ frac {3956 \ left ({\ frac {ba} {8}} \ right) ^ {9}} {14 \, 175}} f ^ {(8)} (\ xi)}$

For true for true${\ displaystyle n = 5}$${\ displaystyle \ textstyle \ sum _ {i = 0} ^ {n} | w_ {i} | = {\ frac {3252} {1440}} = 2 {,} 25833 \, 3 \ dotso}$${\ displaystyle n = 6}$${\ displaystyle \ textstyle \ sum _ {i = 0} ^ {n} | w_ {i} | = {\ frac {9679} {945}} = 10 {,} 24 \ dotso}$

Of these formulas, only the rectangle rule is recommended. With more effort, the formula for has the same order as the rectangular rule, the higher formulas have negative weights. ${\ displaystyle n = 1}$

Example: ${\ displaystyle \ int \ limits _ {1} ^ {3} {\ frac {1} {x}} \, dx = \ ln (3) - \ ln (1) = \ ln (3) = 1 {, } 09861 \, 23 \ dotso}$

Approximation with the formula for . It applies and . ${\ displaystyle n = 2}$${\ displaystyle h = {\ frac {ba} {n + 2}} = {\ frac {2} {4}} = {\ frac {1} {2}}}$${\ displaystyle x_ {0} = a + h = {\ frac {3} {2}}}$

${\ displaystyle \ int \ limits _ {1} ^ {3} p_ {2} (x) \, dx = 2 \ cdot \ left ({\ frac {2} {3}} f \! \ left ({\ frac {3} {2}} \ right) - {\ frac {1} {3}} f \! \ left ({\ frac {4} {2}} \ right) + {\ frac {2} {3 }} f \! \ left ({\ frac {5} {2}} \ right) \ right) = 2 \ cdot \ left ({\ frac {2} {3}} \ cdot {\ frac {2} { 3}} - {\ frac {1} {3}} \ cdot {\ frac {2} {4}} + {\ frac {2} {3}} \ cdot {\ frac {2} {5}}) \ right) = {\ frac {49} {45}} = 1 {,} 0 {\ overline {8}}}$.

Process error: With one receives with . ${\ displaystyle f ^ {(4)} (\ xi) = {\ frac {4!} {\ xi ^ {5}}}}$${\ displaystyle E (f) = {\ frac {14} {45}} \ cdot \ left ({\ frac {2} {4}} \ right) ^ {5} \ cdot {\ frac {4!} { \ xi ^ {5}}} = {\ frac {7} {30}} \ cdot {\ frac {1} {\ xi ^ {5}}}}$${\ displaystyle \ xi \ in [1,3]}$

Error estimation: ${\ displaystyle | E (f) | \ leq {\ frac {7} {30}} \ cdot \ max _ {1 \ leq \ xi \ leq 3} \ left | {\ frac {1} {\ xi ^ { 5}}} \ right | = {\ frac {7} {30}} \ cdot {\ frac {1} {1}} = 0 {,} 2 {\ overline {3}}}$

Exact error: ${\ displaystyle | E (f) | = \ left | \ int \ limits _ {1} ^ {3} {\ frac {1} {x}} \, dx- \ int \ limits _ {1} ^ {3 } p_ {2} (x) \, dx \ right | = \ left | 1 {,} 09861 \, 23 \ dotso -1 {,} 0 {\ overline {8}} \ right | = 0 {,} 00972 \, 33997 \, 79 \ dotso <0 {,} 2 {\ overline {3}}}$

These formulas are named after Colin Maclaurin . The reference points are for the integration interval : . The support points are for a general interval . ${\ displaystyle t_ {i}}$${\ displaystyle [0,1]}$${\ displaystyle t_ {0} = {\ tfrac {1} {2n + 2}}, t_ {i} = {\ tfrac {2i + 1} {2n + 2}}, t_ {n} = {\ tfrac { 2n + 1} {2n + 2}}}$${\ displaystyle [a, b]}$${\ displaystyle x_ {i} = a + t_ {i} \ cdot (ba)}$

${\ displaystyle n}$ Surname Support points ${\ displaystyle t_ {i}}$ Weights ${\ displaystyle w_ {i}}$ ${\ displaystyle E (f)}$
0 Rectangle rule,
center
rule, Tangent trapezoid rule
${\ displaystyle {\ frac {1} {2}}}$ ${\ displaystyle 1 \ quad}$ ${\ displaystyle {\ frac {(ba) ^ {3}} {24}} f '' (\ xi)}$
1 ${\ displaystyle {\ frac {1} {4}} \ quad {\ frac {3} {4}}}$ ${\ displaystyle {\ frac {1} {2}} \ quad {\ frac {1} {2}}}$ ${\ displaystyle {\ frac {\ left ({\ frac {ba} {2}} \ right) ^ {3}} {12}} f '' (\ xi)}$
2 ${\ displaystyle {\ frac {1} {6}} \ quad {\ frac {1} {2}} \ quad {\ frac {5} {6}}}$ ${\ displaystyle {\ frac {3} {8}} \ quad {\ frac {2} {8}} \ quad {\ frac {3} {8}}}$ ${\ displaystyle {\ frac {21 \ left ({\ frac {ba} {3}} \ right) ^ {5}} {640}} f ^ {(4)} (\ xi)}$
3 ${\ displaystyle {\ frac {1} {8}} \ quad {\ frac {3} {8}} \ quad {\ frac {5} {8}} \ quad {\ frac {7} {8}}}$ ${\ displaystyle {\ frac {13} {48}} \ quad {\ frac {11} {48}} \ quad {\ frac {11} {48}} \ quad {\ frac {13} {48}}}$ ${\ displaystyle {\ frac {103 \ left ({\ frac {ba} {4}} \ right) ^ {5}} {1440}} f ^ {(4)} (\ xi)}$
4th ${\ displaystyle {\ frac {1} {10}} \ quad {\ frac {3} {10}} \ quad {\ frac {5} {10}} \ quad {\ frac {7} {10}} \ quad {\ frac {9} {10}}}$ ${\ displaystyle {\ frac {275} {1152}} \ quad {\ frac {100} {1152}} \ quad {\ frac {402} {1152}} \ quad {\ frac {100} {1152}} \ quad {\ frac {275} {1152}}}$ ${\ displaystyle {\ frac {5575 \ left ({\ frac {ba} {5}} \ right) ^ {7}} {193 \, 536}} f ^ {(6)} (\ xi)}$

For true for true${\ displaystyle n = 6}$${\ displaystyle \ sum _ {i = 0} ^ {n} | w_ {i} | = 1 {,} 363 \ dotso}$${\ displaystyle n = 8}$${\ displaystyle \ sum _ {i = 0} ^ {n} | w_ {i} | = 3 {,} 433 \ dotso}$

Example: ${\ displaystyle \ int \ limits _ {1} ^ {3} {\ frac {1} {x}} \, dx = \ ln (3) - \ ln (1) = \ ln (3) = 1 {, } 09861 \, 23 \ dotso}$

Approximation with the formula for . It applies and . ${\ displaystyle n = 2}$${\ displaystyle h = {\ frac {ba} {n + 1}} = {\ frac {2} {3}}}$${\ displaystyle x_ {0} = a + {\ frac {h} {2}} = {\ frac {4} {3}}}$

${\ displaystyle \ int \ limits _ {1} ^ {3} p_ {2} (x) \, dx = 2 \ cdot \ left ({\ frac {3} {8}} f \! \ left ({\ frac {4} {3}} \ right) + {\ frac {2} {8}} f \! \ left ({\ frac {6} {3}} \ right) + {\ frac {3} {8 }} f \! \ left ({\ frac {8} {3}} \ right) \ right) = 2 \ cdot \ left ({\ frac {3} {8}} \ cdot {\ frac {3} { 4}} + {\ frac {2} {8}} \ cdot {\ frac {3} {6}} + {\ frac {3} {8}} \ cdot {\ frac {3} {8}} \ right) = {\ frac {105} {96}} = 1 {,} 09375}$

Process error: With one receives with . ${\ displaystyle f ^ {(4)} (\ xi) = {\ frac {4!} {\ xi ^ {5}}}}$${\ displaystyle E (f) = {\ frac {21} {640}} \ cdot \ left ({\ frac {2} {3}} \ right) ^ {5} \ cdot {\ frac {4!} { \ xi ^ {5}}} = {\ frac {14} {135}} \ cdot {\ frac {1} {\ xi ^ {5}}}}$${\ displaystyle \ xi \ in [1,3]}$

Error estimation: ${\ displaystyle | E (f) | \ leq {\ frac {14} {135}} \ cdot \ max _ {1 \ leq \ xi \ leq 3} \ left | {\ frac {1} {\ xi ^ { 5}}} \ right | = {\ frac {14} {135}} \ cdot {\ frac {1} {1}} = 0 {,} 1 {\ overline {037}}}$

Exact error: ${\ displaystyle | E (f) | = \ left | \ int \ limits _ {1} ^ {3} {\ frac {1} {x}} \, dx- \ int \ limits _ {1} ^ {3 } p_ {2} (x) \, dx \ right | = | 1 {,} 09861 \, 23 \ dotso -1 {,} 09375 | = 0 {,} 00048 \, 6229 \ dotso <0 {,} 1 {\ overline {037}}}$

## Summed Newton-Cotes formulas

Starting with grade 8, many Newton-Cotes formulas have negative weights, which entails the risk of extinction . In addition, one cannot generally expect convergence since the polynomial interpolation is poorly conditioned. In the case of larger integration areas, these are therefore divided into individual sub-intervals and a low-order formula is applied to each individual sub-interval. ${\ displaystyle [a, b]}$

## literature

• Hans R. Schwarz, Norbert Köckler: Numerical Mathematics. 6th edition. Teubner, Stuttgart 2006, ISBN 3-519-42960-8 , pp. 311-316.
• Roland W. Freund, Ronald HW Hoppe: Stoer / Bulirsch: Numerical Mathematics 1st 10th edition. Springer, Berlin 2007, ISBN 978-3-540-45389-5 , pp. 164-169.
• Michael R. Schäferkotter, Prem K. Kythe: Handbook of Computational Methods for Integration. Chapman & Hall, Boca Raton 2005, ISBN 1-58488-428-2 , pp. 54-62, 503-505.
• Günter Bärwolf: Numerics for engineers, physicists and computer scientists. ISBN 978-3-8274-1689-6 , Spektrum, Munich 2007, p. 128.
• Gisela Engeln-Müllges , Klaus Niederdrenk, Reinhard Wodicka: Numerical algorithms: procedures, examples, applications. ISBN 978-3-642-13472-2 , Springer, Berlin and Heidelberg 2011.

## Individual evidence

1. ^ Thomas Weddle ( Newcastle-upon-Tyne ): A new simple and general method of solving numerical equations of all orders . Hamilton, Adams & Co. and J. Philipson, London 1842 ( Internet Archive - 52 pp.).
2. ^ WolframAlpha . wolframalpha.com. Retrieved September 14, 2019.