Error analysis in numerical integration
- 1. FAZAIA COLLEGE OF EDUCATION FOR WOMEN ASSIGNMENT OF NUMERICAL ANALYSIS (Advance Math VIII) TOPIC: NUMERICAL INTEGRATION Submitted to: Ma’am Mehak Submitted by: Amenah Gondal (EDU(S)-2017-F-11) Class: B.S.Ed Hons (VIII)
- 2. ABSTRACT Numerical Integration is the approximation computation of an integral using numerical techniques. The numerical computation of an integral is sometimes called quadrature. We can use numerical integration ti estimate the values of definite integrals when a closed form of the integral is difficult to find or when an approximate value only of integral is needed. The most commonly rules for numerical integration are rectangular, trapezoidal, simpson1/3, simpson 3/8, boole and weddle rules. We get formulas for these rules by using Newton Cotes Quadrature Formula by putting values for ‘n’ i.e., n=1,2,…,6. and neglecting higher derivatives according to the requirement. All these rules can be compared with each other and we can find that the rule with greater value of ‘n’ has more accurate and exact solution. NUMERICAL INTEGRATION 1- RECTANGULAR RULE The formula for rectangular rule is It is also called Mid-Point formula. This rule approximates the area under the curve by rectangles whose height is the mid-point of each sub-interval.
- 3. EXAMPLE: Evaluate the integral using rectangular rule for n=16 and compare with exact value. SOLUTION: Given a=0, b=1, n=16 x 0 0.0625 0.125 0.1875 0.25 0.3125 0.375 0.4375 0.5 0.5625 0.625 0.6875 F(x) 1 0.9961 0.9846 0.9660 0.9412 0.9110 0.8767 0.8393 0.8 0.7596 0.7596 0.6790 0.75 0.8125 0.875 0.9375 1 0.64 0.6024 0.5664 0.5322 0.5 By putting values, we get
- 4. 2-TRAPEZOIDAL RULE: The formula for trapezoidal rule is In this rule, we find area under a curve is evaluated by dividing the total area into little trapezoids rather than rectangles. This rule is used for n=1 and its multiples. EXAMPLE: Calculate the integral where h=0.05 using trapezoidal rule and compare with exact value. SOLUTION: Given a=1.0, b=1.30, h=0.05, n=7 x 1 1.05 1.1 1.15 1.2 1.25 1.30 F(x) 1 1.0246 1.0488 1.072 1.095 1.118 1.14
- 5. By putting values, we get Exact Value: 3-SIMPSON’S RULE: The formula for simpson’s rule is Simpson’s 1/3 rule is an extension of the trapezoidal rule in which the integrand is approximated by a second order polynomial. This rule approximates the definite integral by first approximating the original function using piecewise quadratic functions. This rule is used for n=2 and its multiples.
- 6. EXAMPLE: Use simpson’s 1/3 rule for n=4 to calculate correct to four decimal places. SOLUTION: Given a=0, b=1, n=4 x 0 0.25 0.5 0.75 1 F(x) 0.7071 0.7012 0.6822 0.6500 0.6065 By putting values, we get 4-SIMPSON’S RULE: The formula for simpson’s rule is This rule relies on approximating the curve with a cubic polynomial. This rule is used for n=3 and its multiples.
- 7. EXAMPLE: Evaluate using seven points simpson’s rule. SOLUTION: Given a=0.1, b=0.7, n=6 x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 F(x) 0.0997 0.1974 0.2915 0.3805 0.4636 0.5404 0.6107 By putting values, we get 5-BOOLE’S RULE: The formula for boole’s rule is This rule can be approximated by a polynomial of 4th degree so that 5th and higher derivatives are vanishes. It can be used for the subinterval i.e., n=4. This rule is used for n=4 and its multiples.
- 8. EXAMPLE: Apply nine points Boole’s rule to evaluate . SOLUTION: Given a=0.3, b=0.7, n=8 x 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 F(x) 2.5377 2.5576 2.5791 2.6021 2.6265 2.6525 2.6799 2.7089 2.7893 By putting values, we get 6-WEDDLE’S RULE: The formula for weddle’s rule is This rule approximating the integral of f(x) by giving n=6. It is used to solve multiple integrals. It can only be used for the sub interval i.e., n=6.
- 9. EXAMPLE: Apply seven point weddles’s rule to evaluate SOLUTION: Given a=0, b= , n=6, x 0 F(x) 0 3.2817 0.42303 0.38898 0.27987 0.13820 0 By putting values, we get TWO-POINT GUASSIAN QUADRATURE FORMULA The formula for two point guassian quadrature formula We use two-point guassian quadrature formula for two point interval i.e., -1 to 1. We use this formula for evaluating integral without solving integration. EXAMPLE: Evaluate using two points guassian quadrature formula. SOLUTION: Given a=0, b=1
- 11. ERROR ANALYSIS IN NUMERICAL INTEGRATION Let be the integrand i.e., the function to be integrated within the limits say either or where such that stands for error in the exact value and the approximate value i.e., ERROR TERM IN RECTANGULAR RULE: The amount of error in rectangular rule is
- 12. ERROR TERM IN TRAPEZOIDAL RULE: The magnitude of error in trapezoidal rule is
- 13. ERROR TERM IN SIMPSON’S RULE Using Taylor’s series
- 14. The magnitude of error in simpson’s 1/3 rule is
- 15. SUMMARY OF ERROR ANALYSIS OF NUMERICAL INTEGRATION RULES Sr. # Rule Error 1 Rectangular Rule 2 Trapezoidal Rule 3 Simpson’s 1/3 Rule 4 Simpson’s 3/8 Rule 5 Boole’s Rule 6 Weddle’s Rule
- 16. FORMULAS 1. Rectangular Rule: 2. Trapezoidal Rule: 3. Simpson’s 1/3 Rule: 4. Simpson’s 3/8th Rule: 5. Boole’s Rule: 6. Weddle’s Rule: 7. Two Point Guassian Quadrature Formula
- 17. REFERENCES 1. Iqbal, D. (n.d.). An introduction to Numerical Analysis. Lahore: Ilmi Kitab Khana. 2. Kiran, E. (2015, 5 22). Slideshare. Retrieved from https://www.slideshare.net 3. N.Shah, P. (2021). AtoZ Maths. Retrieved from https://atozmath.com 4. Saleem, M. (n.d.). Numerical Analysis II. Muzammil Tanveer. 5. Wikipedia. (2021, January 12). Retrieved from https://en.wikipedia.org