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Mathematics1 Mathematics 1 Applied Informatics Štefan BEREŽNÝ

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5 th lecture

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MATHEMATICS 1 Applied Informatics 3 Štefan BEREŽNÝ Contents Numerical integration Rectangular Rule Trapezoidal Rule Simpson’s Rule:

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MATHEMATICS 1 Applied Informatics 4 Štefan BEREŽNÝ Numerical integration Definition: The partition D = x 0, x 1, …, x n of interval a, b is given by a = x 0 x 1 … x n-1 x n = b. We will get n sub-intervals x 0, x 1 , x 1, x 2 , x 2, x 3 , …, x n-2, x n-1 , x n-1, x n of the same length h, where x k = a + k h for k = 1, 2, 3, …, n and

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MATHEMATICS 1 Applied Informatics 5 Štefan BEREŽNÝ Numerical integration Notation: We shall denote y k = for k = 1, 2, 3, …, n.

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MATHEMATICS 1 Applied Informatics 6 Štefan BEREŽNÝ Rectangular Rule We approximate function f by a constant function on each sub-interval x k-1, x k . The constant function is uniquely determined by the one point. Then the considered constant function is: y = y k on interval x k 1, x k . Its integral I k in the interval x k 1, x k represents the area of the rectangle and so I k = h y k.

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MATHEMATICS 1 Applied Informatics 7 Štefan BEREŽNÝ Rectangular Rule If we add all the numbers I 1, I 2, …., I n, we obtain:

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MATHEMATICS 1 Applied Informatics 8 Štefan BEREŽNÝ Rectangular Rule S n is the approximate value of the Riemann integral of function f on the interval a, b . It is the sum of the areas of n rectangles constructed on the intervals x 0, x 1 , x 1, x 2 , x 2, x 3 , …, x n-2, x n-1 , x n-1, x n . The accuracy of the approximation should increase with increasing n.

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MATHEMATICS 1 Applied Informatics 9 Štefan BEREŽNÝ Trapezoidal Rule We shall denote y k = f ( x k ). We approximate function f by a linear function on each sub-interval x k-1, x k . The linear function is uniquely determined by the requirement that its graph (a straight line) passes through two chosen points. Let us choose the points x k 1, y k 1 and x k, y k . Then the considered linear function is: y = y k 1 + (( y k y k 1 ) ( x x k 1 ))/ h. Its integral I k in the interval x k 1, x k represents the area of the trapezoid and so I k = ( h ( y k 1, y k ))/2.

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MATHEMATICS 1 Applied Informatics 10 Štefan BEREŽNÝ Trapezoidal Rule If we add all the numbers I 1, I 2, …., I n, we obtain:

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MATHEMATICS 1 Applied Informatics 11 Štefan BEREŽNÝ Trapezoidal Rule S n is the approximate value of the Riemann integral of function f on the interval a, b . It is the sum of the areas of n trapezoids constructed on the intervals x 0, x 1 , x 1, x 2 , x 2, x 3 , …, x n-2, x n-1 , x n-1, x n . It can naturally be expected that the finer the partition of the interval a, b , the better is the approximation of the Riemann integral.

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MATHEMATICS 1 Applied Informatics 12 Štefan BEREŽNÝ Trapezoidal Rule In other words, the accuracy of the approximation should increase with increasing n and with decreasing h. This expectation is correct. It can be proved that if function f has a continuous second derivative f in interval a, b and M 2 is the maximum of f in a, b then the following error estimate holds:

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MATHEMATICS 1 Applied Informatics 13 Štefan BEREŽNÝ Simpson’s Rule Let us now choose an integer n that it is even and let us approximate function f by a quadratic polynomial on each of the sub-intervals x 0, x 2 , x 2, x 4 , x 4, x 6 , …, x n-4, x n-2 , x n-2, x n . The quadratic polynomial on the sub-interval x k 2, x k for k = 2, 4, 6, …, n is uniquely determined if we require that its graph (a parabola) passes trough three chosen points. Let the three chosen points be x k 2, y k 2 , x k 1, y k 1 and x k, y k .

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MATHEMATICS 1 Applied Informatics 14 Štefan BEREŽNÝ Simpson’s Rule The coefficients and the integral of such a quadratic polynomial in the interval x k 2, x k can be relatively easily evaluated - you can verify for yourself that the integral I k = h ( y k 2 + 4 y k 1 + y k )/3. Summing all the numbers I 2, I 4, …., I n, we obtain:

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MATHEMATICS 1 Applied Informatics 15 Štefan BEREŽNÝ Simpson’s Rule S n is the approximate value of the Riemann integral of function f on the interval a, b . It can be proved that if the fourth derivative f (4) of function f is continuous in a, b and M 4 is the maximum of f (4) in a, b then the following error estimate holds:

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MATHEMATICS 1 Applied Informatics 16 Štefan BEREŽNÝ Thank you for your attention.

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