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## Chapters 5 and 6: Numerical Integration

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**Definition of Integral by**Upper and Lower Sums**Application to monotone**decreasing function**As part of assignment #2 (due 2/4/14), derive the**integration and error formulas for upper and lower sums applied to a monotone increasing function.**Derivation of trapezoid rule for any f(x)**Approach: linear approximation Result obtained using upper and lower sums**Use Taylor formula to derive an expression for the**expected error in the trapezoid rule for any function with a continuous second derivative**Single subinterval of size h**Fundamental theorem of calculus**n subintervals**n+1 points Mean value theorem**Assignment 2, Due 2/4/14**• Use the relationship • Integral = (upper sum + lower sum)/2 • to derive a numerical integration formula for a monotone increasing function. • 2. Use the relationship • Error = (upper sum – lower sum)/2 • to derive an error approximation for the formula derived in (1).**Assignment 2, Due 2/4/14 continued**3. Use the error formulas derived in class for the composite trapezoid rule applied to the integral to obtain 2 estimates of the number of sub-intervals needed to have an error less than 10-4 4. Write a MabLab function for the composite trapezoid rule with function handle and number of subintervals as parameters. Use this code to discover which of the error estimates in (3) is more realistic.**Problems from text on application of error formulas**5.1-6 p188 5.2-2 p 200 5.2-4 p200 5.2-7 p200 5.2-12 p210 6.1-2a p227**Problem 5.1-6 text p 188**If the integral is evaluated by upper and lower sums, estimate the number of points needed for accuracy 0.5x10-4**Problem 5.2-2 text p 200**Estimate the error if the integral is evaluated by the composite trapezoid rule with 3 points |error| < (b-a)3|f’’(x)|max /12n2 Plot |f’’(x)| to estimate maximum value on [0, 1]**|f’’(x)|**X**Problem 5.2-2 text p 200**Estimate the error if the integral is evaluated by the composite trapezoid rule with 3 points |error| < (b-a)3|f’’(x)|max /12n2 = (1)(2)/((12)(9)) = 1/54 Correct answer is 1/24. What is wrong?**For given number of points, accuracy of numerical**integration depends on where the integrand is evaluated Accuracy of numerical integration also depends on the variable of integration Consider approximated by the trapezoid rule with (a) equally spaced points in the interval 1<x<5 (b) logarithmically spaced points in the interval 1<x<5 (c) equally spaced points with integration variable y = ln(x) For (b) we need a trap rule that accepts arbitrary spacing**MatLab function: numerical integration with arbitrary**spacing**Use the transformation y = ln(x) to change the variable of**integration. dy = dx/x x(y)= exp(y) Integrand is a function of y through x(y)**3 trapezoid rule approximations to**x(y)= exp(y)**% error**Number of points**Assignment 3, Due 2/11/14**On page 242 of text, the value of is given to high precision, which can be taken as the “exact” value. Defined relative error as re = 100|(TR-exact)/exact|, where TR denotes the trapezoid-rule approximation. Write a MATLAB code to compare the relative error with 10 points when the points are chosen in the following ways: 1. Equally space on [1, 3] 2. Log-spaced on [1, 3] 3. Equally space on [0, ln(3)] in the new integration variable y = ln(x).**Most numerical integration formulas have the form**Example: Trapezoid rule w0 = wn = 1/2n wk = 1/n if 1< k<n-1 n = number of intervals One approach to deriving such formulas: Replace the integrand by the Lagrange interpolation formula**Another example**Why is this different from Simpson’s rule?**Which is likely to**be more accurate?**Problems from text p240 on deriving integration formulas**6.2-5 6.2-6 6.2-9**Example 6.2-6 text p240**Find A and B by requiring formula to be exact for f(x) = 1 and f(x) = x**Problem 6.1-4 text p228**Evaluated by composite Simpson’s rule with h=0.25 Bound the error |error| = (b-a)h4|f(4)(x)|/180 a <x< b (text p221) h = width subintervals**Let {x0, x1, …, xn} be n+1 points on [a, b] where**integrand evaluated. Will be exact for polynomials of degree n Trapezoid rule: (n = 1) exact for a line In general, error proportional to |f’’(z)| Simpson’s rule: (n = 2) exact for a parabola In general, error proportional to |f(4)(z)| How could this happen?**Gauss quadrature extents Simpson’s rule idea to its**logical extreme: Give up all freedom about where the integrand will be evaluated to maximize the degree of polynomial for which formula is exact. nodes {x0, x1, …, xn} exist on [a, b] such that is exact for polynomials of degree at most 2n+1 Nodes are zeros of polynomial g(x) of degree n+1 defined by 0 < k < n**Summary of Gauss Quadrature Theorem**For any set of n+1 nodes on [a,b], is exact for polynomials up to degree n For the special set of node in Gauss quadrature, the formula is correct of polynomials of degree at most 2n+1 Nodes and weights, Ai, depend on both [a,b] and n Scaling property of integrals lessens the impact of these dependences**Degree of g(x) = number of nodes**Number of conditions on g(x) = number of nodes Number of conditions not Sufficient to determine all coefficients Terms with odd powers of x in g(x) do not contribute due to symmetry**Determines ratio of C1 and C3 only.**As noted before, number of conditions on g(x) insufficient to determine all 4 coefficients of a cubic**Nodes and weights determined**for [-1,1] can be used for [a,b] Given values of y from table, this x(y) determines values on [a,b] where integrand will be evaluated**From text p236**Note: n = number of intervals Example: note symmetry of points and weights**(no anti-derivative)**with 2 nodes