Today’s class Numerical Integration Newton-Cotes Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Discuss Mid-term Exam 1 Average = 74.5, Standard dev = 17.6 Histogram of scores Average = 74.5, Standard dev = 17.6 Max = 98 Min = 35 Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Discuss Mid-term Exam 1 1 (30) Use Taylor series to approximate at x=0.5 until t  1%, using a base point x=0. (Use 4 significant digits with rounding, and the true value is 0.4055)   Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Discuss Mid-term Exam 1 2 (30) Use the false-position method with initial guesses of 50 and 70 to determine x to a level of a  0.5%. (Use 4 significant digits with rounding)   Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Discuss Mid-term Exam 1 3 (20) Use the Newton-Raphson method with an initial guess of 6 to determine a root of the following function to a level of a  0.01% (use 4 significant digits with rounding)   Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Discuss Mid-term Exam 1 4 (20) Use the Gauss-Seidel method to solve the following linear equation system to a tolerance of s = 5% with a starting point (0, 0, 0).(Use 4 significant digits with rounding) Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Numerical Integration Numerical technique to solve definite integrals Find the area under the curve f(x) from a to b Another way to put it is that we need to solve the following differential equation Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Numerical Integration Where do we use numerical techniques? When you can’t integrate directly When you have a sampling of points representing f(x) as with the experiments results Graphical techniques Grid approximation Strip approximation Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Graphical Techniques Grid approximation Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Graphical Techniques Strip approximation Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Numerical Integration Numerical techniques Treat integral as a summation Basic idea is to convert a continuous function into discrete function The smaller the interval, the more accurate the solution but also at more computational expense Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Numerical Integration Newton-Cotes formulas Approximate function with a series of polynomials that are easy to integrate Zero-order approximation is equivalent to strip approximation First order approximation is equivalent to trapezoidal approximation Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Newton-Cotes Formulas Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Newton-Cotes Formulas Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12 CSE, UConn

Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12 CSE, UConn

Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12 CSE, UConn

Trapezoidal Rule Evaluate integral Analytical solution Trapezoidal solution Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12 CSE, UConn

Trapezoidal Rule Trapezoidal solution To increase accuracy, shorten up the interval Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12 CSE, UConn

Trapezoidal Rule If you split the interval into n sub-intervals, each sub-interval has a width of h = (b – a)/n Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Trapezoidal Rule More accurate as you increase n Double n and reduce the error by a factor of four However, if n is too large, you will start encountering round-off error and the integral solution can diverge Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Simpson’s Rules Second and third-order polynomial forms of the Newton-Cotes formulas Simpson’s 1/3 Rule Use three points on the curve to form a second order polynomial Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Simpson’s Rules Approximate the function by a parabola Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Simpson’s Rules Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Simpson’s Rules Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Simpson’s 1/3 Rule Error Multiple Application n must be even Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Simpson’s 3/8-Rule Approximate with a cubic polynomial Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Simpson’s 3/8-Rule Numerical Methods Prof. Jinbo Bi Lecture 12 CSE, UConn

Simpson’s 3/8-Rule Error Multiple Application n must be a multiple of 3 Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Simpson’s 3/8-Rule Both the cubic and quadratic approximations are of the same order The cubic approximation is slightly more accurate then the quadratic approximation Usually not worth the extra work required Only use the 3/8 rule if you need odd number of segments Can combine with 1/3 rule on some segments Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn 32

Example Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Integration with unequal segments We have until now assumed that each segment is equal If we are integrating based on experimental or tabular data, this assumption may not be true Newton-Cotes formulas can easily be adapted to accommodate unequal segments Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Trapezoid Rule with Unequal Segments Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12 CSE, UConn

Simpson’s Rule with Unequal Segments Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn

Next class Numerical Integration HW5 due Tuesday Oct 21 Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn