Go over WEB pages Lecture 1 through 8. Work the homework 1 and homework 2. Read the pages in the textbook that are on the syllabus. (Textbook not required this semester.) Open book, open notes, exam. No computers, some do not have laptops. No study guides or copies of study guides. Multiple choice and short answer. Read the instructions. Follow the instructions, else lose points. Read the question carefully. Answer the question that is asked, not the question you want to answer. No programming required. Some code may appear as part of a question. One hour time limit. Lectures follow exam. Example questions and answers will be presented in class. Some things you should know for the exam: IEEE float has 6 or 7 decimal digits precision IEEE double has 15 or 16 decimal digits precision Adding or subtracting numbers of large differences in magnitude causes precision lose. RMS error is root mean square error, a reasonable intuitive measure Maximum error may be the most important measure for some applications Average error is not very useful The machine "epsilon" for a specific floating point arithmetic is the smallest positive number added to exactly 1.0 the results in a sum that is not exactly 1.0 . Be careful reading programs that use "epsilon" or "eps". Sometimes it may be the "machine epsilon" and other times it may be a tolerance on how close the something should be. Most languages include the elementary functions sin and cos, yet many do not include a complete set of reciprocal functions such as cosecant or inverse or hyperbolic functions such as inverse hyperbolic cotangent. If given only a natural logarithm function, log2(x) and be computed as log(x)/log(2.0) and log10(x) as log(x)/log(10.0) Homework 1 used a simple approximation for integration: position s = integral for time=0 to time=t velocity(t) dt as s = sum i=1 to n s_i = s_i-1 + dv_i-1 * dt where n*dt=t In order to guarantee a mathematical unique solution to a set of linear simultaneous equations, two requirements are needed: There must be the same number of equations as unknown variables and the equations must be linearly independent. For two equations to be linearly independent, there must not be two constants p and q such that equation1 * p + q = equation2 The numerical solution of liner simultaneous equations can fail even though the mathematical condition for a unique solution exists. The Gauss Jordan method of solving simultaneous equations A x = y produces the solution x by performing operations of A y, reducing A to the identity matrix such that y is replaced by x at the end of the computation Improved numerical accuracy is obtained in the solution of linear systems of equations by interchanging rows such that the largest magnitude diagonal is used as the pivot element. Some very large systems of equations can be solved accurately. The system of linear equations is solved by the same method when the equations have complex values. It is better to directly solve simultaneous equations rather than invert the "A" matrix and multiply by the "y" vector to find "x". There are small matrices that are very hard to invert accurately. Given data, a least square fit of the data minimizes the RMS error for a given degree polynomial at the data points. Between the data points there may be large variations. When trying to fit large degree polynomials, there may be numerical errors such that the approximation becomes worse. A polynomial of degree n, will exactly fit n+1 points. It may have extreme humps and valleys between the points. Mathematically, a least square fit of n data points should be exactly fit by a n-1 degree polynomial. The numerical computation may not give this result. A least square fit may use terms: powers, sine, cosine or any other smooth function of the data. The basic requirement is that all functions must be linearly independent of each other. A least square fit uses a solution of simultaneous equations Least square fit is the easiest method of fitting data that is given with unequal spacing. A polynomial of degree n has n+1 coefficients. Thus n+1 data points can be exactly fit by an n degree polynomial. A polynomial of degree n has exactly n roots (possibly with multiplicity) Given roots r1, r2, ... rn a polynomial with these roots is created by (x-r1)*(x-r2)* ... *(x-rn) Horner's method of evaluating polynomials provides accuracy and efficiency by never directly computing high powers of the variable. Given the numerical coefficients of a polynomial, the numerical coefficients of the integral, derivative are easily computed. Given the numerical coefficients of two polynomials, the sum, difference, product and ratio are easily computed. Any functions that can be continuously differentiated can be approximated by a Taylor series expansion. It is called "truncation error" when evaluating a Taylor series with a specific number of terms. Orthogonal polynomials are used to fit data and perform numerical integration. Examples of orthogonal polynomials include: Legendre, Chebyshev, Laguerre, Lagrange, Fourier. Chebyshev polynomials are used to approximate smooth functions while minimizing the maximum error. Legendre polynomials are used to approximate smooth functions while minimizing RMS error. Lagrange polynomials are used to approximate smooth functions while exactly fitting a specific set of points. For non smooth functions, including square waves and pulses, a Fourier series approximation may be used. The Fejer approximation can be used to eliminate many oscillations in the Fourier approximation at the expense of a less accurate fit. Numerical integration is typically called numerical quadrature. The Trapezoidal integration method requires a small step size and many function evaluations to get accuracy. (My step size is uniform and the method is easy to code) In general a smaller the step size, usually called "h", will result in a more accurate value for the integral. This implies that more evaluations of the function to be integrated are needed. The Gauss Legendre integration of smooth functions provides good accuracy with a minimum number of function evaluations. (The weights and ordinates are needed for the summation) An adaptive quadrature integration method is needed to get reasonable accuracy of functions with large derivatives. Adaptive quadrature uses a variable step size and at least two methods of integration to determine when the desired accuracy is achieved. This method, as with all integration methods, can fail to give accurate results. Two dimensional and higher dimensions can use simple extensions of one dimensional numerical quadrature methods. The difference between a Taylor Series and a Maclaurin Series is that the Maclaurin Series always expands a function about zero. The two equations for Standard Deviation are mathematically equivalent, yet may give different numerical results. sigma = sqrt((sum(x^2)- sum(x)^2/n)/n) = sqrt(sum((x-mean)^2)/n) For n samples of x with a mean of mean. RMS error is computed from a set of n error measurements using rms_error = sqrt(sum(err^2)/n) Note that this is the same value as Standard Deviation when the mean value of the errors is zero.