1. MA2213 Lecture 12 REVIEW

2. 1.1 on page 10 11. Compute Compute quadratic Taylor polynomials for 12. Compute where g is the function in problems 11 and 12.

3. 1.2 on page 18 1. Bound the error in using on the interval to approximate Use the remainder on page 11 and taketo be the third degree Taylor polynomial about Suggestion Study ‘bounding the error’ on page 15 Review power series and formuli on page 17

4. 1.2 on page 19 14. (a) Obtain the Taylor polynomial for about obtain a Taylor series with remainder for 14. (b) Use the method developed on page 14 to Suggestion: Use part (a) and

5. 2.2 on page 54 Review the elementary concepts in Chapter 2 about errors, particularly the sources of error on pages 45-47, and loss-of-significance errors (and how to reduce such losses) on pages 47-50 5. (a) 5. (d) 6. (c)

6. 3.2 on page 88 Review Newton’s method for finding roots, in particular Example 3.2.2 on pages 81-83 that explains the importance of choosing a sufficiently ‘close’ initial estimate, and the error analysis on pages 83-86 4. a positive integer Read about order of convergence on page 101 : A sequenceconverges towith order of convergenceif there exists a constant such that linear, quadratic, cubic convergence.

7. 3.4 on page 96 4. Experimentally confirm the error estimate where andis a sequence that converges toobtained by the secant method. Question What is the order of convergence of the secant method for this problem ? is the unique positive root of Question What is the order of convergence of Newton’s method, the secant method, for finding the root of the function?

8. 4.1 on pages 134-135 16. As a generalized interpolation problem, find the quadratic polynomial for which Compute coefficientssuch that the function interpolatesat 21. Letbe a polynomial of degree Fordefine Show thatis a polynomial of degree

9. Best Approximation pages 159-165 A. Prove from first principles that if satisfiesand then the best linear approximation toon is the functionwhere Suggestion: draw pictures. B. Prove from first principles that if is convex () andis a linear function that interpolatesat thensatisfies the hypothesis in Prob. A. C. Use A &B to find the best lin. app. toon

10. Least Squares Approximation pages 178-187 A. Compute coefficients approximation of the function such that is the least squares B. Find the linear least squares approximation on the interval [0,1] (see Prob. 1 p. 185)to over the interval and on the interval [-1,1].

11. Trapezoid and Simpson Rules p. 189-201 1. Computeandfor the integral over the interval Read Richardson Extrapolation p. 210-211 2. Do problem 2. (a) on page 200. 17. (page 218) Use Richardson’s extrapolation to estimate the errors in Problem 2(a) on page 200.

12. Gaussian Integration p. 219-229 Determine the values ofsuch that is valid whenever 2. What conditions on 1. Assume that and ensure the equality above for 4. Do problem 1 on page 229. 3. Use these conditions to compute the nodes and weights for n = 1,2,3,4 for a=-1, b = 1 and THEN compare with Table 5.7, p. 223.

13. Matrix Arithmetic p. 248-264 statements is (i) always true, (ii) always false, (iii) sometimes true, sometimes false ? 1. LetWhich of the following a. b. c. d. e. The equationhas 17 solutions. f.

14. Matrix Arithmetic p. 248-264 Which statements below are1. Let logically equivalent ? a. b. c. d. All eigenvalues of is positive definite. has real eigenvalues. are nonzero. e. The equationhas at most 1 solution. f.

15. Matrix Arithmetic p. 248-264 1. (Prob. 21, p, 262) Compute the inverse of the n x n matrix

16. Matrix Arithmetic p. 248-264 1. (Prob. 28, p. 263) Find 2. (Prob. 29, p. 264) such that for whichFind the values of

17. Gaussian Elimination p. 264-283 1. What is the augmented matrix for the following system of linear equations ? 2. What are the elementary row operations ? 3. Performing elementary row operations on an augmented matrix for a system of linear equations corresponds to doing what to the system of equations ?

18. Gaussian Elimination p. 264-283 Augmented Matrix Elementary row operations and correspondences exchange two rows  exchange two equations mult. row by  mult. eqn. by row(j)  row(j) + row(i)  eqn(j)  row(j) + eqn(i) where

19. Gaussian Elimination p. 264-283 1.Use elementary row operations on both the system of equations on slide 17 and on its aug mat so that the resulting matrix of coefficients and augmented matrix is upper triangular. 2. How do elementary row operations on a set of equations change its set of solutions ? 3. How should a system of equations be solved if its matrix of coefficients is upper triangular ? 4. How can the matrix of coefficients of a system of linear equations be found from its aug mat ? 5. Solve the equations on slide 17 using ERO on both the equations and on its aug mat.

20. Gaussian Elimination p. 264-283 1.Use elementary row operations on both the system of equations on slide 17 and on its aug mat so that the resulting matrix of coefficients and augmented matrix is upper triangular. 2. How do elementary row operations on a set of equations change its set of solutions ? 3. How should a system of equations be solved if its matrix of coefficients is upper triangular ? 4. How can the matrix of coefficients of a system of linear equations be found from its aug mat ? 5. Solve the equations on slide 17 using ERO on both the equations and on its aug mat.

21. Direct Solutions p. 281, 298-300 1.Do problem 1(a) on page 281. 2. Read pages 270-272 about partial pivoting then try to do Example 6.3.3 on page 272 both with and without partial pivoting and then compare your answers that are worked out in the textbook on pages 272 and 273. 3. Read pages 273-276 about the computation of inverses then do problem 6(a) on page 282. 4. Read pages 298-300 and do problems 2 and 5 on page 302.

22. Iterative Methods p. 303-318 1.Review the Jacobi and Gauss-Seidel methods on then study the general schema on p 306-308. this splitting of A give the iteration 2.What is the iteration matrix above ? 4.How does the error behave ? a diagrammatic presentation; broadly : a structured framework or plan : outlineoutline 3.What is the splitting for Jacobi and GS ? 5.When does the general iteration converge ? Do prob. 10, 12 on p. 317. Also study the residual correction method on p. 296-297 and 303-318

23. Residual Correction p. 296-297, 303-318 1.Review the Jacobi and Gauss-Seidel methods on then study the general schema on p 306-308. this splitting of A give the iteration 2.What is the iteration matrix above ? 4.How does the error behave ? a diagrammatic presentation; broadly : a structured framework or plan : outlineoutline 3.What is the splitting for Jacobi and GS ? 5.When does the general iteration converge ?

24. The Eigenvalue Problem p. 333-352 statements is (i) always true, (ii) always false, (iii) sometimes true, sometimes false ? 1. LetWhich of the following a. b.and ifis an eigenvalue ofis an eigenvalue ofthenis an eigenvalue of ifis an eigenvector of thenis an eigenvector of c.ifandis an eigenvector of then

25. Nonlinear Systems p. 352-365 1. Read p. 352-365 and do prob. 1,2(a),(b) p.364 of a point that minimizes the sum of squared distances to the points 2. Compute coordinates 3. Compute coordinatesof a point that minimizes the sum of distances to the points 4. Compute exact coordinatesof all points that satisfy: