Introduction to Numerical Math and Matlab ... - Ohio University

More documents

Recommendations

Info

42 LAB 11. ACCURACY, CONDITION NUMBERS AND PIVOTING 11.2 Try > B = [sin(sym(1)) sin(sym(2)); sin(sym(3)) sin(sym(4))] > c = [1; 2] > x = B\c > pretty(x) > double(x) Next input the matrix: Cs = [ 1 2 2 4 symbolically as above. Create a numerical version via Cn = double(Cs) and define the two vectors d1 = [4; 8] and d2 = [1; 1]. Solve the systems Cs*x = d1, Cn*x = d1, Cs*x = d2, and Cn*x = d2. Explain the results. Does the symbolic or numeric way give more information? ]
Lab 12 Eigenvalues and Eigenvectors Suppose that A is a square (n × n) matrix. We say that a nonzero vector v is an eigenvector (ev) and a number λ is its eigenvalue (ew) if Av = λv. (12.1) Geometrically this means that Av is in the same direction as v, since multiplying a vector by a number changes its length, but not its direction. Matlab has a built-in routine for finding eigenvalues and eigenvectors: > A = pascal(4) > [V E] = eig(A) The results are a matrix V that contains eigenvectors as columns and a diagonal matrix E that contains eigenvalues on the diagonal. We can check this by: > v1 = V(:,1) > A*v1 > E(1,1)*v1 Finding Eigenvalues for 2 × 2 and 3 × 3 If A is 2×2 or 3×3 then we can find its eigenvalues and eigenvectors by hand. Notice that Equation (12.1) can be rewritten as: Av − λv = 0. (12.2) It would be nice to factor out the v from the right-hand side of this equation, but we can’t because A is a matrix and λ is a number. However, since Iv = v, we can do the following: Av − λv = Av − λIv = (A − λI)v = 0 (12.3) If v is nonzero, then by Theorem 3 the matrix (A − λI) must be singular. By the same theorem, we must have: This is called the characteristic equation. det(A − λI) = 0. (12.4) 43
Page 1 and 2: Introduction to Numerical Math and
Page 3 and 4: CONTENTS iii Lab 17. Integration: L
Page 5 and 6: Part I Matlab and Solving Equations
Page 7 and 8: 3 > plot(x,y,’*’) > plot(x,y,
Page 9 and 10: Lab 2 Matlab Programs - Input/Outpu
Page 11 and 12: 7 % mygraphs % plots the graphs of
Page 13 and 14: 9 The simplest way to accomplish th
Page 15 and 16: Lab 4 Controlling Error and while L
Page 17 and 18: 13 Exercises 4.1 In Calculus we lea
Page 19 and 20: 15 method and returns not only the
Page 21 and 22: 17 function x = mysecant(f,x0,x1,n)
Page 23 and 24: Lab 7 Symbolic Computations The foc
Page 25 and 26: 21 Other useful symbolic operations
Page 27 and 28: 23 Secant methods are better for ex
Page 29 and 30: Math 344 Sample Test Problems from
Page 31 and 32: Part II Linear Algebra c○Copyrigh
Page 33 and 34: 29 > A + A You can even add a numbe
Page 35 and 36: 31 Exercises 8.1 By hand, calculate
Page 37 and 38: 33 Figure 9.1: An equilateral truss
Page 39 and 40: 35 > b = [4 15 -10]’ > x = A\b Ne
Page 41 and 42: 37 On the other hand, a matrix that
Page 43 and 44: Lab 11 Accuracy, Condition Numbers
Page 45: 41 sensitivity is measured by the c
Page 49 and 50: 45 Larger Matrices For a n × n mat
Page 51 and 52: 47 Likely States of a Markov Matrix
Page 53 and 54: Review of Part II Methods and Formu
Page 55 and 56: Part III Functions and Data c○Cop
Page 57 and 58: 53 When we tried a lower degree pol
Page 59 and 60: Lab 15 Spline Interpolation A chall
Page 61 and 62: Lab 16 Least Squares Interpolation:
Page 63 and 64: 59 Note that whenever you select a
Page 65 and 66: 61 ✻ ✟ ✟ ✟✟✟✟✟✟
Page 67 and 68: 63 efficient, first use x = linspac
Page 69 and 70: 65 ✻ ✟ ✉ ✟ ✟✟✟✟✟
Page 71 and 72: 67 In Matlab there is a built-in co
Page 73 and 74: 69 For another example of how meshg
Page 75 and 76: Review of Part III Methods and Form
Page 77 and 78: Part IV Appendices c○Copyright, T
Page 79 and 80: Appendix B Glossary of Matlab Comma
Page 81 and 82: 77 limit Finds two-sided limit, if
Page 83 and 84: 79 > 2*A ..........................
Page 85: Bibliography [1] Ayyup and McCuen,

Introduction to Numerical Math and Matlab ... - Ohio University

Create successful ePaper yourself

Delete template?

Save as template?