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UCSD | MATH 270
17 sample documents related to MATH 270
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UCSD MATH 270
Math 270C: Numerical Mathematics Spring quarter, 2007 Homework Assignment 3 Due Tuesday, May 1 1. Let 1 dn n x2 - 1 , n = 0, . n n! dxn 2 (a) Show by the definition that Pn Pn for each n. (b) Show by Rolle\'s Theorem that, for n 1, each Pn has n -
UCSD MATH 270
3. Approximation and Numerical ODE In this part, we assume that a, b R with a < b. We also denote by Pn the set of all polynomials of degree n for any integer n 0. Question 3.1. (a) Prove for any f C[a, b] that n qn Pn axb b n qn Pn lim inf ma -
UCSD MATH 270
Math 270C: Numerical Mathematics Spring quarter, 2007 Homework Assignment 1 Due Tuesday, April 10 1. Let Bn f Pn (n = 0, 1, ) be the Bernstein polynomials of f C[0, 1]. (a) Let f0 (x) = 1, f1 (x) = 1, and f2 (x) = x2 . Show that n-1 2 1 x + x, -
UCSD MATH 270
Math 270C: Numerical Mathematics Spring quarter, 2007 Homework Assignment 6 Due Tuesday, May 29 1. Suppose a sequence of nonnegative numbers {ei }n satisfy i=0 for some constants > 0 and B 0. Show that en 1 ei ei e0 + B, i = 0, , n. 2. Consi -
UCSD MATH 270
Math 270C: Numerical Mathematics Spring quarter, 2007 Homework Assignment 5 Due Thursday, May 17 1. Consider the trapezoidal formula 1 f (x) dx (b - a) [f (a) + f (b)] . 2 a (a) Show that the degree of precision of the formula is m = 1. (b) Calcula -
UCSD MATH 270
Math 270C: Numerical Mathematics Spring quarter, 2007 Homework Assignment 2 Due Tuesday, April 17 1. Prove the following properties of the Chebyshev Polynomials of first kind Tn (x) = cos(n arccos x), (a) Expansion. 1 x+ 2 (b) Recursion formula. Tn -
UCSD MATH 270
First Homework Solutions Question 1: a) First column of divided dierence table (xs): (1, 1, 4, 4)T . Second column (f (x)s): (1, 1, 2, 2)T . Third column (rst divided dierences): ( 1 , 1 , 1 )T . Fourth column (second 2 3 4 1 1 divided dierences): ( -
UCSD MATH 270
Math 270C: Numerical Mathematics Spring quarter, 2007 Homework Assignment 4 Due Thursday, May 10 1. Let x0 , , xn be n + 1 distinct points in [a, b] and Ln : C[a, b] Pn the corresponding Lagrange interpolation operator. Show that Ln f where n = -
UCSD MATH 270
Third Homework Question 1: a) Find the natural cubic spline interpolant for the data points (0, 1), (1, 2), (2, 1). b) What additional information on the rst derivative needs to be given so that the piecewise cubic Hermite polynomial interpolating th -
UCSD MATH 270
Math 270C: Numerical Mathematics (Part C) LECTURE NOTES Bo Li Department of Mathematics University of California, San Diego June 11, 2007 Warning! While being expanded with the addition of new material and being carefully polished continuously, th -
UCSD MATH 270
Final Exam Question 1: Let f (x) be a smooth function and h > 0 a constant. a) Let P (x) be the cubic Hermite interpolatory polynomial satisfying P (0) = f (0), P (h) = f (h), P (0) = f (0), P (h) = f (h). Construct P (x) and show P h 2 1 h = [f (0) -
UCSD MATH 270
Homework #1 Question 1: Let f (x) = ex and consider the nodes x0 = 1, x1 = 1, x2 = 2, x3 = 3. a) Use the Lagrange formula to write down the form of the interpolatory polynomial. b) Use Newton divided dierences to write down the form of the interpolat -
UCSD MATH 270
h V H }U oIIs q8 gn @P Bp d G Gx G G x G x Wj6 ~i j6@ii w 6 G(jdgjq@ G x G x G w G x x G hV d jtW4j u i6&~G H 4U o q8 gn P Bp 4` -
UCSD MATH 270
Fourth Homework Question 1: Approximate 1 x 0 xe dx and compare to the exact solution to get the error using the method: a) Trapezoidal Rule. b) Composite Trapezoidal Rule, h = 0.25. c) Simpson\'s Rule. d) Composite Simpson\'s Rule, h = 0.25. e) Midp -
UCSD MATH 270
Fifth(a) Homework Question 1: Gaussian quadrature also works on weighted integrals, with given weight w(x), b w(x)f (x) dx. a In this case, for a given number of nodes n + 1, the formula n aj f (xj ) j=0 chooses node locations xj , j = 0, . . . , -
UCSD MATH 270
Fifth(b) Homework Question 6: a) For nodes x0 , x1 , x2 with equal stepsize h, construct the best possible formula, i.e., error is O(hq ) with q maximized, that is a linear combination of measurements f (x0 ), f (x1 ), f (x2 ) for approximating the s -
UCSD MATH 270
Math 270C: Numerical Mathematics (Part C) LECTURE NOTES Bo Li Department of Mathematics University of California, San Diego June 10, 2009 Warning! While being expanded with the addition of new material and being carefully polished continuously, th
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