ࡱ> &(%M ubjbj== &8WWu l    4F DD p       HJJJb<<$$  `H     H   ]    H  H  \\ pK  \\s0\g g \ EGR 511 NUMERICAL METHODS __________________ LAST NAME, FIRST PROBLEM SET #3 1. (P. 14.5 Chapra) Find the minimum value of f(x, y) = (x ( 2)2 + (y ( 3)2 starting at x = 1 and y = 1, using one iteration of the steepest descent method. Ans: x = 2 and y = 3 ( f(x, y) = 0 2. (P. 14.6 Chapra) Perform one iteration of the steepest ascent method to locate the maximum of f(x, y) = 3.5x + 2y + x2 ( x4 ( 2xy ( y2 using initial guesses x = 0 and y = 0. Ans: x = 0.9766 and y = 0.5581 3. Let f(x) =  EMBED Equation.3 and P2(x) be the interpolation polynomial on x0 = 0, x1 and x2 = 1. Find the largest value of x1 in (0,1) for which f(0.5) P2(0.5) = -0.25. Ans: x1 = 0.872675 4. Let P3(x) be the interpolation polynomial for the data (0,0), (0.5, y), (1,3), and (2,2). Find y if the coefficient of x3 in P3(x) is 6. Ans: y = 4.25 5. Show that the following sequences {pn} converge linearly to p = 0. How large must n be before |pn p|  EMBED Equation.3  5(10-2? a) pn =  EMBED Equation.3 , n ( 1 b) pn =  EMBED Equation.3 , n ( 1 Ans: a) n = 20 and b) n = 5 6. For the given functions f(x), let x0 = 0, x1 = 0.6, and x2 = 0.9. Construct the Lagrange interpolating polynomials of degree 1 and 2 to approximate f(0.45) and find the error bound and the actual error. a) f(x) = cos x b) f(x) =  EMBED Equation.3  c) f(x) = tan x Ans: f(x)cos x EMBED Equation.3 tan xP1(x=0.45) error bound actual error P2(x=0.45) error bound actual error 0.86900 0.03375 3.1445(10-2 0.89810 5.0625(10-3 2.357(10-3 1.19868 4.65(10-2 5.47616(10-2 1.20343 6.98(10-3 7.3573(10-40.5131 0.220 0.03005 0.454614 0.151 0.02844 7. Construct a divided-difference table from: x0.5-0.20.70.10.0f(x)-1.15180.7028-1.4845-0.149430.13534 Use the divided-difference table to estimate f(0.15), using a) a polynomial of degree 2 through the first three points. b) a polynomial of degree 2 through the last three points. c) a polynomial of degree 3 through the first four points. d) a polynomial of degree 3 through the first four points. e) a polynomial of degree 4. Which three points are best to use for constructing the quadratic if we want f) f(0.15)? g) f(-0.1)? h) f(1.2)? Ans: a) ( 0.35871 b) ( 0.28514 c) ( 0.28941 d) ( 0.28938 e) ( 0.28939 f) f(0.15): use 0, 0.1, 0.5 g) f(-0.1): use ( 0.2, 0, 0.1 h) f(1.2): use 0.1, 0.5, 0.7 8. (P. 14.10 Chapra) Develop a one-dimensional equation in the pressure gradient direction at the point (4, 2). 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