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Math 128A - Spring 2000 - Brown - Final

Math 128A - Spring 2000 - Brown - Final - FRI 18:01 FAX...

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Unformatted text preview: 03/23/2001 FRI 18:01 FAX 6434330 MOFFITT LIBRARY 001 Math 128A — Final Exam Spring 2000 — Nate Brown 1) (IOpts) Assume M is a constant, N is a function of h and M = N_(h) + h2 + h3 + 5,4 + - - - _ Use Richardson’s~ extrapolation to find a function N and constants K3, K4, . . . such that M = N(h) +K3h3 +K4h4+v - - . (Hint: Replace h with 2h.) 2) (10pts) Use a divided difference table to find a cubic polynomial Whose graph passes through the points (~1,2), (0,1), (1,0) and (2, 5). 3) (20pts) U‘se Taylor’s theorem to prove that if f E C2[a, b]1 21:0 E [a, b] and h > 0 is such that me + h E [a, b] then there exists .5 E {$0, $0 + h] such that mm) = 1/h(f(a:a + h) — ma) 7 (f”(€)/2)h. 4) Let f(:r) = $2 + Let Qn be the Newton polynomial and H2n+l be the Hermite polynomial which interpolate f at distinct points $0, . . . , :5“ E [2, 3]. a) (lOpts) Find an n such that |f(a:) w Qn(m)| < 10‘6 for all m E [2, 3]. b) (lOpts) Find an n such that |f(z) — H2n+1(32)| < 10*6 for all m E [27 5) (20pts) Use the first Lagrange interpolating polynomial (at the endpoints) to prove the simple Trapezoidal rule. That is, prove that if f E (72 [(1, b] then there exists 5 E [0,, b] such that b —a —a f mumbg (f(a)+f(b))+(512)3f”(€)- (Use the identities ff %dm = f" gate = “Ta and f:{mia)(meb)dm = (1:52.) 130 6)LetA= 010. 203 a)(10pts) Find elementary matrices 241,142 such that AZAIA is upper trian- gular. _I. 1 _\ b)(10pts) Let b: 2 . Find a vector (2 and an upper triangular matrix 3 A such that if 5; is a solution to A a: c then E is also a solution to A E: b. 03/23/2001 FRI 18:02 FAX 6434330 MOFFITT LIBRARY 7) a)(10pts) If T E MAR), —C‘E IR” describe an iterative technique for solving .mx the fixed point type problem 5r. : T E + b)(10pts) Describe the Jacobi method for solving linear systems. 3 2 U c)(5pts) Will the Jacobi method converge if A = f} 2 1 ? 1 1 4 8)(10pts) Assume A E MAR) is invertible and we wish to solve the linear system A In general, the Gauss—Sgdel‘ method applied to this system will not converge. Find a linear system B a): c with the properties that ii) if a: is a solution to the equation B 5:? then 55 is also a solution to the equation A Biz; and ii) the Gauss-Seidel method will converge for the equation B 33:3. 9) Assume y(t) is the unique solution to the initlal value problem :11: 7 f(t,i), ogtgb, y(a):a. Leth=(b—a)/n,ti:a+ih(1Sign), mg :0: and Luz-+1 = 10.; + h.q5(ti,wi, h), 0 S i S n, i 1, for some function Q5. a)(5pts) Define the local truncation error. b)(5pts) Define consistency. c)(5pts) Define convergence. d)(20pts) If f(t,y) : tgeis ~ 37:33], prove that the modified Euler method will converge. (Recall that for this method wit]. : at; + it/2[f(ti,w1;) + fflti_l_1,w,.; + hf(t¢,‘1l}i)).) 10) a)(5pts) Let {ya} be a convergent sequence of numbers with limit y. Define what it means for {yn} to converge quadratically (Le. with order of convergence 2). b)(25pts) Assume f E lea,b], 79 E [(1, b], Hp) : 0 and _f’(p) gt 0. Prove that there exists 6 > 0 such that for any initial guess .170 E [p — 6, p+6], N ewton’s method will converge (at least) quadratically to 73. 002 ...
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