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Unformatted text preview: 04/22/2002 MON 15:38 FAX 6434330 MOFFITT LIBRARY Math 128a, Section 3 — Final Exam — December 17, 2001 Prop Siren}? Problem 1 Let
332/4 + 312/9 — 1 f(w,y)= x_y_1 (a) Deﬁne quadratic convergence of a sequence of vectors :17“ to a limit 33. (b) Compute the Jacobian matrix D f (3:, y). (c) Determine where D f (m,y) is invertible and compute D f (3:, y)‘1 when it exists. ((1) Write down Newton’s .
method for solving f (m,y) = 0. (e) Start with 550 : (2,0)T and compute the first two approximations x1 and 3:2 generated by Newton’s method. (f) Explain why your results demonstrate quadratic convergence. Problem 2 Consider the iteration mg + 3am“ $ 1:: .
71+ 3:13?l + a (a) What is it intended to compute? (b) Given a 2 and 3:0 1,
compute 3:1 and 2:2. (c) Deﬁne and determine the order of convergence of
this iteration. Problem 3 Find the QR factorization of A— 3g —QQoR—3PLP 1 PR
— ""‘ 123 _5l\/§2‘/m3' You don’t need to multiply together the matrices Pi. Problem 4 (a) Derive a numerical integration formula [01mm = were) + w1f(1)+ w2f(2) which is exact for polynomials of as high degree d as possible, and determine
the maximal degree d. (b) Without any additional work, determine an
equally accurate rule of the form fol f(x)da: = u0f(1)+ u1f(0) + u2f(1).
(c) Show that
f sedan = hearse) + wrnh) + WNW!» + 0W1.)
as h e 0. (d) Use (a), (b), and (c) to build a quadrature formula with error 0(hd) on an arbitrary interval [03, (7] divided into n >_ 1 subintervals of length
h = (b — a) 001 04/22/2002 MON__15:38 FAX 6434330 MOFFITT LIBRARY 002 Problem 5 (a) Write down the Newton and Lagrange forms of the quadratic
interpolant p(:r) to a function f at three points a, b and c. (b) Give a formula
for the error p(:c) —— f (3:) if f is a nice function with all derivatives bounded.
Explain why your error formula makes sense in terms of dimensions, zeroes and the derivatives which appear versus the degree of polynomial used. (c)
Specialize to f (as) = R * 1/w and evaluate the coeﬂicients in the Newton representation of p(3:). (d) Use (c) to express p in the power form p($) :
q" + qla: + (123:2. (a) How would you use the formula of (d) to derive an iterative method for ﬁnding I/R? Problem 6 Suppose A is a square invertible matrix. (a) Deﬁne the condi—
tion number (b) Suppose E is a matrix the same size as A and HE” 1 a A ——— < e < —. ( )IIAII   2 Show that A + E is invertible. (c) Show that
“(A + E)‘1  All“ < 26.
"fl—1! _ Extra Credit Problem 7 Given an approximate solution 3; to the linear
system AI = b with a square invertible matrix A, let 7* 2 b — Ay be the
residual of y. (a) Show that if y satisﬁes a perturbed linear system (A +
E)y = b then the perturbation E must satisfy m > uru
IIAH _ IIAIIIIyll'
(b) Show that there is a matrix E such that (A + E)y = b with the norm
of E satisfying
ﬂ§ﬂ_ IMI llAll _ llAllllyll' (Hint: Try a rankone matrix E = admin for some wellchosen scalar as.) (c) Deﬁne the backward relative error in an approximate solution 3; of Am = b. (d) Show that an'approximate solution y of Ax = b has backward relative
 error 0(6) if and only if it has a residual r satisfying urn _ E
“Anny” ‘ 0‘ )‘ Extra Credit Problem 8 Prove that any model of ﬂoating—point arithmetic
which requires that the ﬂoating—point result of the multiplication my be given
by the exact result correctly rounded satisﬁes the relative error bound lwy—ﬂtvwll
I93 * yl
as long as no overﬂow or underﬂow occurs and 3: =1: 3; aé 0. $6 ...
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This note was uploaded on 05/17/2009 for the course MATH 128A taught by Professor Rieffel during the Spring '08 term at University of California, Berkeley.
 Spring '08
 Rieffel
 Math

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