B.Sc. in Mathematics (Ordinary)

Transcription

1 R48/0 DUBLIN INSTITUTE OF TECHNOLOGY KEVIN STREET, DUBLIN 8 B.Sc. in Mthemtics (Ordinry) SUPPLEMENTAL EXAMINATIONS 01 Numericl Methods Dr. D. Mckey Dr. C. Hills Dr. E.A. Cox Full mrks for complete nswers to five questions Mthemtics Tbles

2 1 The Lgrnge polynomil which interpoltes the dt (x i, f i ), where f i = f(x i ) for i = 0, 1,... n, is given by n P n (x) = L n,i (x)f i where L n,i (x) = n i j x x i x j x i. () Show tht if the function f(x) is pproximted by the Lgrnge polynomil P n (x) on n intervl [, b] then the following Newton-Cotes qudrture rule cn be derived for pproximting the integrl of f n I(f) = f(x) dx w i f(x i ), where w i = L n,i (x) dx. [8 mrks] (b) Hence derive the following closed Newton-Cotes qudrture formul for n = I(f) b [ ( ) ] + b f() + 4f + f(b) 6 (Simpson s Rule). [1 mrks] () Let P (x) = (x x 0 )+ (x x 0 )(x x 1 ) denote the Newton interpolting polynomil which pproximtes the function f(x) nd psses through the three points (x 0, f(x 0 )), (x 1, f(x 1 )) nd (x, f(x )). Derive formule for 0, 1 nd nd show tht cn be written in the form = f[x 1, x ] f[x 0, x 1 ] where f[x 1, x ] = f(x ) f(x 1 ), etc. x x 0 x x 1 [10 mrks] (b) Using the formule derived in prt (), write out the Newton form of the interpolting polynomil for f(x) = sin(x) which psses through the points (0, sin(0)), (π/4, sin(π/4)) nd (π/, sin(π/)). [6 mrks] (c) Use the polynomil in prt () to estimte sin(π/3). Wht is the error in this pproximtion? [4 mrks] Pge of 5

3 3 () Consider the Cholesky fctoriztion of mtrix, A = LL T. Wht conditions must A nd L stisfy? In the cse of 3 3 mtrix, derive the equtions stisfied by the entries of L nd indicte the order in which these should be solved. [6 mrks] (b) Find the Cholesky fctoriztion for the mtrix A = [6 mrks] (c) Use the result of prt (b) to solve the system 16x 1 8x = 8 8x x + 10x 3 = 10x + 9x 3 = 38. [8 mrks] 4 Consider the mtrix A = () Sketch the Gerschgorin circles nd hence find n pproximtion for the dominnt eigenvlue of the mtrix A. [8 mrks] (b) Using three itertions of the power method, pproximte the dominnt eigenvlue of A, s well s n ssocited eigenvector. Use x (0) = [1, 1, 1] T s your initil pproximtion. [1 mrks] Pge 3 of 5

4 5 () Using the Euler method nd step size of h = 0.5, find n pproximte solution for the initil vlue problem dx dt = tx3 x, 0 t 1, x(0) = 1. Given tht the exct solution is x(t) =, clculte the pproxi- + 4t + e t mtion error t ech step. [10 mrks] (b) Approximte the solution of the problem in prt (b) using the fourth order Runge-Kutt method with step size of h = 0.5. How do the bsolute errors in this cse compre with those obtined from the Euler s method? [10 mrks] Hint: The fourth-order Runge-Kutt formul is given by w i+1 = w i (k 1 + k + k 3 + k 4 ) where k 1 = hf(t i, w i ), k = hf(t i + h, w i + k 1 ), k 3 = hf(t i + h, w i + k ), k 4 = hf(t i + h, w i + k 3 ), where w i is the pproximte solution t time t i = t 0 + ih. 6 () Derive the three-point Gussin qudrture rule 1 f(x) dx 5 ) ( ) 3 ( 9 f f(0) f. 5 1 [1 mrks] (b) Use the three-point Gussin qudrture rule to pproximte the vlue of the integrl 1 1 e x dx. Wht is the bsolute error in this pproximtion? [8 mrks] Pge 4 of 5

5 7 () Given the trpezoidl rule for pproximting the vlue of the integrl I(f) = f(x) dx, I(f) = b [f() + f(b)] (b )3 1 f (ξ) where < ξ < b, derive the composite trpezoidl rule [ ] T h (f) = h n 1 (b )h f() + f(x i ) + f(b) f ( ξ) 1 i=1 where < ξ < b, h = (b )/n nd x i = + ih for ll i = 0,... n. [1 mrks] (b) Determine the number of intervls needed in the composite trpezoidl rule such tht, when pproximting the vlue of the integrl I(f) = π 0 sin(x) dx, the error is less thn (Do not ttempt to determine this pproximtion.) [8 mrks] 8 () Define the Chebyshev polynomils, T n (x), nd prove the recurrence formul T n+1 (x) + T n 1 (x) = xt n (x), for n 0 nd 1 x 1. (Hint: Use the substitution θ = cos 1 (x).) [5 mrks] (b) Using the definition or the recurrence formul in prt (), clculte the polynomils T 0 (x), T 1 (x), T (x), T 3 (x) nd T 4 (x). [5 mrks] (c) Strting with the fourth-order Mclurin polynomil nd using the modified Chebyshev polynomils T n (x) = T n (x)/ n 1, find the polynomil of lest degree which best pproximtes the function f(x) = e x on [ 1, 1], while keeping the error less thn [10 mrks] Pge 5 of 5