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MATH20602 function

创建日期：2024-04-12 18:08:46

所属分类：数学mathematics

标签： MATH20602

function

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MATH20602

1. (a) Find the Lagrange interpolant for the function f(x) = √ x at the interpolation nodes x0 = 1, x1 =

2. Use this interpolating polynomial to approximate the value of √ 3 2 , and compute the relative error in this approximation. [4 marks]

(b) (i) Suppose we have distinct x0, . . . xn and wish to interpolate a function f ∈ Cn+1([a, b])

at these points. State the formula for the upper bound of the absolute error of the interpolating polynomial,

|f(x)− pn(x)|, being sure to define any terms used. [3 marks] (ii) Compute the upper bound on the error of the approximation

of √ 3 2 from part (a). How does this compare to the true absolute error? [5 marks] (c) Prove that the Lagrangian

basis functions L0(x), . . . , Ln(x) defined in terms of distinct points x0, . . . , xn satisfy L0(x) + · · ·+ Ln(x) = 1. [3 marks]

(d) Suppose that we wish to compute the roots of the quadratic equation given by x2 + 39.7x+ 0.13 = 0,

but we are only able to use arithmetic accurate to 4 significant figures. (i) Compute the approximations of the two roots, and compute the relative errors of each. [2 marks] (ii) Explain the reasons for the differences in the relative errors in the two roots. [1 marks] (iii) Suggest and carry out an alternative method for computing approximations of both of the roots accurately. Hint: consider multiplying the formulae for the two roots. [2 marks] [Total: 20 marks] Page 2 of 5 P.T.O. MATH20602 2. (a) (i) Given the definition of the degree of a quadrature rule. [2 marks] (ii) Give the formula for Simpson’s rule for approximation of the integral of a function f : [0, 1]→ R, and derive its degree. [3 marks] (b) Suppose that we wish to compute an approximation of the integral∫ 2 1 1 1 + x dx. Compute the approximation using the trapezium rule, and using Simpson’s rule. Compute the relative errors of both methods. [5 marks] (c) Consider the function f(x) = e−x/x and the integral∫ 2 1 e−x x dx. We would like to approximate this integral by the composite trapezium and the composite Simpson’s rule on equispaced nodes x0, x1, . . . , xn where n = 2m. (i) How big does n have to be to ensure the approximation error of the composite trapezium rule is below 10−5? [3 marks] (ii) How big does n = 2m have to be to ensure the approximation error of the composite Simpson’s rule is below 10−5? [3 marks] (d) Define the Newton-Cotes quadrature for a function f on n+1 points a = x0 < x1 < · · · < xn = b for approximating the integral ∫ b a f(x) dx. If In(f) = w0f(x0) + · · ·+ wnf(xn) is the Newton-Cotes rule of order n > 0 for f on the interval [a, b], show that w0 + · · ·+ wn = b− a. [4 marks] [Total: 20 marks] Page 3 of 5 P.T.O. MATH20602 3. (a) Carry out two iterations of the Gauss-Seidel method for the following system of equations with initial vector x0 = (1.5, 2.1250,−0.9688)>: 4 −1 0−1 4 −1 0 −1 4 x1x2 x3  =  67 −6  (1) Obtain the solution by rounding the result to two significant figures, and show that this solution does satisfy the given set of equations. [4 marks] (b) A sequence of vectors generated by an iteration of the form xk+1 = Txk + c, converges to a vector x ∈ Rn with x = Tx+c for any starting vector x0 if and only if the eigen- values of the matrix T are all smaller than one in modulus. Use this to prove the convergence of the Gauss-Seidel method for the system (1) for any starting vector. [5 marks] (c) A sequence of vectors xk, k ≥ 0, converges to a vector x∗ ∈ Rn with respect to a norm ‖ · ‖, if for every > 0 there exists an integer N > 0 such that for all k ≥ N , ‖xk − x∗‖ < . Show that a sequence of vectors {xk}k≥0 converges to a vector x∗ with respect to the 1-norm if and only if it converges to x∗ with respect to the ∞-norm. Hint: first show that for any vector x that C1‖x‖∞ ≤ ‖x‖1 ≤ C2‖x‖∞, finding the constants Ci. [7 marks] (d) Let ‖x‖ be a vector norm and ‖A‖ the corresponding operator norm. For n×n matrices A,B, by using properties of vector norms, show that ‖A + B‖ ≤ ‖A‖+ ‖B‖. [4 marks] [Total: 20 marks] Page 4 of 5 P.T.O. MATH20602 4. (a) In order to solve the equation 3 cos(x2)− e−x2/2 = 0, perform two iterations of the bisection method with starting values 0.4, 1.8, followed by one iteration of the Newton-Raphson method on the approximate value found with the bisection method (x measured in radians). [6 marks] (b) Show that the equation x3 + 3x− 7 = 0 has a root between 1 and 2. Show that this is the only real root. [4 marks] (c) State a sufficient condition for an iteration scheme of the form xn+1 = φ(xn), n = 0, 1, . . . to converge to a fixed point α, provided the starting value x0 is sufficiently close to α. [1 marks] (d) Consider the following iteration schemes for computing the real root of the above equation: (i) xn+1 = 2x3n−7 3x2n+3 (ii) xn+1 = (7− 3xn)1/3 (iii) xn+1 = 7−x3n 3 For each scheme, state whether or not it will converge to the required root, if initialised close enough to that root. For each scheme that will converge, state the rate of convergence with justification. [9 marks]

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239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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