Applied Numerical Analysis Quiz #2

Transcription

1 Applied Numerical Analysis Quiz #2 Modules 3 and 4 Name: Student number: DO NOT OPEN UNTIL ASKED Instructions: Make sure you have a machine-readable answer form. Write your name and student number in the spaces above, and on the answer form. Fill in the answer form neatly to avoid risk of incorrect marking. Fill in the version number of your quiz (see bottom right, A-D) on the answer form. Use only pencil on the answer form, and correct with a rubber. This quiz requires a calculator. Each question has exactly one correct answer. Correct answers will receive 5 points, unanswered questions point, incorrect answers 0 points. This quiz has 0 questions and 4 pages in total. Quiz 205, Versie: D

2 Module 3: Advanced interpolation Question Let φ(x) be a cubic spline on x [a, b]. What is the most that can be said about φ (x) in general? B: It is discontunous C: It is continuous, i.e. φ C 0 ([a, b]) D: It is once continuously differentiable, i.e. φ C ([a, b]) E: It is twice continuously differentiable, i.e. φ C 2 ([a, b]) F: It is infinitely differentiable, i.e. φ C ([a, b]) Question 2 Consider the quadratic function f(x) = x 2 on the interval x [, ]. Samples of f(x) are taken at x 0 =, x = 0, and x 2 =. Which of the following methods, when applied to these samples, will give an exact interpolation of f(x) on the interval? a. Polynomial interpolation b. Linear spline interpolation c. Cubic spline interpolation with natural BCs (φ (x 0 ) = φ (x 2 ) = 0) d. Cubic spline interpolation with clamped BCs (φ (x 0 ) = f (x 0 ) and φ (x 2 ) = f (x 2 )) B: None C: All D: a E: a, c, d F: a, c G: a, d H: b Question 3 Again a cubic spline is used to approximate f(x) = x 2 on the interval x [, ] using samples at x 0 =, x = 0, and x 2 =. The two branches of the spline are: { S 0 (x) = a 0 (x + ) 3 + b 0 (x + ) 2 + c 0 (x + ) + d 0 if x < 0, φ(x) = S (x) = a x 3 + b x 2 + c x + d if 0 x Natural BCs are used, i.e. φ (x 0 ) = φ (x 2 ) = 0. Further it is known that a 0 = 2 and b 0 = 0. What is the value of a? B: 3 2 C: D: 2 E: 0 F: 2 G: H: 3 2 Question 4 Consider tensor-product polynomial interpolation in 2d, with grid consisting of the rectangular lattice with nodes X ij = (x i, y j ), with x = (0,, 2, 3, 4) and y = (0, 2, ). Which of the following monomials are represented exactly by the interpolation? a. xy b. x 4 y 2 c. x 2 y 2 d. x 2 B: All C: None D: a, b, c E: d F: a, c, d G: b H: a, b 2 Quiz 205, Versie: D

3 Question 5 In radial-basis function interpolation a different radial function can be used at each node. Consider interpolation in d on the 3-node grid x 0 =, x = 0, and x 2 =. Each node has a corresponding basis function: ϕ 0 (x) = 0 x +, ϕ (x) = x, ϕ 2 (x) = 0 x. By setting up the interpolation conditions for the interpolant φ(x) = a 0 ϕ 0 (x) + a ϕ (x) + a 2 ϕ 2 (x), determine the value of a 0, given that f 0 =, f = 0, and f 2 = 2, and that a 2 = 40. B: 5 2 C: 3 2 D: 2 E: 0 F: 2 G: 3 2 H: 5 2 Module 4: Numerical differentiation and Integration Question 6 When deriving cubic splines we developed the system of equations: h 6 M i + 2h 3 M i + h 6 M i+ = f i+ f i h f i f i, h for the unknowns M i. The right-hand side of this equation (for a fixed i) looks a bit like a difference formula for some derivative of f(x) based on three samples f(x i h), f(x i ) and f(x i + h). What is it approximating? [Hint: Taylor...] B: f (x i ) C: hf (x i ) D: 2hf (x i ) E: f (x i ) F: hf (x i ) G: 2hf (x i ) H: 6h 2 f (x i ) Question 7 Difference rules can be found by differentiating a polynomial interpolant. Consider a 3-node grid (x 0, x, x 2 ) = ( h, 0, h), with corresponding function values (f 0, f, f 2 ). Using a Lagrange basis, first find a quadratic interpolant, and then differentiate it. Obtain an approximation of f (h/3), of the form: ( ) h f 3 h (a 0f 0 + a f + a 2 f 2 ), Which of the following is the resulting approximation of f (x) at x = h 3? B: h ( f 0 + f ) C: 2h ( f 0 + f 2 ) D: 3h ( 2f 0 f + 3f 2 ) E: 6h ( f 0 2f + 3f 2 ) F: 6h ( f 0 4f + 5f 2 ) G: 6h ( 4f 0 f + 5f 2 ) Question 8 Simpson s rule defined on the interval [, ] is: The same rule on a general interval [a, b] is: f(x) dx 3 f( ) f(0) + 3 f() b f(x) dx = a i=0 Determine the weights w and w 2, where h := b a. 2 w i f(x i ). 3 Quiz 205, Versie: D

4 B: w = and w 2 = C: w = h 6 and w 2 = h 6 D: w = 2h 3 and w 2 = h 6 E: w = h 6 and w 2 = 2h 3 F: w = 4h 3 and w 2 = h 3 G: w = h 3 and w 2 = 4h 3 H: w = h and w 2 = 2h Question 9 For a function of two variables f(x, y), we would like a finite-difference approximation of 2 f x y at (x 0, y 0 ). This can be achieved by using forward-differences in x: f x (x 0, y 0 ) φ(x 0, y 0 ) := f(x 0 + x, y 0 ) f(x 0, y 0 ) x and then applying forward-differences in y to the function φ. Using the notation f 00 := f(x 0, y 0 ), f 0 := f(x 0 + x, y 0 ), f 0 := f(x 0, y 0 + y), f := f(x 0 + x, y 0 + y) What is the resulting expression for 2 f x y? B: (f f 00 )/( x y) C: (f 0 f 0 )/( x y) D: (f f 0 f 0 + f 00 )/( x y) E: (f f 0 + f 0 f 00 )/( x y) F: (f + f 0 f 0 f 00 )/( x y) G: (f 0 + f 0 2f 00 )/( x y) Question 0 From polynomial interpolation we know the interpolation error (Cauchy). For f(x) approximated by an N + point polynomial interpolant p N, the error at a point x is: where the nodal polynomial E(f; x) := f(x) p N (x) = f (N+) (ξ) (N + )! ω N+ (x) := N (x x i ). i=0 ω N+ (x), Using this formula, and assuming f (N+) (ξ) = (for all x), which of the following is an estimate for the error in Simpson s rule for an integral on the interval [ h, h]? (Note: Simpson s rule has nodes at x = ( h, 0, h).) B: 0 C: h 3 /2 D: h 3 /2 E: h 3 /48 F: h 4 /2 G: h 4 /2 H: h 4 /48 4 Quiz 205, Versie: D

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