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HOMEWORK FOR ADVANCED NUMERICAL COMPUTING - CS 543

Michael J. Johnson Spring 2008 Abstract. These are the homework assignments for Advanced Numerical Computing – CS 543 taught at Kuwait University Spring Semester 2008.

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CS 543

1. Homework Problem 1.1. Let Ξ1 = {0, 1, 5}, Ξ2 = {0, 1, 2, 5}, and Ξ3 = {0, 1, 3, 5}. a) Is P3,Ξ1 a subspace of P6,Ξ1 ? b) Is P3,Ξ1 a subspace of P3,Ξ2 ? c) Is P3,Ξ2 a subspace of P3,Ξ3 ? Problem 1.2. Let Ξ1 = {0, 1, 2, 3, 5} and Ξ2 = {0, 1, 2, 3, 4, 5}, and consider  4 4x − 5x2 + 1     (x − 1)3 + (x − 1) s(x) :=  2    (x − 3) + 2

if 0 ≤ x < 1 if 1 ≤ x < 2 if 2 ≤ x < 3 if 3 ≤ x ≤ 5.

a) Find the Matlab/Octave representation of s as an element of P4,Ξ1 . b) Find the Matlab/Octave representation of s as an element of P5,Ξ1 . c) Find the Matlab/Octave representation of s as an element of P4,Ξ2 . d) Does s belong to C[a, b]? e) Does s belong to C 1 [a, b]? P∞ Problem 1.3. Let k ∈ N and define f (x) = j=−∞ cj Bjk (x). Determine the largest value of j0 and the smallest value of j1 such that f (x) =

j1 X

cj Bjk (x) for all x ∈ [ξ1 , ξ2 ].

j=j0

Problem 1.4. Let Ξ be the knots a = ξ1 < ξ2 < · · · < ξN +1 = b and let k ∈ N0 . Prove that Sk,Ξ ∩ Sk+1,Ξ = Πk | . [a,b]

Problem 1.5. Let k ∈ N and assume that the P knots {. . . , ξ−1 , ξ0 , ξ1 , . . . } are equispaced ∞ (ie. ξi+1 − ξi = h for all i). Prove that f (x) = j=−∞ ξj Bjk (x) is a polynomial of degree 1.

CS 543

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2. Homework The questions here may involve both written and Octave work. Let me suggest the following format for your answers: Give me your written work (on paper) along with a diskette or an archive (via email). The diskette or archive should contain Octave scripts (*.m files). For each question, your written answer should mention which Octave script file I’m supposed to see from your diskette or archive. You may find the Octave commands disp, keyboard, input useful in Octave scripts. Problem 2.1. Let Ξ = {0, 1, 2, 4}. Find the unique spline s ∈ S1,Ξ which satisfies s(0) = 3, s(1) = 0, s(2) = 1, and s(4) = −1. Problem 2.2. Let Ξ = {0, 1, 3, 6, 10} and k = 3. a) Choose Ξext in an appropriate manner. b) Using Octave, construct the pp representation of f (x) = B13 (x) + B23 (x) + B33 (x) + B43 (x) and plot f on the interval [0, 10]. Problem 2.3. Let Ξ = {−2, 1, 2, 5, 6}. Use Octave to verify the identity d k B (x) = dx i



k ξi+k − ξi



Bik−1 (x)

−



k ξi+k+1 − ξi+1



k−1 Bi+1 (x)

for i = 1, k = 3. (Suggestion: Construct the LHS and the RHS, and then plot both on the same plot). Problem 2.4. Let Ξ = {3, 4, 7, 8, 10, 13}. Use Octave to verify the identity Bik (x)

=



x − ξi ξi+k − ξi



Bik−1 (x)

+



ξi+k+1 − x ξi+k+1 − ξi+1



k−1 Bi+1 (x)

for i = 1, k = 4. Problem 2.5. Use Octave to construct and plot the natural quintic (degree 5) spline which passes thru the points (1, 1), (2, 5), and (5, 5). Problem 2.6. Use Octave to construct and plot the complete quintic spline s which passes thru the points (0, 0), (2, 3), and (3, 1) and satisfies s0 (0) = 1, s00 (0) = −1, s0 (3) = 0, s00 (3) = −2.