Description
Theoretical assignment Consider hyperbolic tangent function
tanh(x) = exp(x) − exp(-x)
exp(x) + exp(-x)
where x ∈ [−2, 2]. Use 5 nodes to uniformly cut the interval [−2, 2] into 4
sub-intervals.
1. Note:
(a) Truncate all numbers to 3 decimal places (5 pts).
(b) Please submit only one single file (Pdf or word), including all the
results (5 pts).
(c) To simplify the evaluation, such as inversion and all arithmetic calculations, you can use Matlab or the like as a calculator.
2. Show these 5 nodes (x,tanh(x)) (5 pts)
3. Use these 5 nodes as given data to form the Van der Monde matrix (10
pts.) and find a polynomial to interpolate these 5 nodes (10 pts). The
specific procedure of formulating the Van der Monde Matrix is required.
4. Write down the the cardinal functions for these 5 nodes (20 pts), and use
the Lagrange polynomials to interpolate these 5 nodes, show the polynomial (15 pts).
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5. Based on these 5 nodes, use linear spline function to approximate f(x) =
tanh(x), x ∈ [−2, 2] (25 pts). The specific liner splines for each subintervals are required.
6. Based on these 5 nodes, use cubic spline function to approximate f(x) =
tanh(x), x ∈ [−2, 2] (25 pts). The specific cubic splines for each subintervals are required.
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