CS 131 FIRST LONG EXAM
SOLVE FOR A TOTAL OF 50 POINTS. PRESENT A COMPLETE SOLUTION FOR EACH PROBLEM. 1. ITERATIVE SCHEMES FOR LINEAR SYSTEMS (/15) (a) Give the general form of the iterative scheme for solving the linear equation problem Ax=b. What are the differences between the Jacobi and Gauss-Seidel schemes in terms of algorithm and convergence. (b) Solve TWO iterations of the schemes on the following problem using ω=0.5 and
[ ] []
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x 0= 1 1
2 −1 x= 1 4 1 2
(c) Give the algorithm that optimizes the Gauss-Seidel iteration. 2. INTERPOLATION VS CURVE FITTING (/15) Given the data below x
1
2
3
4
y
1.89
0.93
0.44
0.31
(a) obtain the polynomial interpolant using Newton's Method (b) obtain the best power fit y=axb by solving the associated normal equations (c) compare the approxiamtions using (i) and (ii) at x= 2.5 . 3. CONCEPTS AND DERIVATIONS 1. Compare the complexities of LU Factorization and Gauss Jordan in solving the inverse of a matrix. (/5) 2. Compare interpolation using monomial basis and Lagrange basis by identifying their PROS and CONS. (/5) 3. Discuss the algorithm for solving linear curve fitting problem using orthogonal transformation. (No need to discuss QR factorization of A.) (/5) n
n
4. The interpolant using Lagrange basis is given by y =∑ i a i l i ( x ) where
l i ( x)=∏ j≠i
ai = yi ∀ i=1, 2, ... , n . (/5)
5. The piecewise polynomial interpolant via cubic spline is given by
[
3 k i ( x− x i+ 1 ) f i,i+ 1 ( x ) = −( x i− x i+ 1 ) ( x− x i+ 1 ) 6 x i −x i+ 1
[
3 k i+ 1 ( x− x i ) − − ( x i −x i+ 1 ) ( x− x i ) 6 ( x i− x i+ 1 )
+y i ( x−x i+ 1 ) − y i+ 1 ( x− x i )
]
]
x i −x i+ 1 Derive the banded linear equation for solving the k i values at the knots. (/10)
x−x j xi −x j . Show that