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midterm08sols

# midterm08sols - L Vandenberghe EE103 Midterm Solutions...

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Unformatted text preview: L. Vandenberghe 10/28/08 EE103 Midterm Solutions Problem 1 (10 points). Express the following problem as a set of linear equations. Find a cubic polynomial f ( t ) = c 1 + c 2 ( t − t 1 ) + c 3 ( t − t 1 ) 2 + c 4 ( t − t 1 ) 2 ( t − t 2 ) that satisfies f ( t 1 ) = y 1 , f ( t 2 ) = y 2 , f ′ ( t 1 ) = s 1 , f ′ ( t 2 ) = s 2 . The numbers t 1 , t 2 , y 1 , y 2 , s 1 , s 2 are given, with t 1 negationslash = t 2 . The unknowns are the coefficients c 1 , c 2 , c 3 , c 4 . Write the equations in matrix-vector form Ax = b , and solve them. Solution. We first derive expressions for f ( t 1 ), f ( t 2 ), f ′ ( t 1 ), f ′ ( t 2 ): f ( t 1 ) = c 1 , f ( t 2 ) = c 1 + c 2 h + c 3 h 2 , f ′ ( t 1 ) = c 2 , f ′ ( t 2 ) = c 2 + 2 c 3 h + c 4 h 2 where h = t 2 − t 1 . In matrix notation, the four interpolation conditions are 1 0 1 h h 2 0 1 0 1 2 h h 2 c 1 c 2 c 3 c 4 = y 1 y 2 s 1 s 2 . If we exchange the second and third rows, we can solve this by forward substitution. The solution is c 1 = y 1 , c 2 = s 1 , c 3 = y 2 − c 1 − hc 2 h 2 = ( y 2 − y 1 ) /h − s 1 h , c 4 = s 2 − c 2 − 2 hc 3 h 2 = s 2 − s 1 − 2(( y 2 − y 1 ) /h − s 1 ) h 2 = s 2 + s 1 − 2( y 2 − y 1 ) /h h 2 . 1 Problem 2 (10 points). A diagonal matrix with diagonal elements +1 or − 1 is called a signature matrix . The matrix S = 1 − 1 − 1 is an example of a 3 × 3 signature matrix. If3 signature matrix....
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midterm08sols - L Vandenberghe EE103 Midterm Solutions...

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