midterm08sols

midterm08sols - L. Vandenberghe 10/28/08 EE103 Midterm...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 5

midterm08sols - L. Vandenberghe 10/28/08 EE103 Midterm...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online