hmwk_3_sol - Discrete Structures Sept 12 2011 Assignment 3:...

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Discrete Structures Sept 12 2011 Assignment 3: Solutions Prof. Hopcroft *******Note: this homework is subject to the style guide on the website. Points will be deducted for homeworks not following the guidelines.******** *******Please print out and staple the grade sheet to the back of your homework!****** 1 Recurrence Relations: Bottom Up For each of the characteristic equations: 1. Find the corresponding recurrence equation. 2. How many boundary conditions are necessary for a complete solution to f ( n )? 3. Show that the roots of the characteristic equations raised to the n th power are solutions of the recurrence equation. Ie f ( n ) = r n , where r is a root of the corresponding characteristic equation, satisfies the recurrence equation. The characteristic equations: 1.1 x - 2 = 0 1. f ( n ) = 2 f ( n - 1) 2. Need 1 boundary condition. 3. The roots are: 2. 2 n = 2(2) n - 1 1.2 x 2 + 2 x - 15 = 0 1. f ( n ) = - 2 f ( n - 1) + 15 f ( n - 2) 2. Need 2 boundary conditions. 3. The roots are: 3 and - 5. ( - 5) n = - 2( - 5) n - 1 + 15( - 5) n - 2 25 = 10 + 15 (1) And: (3) n = - 2(3) n - 1 + 15(3) n - 2 9 = - 6 + 15 (2) 1
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1.3 8 x 3 - 8 x 2 + 4 x - 1 = 0 1. 8 f ( n ) = 8 f ( n - 1) - 4 f ( n - 2) + f ( n - 3) 2. Need 3 boundary conditions. 3. The roots are: 1 2 , 1+ - 3 4 , 1 - - 3 4 . 8 ± 1 2 ² n = 8 ± 1 2 ² n - 1 - 4 ± 1 2 ² n - 2 + ± 1 2 ² n - 3 8 2 3 = 8 2 2 - 4 2 + 1 1 = 2 - 2 + 1 8 ± 1 + - 3 4 ² n = 8 ± 1 + - 3 4 ² n - 1 - 4 ± 1 + - 3 4 ² n - 2 + ± 1 + - 3 4 ² n - 3 8 ± 1 + - 3 4 ² 3 = 8 ± 1 + - 3 4 ² 2 - 4 ± 1 + - 3 4 ² + 1 (1 + - 3) 3 8 = (1 + - 3) 2 2 - - 3 1 + 3 - 3 - 9 - 3 - 3 8 = 1 + 2 - 3 - 3 2 - - 3 8 8 = 2 + 2 - 3 2 - - 3 = 1 In a very similar way the root ³ 1 - - 3 4 ´ n can be shown to satisfy the recurrence. 2
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hmwk_3_sol - Discrete Structures Sept 12 2011 Assignment 3:...

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