HW02_Solution - EML 4507 Finite Element Analysis &...

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EML 4507 Finite Element Analysis & Design Fall 2010 HW02 Solution 0.13. Consider the matrix equation [ A ]{ x } = { b } given by 1 2 3 2 1 0 4 1 2 1 0 0 1 2 4 x x x (a) Construct the quadratic form F ( x ) = { x } T [ A ]{ x } – 2{ x } T { b }. (b) Find { x } = { x * } by minimizing F ( x ). (c) Verify that the vector { x * } satisfies [ A ]{ x } = { b }. Solution: (a) The quadratic form F ( x ) is: 222 1 2 3 1 2 2 3 1 3 44 F x x x x x x x x x (b) Find { x } = { x * } that minimizes F ( x ). 12 1 0 2 4 F xx x 1 2 3 2 02 F x x x x 23 3 0 2 4 F x By solving the above three equations with respect to three unknown, we can obtain: x 1 = 4, x 2 = 4, and x 3 = 4. (c) Verify that the solution { x * } satisfies [ A ]{ x* } = { b }. 2 1 0 4 2 4 1 4 4 1 2 1 4 1 4 2 4 1 4 0 0 1 2 4 1 4 2 4 4 The solution is verified. 0.14. A function f ( x 1 , x 2 ) of two variables x 1 and x 2 is given by 1 1 2 1 2 1 2 2 1 1 0 1 ( , ) 1 1 2 2 2 x f x x x x x x x
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(a) Multiply the matrices and express f as a polynomial in x 1 and x 2. (b) Determine the extreme (maximum or minimum) value of the function and corresponding x 1 and x 2 .
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HW02_Solution - EML 4507 Finite Element Analysis &...

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