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Lecture29 - Lecture 28 Project 7 Gaussian Elimination and...

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Engineering 101 Engineering 101 Lecture 28 Lecture 28 Project 7, Gaussian Elimination and Project 7, Gaussian Elimination and MATLAB MATLAB Prof. Michael Falk University of Michigan, College of Engineering
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Announcements Announcements Exam 3 Monday, Nov 20, 7-9pm Sample Exams now posted, Review Friday Exam Rooms by Last Name: A-B Dow 1018 C-G Chrysler 133 H-K Dow 1017 L-O Dow 1014 P-Z Chrysler 220 (Cheseborough) Early Administration, Fri, Nov 18, 3-5pm in CSE 1670 Contact me today if you need to take the exam early So far: Jack Hall, Alex Lee, Marcus Lewis, Nat Christman, Noah Goodman, Nick Beier, Kaushik Seshan
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Project 7 Project 7 This is an intro to MATLAB with 2 parts Part 1: Write an M-File to solve for the flows in a chemical reactor making methanol Start with the diagram and use it to come up with 15 equations for the 15 unknowns. Compose these into a 15x15 matrix and solve using a MATLAB M-file.
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Project 7 Project 7 Part 2: Write two MATLAB M-files containing functions Function 1 - playit(‘filename’, start, duration) Plays a sample clip from start of length duration. Function 2 - mixit(‘filenameA’, ‘filenameB’, ‘filenameC’, start, duration) Convolve a short clip from B from start of length duration with the sound in A. Play the result and write it to C.
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Tomography Tomography How can we compute the mass through which each beam passes? First row: 8 ρ 0 = m 0 Second row: 2 ρ 0 + 6 ρ 1 = m 1 Third row: 2 ρ 0 + 2 ρ 1 + 4 ρ 2 = m 2 Fourth row: 2 ρ 0 + 2 ρ 1 + 2 ρ 2 + 2 ρ 3 = m 3 ρ 0 = 0.1 ρ 1 = 0.2 ρ 2 = 0.5 ρ 3 = 1.0 m 0 m 1 m 2 m 3 m 3 m 2 m 1 m 0
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Tomography Tomography How can we compute the mass through which each beam passes? 8 ρ 0 = m 0 2 ρ 0 + 6 ρ 1 = m 1 2 ρ 0 + 2 ρ 1 + 4 ρ 2 = m 2 2 ρ 0 + 2 ρ 1 + 2 ρ 2 + 2 ρ 3 = m 3 ρ 0 = 0.1 ρ 1 = 0.2 ρ 2 = 0.5 ρ 3 = 1.0 m 0 m 1 m 2 m 3 m 3 m 2 m 1 m 0
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Tomography Tomography How can we compute the mass through which each beam passes? 8 ρ 0 = m 0 2 ρ 0 + 6 ρ 1 = m 1 2 ρ 0 + 2 ρ 1 + 4 ρ 2 = m 2 2 ρ 0 + 2 ρ 1 + 2 ρ 2 + 2 ρ 3 = m 3 8 0 0 0 ρ 0 m 0 2 6 0 0 ρ 1 m 1 2 2 4 0 ρ 2 m 2 2 2 2 2 ρ 3 m 3 ρ 0 = 0.1 ρ 1 = 0.2 ρ 2 = 0.5 ρ 3 = 1.0 m 0 m 1 m 2 m 3 m 3 m 2 m 1 m 0 =
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Tomography Tomography Of course the problem of finding the mass of a slice is not the problem we are usually interested in solving. We can measure the masses of the slices by tomography. We want to know the density of the rings given the masses of the slices. This is straightforward if the matrix is triangular.
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The Inverse Problem The Inverse Problem 8 0 0 0 ρ 0 m 0 2 6 0 0 ρ 1 m 1 2 2 4 0 ρ 2 m 2 2 2 2 2 ρ 3 m 3 Getting the first density is easy: A 00 ρ 0 = m 0 ρ 0 = m 0 / A 00 =
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The Inverse Problem The Inverse Problem 8 0 0 0 ρ 0 m 0 2 6 0 0 ρ 1 m 1 2 2 4 0 ρ 2 m 2 2 2 2 2 ρ 3 m 3 Getting the second density is a bit harder: A 10 ρ 0 + A 11 ρ 1 = m 1 ρ 1 = (m 1 - A 10 ρ 0 ) / A 11 =
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The Inverse Problem The Inverse Problem 8 0 0 0 ρ 0 m 0 2 6 0 0 ρ 1 m 1 2 2 4 0 ρ 2 m 2 2 2 2 2 ρ 3 m 3 Getting the third density is similar: A 20 ρ 0 + A 21 ρ 1 + A 22 ρ 2 = m 2 ρ 2 = (m 2 – A 20 ρ 0 – A 21 ρ 1 ) / A 22 =
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