Lecture_15_E7_L15_least_squares_F07

# Lecture_15_E7_L15_least_squares_F07 - 1 E7: INTRODUCTION TO...

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E7 L15 1 E7: INTRODUCTION TO COMPUTER E7: INTRODUCTION TO COMPUTER PROGRAMMING FOR SCIENTISTS AND PROGRAMMING FOR SCIENTISTS AND ENGINEERS ENGINEERS Lecture Outline 1. Solutions of linear algebraic equations 2. Least squares solution Copyright 2007, Horowitz, Packard. This work is licensed under the Creative Commons Attribution-Share Alike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

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E7 L15 2 Linear algebraic equations Linear algebraic equations Consider n LINEAR equations and m unknowns. m unknowns n equations 11 12 1 1 21 22 2 2 1 1 2 2 2 1 1 2 m m n n m m n m m n A A A b A A A b A A A b x x x x x x x x x + + + = + + + = + + + = L L M L
E7 L15 3 Linear equations in matrix form Linear equations in matrix form Can be written in matrix form: A   matrix ( n x m ) x   vector ( m x 1) b   vector ( n x 1)

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E7 L15 4 Linear equations in matrix form Linear equations in matrix form = n-by- m m -by-1 n-by-1 = n-by- m m -by-1 n-by-1 A x b A x b
E7 L15 5 Linear equations in matrix form Linear equations in matrix form First row: A   matrix ( n x m ) x   vector ( m x 1) b   vector ( n x 1)

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E7 L15 6 Linear equations in matrix form Linear equations in matrix form Second row: A   matrix ( n x m ) x   vector ( m x 1) b   vector ( n x 1)
E7 L15 7 Linear equations in matrix form Linear equations in matrix form nth row: A   matrix ( n x m ) x   vector ( m x 1) b   vector ( n x 1)

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E7 L15 8 Matrix - vector multiplication along rows Matrix - vector multiplication along rows i’th row of A will be denoted by: ( 29 : 1 m j j j i i A A x x = =
E7 L15 9 Matrix - vector multiplication along rows Matrix - vector multiplication along rows ( 29 ( 29 ( 29 11 1 12 2 1 1: 21 1 22 2 2 2: 1 1 2 2 : m m m m n n nm m n Ax A x A x A x Ax A x A x A x Ax Ax A x A x A x + + + + + + = = + + + L L M M L ( 29 : 1 m j j j i i A A x x = =

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E7 L15 10 Matrix - vector multiplication along columns Matrix - vector multiplication along columns 1 1 2 n j m j j j j A A x x A A = = M j th   column of A
E7 L15 11 Matrix - vector multiplication along columns Matrix - vector multiplication along columns 11 12 13 1 21 22 23 2 1 2 3 m m n n n nm A A A A A A A A A A A A       L L M M M O M L 1 2 3 x x 11 21 1 n A A A M [ ] 1 x 12 22 2 n A A A M [ ] 2 x 23 3 n A A M [ ] 3 x 2 m nm A A M [ ] m x + + + + L add, to give + + + + + + + + + = m nm n n m m m m x A x A x A x A x A x A x A x A x A L M L L 2

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## This note was uploaded on 09/17/2011 for the course ENGINEERIN 7 taught by Professor Patzek during the Spring '08 term at University of California, Berkeley.

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Lecture_15_E7_L15_least_squares_F07 - 1 E7: INTRODUCTION TO...

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