solhw2

# solhw2 - Solutions for Homework 2 Foundations of...

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Solutions for Homework 2 Foundations of Computational Math 2 Spring 2011 Written homework problems are study questions. You need not turn in solutions but you are strongly encouraged to do the problems and read the posted solutions carefully. Problem 2.1 Let p n ( x ) be the unique polynomial that interpolates the data ( x 0 , f 0 ) , . . . , ( x n , f n ) Suppose that we assume the form p n ( x ) = α 0 + α 1 ( x x 0 ) + · · · + α n ( x x 0 )( x x 1 ) · · · ( x x n 1 ) and let a = α 0 . . . α n y = y 0 . . . y n 2.1.a . Show that the constraints yield a linear system of equations La = y where L is lower triangular. 2.1.b . Show that the linear system yields a recurrence for the α i that is equivalent to one of the standard definitions of the divided differences and therefore this is the Newton form of p n ( x ). Solution: 1

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Taking n = 3 suffices to show the pattern. We have p 3 ( x ) = α 0 + α 1 ( x x 0 ) + α 2 ( x x 0 )( x x 1 ) + α 3 ( x x 0 )( x x 1 )( x x 2 ) p 3 ( x 0 ) = α 0 + α 1 ( x 0 x 0 ) + α 2 ( x 0 x 0 )( x x 1 ) + α 3 ( x 0 x 0 )( x x 1 )( x x 2 ) p 3 ( x 1 ) = α 0 + α 1 ( x 1 x 0 ) + α 2 ( x 1 x 0 )( x 1 x 1 ) + α 3 ( x 1 x 0 )( x 1 x 1 )( x 1 x 2 ) p 3 ( x 2 ) = α 0 + α 1 ( x 2 x 0 ) + α 2 ( x 2 x 0 )( x 2 x 1 ) + α 3 ( x 2 x 0 )( x 2 x 1 )( x 2 x 2 ) p 3 ( x 3 ) = α 0 + α 1 ( x 3 x 0 ) + α 2 ( x 3 x 0 )( x 3 x 1 ) + α 3 ( x 3 x 0 )( x 3 x 1 )( x 3 x 2 ) p 3 ( x 0 ) = α 0 = y 0 p 3 ( x 1 ) = α 0 + α 1 ( x 1 x 0 ) = y 1 p 3 ( x 2 ) = α 0 + α 1 ( x 2 x 0 ) + α 2 ( x 2 x 0 )( x 2 x 1 ) = y 2 p 3 ( x 3 ) = α 0 + α 1 ( x 3 x 0 ) + α 2 ( x 3 x 0 )( x 3 x 1 ) + α 3 ( x 3 x 0 )( x 3 x 1 )( x 3 x 2 ) = y 3 1 0 0 0 1 ( x 1 x 0 ) 0 0 1 ( x 2 x 0 ) ( x 2 x 0 )( x 2 x 1 ) 0 1 ( x 3 x 0 ) ( x 3 x 0 )( x 3 x 1 ) ( x 3 x 0 )( x 3 x 1 )( x 3 x 2 ) α 0 . . . α n = y 0 . . . y n La = y Since x i negationslash = x j when i negationslash = j we have nonzero elements on the diagonal of the lower triangular matrix L implying it is nonsingular. The solution a
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