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8 Hand-out on Interpolation

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8 Hand-out on Interpolation
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  MATH 174Numerical Analysis I Section BFirst Semester A.Y. 2014-2015 Mathematics DivisionInstitute of Mathermatical Sciences and PhysicsUniversity of the Philippines Los Ba˜nos September 12, 2014 NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 1 / 11 Chapter Two.  Interpolation of Functions Suppose that a mathematical model for the spread of an epidemicproduces that following table showing the number of deaths due to thedisease on  t  days after the outbreak. t  0 10 25 50 100deaths 0 8 4251 20,677 357What if an expert wants to analyze weekly data? NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 2 / 11 What if we want to get the optimal selling price for a commodity (in termsof gross sales) and the most efficient way to gather data for a model is tosell it at varying prices and observing the number of products sold?Suppose the data gathered is as follows:Price 70 82.5 137.5 187.50Number of Units Sold 1,250 750 550 272.50How will you find the optimal price? NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 3 / 11 A common mathematical problem Given a set of points  ( x i ,f  ( x i ))  for  i  = 0 , 1 , 2 ,...,n  where the nodes  x i ’sare distinct values of the independent variable. Then eitherapproximate the value of   f   at some value of   x  not included in the list;ordetermine a function  g  that mimics the behavior of the data in somesense.These problems give rise to a two different areas of study: Interpolation and Approximation . NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 4 / 11  Interpolation vs Approximation Interpolationthe function  g  is determined by requiring zero errors at the nodes, i.e., f  ( x i )  −  g ( x i ) = 0 , ∀ i  = 0 , 1 ,...,n Approximationthe function  g  is chosen such that some measure of error is minimized, forexample n  i =0 ( f  ( x i )  −  g ( x i )) 2 < . NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 5 / 11 Consider the function  f  ( x ) =  e x .Using the nodes  − 2 , − 1 , 0 , 1 , the interpolating polynomial is P  ( x ) = − x 3 + 3 e  x 2 +  x  −  2  x  +  e 3   x 2 + 3 x  + 2  x 6 e 2 +  − 3 e 2   x 3 + 2 x 2 −  x  −  2   +  x 6 e 2  . While an approximating polynomial for  f   is Q ( x ) = 1 +  x  +  x 2 2! +  x 3 3!  . NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 6 / 11 Interpolation, a special case of Approximation Why focus on interpolation?Common forms of interpolationpolynomialpiecewise polynomialrationaltrigonometricexponential NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 7 / 11 Why polynomials? they are the ’simplest functions’ practical for computerstheir derivatives and integrals easy to compute and are stillpolynomialsfor any required accuracy, there is a polynomial approximating  f  Theorem Weierstrass (First) Approximation Theorem.Let   f   ∈ C  [ a,b ] . Then  ∀  >  0  there is a polynomial   P  ( x )  such that  | f  ( x )  −  p ( x ) |  < , ∀ x  ∈  [ a,b ] . there is one and only one interpolating polynomial for aset of points  ( x i ,f  ( x i ))  for  i  = 0 , 1 , 2 ,...,n  . NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 8 / 11  Algebraic Interpolating Polynomial A polynomial  P  n ( x )  ≡  P  n ( x,f,x 1 ,x 2 ,...,x n +1 )  of degree no greaterthat  n  that has the form P  n ( x ) =  c 0  +  c 1 x  +  ...  +  c n x n and coincides with the data points  ( x i ,f  ( x i ))  for  i  = 1 , 2 ,...,n,n  + 1 is called the algebraic interpolating polynomial of   f  . NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 9 / 11 Unisolvence TheoremTheorem There is a unique algebraic interpolating polynomial  P  n ( x,f,x 1 ,x 2 ,...,x n +1 ) . NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 10 / 11 Ngayon pa lang...The accuracy of the interpolating polynomial isguaranteed ONLY FOR ABSCISSAS between theleast and greatest nodes. NJA Egarguin (IMSP, UPLB) MATH 174 September 12, 2014 11 / 11
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