the coefficients b b 1 b n as The bracketed function evaluations f xi x j are

# The coefficients b b 1 b n as the bracketed function

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the coefficients b0, b1, . . . , bnas: The bracketed function evaluations f [xi, xj] are finite divided differences
Example, the first finite divided difference is represented generally as The second finite divided difference, which represents the difference of two first divided differences, is expressed generally as Similarly, the nth finite divided difference isME 261: Numerical Analysis 10
11 sis 00,211,10,110201011212020112012201010101011000].....,[].....,[],.....,[...)()()()(],[],[],,[)()(][][],[)(][xxxxxfxxxfxxxxfbxxxxxfxfxxxfxfxxxxfxxfxxxfbxxxfxfxxxfxfxxfbxfxfbnnnnnnnnb1: First Finite divided difference [f'(x) ] b2: Second Finite divided difference [f"(x)] bn: nthFinite divided difference
12 sis ijijijiijjxxxxfxxfxxxxf],...,[],...,[],,...,,[1111
13 Construct a 4thorder polynomial in Newton form that passes through the following points: Example i 0 1 2 3 4 xi0 1 -1 2 -2 f(xi)-5 -3 -15 39 -9 )2)(1)(1)(()1)(1)(()1)(()()()2)(1)(1)(0()1)(1)(0()1)(0()0()())()()(())()(())(()()(4321044321043210421031020104xxxxbxxxbxxbxbbxfxxxxbxxxbxxbxbbxfxxxxxxxxbxxxxxxbxxxxbxxbbxf4thorder polynomial:
14 i xif[xi]f[ , ] f[ , , ] f[ , , , ] f[ , , , , ] 0 0 -5 1 1 -3 2 -1 -15 3 2 39 4 -2 -9 To calculate b0, b1, b2, b3, we can construct a divided difference table as )(][iixfxfi 0 1 2 3 4 xi0 1 -1 2 -2 f(xi)-5 -3 -15 39 -9
15 i xif[ ]f[ , ] f[ , , ] f[ , , , ] f[ , , , , ] 0 0 -5 f[x1, x0] f[x2, x1, x0] f[x3, x2, x1, x0] f[x4, x3, x2, x1, x0] 1 1 -3 f[x2, x1] f[x3, x2, x1] f[x4, x3, x2, x1] 2 -1 -15 f[x3, x2] f[x4, x3, x2] 3 2 39 f[x4, x3] 4 -2 -9 ijijijiijjxxxxfxxfxxxxf],...,[],...,[],,...,,[1111First Second Third Fourth Divided Difference (bn)
16 i xif[ ]f[ , ] f[ , , ] f[ , , , ] f[ , , , , ] 0 0 -5 2 1 1 -3 6 2 -1 -15 18 3 2 39 12 4 -2 -9 To calculate b0, b1, b2, b3, we can construct a divided difference table as 201)5(3][][],[010101xxxfxfxxf611)3(15][][],[121212xxxfxfxxf18)1(2)15(39][][],[232323xxxfxfxxf1222)39(9][][],[343434xxxfxfxxf
17 i xif[ ]f[ , ] f[ , , ] f[ , , , ] f[ , , , , ] 0 0 -5 2 -4 1 1 -3 6 12 2 -1 -15 18 6 3 2 39 12 4 -2 -9 To calculate b0, b1, b2, b3

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• Winter '15
• ABM Toufique Hasan
• Numerical Analysis, Polynomial interpolation, b1

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