i	x	f(x)
1	-3	6	C1
			1	C2
2	-3	6		$$- \frac{7}{4}$$	C3
			-2,5		3	C4
3	-1	1		$$\frac{17}{4}$$		$$- \frac{37}{16}$$	C5
			6		$$- \frac{13}{8}$$		$$\frac{41}{144}$$	C
4	-1	1		1		$$- \frac{35}{24}$$		$$\frac{6031}{10584}$$
			6		-6		$$\frac{1723}{756}$$
5	-1	1		-5		$$\frac{176}{27}$$
			1		$$\frac{34}{9}$$
6	0	2		$$\frac{2}{3}$$
			2
7	0,5	3

POLITECHNIKA WARSZAWSKA

WYDZIAŁ INŻYNIERII ŚRODOWISKA

Metody numeryczne

Ćwiczenie 2

Wykonała

Weronika Hejko gr. IW

Rok 2015/2016

Cel ćwiczenia:

Na podstawie następujących danych wyznaczyć wielomian interpolacyjny oraz obliczyć jego wartość w pkt. x=n, oraz wykonać wykres w programie Grapher.

Dane:

i	x	f(x)
1	-3	n
2	-3	n
3	-1	1
4	-1	1
5	-1	1
6	0	2
7	0,5	3

n wynosi 6.

f(x₁,x₂) = 1

f(x₃,x₄) = n=6

f(x₄,x₅) = n=6

f(x₃,x₄,x₅)=1

Obliczenia

$$f\left( x_{2},x_{3} \right) = \frac{f\left( x_{3} \right) - f\left( x_{2} \right)}{x_{3} - x_{2}} = \frac{1 - 6}{- 1 - ( - 3)} = \frac{- 5}{2} = - 2,5$$

$$f\left( x_{5},x_{6} \right) = \frac{f\left( x_{6} \right) - f\left( x_{5} \right)}{x_{6} - x_{5}} = \frac{2 - 1}{0 - ( - 1)} = \frac{1}{1} = 1$$

$$f\left( x_{6},x_{7} \right) = \frac{f\left( x_{7} \right) - f\left( x_{6} \right)}{x_{7} - x_{6}} = \frac{3 - 2}{0,5 - 0} = \frac{1}{0,5} = 2$$

$$f\left( x_{1},x_{2},x_{3} \right) = \frac{f\left( x_{2},x_{3} \right) - f\left( x_{1},x_{2} \right)}{x_{3} - x_{1}} = \frac{- 2,5 - 1}{- 1 - ( - 3)} = - \frac{3,5}{2} = - 1,75$$

$$f\left( x_{2},x_{3},x_{4} \right) = \frac{f\left( x_{3},x_{4} \right) - f\left( x_{2},x_{3} \right)}{x_{4} - x_{2}} = \frac{6 - ( - 2,5)}{- 1 - ( - 3)} = \frac{8,5}{2} = 4,25$$

$$f\left( x_{4},x_{5},x_{6} \right) = \frac{f\left( x_{5},x_{6} \right) - f\left( x_{4},x_{5} \right)}{x_{6} - x_{4}} = \frac{1 - 6}{0 - ( - 1)} = \frac{- 5}{1} = - 5$$

$$f\left( x_{5},x_{6},x_{7} \right) = \frac{f\left( x_{6},x_{7} \right) - f\left( x_{5},x_{6} \right)}{x_{7} - x_{5}} = \frac{2 - 1}{0,5 - ( - 1)} = \frac{1}{1,5} = \frac{2}{3}$$

$$f\left( x_{1},x_{2},x_{3},x_{4} \right) = \frac{f\left( x_{2},x_{3},x_{4} \right) - f\left( x_{1},x_{2},x_{3} \right)}{x_{4} - x_{1}} = \frac{4,25 - ( - 1,75)}{- 1 - ( - 3)} = \frac{6}{2} = 3$$

$$f\left( x_{2},x_{3},x_{4},x_{5} \right) = \frac{f\left( x_{3},x_{4},x_{5} \right) - f\left( x_{2},x_{3},x_{4} \right)}{x_{5} - x_{2}} = \frac{1 - 4,25}{- 1 - ( - 3)} = \frac{- 3,25}{1,5} = - \frac{13}{8}$$

$$f\left( x_{3},x_{4},x_{5},x_{6} \right) = \frac{f\left( x_{4},x_{5},x_{6} \right) - f\left( x_{3},x_{4},x_{5} \right)}{x_{6} - x_{3}} = \frac{- 5 - 1}{0 - ( - 1)} = - \frac{6}{1} = - 6$$

$$f\left( x_{4},x_{5},x_{6},x_{7} \right) = \frac{f\left( x_{5},x_{6},x_{7} \right) - f\left( x_{4},x_{5},x_{6} \right)}{x_{7} - x_{4}} = \frac{\frac{2}{3} - ( - 5)}{0,5 - ( - 1)} = \frac{\frac{17}{3}}{1,5} = \frac{34}{9}$$

$$f\left( x_{1},x_{2},x_{3},x_{4},x_{5} \right) = \frac{f\left( x_{2},x_{3},x_{4},x_{5} \right) - f\left( x_{1},x_{2},x_{3},x_{4} \right)}{x_{5} - x_{1}} = \frac{- \frac{13}{8} - 3}{- 1 - ( - 3)} = - \frac{37}{16}$$

$$f\left( x_{2},x_{3},x_{4},x_{5},x_{6} \right) = \frac{f\left( x_{3},x_{4},x_{5},x_{6} \right) - f\left( x_{2},x_{3},x_{4},x_{5} \right)}{x_{6} - x_{2}} = \frac{- 6 - \frac{- 13}{8}}{0 - ( - 3)} = - \frac{35}{24}$$

$$f\left( x_{3},x_{4},x_{5},x_{6},x_{7} \right) = \frac{f\left( x_{4},x_{5},x_{6},x_{7} \right) - f\left( x_{3},x_{4},x_{5},x_{6} \right)}{x_{7} - x_{3}} = \frac{\frac{34}{9} - ( - 6)}{0,5 - ( - 1)} = \frac{176}{27}$$

$$f\left( x_{1},x_{2},x_{3},x_{4},x_{5},x_{6} \right) = \frac{f\left( x_{2},x_{3},x_{4},x_{5},x_{6} \right) - f\left( x_{1},x_{2},x_{3},x_{4},x_{5} \right)}{x_{6} - x_{1}} = = \frac{- \frac{35}{24} - \frac{37}{16}}{0 - ( - 3)} = \frac{41}{144}$$

$$f\left( x_{2},x_{3},x_{4},x_{5},x_{6},x_{7} \right) = \frac{f\left( x_{3},x_{4},x_{5},x_{6},x_{7} \right) - f\left( x_{2},x_{3},x_{4},x_{5},x_{6} \right)}{x_{7} - x_{2}} = = \frac{\frac{176}{27} - ( - \frac{35}{24})}{0,5 - ( - 3)} = \frac{1723}{756}$$

$$f\left( x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7} \right) = \frac{f\left( x_{2},x_{3},x_{4},x_{5},x_{6},x_{7} \right) - f\left( x_{1},x_{2},x_{3},x_{4},x_{5},x_{6} \right)}{x_{7} - x_{1}} = = \frac{\frac{1723}{756} - \frac{41}{144}}{0,5 - ( - 3)} = \frac{6031}{10584}$$

Wielomian ma postać

$$w\left( x \right) = 6 + 1\left( x + 3 \right) - \frac{7}{4}\left( x + 3 \right)\left( x + 3 \right) + 3\left( x + 3 \right)\left( x + 3 \right)\left( x + 1 \right) - \frac{37}{16}\left( x + 3 \right)\left( x + 3 \right)\left( x + 1 \right)\left( x + 1 \right) + \frac{41}{144}\left( x + 3 \right)\left( x + 3 \right)\left( x + 1 \right)\left( x + 1 \right)\left( x + 1 \right) + \frac{6031}{10584}x\left( x + 3 \right)\left( x + 3 \right)\left( x + 1 \right)\left( x + 1 \right)\left( x + 1 \right)$$

Można również zapisać w postaci:

$$w\left( x \right) = 6 + 1\left( x + 3 \right) - \frac{7}{4}\left( x + 3 \right)^{2} + 3\left( x + 3 \right)^{2}(x + 1) - \frac{37}{16}\left( x + 3 \right)^{2}\left( x + 1 \right)^{2} + \frac{41}{144}\left( x + 3 \right)^{2}\left( x + 1 \right)^{3} + \frac{6031}{10584}x\left( x + 3 \right)^{2}\left( x + 1 \right)^{3}$$

Jak również

w(x) = 0, 57x⁶ + 5, 41x⁵ + 17, 34x⁴ + 19, 25x³ + 0, 28x² − 5, 48x + 2