Find quadratic function which pass through the following points (0,1),(1,1),(2, 3)... 11
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1 Adrew Powuk - Math 49 (Numerical Aalysis) Iterpolatio Polyomial iterpolatio (system of equatio) Lier iterpolatio Fid a lie which pass through (,) (,) Fid a lie which pass through (-,-) (,) Quadratic iterpolatio Fid quadratic fuctio which pass through the followig poits (,),(,),(, ).... Fid quadratic fuctio which pass through the followig poits (,),(,),(, )..... Higher order iterpolatio.... Lagrage iterpolatio Fid a lie which pass through (,) (,) Fid a lie which pass through (, )(, 5) Fid quadratic fuctio which pass through the followig poits (,),(,),(, ) Fid quadratic fuctio which pass through the followig poits (, ),(, ),(, ) Fid quadratic fuctio which pass through the followig poits (,),(,),(, ) Fid quadratic fuctio which pass through the followig poits (, ),(, ),(, ) Fid quadratic fuctio which pass through the followig poits (,),(, ),(,) Fid cubic fuctio which pass through the followig poits (, ),(, ),(, ),(, 7) Fid quadratic fuctio which pass through the followig poits (,),(, 7),(, 5),(, ) Fid cubic fuctio which pass through the followig poits (, ),(, ),(, ),(, 5). 7.. Fid iterpolatio polyomial which pass through the followig poits (,),(, ),(, 5),(, 7),(4, 9)..... Newto divided differece iterpolatio formula..... Fid a lie which pass through (,) (,) Fid a lie which pass through (, )(, 5) Fid a lie which pass through (,),(, 5)... 5
2 Adrew Powuk - Math 49 (Numerical Aalysis)..4 Fid quadratic fuctio which pass through the followig poits (,),(,),(, ) Fid quadratic fuctio which pass through the followig poits (,),(,),(, ) Fid quadratic fuctio which pass through the followig poits (, ),(,),(, ) Fid cubic fuctio which pass through the followig poits (,),(,),(, ),(, 7) Fid cubic fuctio which pass through the followig poits (, ),(,),(, 8),(,7) Fid cubic fuctio which pass through the followig poits (, ),(, ),(, ),(, 7) Fid cubic fuctio which pass through the followig poits (,),(, ),(, 5),(, 7) Fid cubic fuctio which pass through the followig poits (, ),(, ),(, ),(, 5) Fid iterpolatio polyomial which pass through the followig poits (,),(, 7),(, 5),(, ) Fid iterpolatio polyomial which pass through the followig poits (,),(,),(,),(, 6)(4,5) Properties of divided differeces (*) Error i polyomial iterpolatio (*) Hermite iterpolatio (**) Splie iterpolatio Liear splies (*) Example (,), (,),(,) Geeral case Example (,), (,),(,) Quadratic splies (*) Fid quadratic splie which pass through the followig poits (,),(,),(,5) Cubic Splies Fid cubic splie for (,),(,),(,6) Example (,),(,),(,) Example (,),(,),(,) Example (,),(,),(,) Example (,),(,),(,5) Example (,),(,),(,),(,) Example (,),(,),(,),(,)... 9
3 Adrew Powuk - Math 49 (Numerical Aalysis).6..8 Example (,),(,),(,),(,),(4,) Geeral case (**)....7 Review Summer
4 Adrew Powuk - Math 49 (Numerical Aalysis) Iterpolatio. (*) Polyomial iterpolatio (system of equatio) Curve expert (***) 4
5 Adrew Powuk - Math 49 (Numerical Aalysis).. Iterpolatio coditios ( x, y ),( x, y ),...,( x, y ),,,,, 4 y x, a, a, a,..., a y a a x a x a, a, a,..., a x, a, a, a,..., a y a a a x, a, a, a,..., a y a a a a a a x, a, a, a,..., a y a, a, a y x x y x For example y ax bx c ( x, y ),( x, y ),( x, y ) ax bx c y ax bx c y a, b, c ax bx c y (*) Hermitte iterpolatio y x, a, a,..., a ( x, y ),( x, y ),...,( x, y ) () () ( x, y ),...,( x, y )... x, a, a,..., a y m x, a, a,..., a y m... x, a, a,..., a y... x, a, a,..., a y... m () () x, a, a,..., a y m () () m m a, a,..., a m 5
6 Adrew Powuk - Math 49 (Numerical Aalysis).. Zero order iterpolatio... Fid a lie which passes through (,) y... Fid a lie which passes through (4,7) y 7 6
7 Adrew Powuk - Math 49 (Numerical Aalysis).. Lier iterpolatio y ax b ( x, y ) a?, b? ( x, y ) l : y ax b y ax b (, ) l : y ax b y ax b y ax b ( x, y ) l y y x y x y a, b x x x x y ax b y x y l y ax b y y x y x y x x x x x y y y x y y y y y ( x x ) y x x x x x x x y y x x y y y y y x y x ( y x y x ) x y x x x x x x x x x x x x y x y y x x x y x y x y x y y y x y x x x x x x x x y y y y y y y y ( x x ) y x x x x x x x x y y x x y y y y y x y x x y x y x y x x x x x x x x x x x x x x yx y x x y xy y y x y y x y x y x x x x x x x x x Sol=Solve[{a*x+by,a*x+by},{a,b}] a*x+b/.sol {{a-((-y+y)/(x-x)),b-((x y-x y)/(x-x))}} {-((x (-y+y))/(x-x))-(x y-x y)/(x-x)} 7
8 Adrew Powuk - Math 49 (Numerical Aalysis)... Fid a lie which passes through (,) (,), (, ) y ax b a b b b a b a a y x... Fid a lie which passes through (-,-) (,), (, ) y ax b a( ) b ( b) b b b b a b b a b a b a b b b b a a a y x 8
9 Adrew Powuk - Math 49 (Numerical Aalysis)..4 Quadratic iterpolatio y ax bx c ( x, y ),( x, y ),( x, y ) ax bx c y ax bx c y a, b, c ax bx c y Sol=Solve[{a*x^+b*x+cy,a*x^+b*x+cy,a*x^+b*x+cy},{a,b,c}] a*x^+b*x+c/.sol {{a-((-x y+x y+x y-x y-x y+x y)/((x-x) (x -x x-x x+x x))),b-((x y-x y-x y+x y+x y-x y)/((x-x) (x-x) (x-x))),c-((-x x y+x x y+x x y-x x y-x x y+x x y)/((x-x) (x-x) (x-x)))}} {-((x (-x y+x y+x y-x y-x y+x y))/((x-x) (x -x x-x x+x x)))-(x (x y- x y-x y+x y+x y-x y))/((x-x) (x-x) (x-x))-(-x x y+x x y+x x y-x x y-x x y+x x y)/((x-x) (x-x) (x-x))} 9
10 Adrew Powuk - Math 49 (Numerical Aalysis)..4. Fid quadratic fuctio which pass through the followig poits (,),(,4),(,9) x=;y=; x=;y=4; x=;y=9; Sol=Solve[{a*x^+b*x+c==y, a*x^+b*x+c==y, a*x^+b*x+c==y},{a,b,c}] a*x^+b*x+c/.sol {{a->,b->,c->}} {+ x+x } Solutio x x..4. Fid quadratic fuctio which pass through the followig poits (,),(,),(, ). y ax bx c (,),(,),(, ) (,) : a b c c (,) : a( ) b( ) c a b (, ) : a b c a b c a b b a a b a a y ax bx c y x x
11 Adrew Powuk - Math 49 (Numerical Aalysis)..4. Fid quadratic fuctio which pass through the followig poits (,),(,),(, ). y ax bx c (,),(,),(, ) (,) : a b c c c (,) : a b c a b c a b (, ) : a b c a4 b c a4 b a b a b a b a b a4 b 4a b 4a b 4( b) b 4( b) b b b a b ( ) a y ax bx c y x x
12 Adrew Powuk - Math 49 (Numerical Aalysis)..5 Higher order iterpolatio y a x a x... a x a ( x, y ),( x, y ),...,( x, y ) y a x a x... a x a... a, a,..., a y a x a x... a x a y x x... x a y x x... x a y x x... x a y x x... x a a, a,..., a y a x a x... a x a Vadermode matrix x x... x x x... x x x... x x x... x
13 Adrew Powuk - Math 49 (Numerical Aalysis)
14 Adrew Powuk - Math 49 (Numerical Aalysis). Lagrage iterpolatio ( x, y ),( x, y ),...,( x, y ) ( x) y L ( x) y L ( x) y L ( x)... y L ( x) Liear iterpolatio ( x) y L (x ) y L ( x ) Quadratic iterpolatio ( x) y L ( x ) y L ( x) y L ( x) Cubic iterpolatio ( x) y L ( x) y L ( x) y L ( x) y L ( x)... ( x y L x y L x y L x y L ( x) ) ( ) ( )... ( ) i i i 4
15 Adrew Powuk - Math 49 (Numerical Aalysis) Joseph-Louis Lagrage Joseph-Louis (Giuseppe Luigi), comte de Lagrage (5 Jauary 76 April 8 5
16 Adrew Powuk - Math 49 (Numerical Aalysis) Liear iterpolatio ( x, y ),( x, y ) ( x, y ),( x, y ) x x x x L ( x), L ( x) x x x x ( x) y L ( x) y L ( x) ( x) y x x x x y x x x x Quadratic iterpolatio ( x, y )( x, y )( x, y ) ( x) y L ( x) y L ( x) y L ( x) ( x x )( x x )( x x ) L ( x) ( x ) x ( x x )( x x ) ( x x )( x x ) ( x x )( x x ) ( x x )( x x )( x x ) ( x x )( x x ) L ( x) ( x x )( x x )( x x ) ( x x )( x x ) ( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x ) ( x) y L ( x) y L ( x) y L ( x) ( x x )( x x ) ( x x )( x x ) ( x x )( x x ) ( x x )( x x ) ( x x )( x x ) ( x) y y y ( x x )( x x ) ( x x )( x x ) ( x ) ) x ( x x L ( x ) L ( x ) L ( x ) 6
17 Adrew Powuk - Math 49 (Numerical Aalysis) Cubic iterpolatio ( x) y L ( x) y L ( x) y L ( x) y L ( x) ( x, y )( x, y )( x, y )( x, y ) ( x) y L ( x) y L ( x) y L ( x) y L ( x) ( x x ) L ( x) ( x x )( x x )( x x ) ( x x )( x x )( x x ) ( x x )( x x )( x x )( x x ) ( x x )( x x )( x x ) ( x x )( x x )( x x )( x x ) ( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x )( x x ) ( x x )( x x )( x x ) ( x x )( x x )( x x )( x x ) ( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x )( x x ) ( x x )( x x )( x x ) ( x x )( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x )( x x ) ( x) y L ( x) y L ( x) y L ( x) y L ( x) L ( x ) L ( x ) ( x x )( x x )( x x ) ( x x )( x x )( x x ) ( x x )( x x )( x x ) ( x x )( x x )( x x ) ( x) y y ( x x )( x x )( x x ) ( x x )( x x )( x x ) y ( x x )( x x )( x x ) ( x x )( x x )( x x ) y ( x x )( x x )( x x ) ( x x )( x x )( x x ) L ( x ) L ( x ) 7
18 Adrew Powuk - Math 49 (Numerical Aalysis).. Fid a lie which passes through (,) (,) (,),(, ) ( x, y )( x, y ) ( x x )( x x ) L ( x) ( x x )( x x ) ( x x )( x x ) L ( x) ( x x )( x x ) x x x x x x x x x x x x ( x) y L ( x) y L ( x) y y x x x x x x x ( x) x x x x.. Fid a lie which passes through (, )(, 5) (, )(, 5) ( x, y )( x, y ) ( x ) ( x ) ( x ) L ( x) x ( ) ( ) ( ) ( x )( x ) L ( x) ( )( ) ( x ) x ( ) y y L ( x) y L ( x) ( x) 5( x ) 6 x 5x 5 x 8
19 Adrew Powuk - Math 49 (Numerical Aalysis).. Fid quadratic fuctio which passes through the followig poits (,),(,),(, ). (,),(,),(, ) ( x, y ),( x, y ),( x, y ) ( x x )( x x ) ( x x )( x x ) ( x x )( x x ) ( x) y y y ( x x )( x x ) ( x x )( x x ) ( x x )( x x ) ( x ( ))( x x ) ( x )( x x ) ( x ) ( x ( )) ( ( ))( x ) (( ) )(( ) x ) ( x )( x ( )) ( x ) x x x x x x x x x ( x ( ))( x ) ( x )( x ) ( x )( x ( )) ( ( ))( ) (( ) )(( ) ) ( )( ( )) ( x )( x ) x( x ) x( x ) ( ) x x x x 9
20 Adrew Powuk - Math 49 (Numerical Aalysis)..4 Fid quadratic fuctio which passes through the followig poits (, ),(, ),(, ). (, ),(, ),(, ) ( x, y ),( x, y ),( x, y ) ( x) y L ( x) y L ( x) y L ( x) ( x x )( x x )( x x ) ( x x )( x x ) ( x ) ( x )( x ) ( x )( x ) L ( x) ( x x )( x x )( x x ) ( x x )( x x ) ( ) ( )( ) ( )( ) ( x x )( x x )( x x ) ( x x )( x x ) ( x )( x ) ( x ) ( x )( x ) L ( x) ( x x )( x x )( x x ) ( x x )( x x ) ( )( ) ( ) ( )( ) ( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x ) ( x) y L ( x) y L ( x) y L ( x) ( x x )( x x ) ( x )( x )( x ) ( x x )( x x ) ( )( )( ) ( x )( x ) ( x )( x ) ( x )( x ) ( x) ( )( ) ( )( ) ( )( ) ( x )( x ) ( x )( x ) ( )( ) ( ) ( ) x ( x )( x ) ( )( ) ( x )( x ) ( x )( x ) ( x )( x ) x( x ) ( )( ) ( )( ) ( )( ) * x( x ) ( x ) x x x x x x x x
21 Adrew Powuk - Math 49 (Numerical Aalysis)..5 Fid quadratic fuctio which passes through the followig poits (,),(,),(, ). (,),(,),(, ) ( x, y ),( x, y ),( x, y ) ( x) y L ( x) y L ( x) y L ( x) ( x x )( x x )( x x ) ( x x )( x x ) ( x ) ( x )( x ) ( x )( x ) L ( x) ( x x )( x x )( x x ) ( x x )( x x ) ( ) ( )( ) ( )( ) ( x x )( x x )( x x ) ( x x )( x x ) ( x )( x ) ( x ) ( x )( x ) L ( x) ( x x )( x x )( x x ) ( x x )( x x ) ( )( ) ( ) ( )( ) ( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x ) ( x x )( x x ) ( x )( x )( x ) ( x x )( x x ) ( )( )( ) ( x )( x ) ( x )( x ) ( x )( x ) ( x) x x ( )( ) ( )( ) ( )( ) ( x )( x ) ( )( )
22 Adrew Powuk - Math 49 (Numerical Aalysis)..6 Fid quadratic fuctio which passes through the followig poits (,5),(5,),(, 95). (,5),(5,),(, 95) ( x, y ),( x, y ),( x, y ) ( x) y L ( x) y L ( x) y L ( x) ( x x )( x x )( x x ) L ( x) ( x ) ( x x )( x x ) ( x 5)( x ) ( x x )( x x ) ( 5)( ) x ( x x )( x x ) ( x x )( x x )( x x ) ( x x )( x x ) ( x )( x ) L ( x) ( x x )( x x )( x x ) ( x x )( x x ) (5 )(5 ) ( x x )( x x )( x x ) ( x x )( x x ) ( x )( x 5) L ( x) ( x x )( x x )( x x ) ( x x )( x x ) ( )( 5) ( x 5)( x ) ( x )( x ) ( x )( x 5) ( x) x 7x ( 5)( ) (5 )(5 ) ( )( 5)
23 Adrew Powuk - Math 49 (Numerical Aalysis)..7 Fid quadratic fuctio which passes through the followig poits (, ),(, ),(,). (, ),(, ),(,) ( x, y ),( x, y ),( x, y ) ( x) y L ( x) y L ( x) y L ( x) ( x x )( x x )( x x ) ( x ( )) ( x )( x ) ( x )( x ) L ( x) ( x )( )( ) (( ) ( )) (( ) )(( ) ) (( ) )(( ) ) x x x x x ( x x )( x x )( x x ) ( x ( ))( x ) ( x ) ( x ( ))( x ) L ( x) ( x x )( x x )( x x ) ( ( ))( ( )) ( ) ( ( ))( ) ( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x ) ( x) y L ( x) y L ( x) y L ( x) ( x ( ))( x )( x ) ( ( ))( )( ) ( x ( ))( x ) ( ( ))( ) ( x )( x ) ( x ( ))( x ) ( x ( ))( x ) ( x) x (( ) )(( ) ) ( ( ))( ) ( ( ))( ) x..8 Fid quadratic fuctio which passes through the followig poits (,),(, ),(,). (,),(, ),(,) ( x, y ),( x, y ),( x, y ) ( x) y L ( x) y L ( x) y L ( x) L ( x) ( x x )( x x ) ( x )( x ) ( x x )( x x ) ( )( ) ( x x )( x x ) x x x x L ( x) x x ( x x )( x x ) ( x x )( x x ) ( x )( x ) x x L ( x) ( x x )( x x ) ( )( ) x x ( x )( x ) ( x )( x ) ( x) x ( ) ( ) ( )( ) x
24 Adrew Powuk - Math 49 (Numerical Aalysis)..9 Fid quadratic polyomial which passes through the followig poits (, 5),(, ),(5, 6). (, 5),(, ),(5, 6) ( x, y ),( x, y ),( x, y ) (x x )( x x )( x x ) ( x x )( x x ) ( x ( ))( x 5) x x L ( x) (x x )( x x )( x x ) ( x x )( x x ) ( ( ))( 5) ( x x )( x x) ( x x ) ( x x )( x x ) x x 5 5x L ( x) ( x x )(x x )( x x ) ( x x )( x x ) 5 4 ( x x )( x x )(x x ) L ( x) ( x x )( x x )(x x ) ( x) y L ( x) y L ( x) y L ( x) x x 5 5 ( x x )( x x ) ( x )( x ( )) ( x x )( x x ) (5 )(5 ( )) x 4 x x 5 5 ( x ( ))( x 5) ( x )( x ( )) ( x) ( 5) ( ) 6 5 x x ( ( ))( 5) (5 )(5 ( )) 4
25 Adrew Powuk - Math 49 (Numerical Aalysis).. Fid cubic fuctio which passes through the followig poits (, ),(, ),(, ),(, 7). (, ),(, ),(, ),(, 7) ( x, y ),( x, y ),( x, y ),( x, y ) ( x) y L ( x) y L ( x) y L ( x) y L ( x) x x x x x x x ( x x )( x x )( x x )( x x ) ( x )( x )( x ) L ( x) ( x )( )( )( ) (( ) )(( ) )(( ) ) ( x x )( x x ) L ( x) ( x x )( x x ) ( x ( ))( x )( x ) ( x x )( x x )( x x )( x x ) ( ( ))( )( ) ( x x )( x x )( x x )( x x ) ( x ( ))( x )( x ) L ( x) ( x x )( x x )( x x )( x x ) ( ( ))( )( ) ( x x )( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x )( x x ) ( x ( ))( x )( x ) ( ( ))( )( ) ( x )( x )( x ) ( x ( ))( x )( x ) ( x) ( ) ( ) (( ) )(( ) )(( ) ) ( ( ))( )( ) ( x ( ))( x )( x ) ( x ( ))( x )( x ) 7 x ( ( ))( )( ) ( ( ))( )( ) 5
26 Adrew Powuk - Math 49 (Numerical Aalysis).. Fid cubic polyomial which passes through the followig poits (,),(, 7),(, 5),(, ). (,),(, 7),(, 5),(, ) ( x, y ),( x, y ),( x, y ),( x, y ) ( x) y L ( x) y L ( x) y L ( x) y L ( x) x ( x x )( x x )( x x )( x x ) ( x ( ))( x )( x ) L ( x) ( x )( x x )( x x )( x x ) ( ( ))( )( ) ( x x )( x x ) L ( x) ( x x )( x x ) ( x )( x )( x ) ( x x )( x x )( x x )( x x ) ( )( )( ) ( x x )( x x )( x x )( x x ) ( x )( x ( ))( x ) L ( x) ( x x )( x x )( x x )( x x ) ( )( ( ))( ) ( x x )( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x )( x x ) ( x ))( x ( ))( x ) ( )( ( ))( ) ( x ( ))( x )( x ) ( x )( x )( x ) ( x) ( 7) ( ( ))( )( ) ( )( )( ) ( x )( x ( ))( x ) ( x )( x ( ))( x ) 5 ( ) ( )( ( ))( ) ( )( ( ))( ) x x x 6
27 Adrew Powuk - Math 49 (Numerical Aalysis).. Fid cubic fuctio which passes through the followig poits (, ),(, ),(,),(, 5). (, ),(, ),(,),(, 5) ( x, y ),( x, y ),( x, y ),( x, y ) ( x) y L ( x) y L ( x) y L ( x) y L ( x) ( x x )( x x )( x x )( x x ) L ( x) ( x )( )( )( ) x x x x x x x ( x )( x )( x ) ( )( )( ) ( x x )( x x ) L ( x) ( x x )( x x ) ( x )( x )( x ) ( x x )( x x )( x x )( x x ) ( )( )( ) ( x x )( x x )( x x )( x x ) ( x )( x )( x ) L ( x) ( x x )( x x )( x x )( x x ) ( )( )( ) ( x x )( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x )( x x ) ( x )( x )( x ) ( )( )( ) ( x )( x )( x ) ( x )( x )( x ) ( x) ( ) ( )( )( ) ( )( )( ) ( x )( x )( x ) ( x )( x )( x ) 5 x x x ( )( )( ) ( )( )( ) 7
28 Adrew Powuk - Math 49 (Numerical Aalysis).. Fid cubic fuctio which passes through the followig poits (, ),(, ),(,),(,) (, ),(, ),(,),(,) ( x, y ),( x, y ),( x, y ),( x, y ) ( x) y L ( x) y L ( x) y L ( x) y L ( x ) ( x x ) L ( x) ( x x )( x x )( x x ) ( x x )( x x )( x x )( x x ) ( x )( x )( x ) ( )( )( ) ( x x )( x x )( x x )( x x ) ( x )( x )( x ) L ( x) ( x x )( x x )( x x )( x x ) ( )( )( ) ( x x )( x x )( x x )( x x ) ( x )( x )( x ) L ( x) ( x x )( x x )( x x )( x x ) ( )( )( ) ( x x )( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x )( x x ) ( x )( x )( x ) ( )( )( ) ( x )( x )( x ) ( x )( x )( x ) ( x) ( ) ( ) ( )( )( ) ( )( )( ) ( x )( x )( x ) ( x )( x )( x ) x x x ( )( )( ) ( )( )( ) 8
29 Adrew Powuk - Math 49 (Numerical Aalysis)..4 Fid cubic fuctio which passes through the followig poits (, ),(, ),(,),(,) (, ),(, ),(,),(,) ( x, y ),( x, y ),( x, y ),( x, y ) ( x) y L ( x) y L ( x) y L ( x) y L ( x ) ( x x ) L ( x) ( x x )( x x )( x x ) ( x x )( x x )( x x )( x x ) ( x )( x )( x ) ( )( )( ) ( x x )( x x )( x x )( x x ) ( x )( x )( x ) L ( x) ( x x )( x x )( x x )( x x ) ( )( )( ) ( x x )( x x )( x x )( x x ) ( x )( x )( x ) L ( x) ( x x )( x x )( x x )( x x ) ( )( )( ) ( x x )( x x )( x x )( x x ) L ( x) ( x x )( x x )( x x )( x x ) ( x )( x )( x ) ( )( )( ) ( x )( x )( x ) ( x )( x )( x ) ( x) ( ) ( ) ( )( )( ) ( )( )( ) ( x )( x )( x ) ( x )( x )( x ) x x ( )( )( ) ( )( )( ) x 9
30 Adrew Powuk - Math 49 (Numerical Aalysis)..5 Fid iterpolatio polyomial which pass through the followig poits (,),(, ),(, 5),(, 7),(4, 9). (,),(, ),(, 5),(, 7),(4, 9) ( x, y ),( x, y ),( x, y ),( x, y ),( x, y ) 4 4 ( x) y L ( x) y L ( x) y L ( x) y L ( x) y L ( x) 4 4 ( x x )( x x )( x x )( x x )( x x ) 4 L ( x) ( x ) ( x x )( x x )( x x )( x x ) 4 ( x x )( x x )( x x )( x x ) x ( x x )( x x )( x x )( x x ) 4 4 ( x x )( x x )( x x )( x x )( x x ) 4 L ( x) ( x x )( x x )( x x )( x x )( x x ) 4 ( x x )( x x )( x x )( x x ) 4 ( x x )( x x )( x x )( x x ) 4 ( x x )( x x )( x x )( x x )( x x ) 4 ( x x )( x x )( x x )( x x ) 4 L ( x) ( x x )( x x )( x x )( x x )( x x ) ( x x )( x x )( x x )( x x ) 4 4 ( x x )( x x )( x x )( x x )( x x ) 4 ( x x )( x x )( x x )( x x ) 4 L ( x) ( x x )( x x )( x x )( x x )( x x ) ( x x )( x x )( x x )( x x ) 4 ( x x )( x x )( x x )( x x )( x x ) 4 L ( x) 4 ( x x )( x x )( x x )( x x )( x x ) ( x x )( x x )( x x )( x x ) ( x x )( x x )( x x )( x x ) (,),(, ),(, 5),(, 7),(4, 9) ( x, y ),( x, y ),( x, y ),( x, y ),( x, y ) 4 4 ( x )( x )( x )( x 4) L ( x) ( )( )( )( 4) ( x )( x )( x )( x 4) L ( x) ( )( )( )( 4) ( x )( x )( x )( x 4) L ( x) ( )( )( )( 4) ( x )( x )( x )( x 4) L ( x) ( )( )( )( 4) ( x )( x )( x )( x ) L ( x) 4 (4 )(4 )(4 )(4 )
31 Adrew Powuk - Math 49 (Numerical Aalysis) ( x )( x )( x )( x 4) ( x )( x )( x )( x 4) ( x) ( )( )( )( 4) ( )( )( )( 4) ( x )( x )( x )( x 4) ( x )( x )( x )( x 4) ( x )( x )( x )( x ) ( )( )( )( 4) ( )( )( )( 4) (4 ) (4 )(4 )(4 ) x
32 Adrew Powuk - Math 49 (Numerical Aalysis). Newto divided differece iterpolatio formula P ( x) f ( x ) P( x) f ( x ) ( x x ) f [ x, x ] P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] ( x x )( x x )( x x ) f [ x, x, x, x ]... Taylor polyomial ( ) df ( x ) d f ( x ) d f ( x ) f ( x) f ( x ) ( x x ) ( x x )... ( x x ) dx! dx! dx df ( x ) ( ) ( ) d f x d f x f ( x) f ( x ) ( x x ) ( x x )( x x ) ( x x )( x x )( x x )... dx! dx! dx f ( x ) f ( x ) fx [, x ] x x f [ x, x ] f [ x, x ] fx [, x, x ] x x f [ x, x, x ] f [ x, x, x ] fx [, x, x, x ] x x f [ x, x,..., x ] f [ x, x, x, x ] fx [, x, x,..., x ] x x
33 Adrew Powuk - Math 49 (Numerical Aalysis)
34 Adrew Powuk - Math 49 (Numerical Aalysis) Portrait of Newto at 46 i 689 by Godfrey Keller 5 December 64 March 76/7 4
35 Adrew Powuk - Math 49 (Numerical Aalysis).. Fid a lie which passes through (,) (,) (,),(, ) ( x, y )( x, y ) P ( x) f ( x ) ( x x ) f [ x, x ] ( x ) x fx [, x ] f ( x ) y f ( x ) f ( x ) x x.. Fid a lie which passes through (, )(, 5) (, )(, 5) ( x, y )( x, y ) P( x) f ( x ) ( x x ) f [ x, x ] ( x ) x x f f( x ) f( x ) 5 [ x, x ] x x.. Fid a lie which passes through (,),(, 5) (,),(, 5) ( x, y )( x, y ) P( x) f ( x ) ( x x ) f [ x, x ] ( x ) x fx f ( x ) f ( x ) 5 [, x ] x x..4 Fid the quadratic polyomial which passes through the followig poits (,),(,),(, ). (,),(,),(, ) ( x, y ),( x, y ),( x, y ) P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] ( x ) ( x )( x ) x fx f ( x ) f ( x ) [, x ] x x ( ) fx f ( x ) f ( x ) [, x ] x x ( ) fx [ f [ x, x ] f [ x, x ] x, x, x ] x x 5
36 Adrew Powuk - Math 49 (Numerical Aalysis) x f..5 Fid quadratic fuctio which passes through the followig poits (,),(,),(, ). (,),(,),(, ) ( x, y ),( x, y ),( x, y ) P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] ( x ) ( x )( x ) x( x ) x x fx [, x ] x x fx [, x ] x x f [ x, x, x f ( x ) f ( x ) f ( x ) f ( x ) f [ x, x ] f [ x, x ] ] x x..6 Fid quadratic fuctio which passes through the followig poits (, ),(,),(, ). (, ),(,),(, ) ( x, y ),( x, y ),( x, y ) P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] ( x )( ) ( x )( x ) x x x x x fx f ( x ) f ( x ) [, x ] x x f ( x ) f ( x ) fx [, x ] x x f [ x, x ] f [ x, x ] ( ) fx [, x, x ] x x 6
37 Adrew Powuk - Math 49 (Numerical Aalysis)..7 Fid cubic polyomial which passes through the followig poits (,),(,),(, ),(, 7). (,),(,),(, ),(, 7) ( x, y ),( x, y ),( x, y ),( x, y ) P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] ( x x )( x x )( x x ) f [ x, x, x, x ] ( x ) ( x )( x ) ( x )( x )( x ) x( x ) x x fx [, x ] fx f [, x ] x x [ x, x ] 4 x x fx f [, x, x ] x x [ x, x, x ] x x f [ x, x, x, x ] f ( x ) f ( x ) x x f ( x ) f ( x ) f ( x ) f ( x ) 7 4 (,),(,),(, ),(, 7) f [ x, x ] f [ x, x ] f[ x, x ] f[ x, x ] 4 f [ x, x, x ] f [ x, x, x ] x x x fx [ ] x fx [ ] x fx [ ] x fx [ ] 7 f x, x f x, x 7 f x, x 4 f [ x, x, x ] 4 f [ x, x, x ] fx [, x, x, x ] 7
38 Adrew Powuk - Math 49 (Numerical Aalysis)..8 Fid cubic fuctio which passes through the followig poits (, ),(,),(, 8),(,7). (, ),(,),(, 8),(,7) ( x, y ),( x, y ),( x, y ),( x, y ) P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] ( x x )( x x )( x x ) f [ x, x, x, x ] ( x ) ( x )( x ) ( x )( x )( x ) x f ( x ) f ( x ) fx [, x ] x x f ( x ) f ( x ) 8 7 fx [, x ] 7 x x f ( x ) f ( x ) f [ x, x ] 9 x x f [ x, x ] f [ x, x ] 7 fx [, x, x ] x x f [ x, x ] f [ x, x ] 9 7 f [ x, x, x ] 6 x x f [ x, x, x ] fx [, x, x ] 6 fx [, x, x, x ] x x 8
39 Adrew Powuk - Math 49 (Numerical Aalysis)..9 Fid cubic fuctio which passes through the followig poits (,),(,),(,),(, 7). (,),(,),(,),(, 7) ( x, y ),( x, y ),( x, y ),( x, y ) P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] ( x x )( x x )( x x ) f [ x, x, x, x ] ( x ( )) ( x ( ))( x ) ( x ( ))( x )( x ) ( x ) x( x ) x( x ) x x fx [, x ] x x ( ) fx f [, x ] x x [ x, x ] 6 x x fx [, x, x ] x x ( ) f [ x, x, x ] f f ( x ) f ( x ) f ( x ) f ( x ) f ( x ) f ( x ) 7 f [ x, x ] f [ x, x ] f[ x, x ] f[ x, x ] 6 x x f [ x, x, x ] f [ x, x, x ] [ x, x, x, x ] x x ( ) 9
40 Adrew Powuk - Math 49 (Numerical Aalysis).. Fid cubic fuctio which passes through the followig poits (,),(, ),(, 5),(, 7). (,),(, ),(, 5),(, 7) P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] ( x x )( x x )( x x ) f [ x, x, x, x ] P ( x) ( x ) ( x )( x ) ( x )( x )( x ) x fx f ( x ) f ( x ) [, x ] x x f ( x ) f ( x ) 5 fx [, x ] x x f ( x ) f ( x ) 7 5 f [ x, x ] x x f [ x, x ] f [ x, x ] fx [, x, x ] x x f[ x, x ] f[ x, x ] f [ x, x, x ] x x f [ x, x, x ] f [ x, x, x ] f [ x, x, x, x ] x x 4
41 Adrew Powuk - Math 49 (Numerical Aalysis).. Fid cubic fuctio which passes through the followig poits. (,) (,7) (,7) (,9) x x f x, x P x f x 5 x x x x f x, x, x x x x x x x f x, x, x, x x, f x x, f x 7 x, f x 7 x, f x 9 x, f x x, f x 7 f x f x 7 f x, x x x x, f x 7 f x f x 7 7 f x, x x x x, f x 9 f x f x 9 7 f x, x x x
42 Adrew Powuk - Math 49 (Numerical Aalysis) f f f f f x f x 7 x, x x x f x f x 7 7 x, x x x x, x x x 6 f x, x f x, x 6 x, x, x 57 x x f x f x f x, x, x f x, x f x, x 966 x x 4 f f f f x, x f x, x 6 x, x, x 57 x x f x, x f x, x 966 x, x, x 4 x x f x, x, x f x, x, x 4 57 x, x, x, x x x P x f x x x f x, x 5 x x x x f x, x, x x x x x x x f x, x, x, x P x x 6 x x 57 x x x 5 5 x P x x 6 x x 57 x x 9x 9x x 4
43 Adrew Powuk - Math 49 (Numerical Aalysis) Verificatio F[x_]=+9 x-9 x + x ; F[] F[] F[] F[] (,) (,7) (,7) (,9) (,),(,),(, ),(, 7) x fx [ ] x fx [ ] 7 x fx [ ] 7 x fx [ ] 9 7 f x, x f x, x 9 7 f x, x 966 f [ x, x, x ] f[ x, x, x ] x x x x f [ x, x, x ] f[ x, x, x ] f [ x, x, x, x ] 4
44 Adrew Powuk - Math 49 (Numerical Aalysis).. Fid cubic fuctio which passes through the followig poits (, ),(, ),(,),(, 5). (, ),(, ),(,),(, 5) P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] ( x x )( x x )( x x ) f [ x, x, x, x ] P ( x) ( x ) ( x )( x ) ( x )( x )( x ) P ( x) x x x f ( x ) f ( x ) f [ x, x ] x x f ( x ) f ( x ) ( ) fx [, x ] 7 x x f ( x ) f ( x ) 5 f [ x, x ] 5 x x f [ x, x ] f [ x, x ] 7 fx [, x, x ] x x f[ x, x ] f[ x, x ] 5 7 f [ x, x, x ] 4 x x f [ x, x, x ] f [ x, x, x ] 4 fx [, x, x, x ] x x 44
45 Adrew Powuk - Math 49 (Numerical Aalysis).. Fid iterpolatio polyomial which passes through the followig poits (,),(, 7),(, 5),(, ). (,),(, 7),(, 5),(, ) P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] ( x x )( x x )( x x ) f [ x, x, x, x ] P ( x) ( x )4 ( x )( x ( )) ( x )( x ( ))( x ) P ( x) x x x f ( x ) f ( x ) 7 f [ x, x ] 4 x x f ( x ) f ( x ) 5 ( 7) fx [, x ] 4 x x ( ) f ( x ) f ( x ) 5 f [ x, x ] x x f [ x, x ] f [ x, x ] 4 4 fx [, x, x ] x x f[ x, x ] f[ x, x ] 4 f [ x, x, x ] x x ( ) f [ x, x, x ] f [ x, x, x ] f [ x, x, x, x ] x x 45
46 Adrew Powuk - Math 49 (Numerical Aalysis)..4 Fid iterpolatio polyomial which passes through the followig poits (,),(,),(,),(, 6)(4,5). (,),(,),(,),(, 6)(4,5) P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] 5 ( x x )( x x )( x x ) f [ x, x, x, x ] 5 ( x x )( x x )( x x )( x x ) f [ x, x, x, x, x ] 4 P ( x) ( x ) ( x )( x )5 ( x )( x )( x )5 ( x )( x )( x )( x ) x x x x fx 4 f [ x, x, x, x ] fx [, x, x, x ] [, x, x, x, x ] 4 x x 4 4 f [ x x x x f[ x, x, x ] f[ x, x, x ] 47 4,,, ] 9 4 x x 4 4 f [ x, x, x ] f [ x, x, x ] 5 fx [, x, x, x ] x x f [ x, x ] f [ x, x ] fx [, x, x ] 5 x x f[ x, x ] f[ x, x ] 5 f [ x, x, x ] x x f[ x, x ] f[ x, x ] f [ x, x, x ] 47 4 x x 4 4 f [ x ] f [ x ] fx [, x ] x x f[ x ] f[ x ] f [ x, x ] x x f[ x ] f[ x ] 6 f [ x, x ] 5 x x f[ x ] f[ x ] f [ x, x ] 44 4 x x
47 Adrew Powuk - Math 49 (Numerical Aalysis)..5 Fid iterpolatio polyomial which passes through the followig poits (, ),(,),(, 4),(, 9)(4,6),(5,5). (, ),(,),(, 4),(, 9)(4,6),(5,5) x, y, x, y,..., x, y 5 5 x, f x f x f x x, f x, f x, x x x f x f x 4 x, f x 4, f x, x x x f x f x 9 4 x, f x 9, f x, x 5 x x f x f x x 4, f x 6, f x, x x x 4 4 f x f x x 5, f x 5, f x, x x x f x, x f x, x f x, x f x, x, f x, x, x x x f x, x f x, x 5 f x, x 5, f x, x, x x x f x, x f x, x f x, x 7, f x, x, x 4 4 x x 4 4 f x, x f x, x f x, x 9, f x, x, x x x
48 Adrew Powuk - Math 49 (Numerical Aalysis) f x, x, x f x, x, x f x, x, x f x, x, x, f x, x, x, x x x f x, x, x f x, x, x 4 f x, x, x, f x, x, x, x 4 4 x x 4 4 f x, x f x, x, x, f x, x, x, x, x f x, x, x x x 5 5 f x, x, x, x f x, x, x, x f x, x, x, x 4 f x, x, x, x, f x, x, x, x, x 4 4 x x 4 4 f x, x, x, x f x, x, x, x f x, x, x, x, f x, x, x, x, x x x 5 5 f x, x, x, x, x 4 f x, x, x, x, x f x, x, x, x, x f x, x, x, x, x, f x, x, x, x, x, x x x 5 Iterpolatio polyomial x x f x, x x x x x f x, x, x x x x x x x f x, x, x, x P x f x 5 x x x x x x x x f x, x, x, x, x 4 x x x x x x x x x x f x, x, x, x, x, x
49 Adrew Powuk - Math 49 (Numerical Aalysis) P x x x x x x x x x x x x x x x x 5 4 x x( x ) x x x x 49
50 Adrew Powuk - Math 49 (Numerical Aalysis)..6 Fid iterpolatio polyomial which passes through the followig poits. (,) (,7) (,7) (,9) (4,546) (5,95) (6,55987) Iterpolatio polyomial x x f x, x x x x x f x, x, x x x x x x x f x, x, x, x x x x x x x x x f x, x, x, x, x P x f x 5 x x x x x x x x x x f x, x, x, x, x, x x x x x x x x x x x x x f x, x, x, x, x, x, x x, f x x, f x 7 x, f x 7 x, f x 9 x 4, f x x 5, f x x 6, f x
51 Adrew Powuk - Math 49 (Numerical Aalysis) x, f x x, f x 7 f x f x 7 f x, x x x x, f x 7 f x f f x, x x x x, x x x x, f x 9 f x f x 9 7 f x 4, f x f x f x f x x, x 4 x x 4 4 x 5, f x f x f x f 5 4 x, x 4 5 x x x 6, f x f x f x f x, x 5 6 x x
52 Adrew Powuk - Math 49 (Numerical Aalysis) f f f f f x f x 7 x, x x x f x f x 7 7 x, x x x x, x x x 6 f x, x f x, x 6 x, x, x 57 x x f x f x f x, x f x, x 966 f x, x, x x x f x f x f 468 f f 4 x, x 4 x x 4 4 x, x x, x 5 6 x x f x f x x x f x f x
53 Adrew Powuk - Math 49 (Numerical Aalysis).4 (*) Error i polyomial iterpolatio.4. (*) Properties of divided differeces f x x x! f x x Theorem ( ) [,,..., ] ( ), (, ) For pairwise distict poits x, x,..., x i the doaub of a -times diffeetiable fuctio where the -th there exists ad iterior poiot mi x, x,..., x, m ax x, x,..., x derivative of f equals! times the -th divided differece at these poits: f f[ x, x,..., x ] f ( ), mi x, x,..., x, max x, x,..., x! Proof Let of ( ) P x is the iterpolatio polyomial for P x that the highest therm of P x is f [ x, x,..., x ] x x x x... x x f at x, x,..., x. The it follows from the Newto form. Let g be the reimader of the iterpolatio defied by g f P. The g has zeros x, x,..., x. By applyig the Rolle s theorem first to g, the to g ', ad so o util ( ) g, we fid that ( ) g has zero. This meas that ( ) ( ) ( ) g f P f[ x, x,..., x ]! the ( ) f[ x, x,..., x ] f ( ), mi x, x,..., x, max x, x,..., x.! 5
54 Adrew Powuk - Math 49 (Numerical Aalysis) ( ) f [ x, x,..., x ] f ( ), ( x, x )! P ( x) f ( x ) ( x x ) f [ x x ] f f (), ( x ) ( x x ) ( ) fx [ ]! ( x, x ) x x (), x f ( ) lim x lim x x x x () () lim P ( x) lim f ( x ) ( x x ) f f ( x ) ( x x ) f x x x x x ( ) ( ) ( ) f [ x, x,..., x ] f ( ), ( x, x )! P ( x) f ( x ) ( x x ) f [ x, x ] ( x x )( x x ) f [ x, x, x ] () () f ( x ) ( x x ) f ( ) ( x x )( x x ) f ( )! () fx [, x ] f ( ), ( x, x )! () f [ x, x, x ] f ( ), ( x, x )! lim x x x lim x x lim x x x lim x x x x x x x x x x x x x x lim P ( x) () () lim f ( x ) ( x x ) f ( ) ( x x )( x x ) f ( )! () () f ( x ) ( x x ) f ( x ) ( x x )( x x ) f ( x )! () () f( x ) ( x x ) f ( x ) ( x x ) f ( x )! 54
55 Adrew Powuk - Math 49 (Numerical Aalysis).4. (*) Geeralized Rolle s theorem Theorem () - If a real-valued fuctio f is cotiuous o a closed iterval [a,b], differetiable o the ope iterval (a, b), ad f(a) = f(b), the there exists a c i the ope iterval (a, b) such that Or If ) f : ) f is cotiuous o a closed iterval [a,b] ) f is differetiable o a ope iterval (a,b) 4) f(a) = f(b) f ' c. the there exists a c i the ope iterval (a, b) such that f ' c. Proof We kow that f a f b Case Fuctio is costat. ' x c a, b, f ' c f x cost f a f b f 55
56 Adrew Powuk - Math 49 (Numerical Aalysis) For all c ab, f ' c, the the theorem is true. Case Fuctio is ot costat. Case a It is possible to fid x ab, such that f x f a It is possible to fid a maximum of all Fuctio f f x i.e. f such that f x f a is cotiuous the from the Extreme Value Theorem we kow that f ad it is possible to fid x ab, max such that max f max f x f x. max x max a, b max. x max f x ab, max is fiite x f x ab, f ' x. By assumptio the fuctio is differetiable, the accordig to the Fermat s Theorem max The the theorem is true ad c x. max Case b It is possible to fid x ab, such that f x f a It is possible to fid a maximum of all Fuctio f f x such that f x f a i.e. f is cotiuous the from the Extreme Value Theorem we kow that ad it is possible to fid x mi ab, such that mi mi f max f x f x. x mi a, b mi. x f f x a, b mi mi is fiite x f x a, b f ' x. By assumptio the fuctio is differetiable, the accordig to the Fermat s Theorem mi The the theorem is true ad c x. mi 56
57 Adrew Powuk - Math 49 (Numerical Aalysis) Theorem Let us cosider - - cotiuously differetiable fuctio o a closed iterval - -th derivative exists o the ope iterval ab, ab, - there are itervals give by a b a b... a b i ab, such that Proof the f k bk f a there is a umber c for every k i ab, from to. such that the -th derivative of f at c is zero i.e. ( ) f c. From preseted theorem it is possible to get the followig coclusio. 57
58 Adrew Powuk - Math 49 (Numerical Aalysis) Theorem If the fuctio f : a, b is - cotiuous at ab, - differetiable at ab, - has + differet roots x x x... x i ab, the derivative of the fuctio f ' : a, b has at least differet roots c, c,...,,,, ab, c. Proof Let us cosider the roots of the fuctio f i.e. x x x... x. For every iterval x, x i i we have - fuctio - fuctio f f - f xi f xi is cotiuous o x, x i i, is differetiable o x, x i i ( x i are the roots), the accordig to the classical Roll s theorem exists some x, x i i i such that i There are itervals x, x i i the there are f ' i.e. the derivative has differet roots. i differet umbers ab, i f '. such that 58
59 Adrew Powuk - Math 49 (Numerical Aalysis) Theorem (Geeralised Roll s theorem) If the fuctio f : a, b is - cotiuous at ab, - -times differetiable at ab, - has + differet roots x x x... x i ab, the derivative of the fuctio ( f ) : a, b has at least oe root ab, Proof i.e. exist ab, such that ( ) f. We kow that fuctio f : a, b is - cotiuous at ab, - differetiable at ab, - has + differet roots x x x... x i ab, the accordig to previous theorem derivative of the fuctio f ' : a, b has at least differet roots c, c,...,,,, ab, c. 59
60 Adrew Powuk - Math 49 (Numerical Aalysis) c, c a b,,, Because - by assumptio is differetiable at, the ab, f () - c, c,, a, b x ab, f () x is cotiuous at the iterval the the fuctio c,, ab ) () f x is cotiuous at the iterval c, c,,. (because c, ab,, We kow that fuctio () f : c, c,, is - cotiuous at c, c,, - differetiable at, ab i particular c c,,, - has differet roots c c... c i c, c a, b,,,,, the accordig to previous theorem derivative of the fuctio c c... c,,., () f : c, c,, has at least - differet roots By iductio ( ) f x has oe root ab, 6
61 Adrew Powuk - Math 49 (Numerical Aalysis).4. Error theorem Applicatios f ( x) P ( x) error f ( x) P ( x) b a f ( x) dx P ( x) dx b b a error f ( x ) P ( x) dx a d d f ( x) P ( x) dx dx d error f ( x ) P ( x) dx 6
62 Adrew Powuk - Math 49 (Numerical Aalysis) Theorem If f is times cotiuously differetialble o closed iterval polyomial of degree at most that iterpolates iterval. The for each x i the iterval there exists f ad distict poits,,..., at x I a, b i that iterval such that: P x be a x x x i that ( x x )( x x )... ( x x ) f x P x x ( )! Proof ( ) ( ) ( x) f ( ),, x x f : a, b x, x,..., x ab, - grid poits,,,..., i i f x P x i f x P x t f t P t w t wx... i w x x x x x x x x x i 6
63 Adrew Powuk - Math 49 (Numerical Aalysis) P x w x P x w x P x w x f x x f x P x w x f x x x x x... x x f x x x x x f x P x x f x P x w x w x f x P x x x x x... x x w x f x P x x x... x x w x f x P x w x f x P x... w x f x P x x x x x w x x f x P x w x x x x x x x... f x P x w x f x P x f x P x w x w x f x P x f x P x x f x P x w x t Fuctio has + roots at x, x,..., x, x ab,. 6
64 Adrew Powuk - Math 49 (Numerical Aalysis) Case oe root x t f t P t P x w x P x w x f x t f t P t w t f x t f t P t t x P x w x P x w x f x ' t f ' t P ' t t x ' f x ' ' ' Fuctio such that t has two roots i the iterval ' i.e. ab, the accordig to the theorem, exists ab, f P P x wx f x ' ' ' The order of the iterpolatio polyomial is i.e. P x x cost the f x P x P w x f x P x ' w x f x P x f ' w x f x P x f ' w x ' ' f ' ' f f x P x f w x f x P x f x x P ' 64
65 Adrew Powuk - Math 49 (Numerical Aalysis) Case two poits x, x f x P x t f t P t w t w x f x P x t f t P t t x t x w x w t t x t x t f t P f x P x t t x t x w x t f t P f x P x t t x t x t x t x w x t f t P f x P x t t x t x w x t f t P f x P x t t x t x w x t f t P f t x P x w x t f t P f x P x t w x ' ' ' ' ' ' ' ' ' ' ' ' '' '' '' ' ' '' '' '' '' '' '' The order of the iterpolatio polyomial is i.e. P x wx f x '' t f '' t P t at b the P t at b '' '' t Fuctio has three roots ( x, x, x ) i the iterval ab, ab, such that '' i.e. the accordig to the theorem, exists 65
66 Adrew Powuk - Math 49 (Numerical Aalysis) f w x P x w x f x '' '' f '' f x P x P x f '' w x f x Case three poits x, x, x w t t x t x t x w ' t t x ' t x t x t x t x ' t x t x t x t x ' w ' t t x t x t x t x t x t x t x ' t x t x t x ' t x ' t x t x t x ' t x t x t x t x t x t x w '' t t x ' t x t x t x ' t x ' t x ' t x ' t x ' 6 w ' '' t t x ' t x ' 66
67 Adrew Powuk - Math 49 (Numerical Aalysis) f x P x w x w t t x t x t x t f t P t w t t f t P t P x w x f x P x w x f x P x w x f x P x w x f x ' t f ' t P ' t w ' t '' t f '' t P '' t w '' t ''' t f ''' t P ''' t w ''' t ''' ''' ''' 6 The order of the iterpolatio polyomial is P ''' t at bt c ''' i.e. P t at bt c the P x wx f x ''' t f ''' t 6 Fuctio ab, such that t has four roots ( x, x, x, x ) i the iterval ab, ''' i.e. the accordig to the theorem, exists f w x P x w x f x ''' ''' 6 f ''' 6 f x P x P x () w x f x f 6 67
68 Adrew Powuk - Math 49 (Numerical Aalysis) Case - grid poits x, x,..., x f x P x f ( ) w x! 68
69 Adrew Powuk - Math 49 (Numerical Aalysis) f ( x) e ( x, y ),( x, y ),( x, y ) x (,),(, e),(, e ) ( ) x f ( x) e x, x, ( x x )( x x )... ( x x ) ( ) ( x )( x )( x ) () error( x) f ( ) f ( ) ( )! ( )! ( x )( x )( x ) ( x )( x )( x ) e e ( )! ( )! Iterpolatio polyomial (,),(, e),(, e ) y a bx cx y a b c a a () * y() e a b * c * b * c * y() e a b * c * b * c * b c e b e c b 4c e ( e c ) 4c e ( e c ) 4c e e c 4c e c e e e e c e e b e c e e e e e e e e y a bx cx e e x x 69
70 Adrew Powuk - Math 49 (Numerical Aalysis) e=exp[]; Clear[x] Plot[{+(-(/) e +e-/)x+((e -e+)/)x,exp[x]},{x,,}] N[+(-(/) e +e-/)x+((e -e+)/)x ] x e.+.46 x x 7
71 Adrew Powuk - Math 49 (Numerical Aalysis).5 (**) Hermite iterpolatio (Newto form of the iterpolatio polyomial) Give f () i x, i, j k fid p x p m k k i i ( j) ( j) p x f x, j k i i i,... k such that Theorem For give iterpolatio coditios exists a uique iterpolatio polyomial iterpolatio coditios. p pm followig hermite 7
72 Adrew Powuk - Math 49 (Numerical Aalysis).5. Example p Quadratic iterpolatio for p p ' p x a bx cx p x b cx p a b c p ' b c p a b c o solutio p Cubic iterpolatio p p ' p x a bx cx dx p x b cx dx p a b c d p ' b c d p a b c d may solutios 7
73 Adrew Powuk - Math 49 (Numerical Aalysis) Special case () ( k ),,..., f x f x f x p Pk The solutio is the Taylor polyomial Newto divided differece method. x,..., x, x,..., x,..., x,..., x k k k p x f x f x, x x x f x, x, x x x... f x f x f x x f x x f x, lim, lim ' x x x x x x f x, x,..., x f x k! k ( k ) 7
74 Adrew Powuk - Math 49 (Numerical Aalysis) Appropriate table of the Newto s divided differeces require iformatio about derivatives. Example, p' p p 6, p ' 7, p '' 8 Table x f /!= ,,,,,,,,,, f x f f x f x f x x f x x f x x x x x x x 74
75 Adrew Powuk - Math 49 (Numerical Aalysis).5. Example Use the exteded ewto divided differece method to obtai a quartic polyomial that takes the value. X P(x) P (x) -9 4 x () f () f () f () f (4) f ',,,,,,,,, p x f x f x x x 4 f x x x x x x x f x x x x x x x x x x f x x x x x x x x x x x x x x x x x x x x x x p x x x x x 5x ' x 9 x 9x x p x x x x x x p x x p 4 75
76 Adrew Powuk - Math 49 (Numerical Aalysis).6 (**) Hermite iterpolatio (Lagrage form of the iterpolatio polyomial) Lagrage form of the iterpolatio polyomial ' ' p x f x l x x x l x f x x x l x i i i i i i i i i i.6. Example, ',, ' p x y p x d p x y p x d x x x x l x, l x x x x x l ' x, l ' x x x x x ' ' p x f x l x x x l x f x x x l x i i i i i i i i i i ' ' f ' x x x l x f ' x x x l x p x f x l x x x l x f x l x x x l x p x y x x x x x x x y x x x x x x x x x x x x x d x x d x x x x x x 76
77 Adrew Powuk - Math 49 (Numerical Aalysis).7 (*) Multivariable Lagrage iterpolatio (rectagle, hypercube) Iterpolatio coditios for N, N L N, N L N N x x x, L x L x L L x L L L x x L L Lagrage iterpolatio i D x, L. L x x u x u N x u N x u N i i x u u L L i x,y L L. Lagrage iterpolatio i D,,,,,,,, i, j i j u x y u N x N y u N x N y u N x N y u N x N y u x y u N x N y i j xy,, z L L L. Lagrage iterpolatio i D,,,,, i, j, k i j k u x y z u N x N y N z Etc. i j k 77
78 Adrew Powuk - Math 49 (Numerical Aalysis).8 (*) Shape fuctios
79 Adrew Powuk - Math 49 (Numerical Aalysis).9 (*) Trigoometric iterpolatio 79
80 Adrew Powuk - Math 49 (Numerical Aalysis). Splie iterpolatio 8
![Adrew Powuk - http://www.powuk.com- Math 49 (Numerical Aalysis).. (*) Liear splies S ( x) a x b, x [ x, x ] S ( x) a x b, x [ x, x ] Sx ( ).](/docs-images/89/100342761/images/81-0.jpg ".. S ( x) a x b, x [ x, x ] Poits (+) ( x, y ),( x, y ),.")
81 Adrew Powuk - Math 49 (Numerical Aalysis).. (*) Liear splies S ( x) a x b, x [ x, x ] S ( x) a x b, x [ x, x ] Sx ( )... S ( x) a x b, x [ x, x ] Poits (+) ( x, y ),( x, y ),...,( x, y ) I is ecessary to calculate ukows coefficiets Number of equatios = = umber of ukows a i, b i. 8
82 Adrew Powuk - Math 49 (Numerical Aalysis) = itervals S( x) ax b, x [ x, x ] S ( x) S ( x) ax b, x [ x, x ] S( x) ax b, x [ x, x] ( x, y ),( x, y ),( x, y ),( x, y ) +=+=4 poits a, b, a, b, a, b - *=*=6 costats y S ( x ) y a x b y S ( x ) y a x b y S ( x ) y a x b y S ( x ) y a x b y S ( x ) y a x b y S ( x ) y a x b =6 equatio =6 ukows Number of equatios = umber of ukows The solutio is uique. 8
83 Adrew Powuk - Math 49 (Numerical Aalysis)... Example (,), (,),(,) ( x, y ) y=x ( x, y ) S ( x) (,) S ( x ) y=-x+ (,) xx, [,] Sx ( ) x, x [,] (,) ( x, y ) (,), (,),(,) S( x) ax b, x [ x, x ] S( x) ax b, x [,] S( x) S ( x) ax b, x [ x, x ] S ( x) ax b, x [, ] ( x, y ) y a x b a b S( x) ax b, ( x, y) y ax b a b ( x, y ) y a x b a b S ( x) ax b, ( x, y ) y ax b a b a b a b a b a b a x b, x [,] Sx ( ) a x b, x [,] xx, [,] Sx ( ) x, x [,] 8
84 Adrew Powuk - Math 49 (Numerical Aalysis)... Geeral case Iterpolatio coditios Number of poits: + Number of itervals: - ukows - - equatios a, b i i y S ( x ) y S ( x ) y S ( x ) y S ( x ) y S ( x ) y S ( x )... y S ( x ) y S ( x ) y S ( x ) y S ( x ) y a x b y a x b y a x b y a x b y a x b y a x b y a x b y a x b 84
85 Adrew Powuk - Math 49 (Numerical Aalysis)... Example (,), (,),(,) S ( x) a x b, x [,] Sx ( ) S ( x) a x b, x [,] b y S ( x ) S () a b a b y S ( x ) S () a b a b y S ( x ) S () a b a b y S ( x ) S () a b a a b a b a b b a ( ) a S ( x) a x b, x [,] Sx ( ) S ( x) a x b, x [,] S ( x) x, x [,] Sx ( ) S ( x) ( ) x, x [,] S ( x) x, x [,] Sx ( ) S ( x) x, x [,] 85
86 Adrew Powuk - Math 49 (Numerical Aalysis).. (*) Quadratic splies S ( x) a b x c x, x [ x, x ] S ( x) a b x c x, x [ x, x ] Sx ( )... S x a b x c x x x x ( ), [, ] Iterpolatio by usig oly poits Iterpolatio coditios Number of poits: + Number of itervals: - ukows - a, b, c i i i - equatios y S ( x ) y S ( x ) y S ( x ) y S ( x ) y S ( x ) y S ( x )... y S ( x ) y S ( x ) y S ( x ) y S ( x ) y a x b x c y a x b x c y a x b x c y a x b x c y a x b x c y a x b x c y a x b x c y a x b x c There are ifiitely may solutios of this iterpolatio problem. 86
87 Adrew Powuk - Math 49 (Numerical Aalysis) Poits - + ( x, y ),( x, y ),...,( x, y ) Itervals - ukows a, b, c. i i i.. Cotiuity equatios for derivative (- equatios) ' ' S ( x ) S ( x ) ' ' S ( x ) S ( x )... ' ' S ( x ) S ( x ) b c x b c x b c x b c x... b c x b c x Number of equatios ( ) Number of ukows y S ( x ) y S ( x ) ' ' S ( x ) S ( x ) y S ( x ) y S ( x ) ' ' S ( x ) S ( x ) y S ( x ) y S ( x ) ' ' S ( x ) S ( x ) 4... y S ( x ) y S ( x ) ' ' S ( x ) S ( x ) y S ( x ) y S ( x ) 87
88 Adrew Powuk - Math 49 (Numerical Aalysis) Typical assumptios Additioal coditios S '' t a S t b x c Or S '' t a S t b x c 88
89 Adrew Powuk - Math 49 (Numerical Aalysis)... Fid quadratic splie which pass through the followig poits (,),(,),(,5),,,,, 5 ( x, y ),( x, y ),( x, y ) S ( x) a b x c x, x [,] Sx ( ) S ( x) a b x c x, x [,] S ( x ) y S () a b c S ( x ) y S () a b c S '( x ) S '( x ) S '() S '() b c b c S ( x ) y S () a b c S ( x ) y S () 5 a b c 5 a b c a b c b c b c a b c a b c 5 a 89
90 Adrew Powuk - Math 49 (Numerical Aalysis) Clear[x]; Clear[a]; S[x_]=a+b*x+c*x^; S[x_]=a+b*x+c*x^; ds[x_]=b+*c*x; ds[x_]=b+*c*x; Sol=Solve[{S[]==,S[]==,dS[]==dS[],S[]==,S[]= =5},{a,a,b,b,c,c}] a:=; f[x_]=extract[s[x]/.sol,] f[x_]=extract[s[x]/.sol,] f[x_]=if[x==,f[x],f[x]]; Plot[f[x],{x,,}] Solutio {{a,b-(/)+a/,b/-( a)/,c/-a/,c/+a/}} +x +x
91 Adrew Powuk - Math 49 (Numerical Aalysis).. Cubic Splies ( x x) M ( x x ) M ( x x) y ( x x ) y j j j j j j j j S ( x) j 6( x x ) x x j j j j x x ( x x) M ( x x ) M 6 M j j j j j j M x x x x x x j j j j j j M M M j j j 6 6 y y y y j j j j, j,..., x x x x j j j j 9
92 Adrew Powuk - Math 49 (Numerical Aalysis) Equatios S ( x) a b x c x d x, x [ x, x ] S ( x) a b x c x d x, x [ x, x ] Sx ( )... S x a b x c x d x x x x ( ), [, ] Poits (+) ( x, y ),( x, y ),...,( x, y ) y S ( x ) y S ( x ) y S ( x ) y S ( x ) y S ( x ) y S ( x )... y S ( x ) y S ( x ) y S ( x ) y S ( x ) y S ( x ) y S ( x ) y S ( x ) y S ( x ) y ( ) S x y S ( x )... y S ( x ) y S ( x ) y S ( x ) y ( ) S x y a b x c x d x y a b x c x d x y a b x c x d x y a b x c x d x y a b x c x d x y a b x c x d x y a b x c x d x y a b x c x d x Cotiuity equatios for derivative (- equatios) ' ' S ( x ) S ( x ) ' ' S ( x ) S ( x )... ' ' S ( x ) S ( x ) b c x d x b c x d x b c x d x b c x d x... b c x d x b c x d x 9
93 Adrew Powuk - Math 49 (Numerical Aalysis) Cotiuity equatios for secod order derivative (- equatios) '' '' S ( x ) S ( x ) '' '' S ( x ) S ( x )... '' '' S ( x ) S ( x ) c 6d x c 6d x c 6d x c 6d x... c 6d x c 6d x 4 To complete the system, we are addig the followig equatios: S '' ( x ) S '' ( x ) y '' y' C y Cx D it is a lie !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ( x x) M ( x x ) M ( x x) y ( x x ) y j j j j j j j j S ( x) j 6( x x ) x x j j j j x x ( x x) M ( x x ) M 6 j j j j j j M M x x x x x x j j j j j j M M M j j j 6 6 y y y y j j j j, j,..., x x x x j j j j!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 9
94 Adrew Powuk - Math 49 (Numerical Aalysis)... Fid cubic splie for (,),(,),(,6) x=;x=;x=; y=;y=;y=6; S[x_]=a+b*x+c*x^+d*x^; S[x_]=a+b*x+c*x^+d*x^; ds[x_]=b+*c*x+*d*x^; ds[x_]=b+*c*x+*d*x^; ds[x_]=*c+6*d*x; ds[x_]=*c+6*d*x; Sol=Solve[{S[x]==y,S[x]==y,dS[x]==dS[x],dS[x]==dS[x],S[x]==y,S[x ]==y,ds[x]==,ds[x]==},{a,a,b,b,c,c,d,d}] f[x_]=extract[s[x]/.sol,] f[x_]=extract[s[x]/.sol,] f[x_]=if[x<,f[x],f[x]]; Plot[f[x],{x,,}] Solutio {{a,a,b,b-6,c,c6,d,d-}} S=x S=-6 x+6 x -x Plot[Piecewise[{{x^,x<},{-6 x+6 x^-x^,x>}}],{x,,}] Solutio
95 Adrew Powuk - Math 49 (Numerical Aalysis) Method x=;x=;x=; y=;y=;y=6; x ; x ; x ; y ; y ; y 6; ( x x) M ( x x ) M ( x x) y ( x x ) y j j j j j j j j S ( x) j 6( x x ) x x 6 j j j j x x ( x x) M ( x x ) M j j j j j j M M j j j j j j M M M j j j 6 6 j x x x x x x y y y y j j j x x x x j j j j, j,..., ( x x) M ( x x ) M ( x x) y ( x x ) y j : S ( x) 6( x x ) x x x x ( x x) M ( x x ) M 6 ( x x) M ( x x ) M ( x x) y ( x x ) y j : S ( x) 6( x x ) x x x x ( x x) M ( x x ) M 6 x = ; x = ; x = ; y ; y ; y 6; ( x) M ( x ) M ( x) ( x ) S ( x) ( x) M ( x ) M 6( ) 6 ( x) M ( x ) M ( x) ( x )6 S ( x) ( x) M ( x ) M 6( ) 6 95
96 Adrew Powuk - Math 49 (Numerical Aalysis) M, M, M M M M M M,? j x x x x x x y y y y M M M j j j 6 6 x x x x j j j j j j j j j j j j j x x x x x x y y y y j : M M M : 6 6 x x x x 6 j : M j 6 6 M 5 j : M 4 j : M 4 6 ( x) ( x ) 6 ( x) ( x ) S ( x) ( x) ( x )6 x 6( ) 6 6( ) 6 ( x) 6 ( x ) ( x) ( x )6 S ( x) ( x)6 ( x ) 6x 6x x j, 96
97 Adrew Powuk - Math 49 (Numerical Aalysis)... Example (,),(,),(,) x, x, x y, y, y ( x x) M ( x x ) M ( x x) y ( x x ) y j j j j j j j j S ( x) j 6( x x ) x x j j j j x x ( x x) M ( x x ) M 6 j j j j j j M M j x x x x x x j j j j j j M M M j j j 6 6 y y y y j j j j, j,..., x x x x j j j ( x x) M ( x x ) M ( x x ) y ( x x ) y j : S ( x) 6( x x ) x x x x ( x x) M ( x x ) M 6 ( x x) M ( x x ) M ( x x) y ( x x ) y j : S ( x) 6( x x ) x x x x ( x x) M ( x x ) M 6 x, x, x y, y, y ( x) ( x ) M ( x) ( x ) : ( ) ( ) ( ) j S x x x M 6( ) 6 ( x) M ( x ) ( x) ( x ) j : S ( x) ( x) M ( x ) 6( ) 6 x x x x x x j j j j j j : M M j j 6 6 j j j j j x x x x x x y y y y j : M M M j 6 6 x x x x M y y y y x x x x j j j j 97
Math 128A: Homework 1 Solutions
Math 8A: Homework Solutios Due: Jue. Determie the limits of the followig sequeces as. a) a = +. lim a + = lim =. b) a = + ). c) a = si4 +6) +. lim a = lim = lim + ) [ + ) ] = [ e ] = e 6. Observe that
*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully
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