Методы решения нелинейных уравнений

9 .46 9 f (x) = 0 , : f (x) – . x1* , x *2 , x *3 ...x *n , , . . ! . f(x) x x1* x* - ." f (x) ! x3* x2* $ ( k 1) % ! f ( x * ) = f ( x * ) = f ( x * ) = ... = f ( k & k =1 $ % ' f1 ( x ) f 2 ( x ) = 0 , , ) ) ( $ x *2 - + . ' ( $. . . . (x* ) = 0 ( % , f (x) f1 ( x ), f 2 ( x ) . + $ ) % % . , ) % . : $!. ( % .- $ f (x) $! Ox , : [a , b] , % % $ f1 ( x ), f 2 ( x ) -) ) , . .: . - * * x *3 - ( $ 1) $ . f1 ( x ) = f 2 ( x ) , – " , ! x1* , x = x* k, $ . . ., . 805 !" , 2005 $. % 9 .47 ! f (x) , ! , . . f ( a ) f ( b) < 0 , f (x) [a, b] , . . sign f ( x ) = const , . / % % ( % $ ./ % . % , ( [a , b] % [a , b] % % [a , b] ( ( $ $ [ a ,b ] . * 1) 2) 3) % $ $% ! ! % + : $! ; ; $ , % . !" # 1) , ! f (x) . $ 2) ' , : 3) 6 [a , b] f ( a ) f ( b) < 0 . % f ( x i ) f ( x i +1 ) < 0 , [ x i , x i +1 ] . / % [ x i , x i +1 ] % ( $ ] , x1 ] [x m , [ - % # [a 0 , b 0 ] % [a , b] ( x i , x i +1 ) . / . $ , ! x1 , x 2 , x 3 ...x m . $ 4) 8 f (x) = 0 . 5 + $ $! ( $ , ) . 8 % . ) [a 0 , b 0 ] % ! . , % , % ( % ! $ $ ( ( ! % ( % y % . [a0, b0] y = f(x) [a1, b1] [a2, b2] a0 c1 c0 b0 x* . . ., ( . 805 !" , 2005 $. x $ 9 .48 , % % $+ . 9 % % . ' # ( 1. 8 $ $ a k + bk , k = 0,1,2... 2 f (a k ) f (c k ) < 0 , [a k +1 , b k +1 ] = [a k , c k ] , [a k +1 , b k +1 ] = [c k , b k ] a + b k +1 x * = k +1 $ 2-4 ( b k +1 a k +1 < , 2 2. ' ck = % 3. / 4. , : !, . ' $+ % ( ! ( $. / , + $ % ! $% . $ 9 ) ) ( ( $ $! $! $ ( $ $+ . y y = f(x) x* x x2 8 % x k +1 = x k : ( $ ( ( ( ( $ ! x0 ! k f (x ) f (x k ) + 8 % , % : , k = 0,1,2..... $), ) f (x ) . ( ( ( '$! ! ) : f (x ) < ' x1 ! ( ( [ a , b] . [ a , b] $ f ( x ), f ( x ) ( $ x*, f (a ) f ( b ) < 0 . [ a , b] . ! % f (x 0 ) f (x 0 ) > 0 . $ . x0 . * y = f (x) $ ) $ % ( x0 . $ ! , , ! $ . ., . 805 !" , 2005 $. % . , 9 .49 $! ' # 1. ' , '$! $ $ ! $!. ( [ a , b] % % $ 2. * $ 3. , $ x0 . $ f (x 0 ) f (x 0 ) > 0 ( % $ $ $ , . f (x k ) x k +1 = x k 2 f (x k ) x * = x k +1 x k +1 x k < , ( , k = 0,1,2..... $ , ( ( x = ( x ) (3) $ $% % ! % + . 9 . . ( y=x ! = % % $ y : x= y=x y= 1 (x) x x* x2 x1 x0 y y=x y= 2 (x) x x* x1 x0 x2 . . ., . 805 !" , 2005 $. x k +1 = ( x k ) y = (x) 1 (x ) x= 2 (x ) 9 .50 & % . . ( x = (x) % ( ( ( ) : (x) < % $ k ( < 1, (0 5 . .x . , < 1. ( $ ( % (x) $ , , $ x* = const ) , (x) % $ [ a , b] . % ) $ $ 4. , $ * (x) $ $ ( [ a , b] . / % x . x k +1 = ( x k ), k = 0,1,2 3 .* ! , $ 3. * ! x = (x) . % 2. 9 + x = (x) . , $ (x) x [ a , b] . ( 1. , ( x k +1 = ( x k ) ( $, k ' # x [ a , b] ) % : x * = x k +1 x k +1 x k < , ( ( f (x) = 0 % x = x+ $ f (x) 1. > $ x2 1 = 0 $ $ = 0 .1 a b % [0.5, 3] . = + k 1 2 3 4 5 0.5 0.5 0.5 0.8125 0.96875 0.96875 , + f (a ) 3 1.75 1.125 1.125 1.125 1.046875 x* c f ( b) -0.75 8 -0.75 2.0625 -0.75 0.265625 -0.33984375 0.265625 -0.06152344 0.265625 -0.06152344 0.095947266 x=b a f ( c) 1.75 1.125 0.8125 0.96875 1.046875 1.007813 2.0625 0.265625 -0.33984 -0.06152 0.095947 0.015686 1.007813 2. > $ x2 1 = 0 $ % [0.5, 3] . = 0 .1 $ = + * , % $ ( % f (3) = 9 1 = 8, f '$! : (3) = 2 , % . x0 = 3 , ( % ., . 805 , f (3) f (3) > 0 . !" , 2005 $. 1)-3) 2.5 1.25 0.625 0.3125 0.15625 0.078125 9 .51 x k 1 2 3 4 , x* + f (x) f (x) x 3 8 6 1.666666667 1.777777778 3.333333333 1.333333333 1.133333333 0.284444444 2.266666667 0.533333333 1.007843137 0.015747789 2.015686275 0.125490196 1.000030518 6.1037E-05 2.000061036 0.007812619 1.000030518 3. > $ x2 1 = 0 $ ( % [0.5, 1.5] . = 0 .1 $ = + , % x = x 0.3 ( x 2 1) ( x ) : ( x ) = 1 0 .6 x ( x ) = 1 0 .6 x ( ' % : ! , % [0.5, 1.5] (0.5) = 1 0.3 = 0.7 (1.5) = 1 0.9 = 0.1 '(x) x 0.5 8 1.5 ' (x) , * 0.7 < 1 - % ( $ 0.5 x k 1 2 3 4 5 6 , + x* . 0,5 0,725 0,867313 0,941643 0,975636 0,990076 0,996001 (x) x 0,725 0,867313 0,225 0,941643 0,142313 0,975636 0,07433 0,990076 0,033993 0,996001 0,01444 0,998396 0,005925 0.996001 . . ., . 805 !" , 2005 $.