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الكلية كلية تكنولوجيا المعلومات
القسم قسم البرامجيات
المرحلة 2
أستاذ المادة حيدر كاظم زغير الجبوري
17/11/2014 05:20:10
Solutions of Equations in one Variable.
Locating the position of roots (programming method):
To locate the position of roots of the function (equation) f(x)=0 by using programming method, let f(x) be continuous function on the interval [a,b]. We divide the interval [a,b] into n subintervals a=x0Example 1:
Find the approximate location of the function
f(x)=x4-7x3+3x2+26x-10=0 on the interval [-8,8] with n=4 and n=8..
f(x)=x3+4x2-10=0 on the interval [1,2] with n=5.
f(x)=x3-4x+1=0 on the interval [-1,4] with n=5.
Solution: (1): Let n=4, h=
x -8 -4 0 4 8
f(x) + +
- -
+
There is a root between (-4,0) and (4,8).
If n=8, h=2:
x -8 -6 -4 -2 0 2 4 6 8
f(x) + + + +
-
+
-
+ +
There is a root between (-2,0), (0,2), (2,4) and (4,6).
Solution: (2): Let n=5, h=0.2
x 1 1.2 1.4 1.6 1.8 2
f(x) - -
+ + + +
There is a root between (1.2,1.4).
Solution: (3): Let n=5, h=1
x -1 0 1 2 3 4
f(x) + +
-
+ + +
There is a root between (0,1) and (1,2).
It s clear that we study ( in this chapter ) numerical methods for solving equation : f(x)=0…(1)
where x and f(x) are real . The values of x for which (1) holds are called roots of equation for example :-
x^2-x-2=0 …(1-1)
x^3+x^2-3x-2=0 ...(1-2)
2^x-5x+2=0 ...(1-3)
e^x-3x=0 …(1-4)
Equation (1-1) is a second degree polynomial whose roots are x=2,x=-1 , but for the other equation (1-2,1-3,1-4) it is difficult ( if it is not possible ) to be able to express the roots through a formula as equation (1-1) , hence we have to obtain the roots of the equation(1-2,1-3,1-4) numerically we study some methods
1- False-Position method (Regula falsi method):
Suppose a continuous function f defined on the interval [a,b] is given with f(a) and f(b) of opposite sign (i.e. f(a)?f(b)<0). To derive a formula for false-position method, approximate the graph of f by a straight line on [a,b] connecting (a, f(a)) and (b, f(b)) which intersect x-axis at (c,0) where c is more approximate to the exact (actual) root than a and b.
To obtain a formula for c we use the slop equality:
? ?
To find another approximation:
<0 there is a root between a, c? d=
If f(a)?f(c) >0 there is a root between b, c? d=
=0 c is exact root ((Stop)).
We stop iteration if the interval width is as small as desired i.e. for any i.
Straight line
• • •
Example :
Use an approximate root of f(x)=xlogx-1=0 by using False position method with ?=0.005 for five digit
Solution :
Let x_1=2 , x_2=3
f(x_1 )=-0.39791 , f(x_2 )=0.43136?f(x_1 )× f(x_2 )<0
x_3=(x_1 f(x_2 )-x_2 f(x_1 ))/(f(x_2 )-f(x_1 ) ) = (2(0.43136)-3(-0.39791))/(0.43136-0.39791)=2.47985 , ?|x_3-x_2 |>?
f(x_3 )=-0.02188 find x_4 ?f(x_2 )× f(x_3 )<0
?x_4=(x_2 f(x_3 )-x_3 f(x_2 ))/(f(x_3 )-f(x_2 ) )=(3(-0.02188)-2.47985(0.43136))/(-0.02188-0.43136)
=2.50496 , ?|x_4-x_3 |>?
f(x_4 )=-0.00101 find x_5?f(x_2 )× f(x_4 )<0
?x_5=(x_2 f(x_4 )-x_4 f(x_2 ))/(f(x_4 )-f(x_2 ) )=(3(-0.00101)-2.50496(0.43136))/(-0.00101-0.43136)
=2.50611
?|2.50611-2.50496|=0.001? The root is 2.50611
H.W:
Use false position method to find the root of x^5-x-0.2 ,?=0.005
Solutions of Equations in one Variable.
Locating the position of roots (programming method):
To locate the position of roots of the function (equation) f(x)=0 by using programming method, let f(x) be continuous function on the interval [a,b]. We divide the interval [a,b] into n subintervals a=x0Example 1:
Find the approximate location of the function
f(x)=x4-7x3+3x2+26x-10=0 on the interval [-8,8] with n=4 and n=8..
f(x)=x3+4x2-10=0 on the interval [1,2] with n=5.
f(x)=x3-4x+1=0 on the interval [-1,4] with n=5.
Solution: (1): Let n=4, h=
x -8 -4 0 4 8
f(x) + +
- -
+
There is a root between (-4,0) and (4,8).
If n=8, h=2:
x -8 -6 -4 -2 0 2 4 6 8
f(x) + + + +
-
+
-
+ +
There is a root between (-2,0), (0,2), (2,4) and (4,6).
Solution: (2): Let n=5, h=0.2
x 1 1.2 1.4 1.6 1.8 2
f(x) - -
+ + + +
There is a root between (1.2,1.4).
Solution: (3): Let n=5, h=1
x -1 0 1 2 3 4
f(x) + +
-
+ + +
There is a root between (0,1) and (1,2).
It s clear that we study ( in this chapter ) numerical methods for solving equation : f(x)=0…(1)
where x and f(x) are real . The values of x for which (1) holds are called roots of equation for example :-
x^2-x-2=0 …(1-1)
x^3+x^2-3x-2=0 ...(1-2)
2^x-5x+2=0 ...(1-3)
e^x-3x=0 …(1-4)
Equation (1-1) is a second degree polynomial whose roots are x=2,x=-1 , but for the other equation (1-2,1-3,1-4) it is difficult ( if it is not possible ) to be able to express the roots through a formula as equation (1-1) , hence we have to obtain the roots of the equation(1-2,1-3,1-4) numerically we study some methods
1- False-Position method (Regula falsi method):
Suppose a continuous function f defined on the interval [a,b] is given with f(a) and f(b) of opposite sign (i.e. f(a)?f(b)<0). To derive a formula for false-position method, approximate the graph of f by a straight line on [a,b] connecting (a, f(a)) and (b, f(b)) which intersect x-axis at (c,0) where c is more approximate to the exact (actual) root than a and b.
To obtain a formula for c we use the slop equality:
? ?
To find another approximation:
<0 there is a root between a, c? d=
If f(a)?f(c) >0 there is a root between b, c? d=
=0 c is exact root ((Stop)).
We stop iteration if the interval width is as small as desired i.e. for any i.
Straight line
• • •
Example :
Use an approximate root of f(x)=xlogx-1=0 by using False position method with ?=0.005 for five digit
Solution :
Let x_1=2 , x_2=3
f(x_1 )=-0.39791 , f(x_2 )=0.43136?f(x_1 )× f(x_2 )<0
x_3=(x_1 f(x_2 )-x_2 f(x_1 ))/(f(x_2 )-f(x_1 ) ) = (2(0.43136)-3(-0.39791))/(0.43136-0.39791)=2.47985 , ?|x_3-x_2 |>?
f(x_3 )=-0.02188 find x_4 ?f(x_2 )× f(x_3 )<0
?x_4=(x_2 f(x_3 )-x_3 f(x_2 ))/(f(x_3 )-f(x_2 ) )=(3(-0.02188)-2.47985(0.43136))/(-0.02188-0.43136)
=2.50496 , ?|x_4-x_3 |>?
f(x_4 )=-0.00101 find x_5?f(x_2 )× f(x_4 )<0
?x_5=(x_2 f(x_4 )-x_4 f(x_2 ))/(f(x_4 )-f(x_2 ) )=(3(-0.00101)-2.50496(0.43136))/(-0.00101-0.43136)
=2.50611
?|2.50611-2.50496|=0.001? The root is 2.50611
H.W:
Use false position method to find the root of x^5-x-0.2 ,?=0.005
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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