محاسبات عددی علم و هنر محاسبه است. محاسبات عددی (یا آنالیز عددی) به مطالعه ی روش ها و الگوریتم هایی گفته می شود که تقریب های عددی (در مقابل جواب های تحلیلی) را برای مسائل ریاضی بکار می برند. محاسبات عددی با اعمال شیوه های تقریبی محاسباتی به حل مسائلی از ریاضیات پیوسته می پردازد که به روش تحلیلی قابل حل نبوده و یا به سختی قابل حل تحلیلی هستند. سرفصل هایی که در این آموزش به آن پرداخته شده است: درس اول: خطاها و اشتباهات درس دوم: حل دستگاه های معادلات خطی درس سوم: درون یابی و برازش درس چهارم: مشتق گیری و انتگرال گیری عددی درس پنجم: حل عددی معادلات دیفرانسیل معمولی ... برای توضیحات بیشتر و تهیه این آموزش لطفا به لینک زیر مراجعه بفرمائید: http://faradars.org/courses/fvmth102

1. ‫عددی‬ ‫محاسبات‬
‫کمک‬ ‫به‬MATLAB
faradars.org/fvmth102
‫افزار‬ ‫نرم‬ ‫کمک‬ ‫به‬ ‫عددی‬ ‫محاسبات‬MATLAB
«‫های‬ ‫روش‬‫خطی‬ ‫غیر‬ ‫معادالت‬ ‫یابی‬ ‫ریشه‬ ‫عددی‬»
‫مدرس‬:
‫تاشک‬ ‫اشکان‬
‫تحصیلی‬ ‫رشته‬ ‫و‬ ‫درجه‬
‫برق‬ ‫مهندسی‬ ‫دکترای‬-‫دانشگاه‬ ‫مدرس‬ ‫و‬ ‫سیستم‬ ‫گرایش‬ ‫مخابرات‬
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

2. ‫ﻓﺼﻞ‬ ‫ﻋﻨﺎوﻳﻦ‬‫ﺷﺸﻢ‬
‫ﻓﺼﻞ‬‫ﺷﺸﻢ‬:‫رﻳﺸﻪ‬‫ﻳﺎﺑﻲ‬‫ﻋﺪدي‬‫ﻣﻌﺎدﻻت‬‫ﻏﻴﺮﺧﻄﻲ‬
1-‫ﻣﻘﺪﻣﻪ‬‫ﻣﺴﺄﻟﻪ‬‫رﻳﺸﻪ‬‫ﻳﺎﺑﻲ‬
-‫اﻧﻮاع‬‫رﻫﻴﺎﻓﺖ‬‫ﻫﺎي‬‫ﻋﺪدي‬‫ﺑﺮاي‬‫رﻳﺸﻪ‬‫ﻳﺎﺑﻲ‬‫ﻣﻌﺎدﻻت‬‫ﻏﻴﺮ‬‫ﺧﻄﻲ‬
2-‫روش‬‫ﻫﺎي‬‫رﻳﺸﻪ‬‫ﻳﺎﺑﻲ‬‫ﻋﺪدي‬‫داﻣﻨﻪ‬‫ﻣﺤﺪود‬‫ﻳﺎ‬‫ﺑﺴﺘﻪ‬)‫ﺑﻪ‬‫ﻫﻤﺮاه‬‫ﺑﺮﻧﺎﻣﻪ‬MATLAB(
-‫روش‬‫ﻧﺼﻒ‬‫ﻛﺮدن‬‫ﻳﺎ‬Bisection
-‫روش‬‫ﻧﺎﺑﺠﺎﻳﻲ‬‫ﻳﺎ‬‫ﻣﻮﻗﻌﻴﺖ‬‫ﺧﻄﺎ‬‫ﻳﺎ‬Regula Falsi‫ﻳﺎ‬False Position
3-‫روش‬‫ﻫﺎي‬‫رﻳﺸﻪ‬‫ﻳﺎﺑﻲ‬‫ﻋﺪدي‬‫داﻣﻨﻪ‬‫ﻧﺎﻣﺤﺪود‬‫ﻳﺎ‬‫ﺑﺎز‬)‫ﺑﻪ‬‫ﻫﻤﺮاه‬‫ﺑﺮﻧﺎﻣﻪ‬MATLAB(
-‫روش‬‫ﺳﻜﺎﻧﺖ‬)Secant(
-‫روش‬‫ﻧﻴﻮﺗﻮن‬-‫راﻓﺴﻮن‬)Newton-Raphson(‫ﺳﺎده‬‫و‬‫ﺗﻌﻤﻴﻢ‬‫ﻳﺎﻓﺘﻪ‬
-‫روش‬‫ﻧﻘﻄﻪ‬‫ﺛﺎﺑﺖ‬)Fixed-Point(
4-‫ﺣﻞ‬‫دﺳﺘﮕﺎه‬‫ﻫﺎي‬‫ﻣﻌﺎدﻻت‬‫ﻏﻴﺮﺧﻄﻲ‬‫ﺑﻪ‬‫روش‬‫ﻧﻴﻮﺗﻮن‬)‫ﺑﻪ‬‫ﻫﻤﺮاه‬‫ﺑﺮﻧﺎﻣﻪ‬Matlab(
2
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

3. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﻣﻘﺪﻣﻪ‬
‫اﻫﺪاف‬:
•‫ﻣﺤﺎﺳﺒﻪ‬‫رﻳﺸﻪ‬‫ﻣﻌﺎدﻻت‬‫ﻏﻴﺮ‬‫ﺧﻄﻲ‬
•‫رﻳﺸﻪ‬‫ﻫﺎي‬‫ﭼﻨﺪﺟﻤﻠﻪ‬‫اي‬
•‫دﺳﺘﮕﺎه‬‫ﻣﻌﺎدﻻت‬‫ﻏﻴﺮﺧﻄﻲ‬
‫دﺳﺘﻪ‬‫ﺑﻨﺪي‬‫روش‬‫ﻫﺎي‬‫ﻋﺪدي‬
•‫ﻋﺪم‬‫واﺑﺴﺘﮕﻲ‬‫ﺑﻪ‬‫ﻣﺸﺘﻖ‬‫ﺗﺎﺑﻊ‬
•‫واﺑﺴﺘﻪ‬‫ﺑﻪ‬‫ﻣﺸﺘﻖ‬‫ﺗﺎﺑﻊ‬
3
‫ﻫﺎي‬ ‫روش‬ ‫دوم‬ ‫ﺑﻨﺪي‬ ‫دﺳﺘﻪ‬
‫ﻋﺪدي‬
•‫ﻣﺤﺪود‬ ‫داﻣﻨﻪ‬
•‫داﻣﻨﻪ‬‫ﻧﺎﻣﺤﺪود‬
‫ﻣﺤﺎﺳﺒﺎﺗﻲ‬ ‫ﻫﺎي‬ ‫رﻫﻴﺎﻓﺖ‬:
•‫ﺧﻄﺎ‬ ‫و‬ ‫ﺳﻌﻲ‬
•‫ﻣﻨﺤﻨﻲ‬ ‫رﺳﻢ‬
•‫ﻣﺤﺪود‬ ‫داﻣﻨﻪ‬)‫ﺑﺴﺘﻪ‬(
•‫ﻧﺎﻣﺤﺪود‬ ‫داﻣﻨﻪ‬
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

4. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﻣﺤﺎﺳﺒﺎﺗﻲ‬ ‫ﻫﺎي‬ ‫رﻫﻴﺎﻓﺖ‬
•‫ﺧﻄﺎ‬ ‫و‬ ‫ﺳﻌﻲ‬ ‫ﻫﺎي‬ ‫روش‬:‫ﺟﻮاب‬ ‫ﻓﺮض‬x = α‫ﻣﺤﺎﺳﺒﻪ‬ ،f(α)‫ﺗﺎ‬ ‫آﻟﮕﻮرﻳﺘﻢ‬ ‫ﺗﻜﺮار‬ ،f(α)≈0‫اﻧﺘﻬﺎ‬ ‫در‬ ‫و‬
‫ﻣﺤﺎﺳﺒﻪ‬ ‫ﺗﻮﻗﻒ‬.
•‫روش‬‫رﺳﻢ‬‫ﻣﻨﺤﻨﻲ‬:‫رﺳﻢ‬)x(f‫و‬‫ﻳﺎﻓﺘﻦ‬‫ﻣﺤﻞ‬‫ﺗﻘﺎﻃﻊ‬‫آن‬‫ﺑﺎ‬‫ﻣﺤﻮر‬x،‫ﻫﺎ‬‫در‬‫ﻏﻴﺮ‬‫اﻳﻦ‬‫ﺻﻮرت‬‫ﺗﺒﺪﻳﻞ‬
)x(f=)x(1f-)x(2f‫و‬‫رﺳﻢ‬‫آﻧﻬﺎ‬‫و‬‫ﺗﻼﻗﻲ‬‫دادن‬‫ﺑﺎﻳﻜﺪﻳﮕﺮ‬
•‫ﻣﺤﺪود‬ ‫داﻣﻨﻪ‬)‫ﺑﺴﺘﻪ‬(
•‫ﻓﺮض‬‫ﺟﻮاب‬‫در‬‫ﻣﺤﺪوده‬a < x < b،‫ﻛﺎﻫﺶ‬‫داﻣﻨﻪ‬‫ﺑﺼﻮرت‬،‫ﺳﻴﺴﺘﻤﺎﺗﻴﻚ‬‫ﺑﺎ‬‫ﺣﻔﻆ‬‫ﺟﻮاب‬‫در‬
‫داﺧﻞ‬‫داﻣﻨﻪ‬.
•‫ﻣﺘﺪاول‬ ‫روش‬ ‫دو‬:
1(‫ﻛﺮدن‬ ‫ﻧﺼﻒ‬‫ﺗﺼﻨﻴﻒ‬ ‫ﻳﺎ‬ ‫ﺑﺎزه‬(bisection method)
2(‫ﺧﻄﺎ‬ ‫ﻣﻮﻗﻌﻴﺖ‬(False position or regula falsi)
4
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

5. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﻣﺤﺎﺳﺒﺎﺗﻲ‬ ‫ﻫﺎي‬ ‫رﻫﻴﺎﻓﺖ‬ ‫اداﻣﻪ‬
•‫داﻣﻨﻪ‬‫ﻧﺎﻣﺤﺪود‬
•‫ﻋﺪم‬‫ﺣﻔﻆ‬‫رﻳﺸﻪ‬‫در‬‫ﻣﺤﺪوده‬‫ﻣﺸﺨﺺ‬.
•‫اﺳﺘﻔﺎده‬‫از‬‫اﻃﻼﻋﺎت‬‫ﺗﺎﺑﻊ‬‫ﺑﺮاي‬‫ﺗﺨﻤﻴﻦ‬‫ﺟﻮاب‬.
•‫ﭼﻬﺎر‬‫روش‬‫ﻣﺘﺪاول‬:
1.‫روش‬‫ﻧﻘﻄﻪ‬،‫ﺛﺎﺑﺖ‬
2.‫روش‬،‫ﻧﻴﻮﺗﻦ‬
3.‫روش‬‫ﺳﻜﺎﻧﺖ‬.
4.‫روش‬‫ﻣﻮﻟﺮ‬.
*‫در‬‫اﻳﻨﺠﺎ‬‫ﺗﻨﻬﺎ‬‫ﺳﻪ‬‫روش‬‫اول‬‫از‬‫ﺷﻴﻮه‬‫ﻫﺎي‬‫رﻳﺸﻪ‬‫ﻳﺎﺑﻲ‬‫داﻣﻨﻪ‬‫ﻧﺎﻣﺤﺪود‬‫ﻣﻮرد‬‫ﻧﻈﺮ‬‫ﻫﺴﺘﻨﺪ‬.
5
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

6. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﻫﺎي‬ ‫روش‬
‫ﻣﺤﺪود‬ ‫داﻣﻨﻪ‬
)‫ﺑﺴﺘﻪ‬(
6
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

7. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺑﺎزه‬ ‫ﻛﺮدن‬ ‫ﻧﺼﻒ‬ ‫روش‬)Bisection(
•‫رﻳﺎﺿﻲ‬ ‫ﺑﻴﺎن‬:f(x)=0
•‫روش‬ ‫ﻫﺎي‬ ‫ﻣﺤﺪودﻳﺖ‬:‫ﻣﻌﻴﻦ‬ ‫ﺑﺎزه‬ ‫در‬ ‫رﻳﺸﻪ‬ ‫ﻳﻚ‬.
•‫رﻳﺎﺿﻴﺎﺗﻲ‬ ‫ﻓﺮﺿﻴﺎت‬:
•y = f(x)‫درﺑﺎزه‬[a,b]‫اﺳﺖ‬ ‫ﭘﻴﻮﺳﺘﻪ‬.
f(a).f(b) < 0 •
f’(x) ≠ 0 •
•‫ﻣﻨﺎﺳﺐ‬ ‫ﺣﺎﻟﺖ‬:
7
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

8. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺑﺎزه‬ ‫ﻛﺮدن‬ ‫ﻧﺼﻒ‬ ‫روش‬ ‫اﻟﮕﻮرﻳﺘﻢ‬)Bisection(
•‫اول‬ ‫ﮔﺎم‬:
•‫دوم‬ ‫ﮔﺎم‬:
•‫ﺳﻮم‬ ‫ﮔﺎم‬:
•‫ا‬ ‫ﮔﺎم‬ ‫ﺑﻪ‬ ‫ﺑﺎزﮔﺮد‬ ‫ﻏﻴﺮاﻳﻨﺼﻮرت‬ ‫در‬ ،‫ﺑﺮو‬ ‫ﭼﻬﺎر‬ ‫ﻣﺮﺣﻠﻪ‬ ‫ﺑﻪ‬ ،‫ﺑﺎﺷﺪ‬ ‫ﺑﺮﻗﺮار‬ ‫ﻓﻮق‬ ‫ﺷﺮط‬ ‫اﮔﺮ‬‫ول‬
•‫ﭼﻬﺎرم‬ ‫ﮔﺎم‬:‫ﻧﻤﺎﻳﺶ‬c‫اﻟﮕﻮرﻳﺘﻢ‬ ‫ﭘﺎﻳﺎن‬ ‫و‬ ‫رﻳﺸﻪ‬ ‫ﻋﻨﻮان‬ ‫ﺑﻪ‬.
8
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

9. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺑﺎزه‬ ‫ﻛﺮدن‬ ‫ﻧﺼﻒ‬ ‫روش‬ ‫اﻟﮕﻮرﻳﺘﻢ‬ ‫اداﻣﻪ‬)Bisection(
9
‫ﻣﺰاﻳﺎ‬:‫ﻣﻌﺎﻳﺐ‬:
*‫آﺳﺎن‬ ‫ﺧﻄﺎ‬ ‫ﻣﺤﺎﺳﺒﻪ‬ ‫و‬ ‫ﺳﺎده‬ ‫روش‬*‫ﻛﻨﺪ‬ ‫ﻫﻤﮕﺮاﺋﻲ‬ ‫ﻧﺮخ‬.
*‫رﺳﺪ‬ ‫ﻧﻤﻲ‬ ‫ﺟﻮاب‬ ‫ﺑﻪ‬ ‫روش‬ ‫ﺧﺎص‬ ‫ﻣﻮارد‬ ‫در‬.
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

10. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺑﺮﻧﺎﻣﻪ‬matlab‫ﺑﺎزه‬ ‫ﻛﺮدن‬ ‫ﻧﺼﻒ‬ ‫روش‬)Bisection(
10
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

11. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺑﺮﻧﺎﻣﻪ‬matlab‫ﺑﺎزه‬ ‫ﻛﺮدن‬ ‫ﻧﺼﻒ‬ ‫روش‬)Bisection(
11
function bisect(f,a,b,tol,n)
% Bisection method for solving the nonlinear
%equation f(x)=0.
a0=a;
b0=b;
iter=0;
u=feval(f,a);
v=feval(f,b);
c=(a+b)*0.5;
err=abs(b-a)*0.5;
disp('_______________________________________')
disp(' iter a b c f(c) |b-a|/2 ')
disp('_______________________________________')
fprintf('n')
if (u*v<=0)
while (err>tol)&(iter<=n
w=feval(f,c);
fprintf('%2.0f %10.4f %10.4f %12.6f %10.6f,
%10.6fn',iter,a,b,c,w,err)
if (w*u<0)
b=c;v=w;
end
if (w*u>0)
a=c;u=w;
end
iter=iter+1;
c=(a+b)*0.5;
err=abs(b-a)*0.5;
end
if (iter>n)
disp(' Method failed to converge')
end
else
disp(' The method cannot be applied f(a)f(b)>0')
end
% Plot f(x) in the interval [a,b].
fplot(f, [a0 b0])
xlabel('x');ylabel('f(x)');grid
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

12. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﺜﺎل‬Bisection
12
:
‫ﭘﺲ‬f(a)f (b)<0
===========================================================================================================================
‫ﺟﺪول‬ ‫در‬ ‫ﻧﺘﺎﻳﺞ‬ ‫ﻛﻪ‬
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

13. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬Bisection
•‫ﺗﺎﺑﻊ‬ ‫رﺳﻢ‬1-2x-3x)=x(f‫ﺑﺎزه‬ ‫در‬]2,1[
13
>>bisect('f1',1,2,10^(‐4),40)
iter a b c f( c) |b‐a|/2
________________________________________________
0 1.0000 2.0000 1.500000 0.125000 0.500000
1 1.0000 1.5000 1.250000 ‐0.609375 0.250000
2 1.2500 1.5000 1.375000 ‐0.291016 0.125000
3 1.3750 1.5000 1.437500 ‐0.095947 0.062500
4 1.4375 1.5000 1.468750 0.011200 0.031250
5 1.4375 1.4688 1.453125 ‐0.043194 0.015625
6 1.4531 1.4688 1.460938 ‐0.016203 0.007813
7 1.4609 1.4688 1.464844 ‐0.002554 0.003906
8 1.4648 1.4688 1.466797 0.004310 0.001953
9 1.4648 1.4668 1.465820 0.000875 0.000977
10 1.4648 1.4658 1.465332 ‐0.000840 0.000488
11 1.4653 1.4658 1.465576 0.000017 0.000244
12 1.4653 1.4656 1.465454 ‐0.000411 0.000122
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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14. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺑﺎزه‬ ‫ﻛﺮدن‬ ‫ﻧﺼﻒ‬ ‫روش‬ ‫ﻫﻤﮕﺮاﻳﻲ‬ ‫ﻣﻮرد‬ ‫در‬ ‫ﻣﻬﻢ‬ ‫ﻗﻀﻴﻪ‬)Bisection(
14
===========================================================================================================================
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

15. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫دﻗﺖ‬ ‫از‬ ‫ﻣﺜﺎل‬Bisection
15
‫ﻣﺜﺎل‬:‫ﺻﻔﺮ‬ ‫ﻳﺎﻓﺘﻦ‬ ‫ﺑﺮاي‬ ‫ﻛﺮدن‬ ‫ﻧﺼﻒ‬ ‫ﻋﻤﻞ‬ ‫ﺗﻜﺮار‬ ‫ﻻزم‬ ‫دﻓﻌﺎت‬ ‫ﺗﻌﺪاد‬‫ﺗﺎﺑﻊ‬f(x)‫را‬ ‫ﻗﺒﻞ‬ ‫ﻣﺜﺎل‬ ‫در‬ ‫ﺷﺪه‬ ‫ﺗﻌﺮﻳﻒ‬
‫ﺧﻄﺎي‬ ‫از‬ ‫ﻛﻤﺘﺮ‬ ‫دﻗﺖ‬ ‫ﺑﺎ‬‫ﻣﻄﻠﻖ‬6-10‫آورﻳﺪ‬ ‫ﺑﺪﺳﺖ‬.
=========================================================================================================
‫ﺟﻮاب‬(‫ﺧﻮاﻫﻴﻢ‬‫ﻣﻲ‬n‫آورﻳﻢ‬ ‫ﺑﺪﺳﺖ‬ ‫اي‬‫ﮔﻮﻧﻪ‬ ‫ﺑﻪ‬ ‫را‬‫ﺑﺎﺷﻴﻢ‬ ‫داﺷﺘﻪ‬ ‫ﻛﻪ‬:
‫ﺻﻮرت‬ ‫اﻳﻦ‬ ‫در‬:
‫ﺣﺪاﻛﺜﺮ‬ ‫ﺑﻨﺎﺑﺮاﻳﻦ‬19‫ﻳﻚ‬ ‫آوردن‬ ‫ﺑﺪﺳﺖ‬ ‫ﺑﺮاي‬ ،‫ﻛﺮدن‬‫ﻧﺼﻒ‬ ‫ﻋﻤﻞ‬ ‫ﺗﻜﺮار‬ ‫ﺑﺎر‬‫ﺣﺪود‬ ‫ﺗﻘﺮﻳﺐ‬6-10‫ﺑﻮد‬ ‫ﺧﻮاﻫﺪ‬ ‫ﻻزم‬.
6
1 1 1 1
2 1
| | ( ) 10
2 2
n n n n n
b a
c b a 
   
 
     
6
( 1) log(2) 6 1 19
log 2
n or n      
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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16. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺑﺎزه‬ ‫ﻛﺮدن‬ ‫ﻧﺼﻒ‬ ‫روش‬ ‫ﺧﺎﺗﻤﻪ‬ ‫ﺑﺮاي‬ ‫ﺧﻄﺎ‬ ‫ﻣﻌﻴﺎرﻫﺎي‬)Bisection(
•‫ﻧﻜﺘﻪ‬:‫ﺑﺮاي‬‫ﻣﺤﺎﺳﺒﻪ‬،‫ﺧﻄﺎ‬‫ﻣﻲ‬‫ﺗﻮان‬‫از‬‫ﻓﺮﻣﻮل‬‫ﻫﺎﻳﻲ‬‫ﺑﻪ‬‫ﻗﺮار‬‫زﻳﺮ‬‫اﺳﺘﻔﺎده‬‫ﻛ‬‫ﺮد‬:
‫ﻳﺎ‬‫ﻳﺎ‬
16
| |n n
n
a b
a


 | |n na b  
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

17. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﻏﻴﺮﺧﻄﻲ‬ ‫ﺗﺎﺑﻊ‬f (x)‫ﺑﺎزه‬ ‫در‬[a,b]‫ﺷﻮد‬ ‫ﻣﻲ‬ ‫زده‬ ‫ﺗﻘﺮﻳﺐ‬ ‫ﺧﻄﻲ‬ ‫ﺑﺼﻮرت‬.
‫روش‬False Position‫ﻛﻪ‬‫ﺑﻪ‬‫ﻧﺎم‬Regula Falsi‫ﻧﻴﺰ‬‫ﻣﺸﻬﻮر‬،‫اﺳﺖ‬‫ﻣﺸﺎﺑﻪ‬‫ي‬‫ﺷﻴﻮه‬
Bisection،‫ﺑﺎﺷﺪ‬‫ﻣﻲ‬‫ﺑﺎ‬‫اﻳﻦ‬‫ﺗﻔﺎوت‬‫ﻛﻪ‬‫داراي‬‫اﻳﻦ‬‫ﺑﺮﺗﺮي‬‫اﺳﺖ‬‫ﻛﻪ‬‫ﺑﺎ‬‫ﺳﺮﻋﺖ‬‫ﻛﻤﻲ‬‫ﺑﻴﺸ‬‫ﺘﺮ‬‫از‬
‫ي‬‫ﺷﻴﻮه‬،‫ﻗﺒﻞ‬‫ﺟﻮاب‬‫را‬‫ﺑﺪﺳﺖ‬‫دﻫﺪ‬‫ﻣﻲ‬.
‫ﻣﻌﺎدﻟﻪ‬3x‫ﺟﺪﻳﺪ‬‫ﻛﻪ‬‫از‬‫ﻣﺤﻞ‬‫ﺗﻼﻗﻲ‬‫ﺧﻂ‬‫واﺻﻞ‬‫ﺑﻴﻦ‬
‫دو‬‫ﻧﻘﻄﻪ‬‫و‬‫ﺣﺎﺻﻞ‬‫ﻣﻲ‬‫ﺷﻮد‬:
‫ﻳﺎ‬ ‫ﻧﺎﺑﺠﺎﻳﻲ‬ ‫روش‬False Position
17
2 1
3 2 2
2 1
( )
( ) ( )
x x
x x f x
f x f x

 

1 1( , ( ))x f x2 2( , ( ))x f x
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18. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺧﻄﺎ‬ ‫ﻣﻮﻗﻌﻴﺖ‬ ‫روش‬ ‫اﻟﮕﻮرﻳﺘﻢ‬)False Position(
•‫اول‬ ‫ﮔﺎم‬:
•‫دوم‬ ‫ﮔﺎم‬:
•‫ﺳﻮم‬ ‫ﮔﺎم‬:
•‫ا‬ ‫ﮔﺎم‬ ‫ﺑﻪ‬ ‫ﺑﺎزﮔﺮد‬ ‫ﻏﻴﺮاﻳﻨﺼﻮرت‬ ‫در‬ ،‫ﺑﺮو‬ ‫ﭼﻬﺎر‬ ‫ﻣﺮﺣﻠﻪ‬ ‫ﺑﻪ‬ ،‫ﺑﺎﺷﺪ‬ ‫ﺑﺮﻗﺮار‬ ‫ﻓﻮق‬ ‫ﺷﺮط‬ ‫اﮔﺮ‬‫ول‬
•‫ﭼﻬﺎرم‬ ‫ﮔﺎم‬:‫ﻧﻤﺎﻳﺶ‬c‫اﻟﮕﻮرﻳﺘﻢ‬ ‫ﭘﺎﻳﺎن‬ ‫و‬ ‫رﻳﺸﻪ‬ ‫ﻋﻨﻮان‬ ‫ﺑﻪ‬.
18
   
n
n
n
n
bc
bf
ac
af



  
   nn
nnn
n
bfaf
babf
bc



an
bn
f(an)
next estimate,
c
f(bn)
F(c)
nnn
nnnn
bbcaotherwise
aacbafcfif




11
11
,
,0)().(
2111 ||/||    ceorandabe nnnn
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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19. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫روش‬ ‫اﻟﮕﻮرﻳﺘﻢ‬ ‫ﻛﺪ‬ ‫ﺷﺒﻪ‬False position
19
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20. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫روش‬False position
function false(f,a,b,tol,n)
% False position method for solving the nonlinear
% equation f(x)=0.
a0=a;
b0=b;
iter=0;
u=feval(f,a);
v=feval(f,b);
c=(v*a-u*b)/(v-u);
w=feval(f,c);
disp('_________________________________________')
disp(' iter a b c f(c) |b-a|')
disp('_________________________________________')
fprintf('n')
if (u*v<=0)
while (abs(w)>tol)&(abs(b-a)>tol)&(iter<=n)&((v-
u)~=0)
w=feval(f,c);
fprintf('%2.0f %12.4f %12.4f %12.6f %10.6f
%10.6fn',iter,a,b,c,w,abs(b-a))
20
if (w*u<0)
b=c;v=w;
end
if (w*u>0)
a=c;u=w;
end
iter=iter+1;
c=(v*a-u*b)/(v-u);
end
if (iter>n)
disp(' Method failed to converge')
end
if (v-u==0)
disp(' Division by zero')
end
else
disp(' The method cannot be applied f(a)f(b)>0')
end
% Plot f(x) in the interval [a,b].
fplot(f,[a0 b0])
xlabel('x');ylabel('f(x)'); grid
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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21. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﺜﺎل‬False Position
21
 11 ba
:
===========================================================================================================================
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

22. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﻣﻮﻗﻌﻴﺖ‬ ‫ﻳﺎ‬ ‫ﻧﺎﺑﺠﺎﻳﻲ‬ ‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬‫ﺧﻄﺎ‬
•‫ﺗﺎﺑﻊ‬ ‫رﺳﻢ‬1-2x-3x)=x(f‫ﺑﺎزه‬ ‫در‬]2,1[
22
>> fals('f1',1,2,10^(‐4),40)
________________________________________________
iter a b c f(c) |b‐a|
________________________________________________
0 1.0000 2.0000 1.250000 ‐0.609375 1.000000
1 1.2500 2.0000 1.376623 ‐0.286264 0.750000
2 1.3766 2.0000 1.430925 ‐0.117660 0.623377
3 1.4309 2.0000 1.452402 ‐0.045671 0.569075
4 1.4524 2.0000 1.460613 ‐0.017331 0.547598
5 1.4606 2.0000 1.463712 ‐0.006520 0.539387
6 1.4637 2.0000 1.464875 ‐0.002445 0.536288
7 1.4649 2.0000 1.465310 ‐0.000916 0.535125
8 1.4653 2.0000 1.465474 ‐0.000343 0.534690
9 1.4655 2.0000 1.465535 ‐0.000128 0.534526
10 1.4655 2.0000 1.465558 ‐0.000048 0.534465
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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23. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺧﻄﺎ‬ ‫ﻣﻮﻗﻌﻴﺖ‬ ‫روش‬)False Position(
•‫ﻧﺎﺑﺠﺎﻳﻲ‬ ‫روش‬ ‫اﻟﮕﻮرﻳﺘﻢ‬ ‫و‬ ‫ﻛﺪ‬ ‫ﺷﺒﻪ‬)False Position(
23
1 1
1 1
0,1,...,
( ) ( )
( ) ( )
( ). ( ) 0, ,
, , .
n n n n
n n
n n n n
n n n
for n ITMAX
f b a f a b
c
f b f a
if f a f c set a a b c
otherwise set a c b b
 
 




  
 
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

24. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﻳﺎﻓﺘﻪ‬ ‫ﺑﻬﺒﻮد‬ ‫ﺧﻄﺎي‬ ‫ﻣﻮﻗﻌﻴﺖ‬ ‫روش‬ ‫اﻟﮕﻮرﻳﺘﻢ‬)Modified False Position(
•‫اوﻟﻴﻪ‬ ‫ﻣﻔﺮوﺿﺎت‬:)nb(f=G‫و‬)na(f=F‫و‬0a=0c
•‫اول‬ ‫ﮔﺎم‬:
•‫دوم‬ ‫ﮔﺎم‬:
•‫ﺳﻮم‬ ‫ﮔﺎم‬:
•‫اول‬ ‫ﮔﺎم‬ ‫ﺑﻪ‬ ‫ﺑﺎزﮔﺮد‬ ‫ﻏﻴﺮاﻳﻨﺼﻮرت‬ ‫در‬ ،‫ﺑﺮو‬ ‫ﭼﻬﺎر‬ ‫ﻣﺮﺣﻠﻪ‬ ‫ﺑﻪ‬ ،‫ﺑﺎﺷﺪ‬ ‫ﺑﺮﻗﺮار‬ ‫ﻓﻮق‬ ‫ﺷﺮط‬ ‫اﮔﺮ‬
•‫ﭼﻬﺎرم‬ ‫ﮔﺎم‬:‫ﻧﻤﺎﻳﺶ‬c‫اﻟﮕﻮرﻳﺘﻢ‬ ‫ﭘﺎﻳﺎن‬ ‫و‬ ‫رﻳﺸﻪ‬ ‫ﻋﻨﻮان‬ ‫ﺑﻪ‬.
24
  
    FG
FbGa
bfaf
babf
bc
nn
nnn
nn






][
1
2/,)(,
2/,)(,0)().(
1111
11111
GGafFbbcaotherwise
FFbfGcbaacfcfif
nnnnn
nnnnnnn




21111 ||/||    nnnnn ceorandabe
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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25. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫روش‬Modified False position
function Modified_fals(f,a,b,tol,n)
% False position method for solving the nonlinear
% equation f(x)=0.
a0=a;
b0=b;
iter=0;
F=feval(f,a);
G=feval(f,b);
c=(G*a-F*b)/(G-F);
W=feval(f,c);
disp('______________________________________________')
disp(' iter a b c f(c) |b-a|')
disp('______________________________________________')
fprintf('n')
if (F*G<=0)
while (abs(W)>tol)&(abs(b-a)>tol)&(iter<=n)&((G-F)~=0)
W=feval(f,c);
25
fprintf('%2.0f %12.4f %12.4f %12.6f %10.6f
%10.6fn',iter,a,b,c,w,abs(b-a))
if (W*F<0)
b=c;G=W;F=F/2;
end
if (W*F>0)
a=c;F=W;G=G/2;
end
iter=iter+1;
c=(G*a-F*b)/(G-F);
end
if (iter>n)
disp(' Method failed to converge')
end
if (G-F==0)
disp(' Division by zero')
end
else
disp(' The method cannot be applied f(a)f(b)>0')
end
% Plot f(x) in the interval [a,b].
fplot(f,[a0 b0])
xlabel('x');ylabel('f(x)'); grid
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

26. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﺜﺎل‬Modified False Position
26
:
===========================================================================================================================
‫ﻣﺜﺎل‬ ‫ﺟﻮاب‬(‫ﺷ‬ ‫اﺻﻼح‬ ‫ﻧﺎﺑﺠﺎﻳﻲ‬ ‫ﻋﺪدي‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫روش‬ ‫اﻟﮕﻮرﻳﺘﻢ‬ ‫در‬ ‫اوﻟﻴﻪ‬ ‫ﻓﺮﺿﻴﺎت‬ ‫ﻋﻨﻮان‬ ‫ﺑﻪ‬‫دارﻳﻢ‬ ‫ﺪه‬:
3)=0b(f=G‫و‬1-)=0a(f=F
‫ﻧﺨﺴﺖ‬ ‫ﺗﻜﺮار‬ ‫از‬ ‫اول‬ ‫ﮔﺎم‬-‫ﻣﺤﺎﺳﺒﻪ‬ ‫ﺑﺮاي‬c‫دارﻳﻢ‬ ‫ﺟﺪﻳﺪ‬:
‫ﻧﺨﺴﺖ‬ ‫ﺗﻜﺮار‬ ‫از‬ ‫دوم‬ ‫ﮔﺎم‬-‫در‬‫اﻳﻦ‬‫داﺷﺖ‬ ‫ﺧﻮاﻫﻴﻢ‬ ‫ﺻﻮرت‬:
‫ﻧﺨﺴﺖ‬ ‫ﺗﻜﺮار‬ ‫از‬ ‫ﺳﻮم‬ ‫ﮔﺎم‬-‫ﻳﺎ‬ ‫ﺷﺮط‬ ‫ﻛﺮدن‬ ‫ﭼﻚ‬
‫روﻳﻢ‬ ‫ﻣﻲ‬ ‫دوم‬ ‫ﺗﻜﺮار‬ ‫از‬ ‫اول‬ ‫ﮔﺎم‬ ‫ﺑﻪ‬ ،‫ﻓﻮق‬ ‫ﺷﺮط‬ ‫دو‬ ‫از‬ ‫ﻛﺪام‬ ‫ﻫﺮ‬ ‫ﺑﺮﻗﺮاري‬ ‫ﻋﻠﺖ‬ ‫ﺑﻪ‬.
  
   
25.1
)1(3
]1231[][
1
00
00
000
01 








 c
FG
FbGa
bfaf
babf
bc
5.12/,6094.0)(
25.1,20)()(1&6094.0)(
11
110110001


GGcafF
cabbcfcfaccf
tolcf )( 1tol
a
ab


1
11 ||
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

27. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬Modified False Position
27
‫ﻣﺜﺎل‬ ‫ﺟﻮاب‬ ‫اداﻣﻪ‬(‫دوم‬ ‫ﺗﻜﺮار‬ ‫از‬ ‫اول‬ ‫ﮔﺎم‬-
‫ﺗﻜﺮار‬ ‫از‬ ‫دوم‬ ‫ﮔﺎم‬‫دوم‬-‫داﺷﺖ‬ ‫ﺧﻮاﻫﻴﻢ‬ ‫ﺻﻮرت‬ ‫اﻳﻦ‬ ‫در‬:
‫ﺗﻜﺮار‬ ‫از‬ ‫ﺳﻮم‬ ‫ﮔﺎم‬‫دوم‬-‫ﺷﺮط‬ ‫ﻛﺮدن‬ ‫ﭼﻚ‬‫ﻳﺎ‬
‫ﺑﺎ‬‫ﺗﻮﺟﻪ‬‫ﺑﻪ‬‫اﻳﻨﻜﻪ‬‫ﻫﻤﭽﻨﺎن‬‫ﺷﺮاﻳﻂ‬‫ﮔﺎم‬‫ﺳﻮم‬‫ﺑﺮﻗﺮار‬،‫ﻫﺴﺘﻨﺪ‬‫ﻣﺠﺪد‬‫ﺑﺎﻳﺴﺘﻲ‬‫ﺑﻪ‬‫ﮔﺎم‬‫اول‬‫از‬‫ﺗﻜﺮار‬‫ﺑ‬‫ﻌﺪ‬‫رﻓﺖ‬.
‫ﻳﺎﺑﻨﺪ‬ ‫ﻣﻲ‬ ‫اداﻣﻪ‬ ‫ﺷﻮﻧﺪ‬ ‫ﻧﻘﺾ‬ ‫ﺳﻮم‬ ‫ﮔﺎم‬ ‫ﺷﺮاﻳﻂ‬ ‫ﻛﻪ‬ ‫زﻣﺎﻧﻲ‬ ‫ﺗﺎ‬ ‫ﺗﻜﺮار‬ ‫ﻋﻤﻠﻴﺎت‬ ،‫ﺗﺮﺗﻴﺐ‬ ‫اﻳﻦ‬ ‫ﺑﻪ‬.
tolcf )( 2
  
   
4667.1
)6094.0(5.1
)]6094.0(25.125.1[][
2
11
11
111
12 








 c
FG
FbGa
bfaf
babf
bc
00397.0)(,3047.02/
25.1,4667.10)()(25.1&00397.0)(
22
12221212


cbfGFF
aacbcfcfccf
tol
a
ab


1
11 ||
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

28. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺷﺪ‬ ‫اﺻﻼح‬ ‫ﻧﺎﺑﺠﺎﻳﻲ‬ ‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬‫ه‬
•‫ﺗﺎﺑﻊ‬ ‫رﺳﻢ‬1-2x-3x)=x(f‫ﺑﺎزه‬ ‫در‬]2,1[
28
_________________________________________________
iter a b c f(c) |b‐a|
_________________________________________________
0 1.0000 2.0000 1.250000 ‐0.609375 1.000000
1 1.2500 2.0000 1.466667 0.003852 0.750000
2 1.2500 1.4667 1.463962 ‐0.005645 0.216667
3 1.4640 1.4667 1.465979 0.001431 0.002705
4 1.4640 1.4660 1.465300 ‐0.000953 0.002017
5 1.4653 1.4660 1.465687 0.000408 0.000679
6 1.4653 1.4657 1.465509 ‐0.000220 0.000388
7 1.4655 1.4657 1.465601 0.000106 0.000179
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

29. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﻫﺎي‬ ‫روش‬
‫داﻣﻨﻪ‬‫ﻧﺎﻣﺤﺪود‬
)‫ﺑﺎز‬(
29
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

30. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺳﻜﺎﻧﺖ‬ ‫روش‬
‫ي‬‫ﺷﻴﻮه‬Secant‫ﻧﺎم‬ ‫ﺑﻪ‬ ‫ﺗﺎﺑﻊ‬ ‫ﺻﻔﺮ‬ ‫ﺣﻮل‬ ‫آن‬ ‫اوﻟﻴﻪ‬ ‫ﺗﻘﺮﻳﺐ‬ ‫دو‬ ‫و‬α
30
‫از‬‫آﻧﺠﺎﺋﻴﻜﻪ‬‫ﻫﺎي‬‫روش‬Bisection‫و‬False Position‫ﺑﺎ‬
‫ﺳﺮﻋﺖ‬‫ﻛﻤﻲ‬‫ﺑﻪ‬‫ﺳﻤﺖ‬‫رﻳﺸﻪ‬‫ﻣﻴﻞ‬،‫ﻛﻨﻨﺪ‬‫ﻣﻲ‬‫ﻟﺬا‬‫ﺷ‬‫ﻴﻮه‬
‫اي‬‫ﺳﺮﻳﻌﺘﺮ‬‫ﺑﺮاي‬‫ﻳﺎﻓﺘﻦ‬‫رﻳﺸﻪ‬‫ﻧﻴﺎز‬‫اﺳﺖ‬.‫ﻳﻚ‬‫ﭼﻨ‬‫ﻴﻦ‬
،‫اي‬‫ﺷﻴﻮه‬‫ي‬‫ﺷﻴﻮه‬Secant‫ﻧﺎم‬‫دارد‬.‫ﻣﺸﺎﺑﻪ‬‫ي‬‫ﺷﻴﻮه‬
False Position،‫اﺳﺎس‬‫اﻳﻦ‬‫روش‬‫ﻧﻴﺰ‬‫ﺑﺮ‬‫ﺗﻘﺮﻳﺐ‬‫زدن‬
‫رﻳﺸﻪ‬‫ﺗﺎﺑﻊ‬‫از‬‫ﻃﺮﻳﻖ‬‫ﻳﻚ‬‫ﺧﻂ‬‫ﻣﺴﺘﻘﻴﻢ‬‫ﻗﺮار‬‫دارد‬‫ﻛ‬‫ﻪ‬‫دو‬
‫ﻧﻘﻄﻪ‬‫از‬‫ﻧﻤﻮدار‬‫ﺗﺎﺑﻊ‬‫را‬‫ﺑﻪ‬‫ﻳﻜﺪﻳﮕﺮ‬‫وﺻﻞ‬،‫ﻛﻨﺪ‬‫ﻣﻲ‬‫اﻣﺎ‬
‫ﻧﻴﺎزي‬‫ﻧﻴﺴﺖ‬‫ﻛﻪ‬‫ﻧﻘﺎط‬‫ﺣﺪس‬‫اوﻟﻴﻪ‬ً‫ﺎ‬‫ﺣﺘﻤ‬‫داراي‬
‫ﻋﻼﻣﺖ‬‫ﻣﺨﺎﻟﻒ‬‫ﺑﺎﺷﻨﺪ‬.
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

31. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺳﻜﺎﻧﺖ‬ ‫روش‬ ‫اﻟﮕﻮرﻳﺘﻢ‬)Secant(
•‫اول‬ ‫ﮔﺎم‬:
•‫دوم‬ ‫ﮔﺎم‬:
•‫ﺳﻮم‬ ‫ﮔﺎم‬:،‫ﺑﺮو‬ ‫ﭼﻬﺎر‬ ‫ﻣﺮﺣﻠﻪ‬ ‫ﺑﻪ‬ ،‫ﺑﺎﺷﺪ‬ ‫ﺑﺮﻗﺮار‬ ‫ﻓﻮق‬ ‫ﺷﺮط‬ ‫اﮔﺮ‬
‫دارﻳﻢ‬ ‫ﻏﻴﺮاﻳﻨﺼﻮرت‬ ‫در‬:
•‫ﭼﻬﺎرم‬ ‫ﮔﺎم‬:‫ﻧﻤﺎﻳﺶ‬1i+x‫اﻟﮕﻮرﻳﺘﻢ‬ ‫ﭘﺎﻳﺎن‬ ‫و‬ ‫رﻳﺸﻪ‬ ‫ﻋﻨﻮان‬ ‫ﺑﻪ‬.
31
iiii xxxxSet   11 ,
2111 |)(|/||    iniin xfeorandxxe
  
   ii
iii
ii
xfxf
xxxf
xx






1
1
1
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

32. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺑﺮﻧﺎﻣﻪ‬ ‫و‬ ‫ﻛﺪ‬ ‫ﺷﺒﻪ‬Matlab‫ﺳﻜﺎﻧﺖ‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬
•‫ﻛﺪ‬ ‫ﺷﺒﻪ‬
32
‫ﺗﺎﺑﻊ‬Matlab‫ﺳﻜﺎﻧﺖ‬ ‫روش‬ ‫ﺑﺮاي‬‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

33. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫روش‬‫ﺳﻜﺎﻧﺖ‬
function secant(f,x0,x1,tol,n)
% The secant method for solving the
nonlinear
% equation f(x)=0.
iter=0;
u=feval(f,x0);
v=feval(f,x1);
err=abs(x1‐x0);
disp('______________________________
____')
disp('iter xn f(xn) f(xn+1)‐f(xn)
|xn+1‐xn|')
disp('______________________________
____')
fprintf('%2.0f %12.6f %12.6fn',iter,x0,u)
33
fprintf('%2.0f %12.6f %12.6f %12.6f
%12.6fn',iter,x1,v,v‐u,err)
while (err>tol)&(iter<=n)&((v‐u)~=0)
x=x1‐v*(x1‐x0)/(v‐u);
x0=x1;
u=v;
x1=x;
v=feval(f,x1);
err=abs(x1‐x0);
iter=iter+1;
fprintf('%2.0f %12.6f %12.6f %12.6f
%12.6fn',iter,x1,v,v‐u,err)
end
if ((v‐u)==0)
disp(' Division by zero')
end
if (iter>n)
disp(' Method failed to converge')
end
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

34. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺳﻜﺎﻧﺖ‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﻌﺎﻳﺐ‬ ‫و‬ ‫ﻣﺰاﻳﺎ‬
•‫ﻣﺰاﻳﺎ‬:
1(‫ﺳﺮﻳﻊ‬ ‫ﻫﻤﮕﺮاﻳﻲ‬
2(‫ﻧﺪارد‬ ‫ﻧﻴﺎزي‬ ‫ﻣﺸﺘﻖ‬ ‫ﺑﻪ‬
34
‫ﻣﻌﺎﻳﺐ‬
1(‫ﭼﻨﺎﻧﭽﻪ‬‫ﺗﺎﺑﻊ‬‫ﺑﻴﺶ‬‫از‬‫ﺣﺪ‬‫ﻏﻴﺮ‬
‫ﺧﻄﻲ‬،‫ﺑﺎﺷﺪ‬‫در‬‫ﭼﻨﻴﻦ‬،‫ﺣﺎﻟﺘﻲ‬
‫رﻳﺸﻪ‬‫ﻫﺎي‬‫ﻳﺎﻓﺖ‬‫ﺷﺪه‬‫در‬‫ﮔﺎم‬
‫ﻫﺎي‬،‫ﺑﻌﺪي‬‫ﺑﻪ‬‫ﺟﺎي‬‫ﻫﻤﮕﺮاﻳﻲ‬‫ﺑﻪ‬
‫ﺳﻤﺖ‬،‫رﻳﺸﻪ‬‫ﻣﻄﺎﺑﻖ‬،‫ﺷﻜﻞ‬‫واﮔ‬‫ﺮا‬
‫ﻣﻲ‬‫ﮔﺮدﻧﺪ‬.
x0 x1r x2
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

35. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﺜﺎل‬Secant
35
:
===========================================================================================================================
‫ﺟﺪو‬ ‫در‬ ‫ﻧﺘﺎﻳﺞ‬ ‫ﻛﻪ‬‫ل‬
f (x1)=3
x1
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

36. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬‫ﺳﻜﺎﻧﺖ‬
•‫ﺗﺎﺑﻊ‬ ‫رﺳﻢ‬1-2x-3x)=x(f‫ﺑﺎزه‬ ‫در‬]2,1[
36
>> secant('f1',1,2,10^(‐4),40)
________________________________________________
iter xn f(xn) f(xn+1)‐f(xn) |xn+1‐xn|
________________________________________________
0 1.000000 ‐1.000000
0 2.000000 3.000000 4.000000 1.000000
1 1.250000 ‐0.609375 ‐3.609375 0.750000
2 1.376623 ‐0.286264 0.323111 0.126623
3 1.488807 0.083463 0.369727 0.112184
4 1.463482 ‐0.007322 ‐0.090786 0.025325
5 1.465525 ‐0.000163 0.007160 0.002043
6 1.465571 ‐0.000163 0.000163 0.002043
7 1.465571 ‐3.20E‐07 ‐0.000000 0.000000
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

37. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺳﻜﺎﻧﺖ‬ ‫روش‬)Secant(
37
for n=1,2,…, ITMAX
1
1
1
( )[ ]
( ) ( )
n n
n n n
n n
x x
x x f x
f x f x




 

1
1
1
| |
| ( ) | , | |
| |
n n
n n n
n
x x
f x x x or
x
  



   
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

38. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫روش‬‫ﻧﻴﻮﺗﻦ‬-‫ﺳﺎده‬ ‫راﻓﺴﻮن‬
•‫روﺷﻲ‬‫ﻗﻮي‬‫ﻛﻪ‬ً‫ﺎ‬‫ﺗﻘﺮﻳﺒ‬‫در‬‫ﻛﻠﻴﻪ‬‫ﻣﻮاردي‬‫ﻛﻪ‬‫ﻓﺮض‬‫اوﻟﻴﻪ‬‫ﻧﺰدﻳﻚ‬‫ﺟﻮاب‬‫اﺳﺖ‬‫ﻫﻤﮕﺮاﺳﺖ‬.
•‫آﻟﮕﻮرﻳﺘﻢ‬‫ﺑﺮ‬‫ﻣﺒﻨﺎي‬‫اﺳﺘﻔﺎده‬‫از‬‫رﻳﺸﻪ‬‫ﺗﺎﺑﻊ‬‫ﻣﻤﺎس‬‫ﺑﺮ‬‫ﻣﻨﺤﻨﻲ‬
38
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39. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫اﻟﮕﻮرﻳﺘﻢ‬)Newton-Raphson(
•‫اول‬ ‫ﮔﺎم‬:
•‫دوم‬ ‫ﮔﺎم‬:
•‫ﺳﻮم‬ ‫ﮔﺎم‬:،‫ﭼﻬﺎر‬ ‫ﻣﺮﺣﻠﻪ‬ ‫ﺑﺮو‬ ،‫ﺑﺎﺷﺪ‬ ‫ﺑﺮﻗﺮار‬ ‫ﻓﻮق‬ ‫ﺷﺮط‬ ‫اﮔﺮ‬
‫دارﻳﻢ‬ ‫ﻏﻴﺮاﻳﻨﺼﻮرت‬ ‫در‬:
•‫ﭼﻬﺎرم‬ ‫ﮔﺎم‬:‫ﻧﻤﺎﻳﺶ‬1i+x‫اﻟﮕﻮرﻳﺘﻢ‬ ‫ﭘﺎﻳﺎن‬ ‫و‬ ‫رﻳﺸﻪ‬ ‫ﻋﻨﻮان‬ ‫ﺑﻪ‬.
39
11   iixxSet ii
2111 |)(|/||    iniin xfeorandxxe
 
 i
i
ii
xf
xf
xx

1
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40. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫روش‬‫ﻧﻴﻮﺗﻦ‬-‫راﻓﺴﻮن‬
function newton(f,df,x0,tol,n)
% Newton's method for solving the
nonlinear
% equation f(x)=0.
iter=0;
u=feval(f,x0);
v=feval(df,x0);
err=abs(u/v);
disp('_____________________________')
disp(' iter x f(x) df(x)
|xn+1‐xn|')
disp('_____________________________')
fprintf('%2.0f %12.6f %12.6f
%12.6fn',iter,x0,u,v)
while (err>tol)&(iter<=n)&(v~=0)
x1=x0‐u/v;
40
err=abs(x1‐x0);
x0=x1;
u=feval(f,x0);
v=feval(df,x0);
iter=iter+1;
fprintf('%2.0f %12.6f %12.6f %12.6f
%12.6fn',iter,x0,u,v,err)
end
if (v==0)
disp(' division by zero')
end
if (iter>n)
disp(' Method failed to converge')
end
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41. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫اﻳﺮادات‬
•‫واﮔﺮاﻳﻲ‬‫در‬‫ﺻﻮرت‬‫ﻋﺪم‬‫اﻧﺘﺨﺎب‬
‫ﻧﻘﻄﻪ‬‫ﺷﺮوع‬،‫ﻣﻨﺎﺳﺐ‬ً‫ﻼ‬‫ﻣﺜ‬‫ﺑﺮ‬‫اي‬
‫ﺣﺎﻟﺖ‬‫ﻧﺸﺎن‬‫داده‬،‫ﺷﺪه‬‫ﻫﻴﭽﮕﺎه‬
‫ﺧﻂ‬‫ﺣﺎﺻﻞ‬‫از‬‫ﻣﺸﺘﻖ‬‫ﻣﺤﻮر‬x‫ﻫﺎ‬
‫را‬‫ﻗﻄﻊ‬‫ﻧﻤﻲ‬‫ﻛﻨﺪ‬.
•‫ﻣﺸﻜﻞ‬‫و‬‫ﻳﺎ‬‫ﻏﻴﺮﻗﺎﺑﻞ‬‫ﻣﺤﺎﺳﺒ‬‫ﻪ‬
‫ﺑﻮدن‬‫ﻣﺸﺘﻖ‬‫ﺗﺎﺑﻊ‬‫در‬‫ﻧﻘﺎط‬‫ﺑ‬‫ﺎزه‬
‫اي‬‫داده‬‫ﺷﺪه‬.
41
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42. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﻧﻴﻮﺗﻦ‬ ‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﺜﺎل‬)Newton-Raphson(
42
:
‫ﻛﻪ‬
===========================================================================================================================
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43. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﻧﻴﻮﺗﻦ‬ ‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬-‫راﻓﺴﻮن‬
•‫ﺗﺎﺑﻊ‬ ‫رﺳﻢ‬1-2x-3x)=x(f‫ﺑﺎزه‬ ‫در‬]2,1[
43
>> newton('f1','df1',1,10^(‐4),40)
________________________________________________
iter xn f(xn) df(xn) |xn+1‐xn|
________________________________________________
0 1.000000 ‐1.000000 1.000000
1 2.000000 3.000000 8.000000 1.000000
2 1.625000 0.650391 4.671875 0.375000
3 1.485786 0.072402 3.651108 0.139214
4 1.465956 0.001352 3.515168 0.019830
5 1.465571 0.000001 3.512556 0.000385
6 1.465571 0.000000 3.512555 143E‐7
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44. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫روش‬‫ﻧﻴﻮﺗﻦ‬-‫راﻓﺴﻮن‬‫ﭼﻨﺪ‬ ‫ﻫﺎي‬‫رﻳﺸﻪ‬ ‫و‬ ‫ﻳﺎﻓﺘﻪ‬ ‫ﺗﻌﻤﻴﻢ‬‫ﮔﺎﻧﻪ‬
•‫در‬‫ﻗﺴﻤﺖ‬،‫ﻗﺒﻞ‬‫اﻟﮕﻮرﻳﺘﻢ‬‫ﻧﻴﻮﺗﻦ‬‫را‬‫ﺗﺤﺖ‬‫ﺷﺮط‬f’ (α) ≠ 0‫ﺑﺪﺳﺖ‬‫آوردﻳﻢ‬‫و‬α‫را‬‫ﻧﻴﺰ‬‫ﺑﻪ‬‫ﻋﻨﻮان‬
‫ي‬‫رﻳﺸﻪ‬‫ﺳﺎده‬‫ﻓﺮض‬‫ﻧﻤﻮدﻳﻢ‬.
•‫ﻣﻄﺎﺑﻖ‬‫ﺷﻜﻞ‬،‫روﺑﺮو‬‫ﻣﺸﻜﻞ‬‫در‬‫اﺳﺘﻔﺎده‬‫از‬‫روش‬‫ﻧﻴﻮﺗﻦ‬‫زﻣﺎﻧﻲ‬
‫ﻇﺎﻫﺮي‬‫ﺷﻮد‬‫ﻣﻲ‬‫ﻛﻪ‬α‫دﻳﮕﺮ‬‫رﻳﺸﻪ‬‫ي‬‫ا‬‫ﺳﺎده‬‫ﻧﺒﺎﺷﺪ‬.
•α‫را‬‫رﻳﺸﻪ‬‫ي‬‫ﻣﻜﺮر‬‫ي‬‫ﻣﺮﺗﺒﻪ‬k‫ام‬‫از‬‫ﺗﺎﺑﻊ‬f‫ﮔﻮﺋﻴﻢ‬‫اﮔﺮ‬:
44
0)(0)()()( )()1(
 
 kk
fbutfff  0)(0)()( )()1(
 
 kk
fbutfff 
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45. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫اداﻣﻪ‬-‫راﻓﺴﻮن‬‫ﭼﻨﺪﮔﺎﻧ‬ ‫ﻫﺎي‬‫رﻳﺸﻪ‬ ‫و‬ ‫ﻳﺎﻓﺘﻪ‬ ‫ﺗﻌﻤﻴﻢ‬‫ﻪ‬
‫ﻣﺸﻜﻼت‬‫ﻧﺎﺷﻲ‬‫از‬‫ﻫﺎي‬‫رﻳﺸﻪ‬‫ﻣﻜﺮر‬،‫ﻣﺎ‬‫را‬‫ﻣﺠﺒﻮر‬‫ﺑﻪ‬‫اﺻﻼح‬‫روش‬‫ﻧﻴﻮﺗﻦ‬‫ﻛﻨﺪ‬‫ﻣﻲ‬.‫ﻓﺮض‬‫ﻛﻨﻴﻢ‬‫ﻣﻲ‬‫ﻛﻪ‬f‫داراي‬‫ﻳﻚ‬
‫رﻳﺸﻪ‬‫ﻣﺎﻧﻨﺪ‬α‫ﺑﺎ‬m‫ﺑﺎر‬‫ﺗﻜﺮار‬‫اﺳﺖ‬.‫اﺑﺘﺪا‬‫ﺗﺎﺑﻊ‬‫ﺟﺪﻳﺪ‬‫زﻳﺮ‬‫را‬‫ﺗﻌﺮﻳﻒ‬‫ﻣﻲ‬‫ﻛﻨﻴﻢ‬:
‫ﺣﺎل‬‫ﺑﺎ‬‫اﺳﺘﻔﺎده‬‫از‬‫ﻣﻌﺎدﻟﻪ‬‫ﺑﺴﻂ‬‫ﺗﻴﻠﻮر‬‫ﺗﺎﺑﻊ‬f (x)‫ﺣﻮل‬
α‫ﺑﻪ‬‫ﺻﻮرت‬‫زﻳﺮ‬‫در‬‫آﻳﺪ‬‫ﻣﻲ‬:
‫ﺑﮕﻮﻧﻪ‬‫اي‬‫ﻛﻪ‬:
‫ﺑﺎ‬‫ﻣﺸﺘﻖ‬‫ﮔﻴﺮي‬‫از‬‫ﻣﻌﺎدﻟﻪ‬،‫ﻓﻮق‬‫ﺑﺪﺳﺖ‬‫آورﻳﻢ‬‫ﻣﻲ‬:
45
( )
( )
'( )
f x
u x
f x

( ) ( ) ( )m
f x x h x 
( ) ( 1)
( ) ( )
( ) ( ) ...
! ( 1)!
m m
f f
h x x
m m
 


   

1 1
'( ) ( ) '( ) ( ) ( ) ( ) [( ) '( ) ( )]m m m
f x x h x m x h x x x h x mh x    
       
0)(0)()()( )()1(
 
 mm
fbutfff 
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46. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫اداﻣﻪ‬-‫راﻓﺴﻮن‬‫ﭼﻨﺪﮔﺎﻧ‬ ‫ﻫﺎي‬‫رﻳﺸﻪ‬ ‫و‬ ‫ﻳﺎﻓﺘﻪ‬ ‫ﺗﻌﻤﻴﻢ‬‫ﻪ‬
‫ﺑﻨﺎﺑﺮاﻳﻦ‬‫از‬‫ﻣﻌﺎدﻻت‬‫ﻗﺒﻞ‬‫ﺗﻮان‬‫ﻣﻲ‬‫ﻣﻌﺎدﻟﻪ‬‫را‬‫ﺑﻪ‬‫ﻓﺮم‬‫زﻳﺮ‬‫ﻧﻮﺷﺖ‬:
‫ﻛﻪ‬‫در‬‫آن‬:
‫ﺑﺎﺷﺪ‬.‫از‬‫آﻧﺠﺎﻳﻲ‬‫ﻛﻪ‬‫ﻣﻲ‬‫ﺑﺎﺷﺪ‬.‫دارﻳﻢ‬:
‫و‬
‫ﺑﺎ‬‫ﺗﻮﺟﻪ‬‫ﺑﻪ‬‫رواﺑﻂ‬،‫ﻓﻮق‬‫ﺗﻮان‬‫ﻣﻲ‬‫اﺳﺘﺒﺎط‬‫ﻛﺮد‬‫ﻛﻪ‬u(x)‫داراي‬‫ﻳﻚ‬‫ﺳﺎده‬‫رﻳﺸﻪ‬‫در‬x=α‫اﺳﺖ‬.‫ﺣﺎل‬‫ﺑﺎ‬‫اﻋﻤﺎل‬‫روش‬
‫ﻧﻴﻮﺗﻦ‬‫ﺑﻪ‬‫ﺗﺎﺑﻊ‬u(x)‫و‬‫ﺑﺎ‬‫اﺳﺘﻔﺎده‬‫از‬‫ﻧﺘﺎﻳﺞ‬‫ﻓﻮق‬‫ﺣﺎﺻﻞ‬‫ﺷﻮد‬‫ﻣﻲ‬:
‫ﻛﻪ‬‫در‬‫اﻳﻦ‬‫راﺑﻄﻪ‬،m‫درﺟﻪ‬‫ﭼﻨﺪﮔﺎﻧﮕﻲ‬‫رﻳﺸﻪ‬‫را‬‫ﻣﺸﺨﺺ‬‫ﻛﻨﺪ‬‫ﻣﻲ‬.‫ي‬‫ﻣﻌﺎدﻟﻪ‬‫ﻓﻮق‬‫را‬‫روش‬‫ﺗﻌﻤﻴﻢ‬‫ﻳﺎﻓﺘﻪ‬‫ﻧﻴﻮﺗﻦ‬
)Modified Newton Method(‫ﻣﻲ‬‫ﻧﺎﻣﻨﺪ‬.
46
( )
( )
'( )
f x
u x
f x

( ) ( )
( ) ( ) ( )
( ) '( ) ( )
x h x
u x x x
x h x mh x

 


  
 
( )
( )
( ) '( ) ( )
h x
x
x h x mh x



 
1
lim ( )
x
x
m



( )
( )
( )
!
m
f
h
m

 
1
lim '( ) lim[( ) '( ) ( )]
x x
u x x x x
m 
  
 
   
1
( )
'( )
n
n n
n
f x
x x m
f x
  
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47. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫ﺑﺮاي‬ ‫ﻣﺜﺎل‬-‫راﻓﺴﻮن‬‫ﭼ‬ ‫ﻫﺎي‬‫رﻳﺸﻪ‬ ‫و‬ ‫ﻳﺎﻓﺘﻪ‬ ‫ﺗﻌﻤﻴﻢ‬‫ﻨﺪﮔﺎﻧﻪ‬
‫ﺻﻮرت‬‫ﻣﺜﺎل‬1(‫ﺗﺎﺑﻊ‬‫داراي‬‫ﻳﻚ‬‫ي‬‫رﻳﺸﻪ‬‫ﻣﻀﺎﻋﻒ‬‫در‬1=x‫ﺑﺎﺷﺪ‬‫ﻣﻲ‬.‫ﺑﺎ‬‫در‬‫ﻧﻈﺮ‬
‫ﮔﺮﻓﺘﻦ‬0=0x‫و‬‫ﺑﺎ‬‫اﺳﺘﻔﺎده‬‫از‬‫روش‬‫ﻧﻴﻮﺗﻦ‬‫اﻳﻦ‬‫رﻳﺸﻪ‬‫را‬‫ﺗﺨﻤﻴﻦ‬‫و‬‫ﺗﻘﺮﻳﺐ‬‫ﺑﺰﻧﻴﺪ‬.
==========================================================================================
‫ﭘﺎﺳﺦ‬‫ﻣﺜﺎل‬(‫ﻧﻤﻮداري‬‫از‬f (x)‫رﺳﻢ‬‫ﺷﺪه‬‫ﺗﻮﺳﻂ‬‫ﻧﺮم‬‫اﻓﺰار‬MATLAB‫در‬
‫ﺷﻜﻞ‬‫روﺑﺮو‬‫ﻧﺸﺎن‬‫داده‬‫ﺷﺪه‬‫اﺳﺖ‬.
‫ﺗﻌﺮﻳﻒ‬:‫ﻓﺮض‬‫ﻛﻨﻴﻢ‬‫ﻣﻲ‬‫ﻛﻪ‬…,1x,0x‫ﻳﻚ‬‫ﺳﺮي‬‫از‬‫اﻋﺪاد‬‫ﻫﺴﺘﻨﺪ‬‫ﻛﻪ‬‫ﺑﻪ‬
‫ﻋﺪدي‬‫ﻣﺜﻞ‬α‫ﻫﻤﮕﺮا‬‫ﺑﺎﺷﻨﺪ‬‫و‬‫ﻣﺠﻤﻮﻋﻪ‬nx-α=ne‫را‬‫در‬‫ﻧﻈﺮ‬‫ﺑﮕﻴﺮﻳﻢ‬.
‫ﺣﺎل‬‫اﮔﺮ‬‫ﻋﺪدي‬‫ﻣﺜﻞ‬k‫و‬‫ﻋﺪد‬‫ﻣﺜﺒﺖ‬‫ﺛﺎﺑﺖ‬‫دﻳﮕﺮي‬‫ﻣﺜﻞ‬C‫ﻣﻮﺟﻮد‬‫ﺑﺎﺷﻨﺪ‬
‫ﺑﻪ‬‫اي‬‫ﮔﻮﻧﻪ‬‫ﻛﻪ‬:
‫آﻧﮕﺎه‬k‫را‬‫ﻣﺮﺗﺒﻪ‬‫ﻳﺎ‬‫ي‬‫درﺟﻪ‬‫ﻫﻤﮕﺮاﺋﻲ‬‫و‬C‫را‬‫ﺛﺎﺑﺖ‬‫ﺧﻄﺎي‬‫ﻣﺠﺎﻧﺒﻲ‬‫ﻧﺎﻣﻨﺪ‬‫ﻣﻲ‬.
47
3 2
( ) 7 11 5f x x x x   
1|
lim n
kx
n
e
C
e


0 1 2 3 4 5
-15
-10
-5
0
5
10
15
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

48. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫ﺑﺮاي‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬-‫راﻓﺴﻮن‬‫ﻫ‬‫رﻳﺸﻪ‬ ‫و‬ ‫ﻳﺎﻓﺘﻪ‬ ‫ﺗﻌﻤﻴﻢ‬‫ﺎي‬
‫ﭼﻨﺪﮔﺎﻧﻪ‬
‫اداﻣﻪ‬‫ﭘﺎﺳﺦ‬‫ﻣﺜﺎل‬(‫ﺑﺎ‬‫ﺗﻮﺟﻪ‬‫ﺑﻪ‬‫ﺗﻌﺮﻳﻒ‬‫ﻓﻮق‬‫و‬‫ﻧﺘﺎﻳﺞ‬‫ﻋﺪدي‬‫داده‬‫ﺷﺪه‬‫در‬‫ﺟﺪول‬،‫زﻳﺮ‬‫ﻣﺸﺎﻫﺪه‬‫ﺷﻮد‬‫ﻣﻲ‬‫ﻛﻪ‬‫روش‬‫ﻧﻴ‬‫ﻮﺗﻦ‬
‫ﺑﻪ‬‫ﻃﻮر‬‫ﺧﻄﻲ‬‫اﻣﺎ‬‫ﺑﺴﻴﺎر‬،‫ﻛﻨﺪ‬‫ﺑﻪ‬‫ﺳﻤﺖ‬=1α‫ﻫﻤﮕﺮا‬‫ﻣﻲ‬‫ﺑﺎﺷﺪ‬.
48
>> newton('f12','df12',0,10^(-4),40)
____________________________________________________
iter x f(x) df(x) |xn+1-xn|
____________________________________________________
0 0.000000 -5.000000 11.000000 ---
1 0.454545 -1.352367 5.256198 0.454545
2 0.711835 -0.356084 2.554434 0.257290
3 0.851234 -0.091818 1.256523 0.139399
4 0.924307 -0.023352 0.622734 0.073073
5 0.961805 -0.005891 0.309935 0.037498
6 0.980813 -0.001480 0.154603 0.019008
7 0.990384 -0.000371 0.077209 0.009571
8 0.995186 -0.000093 0.038582 0.004802
9 0.997592 -0.000023 0.019285 0.002406
10 0.998795 -0.000006 0.009641 0.001204
11 0.999398 -0.000001 0.004820 0.000602
12 0.999699 -0.000000 0.002410 0.000301
13 0.999849 -0.000000 0.001205 0.000151
14 0.999925 -0.000000 0.000602 0.000075
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

49. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫روش‬‫ﻳﺎﻓﺘﻪ‬ ‫ﺗﻌﻤﻴﻢ‬ ‫ﻧﻴﻮﺗﻦ‬
49
function newton2(f,df,x0,m,tol,n)
% The Modified Newton's method for solving the nonlinear
% equation f(x)=0.
iter=0;
u=feval(f,x0);
v=feval(df,x0);
err=abs(m*u/v);
disp('___________________________________________')
disp('iter x f(x) df(x) |xn+1-xn|')
disp('___________________________________________')
fprintf('%2.0f %12.6f %12.6f %12.6fn',iter,x0,u,v)
while (err>tol)&(iter<=n)&(v~=0)
x1=x0-m*u/v;
err=abs(x1-x0);
x0=x1;
u=feval(f,x0);
v=feval(df,x0);
iter=iter+1;
fprintf('%2.0f %12.6f %12.6f %12.6f
%12.7fn',iter,x0,u,v,err)
pause(1)
y(iter)=err;
end
if (v==0)
disp(' division by zero')
end
if (iter>n), disp(' Method failed to converge') ; end
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

50. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫ﺑﺮاي‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬-‫راﻓﺴﻮن‬‫ﻫ‬‫رﻳﺸﻪ‬ ‫و‬ ‫ﻳﺎﻓﺘﻪ‬ ‫ﺗﻌﻤﻴﻢ‬‫ﺎي‬
‫ﭼﻨﺪﮔﺎﻧﻪ‬
‫اداﻣﻪ‬‫ﭘﺎﺳﺦ‬‫ﻣﺜﺎل‬(‫ﺣﺎل‬‫ﺑﺎ‬‫اﺳﺘﻔﺎده‬‫از‬‫ي‬‫ﺷﻴﻮه‬Modified Newton Method،‫و‬‫اﻋﻤﺎل‬‫داده‬‫ﻫﺎي‬‫ﻫﻤﺎن‬‫ﻣﺜﺎل‬‫ﻗﺒﻞ‬‫ﺑﻪ‬
‫اﻳﻦ‬‫روش‬،‫ﻧﻴﻮﺗﻦ‬‫ﻣﻲ‬‫ﺗﻮاﻧﻴﺪ‬‫ﺑﺒﻴﻨﻴﺪ‬‫ﻛﻪ‬‫ﭘﺎﺳﺦ‬‫ﻫﺎ‬‫ﺑﻪ‬‫ﻃﻮر‬‫ﺧﻄﻲ‬‫و‬‫ﺑﺴﻴﺎر‬،‫ﺳﺮﻳﻊ‬‫ﺑﻪ‬‫ﺳﻤﺖ‬=1α‫ﻫﻤﮕﺮا‬‫ﻣﻲ‬‫ﺷﻮد‬.
50
>> newton2('f12','df12',0,2,10^(‐2),40)
________________________________________________
iter xn f(xn) df(xn) |xn+1‐xn|
___________________________________________________
0 0.000000 ‐5.000000 11.000000
1 0.909091 ‐0.033809 0.752066 0.9090909
2 0.999001 ‐0.000004 0.007995 0.0899101
3 1.000000 ‐0.000000 0.000001 0.0009989
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

51. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫ﺑﺮاي‬ ‫ﻣﺜﺎل‬-‫راﻓﺴﻮن‬‫ﭼ‬ ‫ﻫﺎي‬‫رﻳﺸﻪ‬ ‫و‬ ‫ﻳﺎﻓﺘﻪ‬ ‫ﺗﻌﻤﻴﻢ‬‫ﻨﺪﮔﺎﻧﻪ‬
‫ﺻﻮرت‬‫ﻣﺜﺎل‬2(‫ﺗﺎﺑﻊ‬‫داراي‬‫ﻳﻚ‬‫ي‬‫رﻳﺸﻪ‬‫ﻣﻀﺎﻋﻒ‬‫در‬3=x‫ﺑﺎﺷﺪ‬‫ﻣﻲ‬.‫ﺑﺎ‬‫در‬‫ﻧﻈﺮ‬
‫ﮔﺮﻓﺘﻦ‬0=0x‫و‬2m=‫و‬‫اﺳﺘﻔﺎده‬‫از‬‫روش‬‫ﻧﻴﻮﺗﻦ‬‫ﺑﺴﻂ‬‫ﻳﺎﻓﺘﻪ‬‫رﻳﺸﻪ‬‫را‬‫ﺗﺨﻤﻴﻦ‬‫و‬‫ﺗﻘﺮﻳﺐ‬‫ﺑﺰﻧﻴﺪ‬.
==========================================================================================
‫ﭘﺎﺳﺦ‬‫ﻣﺜﺎل‬(‫ﻧﻤﻮداري‬‫از‬f (x)‫ﺗﻮﺳﻂ‬‫ﻧﺮم‬‫اﻓﺰار‬MATLAB‫در‬‫ﺷﻜﻞ‬‫روﺑﺮو‬
‫ﻧﺸﺎن‬‫داده‬‫ﺷﺪه‬‫اﺳﺖ‬.
‫ﺣﺎل‬‫ﺑﺎ‬‫اﺳﺘﻔﺎده‬‫از‬‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫ﻣﻲ‬‫ﺗﻮاﻧﻴﻢ‬‫ﺗﻮاﺑﻌﻲ‬‫ﺑﻪ‬‫ﻗﺮار‬‫ﺗﻌﺮﻳﻒ‬‫و‬‫ﺳﭙﺲ‬
‫در‬‫ﻗﺎﻟﺐ‬‫ﺑﺮﻧﺎﻣﻪ‬‫ﺗﺎﺑﻊ‬newton2‫ﻛﻪ‬‫ﻫﻤﺎن‬
‫ﻓﺮم‬‫ﺗﻌﻤﻴﻢ‬‫ﻳﺎﻓﺘﻪ‬‫رﻳﺸﻪ‬‫ﻳﺎﺑﻲ‬‫ﻧﻴﻮﺗﻦ‬،‫اﺳﺖ‬
‫اﺳﺘﻔﺎده‬‫ﻛﻨﻴﻢ‬:
51
1834)( 23
 xxxxf
-2 -1 0 1 2 3 4 5
-20
-15
-10
-5
0
5
10
15
20
25
30
13( );
^3 4* ^2 3* 18;
13( );
3* ^2 8* 3;
function f f x
f x x x
function f df x
f x x

   

  
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

52. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫ﺑﺮاي‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬-‫راﻓﺴﻮن‬‫ﻫ‬‫رﻳﺸﻪ‬ ‫و‬ ‫ﻳﺎﻓﺘﻪ‬ ‫ﺗﻌﻤﻴﻢ‬‫ﺎي‬
‫ﭼﻨﺪﮔﺎﻧﻪ‬
‫اداﻣﻪ‬‫ﭘﺎﺳﺦ‬‫ﻣﺜﺎل‬(‫ﺣﺎل‬‫ﺑﺎ‬‫اﺳﺘﻔﺎده‬‫از‬‫ي‬‫ﺷﻴﻮه‬Modified Newton Method،‫و‬‫اﻋﻤﺎل‬‫ﺗﻮاﺑﻊ‬‫ﺻﻔﺤﻪ‬‫ﻗﺒﻞ‬‫ﺑﻪ‬‫روش‬
‫ﻧﻴﻮﺗﻦ‬‫ﺗﻌﻤﻴﻢ‬،‫ﻳﺎﻓﺘﻪ‬‫ﻣﻲ‬‫ﺗﻮاﻧﻴﺪ‬‫ﺑﺒﻴﻨﻴﺪ‬‫ﻛﻪ‬‫ﭘﺎﺳﺦ‬‫ﻫﺎ‬‫ﺑﻪ‬‫ﻃﻮر‬‫ﺧﻄﻲ‬‫و‬‫ﺑﺴﻴﺎر‬،‫ﺳﺮﻳﻊ‬‫ﺑﻪ‬‫ﺳﻤﺖ‬=3α‫ﻫﻤﮕﺮا‬‫ﻣﻲ‬‫ﺷﻮﻧﺪ‬.
52
>> newton2('f13','df13',0,2,10^(‐2),40)
__________________________________________________
iter xn f(xn) df(xn) |xn+1‐xn|
__________________________________________________
0 0.000000 18.000000 ‐3.000000 ‐‐‐
1 12.000000 1134.000000 333.000000 12.0000000
2 5.189189 34.454544 36.269540 6.8108108
3 3.289273 0.442600 3.143766 1.8999162
4 3.007700 0.000297 0.077175 0.2815733
5 3.000006 0.000000 0.000059 0.0076938
>> newton('f13','df13',1,10^(‐2),40)
___________________________________________
iter x f(x) df(x) |xn+1‐xn|
____________________________________________
0 1.000000 12.000000 ‐8.000000
1 2.500000 1.125000 ‐4.250000 1.500000
2 2.764706 0.263790 ‐2.186851 0.264706
3 2.885331 0.064237 ‐1.107240 0.120625
4 2.943347 0.015866 ‐0.556906 0.058015
5 2.971837 0.003944 ‐0.279255 0.028490
6 2.985958 0.000983 ‐0.139826 0.014122
7 2.992989 0.000245 ‐0.069962 0.007031
8 2.996497 0.000061 ‐0.034993 0.003508
9 2.998249 0.000015 ‐0.017500 0.001752
10 2.999125 0.000004 ‐0.008751 0.000876
11 2.999562 0.000001 ‐0.004376 0.000438
12 2.999781 0.000000 ‐0.002188 0.000219
13 2.999891 0.000000 ‐0.001094 0.000109
14 2.999945 0.000000 ‐0.000547 0.000055
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

53. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺳﻜﺎﻧﺖ‬ ‫و‬ ‫ﻧﻴﻮﺗﻦ‬ ‫ﻫﺎي‬ ‫روش‬ ‫ﻫﻤﮕﺮاﻳﻲ‬ ‫ﺑﺮرﺳﻲ‬
•‫در‬‫ﺟﻬﺖ‬‫ﻣﺸﺨﺺ‬‫ﻛﺮدن‬‫ي‬‫درﺟﻪ‬‫ﻫﻤﮕﺮاﺋﻲ‬‫در‬‫ﻫﺎي‬‫روش‬Newton‫و‬Secant،‫ﺗﻌﺮﻳﻒ‬‫زﻳﺮ‬‫را‬‫اراﺋﻪ‬
‫دﻫﻴﻢ‬‫ﻣﻲ‬:
‫ﺗﻌﺮﻳﻒ‬:‫ﻓﺮض‬‫ﻛﻨﻴﻢ‬‫ﻣﻲ‬‫ﻛﻪ‬…,1x,0x‫ﻳﻚ‬‫ﺳﺮي‬‫از‬‫اﻋﺪاد‬‫ﻫﺴﺘﻨﺪ‬‫ﻛﻪ‬‫ﺑﻪ‬‫ﻋﺪدي‬‫ﻣﺜﻞ‬α‫ﻫﻤﮕﺮا‬
‫ﺑﺎﺷﻨﺪ‬‫و‬‫ﻣﺠﻤﻮﻋﻪ‬nx-α=ne‫را‬‫در‬‫ﻧﻈﺮ‬‫ﺑﮕﻴﺮﻳﻢ‬.‫ﺣﺎل‬‫اﮔﺮ‬‫ﻋﺪدي‬‫ﻣﺜﻞ‬k‫و‬‫ﻋﺪد‬‫ﻣﺜﺒﺖ‬‫ﺛﺎﺑﺖ‬‫دﻳﮕﺮي‬
‫ﻣﺜﻞ‬C‫ﻣﻮﺟﻮد‬‫ﺑﺎﺷﻨﺪ‬‫ﺑﻪ‬‫اي‬‫ﮔﻮﻧﻪ‬‫ﻛﻪ‬:
‫آﻧﮕﺎه‬k‫را‬‫ﻣﺮﺗﺒﻪ‬‫ﻳﺎ‬‫ي‬‫درﺟﻪ‬‫ﻫﻤﮕﺮاﺋﻲ‬‫و‬C‫را‬‫ﺛﺎﺑﺖ‬‫ﺧﻄﺎي‬‫ﻣﺠﺎﻧﺒﻲ‬‫ﻧﺎﻣﻨﺪ‬‫ﻣﻲ‬.
•‫ﺣﺎل‬‫ﻣﻴﺰان‬‫ﻫﻤﮕﺮاﻳﻲ‬‫را‬‫ﺑﺮاي‬‫ﻫﺎي‬‫رﻳﺸﻪ‬‫ﺳﺎده‬‫ﻳﻚ‬‫ﻣﻌﺎدﻟﻪ‬‫ﺑﺮاي‬‫ﻫﺮ‬‫دو‬‫روش‬Newton‫و‬Secant
‫اﻣﺘﺤﺎن‬‫ﻛﻨﻴﻢ‬‫ﻣﻲ‬.‫اﺑﺘﺪا‬‫روش‬Newton‫را‬‫در‬‫ﻧﻈﺮ‬‫ﮔﻴﺮﻳﻢ‬‫ﻣﻲ‬.
53
1|
lim n
kx
n
e
C
e



‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

54. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺳﻜﺎﻧﺖ‬ ‫و‬ ‫ﻧﻴﻮﺗﻦ‬ ‫ﻫﺎي‬ ‫روش‬ ‫ﻫﻤﮕﺮاﻳﻲ‬ ‫ﺑﺮرﺳﻲ‬
•‫ﺣﺎل‬‫ﻣﻴﺰان‬‫ﻫﻤﮕﺮاﻳﻲ‬‫را‬‫ﺑﺮاي‬‫ﻫﺎي‬‫رﻳﺸﻪ‬‫ﺳﺎده‬‫ﻳﻚ‬‫ﻣﻌﺎدﻟﻪ‬‫ﺑﺮاي‬‫ﻫﺮ‬‫دو‬‫روش‬Newton‫و‬Secant
‫اﻣﺘﺤﺎن‬‫ﻛﻨﻴﻢ‬‫ﻣﻲ‬.‫اﺑﺘﺪا‬‫روش‬Newton‫را‬‫در‬‫ﻧﻈﺮ‬‫ﮔﻴﺮﻳﻢ‬‫ﻣﻲ‬.
•‫ﻃﺒﻖ‬‫ﻣﻌﺎدﻟﻪ‬‫دارﻳﻢ‬:
•‫اﮔﺮ‬‫ﺗﺎﺑﻊ‬f‫دوﺑﺎر‬‫ﻣﺸﺘﻖ‬‫ﭘﺬﻳﺮ‬،‫ﺑﺎﺷﺪ‬‫ﺳﺮي‬‫ﺗﻴﻠﻮر‬f‫ﺣﻮل‬nx‫ﻋﺒﺎرت‬‫زﻳﺮ‬‫را‬‫ﺑﺪﺳﺖ‬‫ﻣﻲ‬‫دﻫﺪ‬:
‫ﻛﻪ‬‫در‬‫آن‬ξ‫ﺑﻴﻦ‬nx‫و‬α‫ﻗﺮار‬‫دارد‬.
•‫ﭼﻮن‬f (α) = 0‫و‬f’ (α) ≠ 0‫ﺑﺮﻗﺮار‬‫ﻫﺴﺘﻨﺪ؛‬‫ﺑﻨﺎﺑﺮاﻳﻦ‬‫از‬‫ﺗﺮﻛﻴﺐ‬‫ﻛﺮدن‬‫دو‬‫راﺑﻄﻪ‬‫ﻓﻮق‬‫دارﻳﻢ‬:
54
1
( )
, 0
'( )
n
n n
n
f x
x x n
f x
   1 1
( )
'( )
n
n n n
n
f x
e x x
f x
      
2
( )
( ) ( ) ( ) '( ) ''( )
2
n
n n n n
x
f a f x x f x f

 

   
2 2
1
''( ) ''( )
( )
2 '( ) '( )
n n
n n n
n n
f f
e x e
f x f x
 
     
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

55. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺳﻜﺎﻧﺖ‬ ‫و‬ ‫ﻧﻴﻮﺗﻦ‬ ‫ﻫﺎي‬ ‫روش‬ ‫ﻫﻤﮕﺮاﻳﻲ‬ ‫ﺑﺮرﺳﻲ‬
•‫در‬‫ﻧﺘﻴﺠﻪ‬‫اﮔﺮ‬α‫ﻳﻚ‬‫رﻳﺸﻪ‬‫ﺳﺎده‬،‫ﺑﺎﺷﺪ‬‫ﻳﺎ‬‫ﺑﻪ‬‫ﻋﺒﺎرت‬‫دﻳﮕﺮ‬f’ (α) ≠ 0‫آﻧﮕﺎه‬‫ﻣﻌﺎدﻟﻪ‬
‫ﻧﺸﺎن‬‫دﻫﺪ‬‫ﻣﻲ‬‫ﻛﻪ‬‫روش‬Newton‫ﺑﻪ‬‫ﺻﻮرت‬‫ﻣﺮﺗﺒﻪ‬‫دو‬
‫ﻫﻤﮕﺮا‬‫ﺧﻮاﻫﺪ‬‫ﺑﻮد‬.‫ﺑﻪ‬‫ﻃﻮر‬‫ﻣﺸﺎﺑﻪ‬‫اﻳﻦ‬‫ﻣﻮﺿﻮع‬‫ﺗﻮاﻧﺪ‬‫ﻣﻲ‬‫ﺑﻪ‬‫اﻳﻦ‬‫ﺷﻜﻞ‬‫ﻧﺸﺎن‬‫داده‬‫ﺷﻮد‬‫ﻛﻪ‬‫ﺧﻄﺎي‬
‫روش‬Secant‫ي‬‫ﻣﻌﺎدﻟﻪ‬‫زﻳﺮ‬‫را‬‫ارﺿﺎ‬‫ﻛﻨﺪ‬‫ﻣﻲ‬:
•‫ﻛﻪ‬‫در‬‫آن‬‫و‬‫ﻛﻮﭼﻜﺘﺮﻳﻦ‬‫ﻫﺎي‬‫ﺑﺎزه‬‫ﺷﺎﻣﻞ‬nx،α‫و‬1n+x‫ﻫﺴﺘﻨﺪ‬.‫ﺑﻪ‬‫ﻫﻤﻴﻦ‬،‫ﺧﺎﻃﺮ‬‫روش‬
Secantً‫ﺎ‬‫ﺗﻘﺮﻳﺒ‬‫داراي‬‫ﺳﺮﻋﺖ‬‫ﻫﻤﮕﺮاﺋﻲ‬‫ﻣﺸﺎﺑﻪ‬‫ﺑﺎ‬‫روش‬Newton‫ﺑﺎﺷﺪ‬‫ﻣﻲ‬.
55
2 2
1
''( ) ''( )
( )
2 '( ) '( )
n n
n n n
n n
f f
e x e
f x f x
 
     
1 1
''( )
2 '( )
n
n n n
n
f
e e e
f


 
nn
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

56. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺛﺎﺑﺖ‬ ‫ﻧﻘﻄﻪ‬ ‫ﺗﻜﺮاري‬ ‫روش‬)Fixed-Point(
56
‫ﺗﻤﺎﻣﻲ‬‫ﻫﺎ‬‫روش‬‫ﻳﻲ‬‫ﻛﻪ‬‫ﺗﺎ‬‫ﺑﻪ‬‫ﺣﺎل‬‫ﻣﻮرد‬‫ﺑﺤﺚ‬‫و‬‫ﺑﺮرﺳﻲ‬‫ﻗﺮار‬‫ﮔﺮﻓﺘﻨﺪ‬‫ﻣﺜﺎل‬‫ﻫﺎﻳﻲ‬
‫از‬‫ﻫﺎي‬‫روش‬‫ﺗﻜﺮار‬‫ﻋﻤﻠﻴﺎت‬‫ﺑﻮدﻧﺪ‬.‫در‬‫اﻳﻦ‬‫ﺑﺨﺶ‬‫ﻳﻚ‬‫ﺷﻴﻮه‬‫ي‬‫ﻣﻬﻢ‬‫دﻳﮕﺮ‬‫از‬
‫ﻋﻤﻠﻴﺎت‬‫ﺗﻜﺮار‬‫را‬‫ﺑﺮاي‬‫ﺣﻞ‬‫ﻣﻌﺎدﻻت‬‫ﺑﻪ‬‫ﻓﺮم‬g(x)=x‫ﻣﻌﺮﻓﻲ‬‫ﻣﻲ‬‫ﻧﻤﺎﻳﻴﻢ‬.‫روش‬
‫ﻧﻘﻄﻪ‬‫ﺛﺎﺑﺖ‬‫ﻛﻪ‬‫ﺑﺮاي‬‫ﺗﻌﻴﻴﻦ‬‫ﻧﻘﻄﻪ‬‫اي‬‫ﺛﺎﺑﺖ‬‫از‬‫ﺗﺎﺑﻊ‬g‫ﺑﻪ‬‫ﻛﺎر‬‫رود‬‫ﻣﻲ‬،‫در‬‫اﺑﺘﺪا‬‫از‬
‫ﻳﻚ‬‫ﺣﺪس‬‫اوﻟﻴﻪ‬‫ﻣﺜﻞ‬0p‫آﻏﺎز‬‫ﺷﻮد‬‫ﻣﻲ‬‫ﻛﻪ‬‫اﻳﻦ‬‫ﺣﺪس‬‫اوﻟﻴﻪ‬،‫ﻳﻚ‬‫ﺳﺮي‬‫از‬
‫ﻣﻘﺎدﻳﺮ‬‫ﺑﺪﺳﺖ‬‫آﻣﺪه‬‫از‬‫ﻃﺮﻳﻖ‬‫روش‬
‫را‬‫ﻣﺸﺨﺺ‬‫ﻣﻲ‬‫ﻛﻨﺪ‬.
‫اﮔﺮ‬g‫ﻳﻚ‬‫ﺗﺎﺑﻊ‬‫ﭘﻴﻮﺳﺘﻪ‬‫و‬‫ﭘﺮوﺳﻪ‬‫ﻓﻮق‬‫ﺑﻪ‬‫ﻣﻘﺪار‬α‫ﻫﻤﮕﺮا‬،‫ﺷﻮد‬‫ﻳﻌﻨﻲ‬‫اﮔﺮ‬n‫ﺑﻪ‬
‫ﺳﻤﺖ‬∞،‫ﻣﻴﻞ‬‫ﻛﺮده‬‫و‬α=np‫ﮔﺮدﻳﺪ‬‫ﺟﻮاب‬‫ﻣﻌﺎدﻟﻪ‬‫ﺑﻪ‬‫ﻓﺮم‬‫ﻣﻌﺎدﻟﻪ‬‫ﻫﻤﻴﻦ‬
‫ﺻﻔﺤﻪ‬‫ﺑﺪﺳﺖ‬‫ﻣﻲ‬‫آﻳﺪ‬.‫ﺑﻄﻮر‬‫ﻫﻨﺪﺳﻲ‬،‫ﻧﻘﻄﻪ‬‫ﺛﺎﺑﺖ‬p‫از‬‫ﺗﺎﺑﻊ‬g‫ﻣﺤﻞ‬‫ﺑﺮﺧﻮرد‬
‫ﺧﻂ‬y=x‫و‬‫ﺗﺎﺑﻊ‬y=g(x)‫ﻣﻲ‬‫ﺑﺎﺷﺪ‬.
1 ( ), 0,1, 2...n np g p n  
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

57. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺛﺎﺑﺖ‬ ‫ﻧﻘﻄﻪ‬ ‫ﺗﻜﺮاري‬ ‫روش‬ ‫اداﻣﻪ‬
‫ﺷﻜﻞ‬2-‫ﺗﻜﺮار‬Fixed-Point‫واﮔﺮاﻳﻲ‬ ‫و‬
‫ﻧﻘﻄﻪ‬ ‫از‬ ‫آن‬
57
‫ﺷﻜﻞ‬1-‫ﺗﻜﺮار‬ ‫روش‬Fixed-Point‫و‬
‫ﻧﻘﻄﻪ‬ ‫ﺑﻪ‬ ‫آن‬ ‫ﻫﻤﮕﺮاﻳﻲ‬
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

58. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
58
‫ﺛﺎﺑﺖ‬ ‫ﻧﻘﻄﻪ‬ ‫ﺗﻜﺮاري‬ ‫روش‬ ‫اداﻣﻪ‬
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

59. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
59
‫ﺛﺎﺑﺖ‬ ‫ﻧﻘﻄﻪ‬ ‫ﺗﻜﺮاري‬ ‫روش‬ ‫اداﻣﻪ‬
‫ﻗﻀﻴﻪ‬‫ﻣﻘﺪار‬‫ﻣﺘﻮﺳﻂ‬:‫اﮔﺮ‬‫ﺗﺎﺑﻊ‬f‫ﺑﺮ‬‫روي‬‫ﺑﺎزه‬‫ﺑﺴﺘﻪ‬[a b]‫ﭘﻴﻮﺳﺘﻪ‬‫ﺑﻮده‬‫و‬‫در‬‫ﺑﺎزه‬‫ﺑﺎز‬(a b)‫ﭘﺬﻳﺮ‬‫ﻣﺸﺘﻖ‬،‫ﺑﺎﺷﺪ‬
‫در‬‫اﻳﻨﺼﻮرت‬‫ﻋﺪدي‬‫ﻣﺜﻞ‬c‫ﻣﺘﻌﻠﻖ‬‫ﺑﻪ‬‫ي‬‫ﺑﺎزه‬(a b)‫وﺟﻮد‬‫ﺧﻮاﻫﺪ‬‫داﺷﺖ‬‫ﺑﻪ‬‫ﮔﻮﻧﻪ‬‫اي‬‫ﻛﻪ‬:
))(()()( abcfafbf 
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

60. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
60
‫ﺛﺎﺑﺖ‬ ‫ﻧﻘﻄﻪ‬ ‫ﺗﻜﺮاري‬ ‫روش‬ ‫اداﻣﻪ‬
‫ﻗﻀﻴﻪ‬‫زﻳﺮ‬‫اﺑﺰاري‬‫ﻣﻬﻢ‬‫ﺑﺮاي‬‫ﻛﺮدن‬‫ﻣﺸﺨﺺ‬‫وﺟﻮد‬‫ﻳﺎ‬‫ﻋﺪم‬‫وﺟﻮد‬‫ﺟﻮاب‬‫ﺑﺮاي‬‫ﻳﻚ‬‫ﻣﺴﺌﻠﻪ‬‫ﺑﺎﺷﺪ‬‫ﻣﻲ‬.
‫ﻗﻀﻴﻪ‬‫ﻣﻘﺪار‬‫ﻣﻴﺎﻧﻲ‬:‫ﺗﺎﺑﻊ‬‫ﭘﻴﻮﺳﺘﻪ‬f‫ﺑﺮ‬‫روي‬‫ﺑﺎزه‬‫ﺑﺴﺘﻪ‬[a b]‫ﻣﻔﺮوض‬‫اﺳﺖ‬.‫اﮔﺮ‬w‫ﻫﺮ‬‫ﻋﺪدي‬‫ﺑﻴﻦ‬f(a)‫و‬f(b)
‫ﺑﺎﺷﺪ‬،‫در‬‫اﻳﻨﺼﻮرت‬‫ﺣﺪاﻗﻞ‬‫ﻋﺪدي‬‫ﻣﺜﻞ‬c‫ﻣﺘﻌﻠﻖ‬‫ﺑﻪ‬‫ي‬‫ﺑﺎزه‬‫ﺑﺎز‬(a b)‫وﺟﻮد‬‫ﺧﻮاﻫﺪ‬‫داﺷﺖ‬‫ﺑﻪ‬‫ﮔﻮﻧﻪ‬‫اي‬‫ﻛﻪ‬
‫ﺑﺮﻗﺮار‬‫ﺧﻮاﻫﺪ‬‫ﺑﻮد‬. wcf )(
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

61. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
61
‫ﺛﺎﺑﺖ‬ ‫ﻧﻘﻄﻪ‬ ‫ﺗﻜﺮاري‬ ‫روش‬ ‫اداﻣﻪ‬
‫ﻗﺒﻞ‬ ‫ﻗﻀﻴﻪ‬ ‫دو‬
‫و‬
‫ﺑﺎ‬
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

62. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
62
‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫از‬ ‫ﻫﺎﻳﻲ‬ ‫ﻣﺜﺎل‬‫ﺛﺎﺑﺖ‬ ‫ﻧﻘﻄﻪ‬ ‫ﺗﻜﺮاري‬ ‫روش‬
1:
‫ﻗﺒ‬ ‫ﻗﻀﯿﻪ‬‫ﻞ‬
‫و‬ ‫ﻧﻤﻮده‬ ‫ﺻﺪق‬ ‫ﻗﺒﻞ‬
==============================================================================================================================
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

63. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
63
‫ﺛﺎﺑﺖ‬ ‫ﻧﻘﻄﻪ‬ ‫ﺗﻜﺮاري‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫از‬ ‫ﻫﺎﻳﻲ‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬
2:
==============================================================================================================================
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

64. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
64
‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫از‬ ‫ﻫﺎﻳﻲ‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬‫ﺛﺎﺑﺖ‬ ‫ﻧﻘﻄﻪ‬ ‫ﺗﻜﺮاري‬ ‫روش‬
3:
===================================================================================================================================
‫ب‬‫اﻟﻒ‬n
1
4
1
4
1
4
1
4
1
4
1
4
1
1.00000000
1.30000000
1.53100000
1.69660000
1.80875742
1.88159708
1.92755632
1.95600899
1.97341187
1.98397643
1.99036018
1.99420682
1.99652073
0
1
2
3
4
5
6
7
8
9
10
11
12
‫ﺟﺪول‬‫روﺑﺮو‬‫ﻧﺘﺎﻳﺞ‬‫ﺣﺎﺻﻞ‬‫از‬‫اﻋﻤﺎل‬‫روش‬‫ﻧﻘﻄﻪ‬‫ﺛﺎﺑﺖ‬‫را‬‫ﺑﺮاي‬‫رﻳﺸﻪ‬‫ﻳﺎﺑﻲ‬‫ﻃ‬‫ﺒﻖ‬
‫ﺣﺎﻟﺖ‬‫ﻫﺎي‬)‫اﻟﻒ‬(‫و‬)‫ب‬(‫ﻧﺸﺎن‬‫ﻣﻲ‬‫دﻫﺪ‬.‫از‬‫آﻧﺠﺎﺋﻴﻜﻪ‬‫ﺟﻮاب‬‫دﻗﻴﻖ‬‫ﺣﻞ‬‫ﻣﻌﺎدﻟﻪ‬‫ذﻛﺮ‬
‫ﺷﺪه‬‫در‬‫ﺻﻮرت‬‫ﻣﺜﺎل‬‫ﺑﺮاﺑﺮ‬2.0‫ﻣﻲ‬،‫ﺑﺎﺷﺪ‬‫ﺑﺎ‬‫ﺗﻮﺟﻪ‬‫ﺑﻪ‬‫اﻧﺘﺨﺎب‬g(x)،‫دﻳﺪه‬‫ﻣﻲ‬‫ﺷﻮد‬
‫ﻛﻪ‬‫ﭘﺎﺳﺦ‬‫ﻫﺎي‬‫ﺣﺎﻟﺖ‬)‫اﻟﻒ‬(‫ﻫﻤﮕﺮا‬‫و‬‫ﭘﺎﺳﺦ‬‫ﻫﺎي‬‫ﺣﺎﻟﺖ‬)‫ب‬(‫واﮔﺮا‬‫ﻣﻲ‬‫ﺑﺎﺷﻨﺪ‬.
1
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65. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺛﺎﺑﺖ‬ ‫ﻧﻘﻄﻪ‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﻛﺪ‬ ‫ﺷﺒﻪ‬
•‫ﻛﺪ‬ ‫ﺷﺒﻪ‬
65
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66. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫روش‬‫ﺛﺎﺑﺖ‬ ‫ﻧﻘﻄﻪ‬)Fixed-point(
66
function fixed(g,x0,tol,n)
% The fixed point iteration method for solving the nonlinear
% equation f(x)=0.
iter=0;
u=feval(g,x0);
err=abs(u-x0);
disp('_________________________________________')
disp('iter x g(x) |xn+1-xn|')
disp('_________________________________________')
fprintf('%2.0f %12.6f %12.6fn',iter,x0,u)
while (err>tol)&(iter<=n)
x1=u;
err=abs(x1-x0);
x0=x1;
u=feval(g,x0);
iter=iter+1;
fprintf('%2.0f %12.6f %12.6f %12.8fn',iter,x0,u,err)
end
if (iter>n)
disp('Method failed to converge')
end
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67. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺣﻞ‬‫ﻫﺎي‬ ‫دﺳﺘﮕﺎه‬‫ﺧﻄﻲ‬ ‫ﻏﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫ﺑﻪ‬
67
•‫ﺑﺴﻂ‬‫روش‬‫ﻧﻴﻮﺗﻦ‬‫ﻣﻌﺮﻓﻲ‬‫ﺷﺪه‬‫در‬‫ﻗﺴﻤﺖ‬‫ﻫﺎي‬‫ﻗﺒﻞ‬‫ﺑﺮاي‬‫ﻳﺎﻓﺘﻦ‬‫ﻫﺎي‬‫رﻳﺸﻪ‬‫ﻳﻚ‬‫ﺗﺎﺑﻊ‬‫ﻏﻴﺮ‬‫ﺧﻄﻲ‬‫ﺗﻚ‬‫ﻣﺘﻐﻴﺮ‬‫ه‬‫ﺑﻪ‬‫رﻳﺸﻪ‬
‫ﻳﺎﺑﻲ‬‫ﺗﻮاﺑﻊ‬‫ﻏﻴﺮﺧﻄﻲ‬‫ﭼﻨﺪ‬‫ﻣﺘﻐﻴﺮه‬.
•‫ﻣﺸﻜﻞ‬‫اﺻﻠﻲ‬‫ﭘﻴﭽﻴﺪﮔﻲ‬‫ﺑﻴﺸﺘﺮ‬‫ﻳﺎﻓﺘﻦ‬‫ﺻﻔﺮﻫﺎي‬‫ﺗﻮاﺑﻊ‬‫ﻏﻴﺮ‬‫ﺧﻄﻲ‬‫اﺳﺖ‬.
•‫ﺑﺎ‬‫ﺷﺮوع‬‫از‬‫ﻳﻚ‬‫ﺗﺨﻤﻴﻦ‬‫اوﻟﻴﻪ‬‫ﺑﺮاي‬‫ﻫﺮ‬‫ﺗﺎﺑﻊ‬‫ﻏﻴﺮ‬‫ﺧﻄﻲ‬‫،روش‬‫ﻧﻴﻮﺗﻦ‬‫ﺗﻮاﺑﻊ‬‫ﻏﻴﺮﺧﻄﻲ‬‫را‬‫ﺑﺎ‬‫ي‬‫ﺻﻔﺤﻪ‬‫ﻣﻤﺎس‬‫ﺑﺮ‬‫آﻧﻬﺎ‬
‫ﺗﺨﻤﻴﻦ‬‫ﻣﻲ‬‫زﻧﺪ‬‫و‬‫رﻳﺸﻪ‬‫ﻣﺸﺘﺮك‬‫ﺣﺎﺻﻞ‬‫از‬‫ﺣﻞ‬‫ﻣﻌﺎدﻻت‬‫ﻏﻴﺮﺧﻄﻲ‬‫ﻣﻘﺪار‬‫ﺗﻘﺮﻳﺒﻲ‬‫ﺑﻌﺪي‬‫را‬‫اﻳﺠﺎد‬‫ﻛﻨﺪ‬‫ﻣﻲ‬.
•‫ﺑﺮاي‬‫ﻓﻬﻤﻴﺪن‬‫اﻳﻨﻜﻪ‬‫ﭼﻄﻮر‬‫اﻳﻦ‬‫اﻟﮕﻮرﻳﺘﻢ‬‫ﻛﺎر‬،‫ﻛﻨﺪ‬‫ﻣﻲ‬‫اﺑﺘﺪا‬‫دﺳﺘﮕﺎﻫﻲ‬‫از‬‫دو‬‫ﻣﻌﺎدﻟﻪ‬‫ي‬‫ﻏﻴﺮ‬‫ﺧﻄﻲ‬‫ﺑﻪ‬‫ﺻﻮرت‬‫زﻳﺮ‬‫در‬‫ﻧﻈﺮ‬
‫ﻣﻲ‬‫ﮔﻴﺮﻳﻢ‬:
1 1 2
2 1 2
( , ) 0
( , ) 0
f x x
f x x


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68. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺣﻞ‬ ‫اداﻣﻪ‬‫ﻫﺎي‬ ‫دﺳﺘﮕﺎه‬‫ﺧﻄﻲ‬ ‫ﻏﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫ﺑﻪ‬
68
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69. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺣﻞ‬ ‫اداﻣﻪ‬‫ﻫﺎي‬ ‫دﺳﺘﮕﺎه‬‫ﺧﻄﻲ‬ ‫ﻏﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫ﺑﻪ‬
69
‫ﺑﺎﻻ‬
2
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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70. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺣﻞ‬ ‫اداﻣﻪ‬‫ﻫﺎي‬ ‫دﺳﺘﮕﺎه‬‫ﺧﻄﻲ‬ ‫ﻏﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫ﺑﻪ‬
70
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71. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺣﻞ‬ ‫اداﻣﻪ‬‫ﻫﺎي‬ ‫دﺳﺘﮕﺎه‬‫ﺧﻄﻲ‬ ‫ﻏﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫ﺑﻪ‬
71
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72. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺣﻞ‬ ‫ﺑﺮاي‬ ‫ﻣﺜﺎل‬‫دﺳﺘﮕﺎه‬‫ﺧﻄﻲ‬ ‫ﻏﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬‫ﻧﻴﻮﺗﻦ‬ ‫روش‬ ‫ﺑﻪ‬
72
‫ﺻﻮرت‬‫ﻣﺜﺎل‬(‫دﺳﺘﮕﺎه‬‫ﻏﻴﺮ‬‫ﺧﻄﻲ‬‫زﻳﺮ‬‫را‬‫ﺑﺎ‬‫اﺳﺘﻔﺎده‬‫از‬‫روش‬‫ﻧﻴﻮﺗﻦ‬‫و‬‫آﻏﺎز‬‫از‬)1-,1=()0(x‫ﺣﻞ‬‫ﻛﻨﻴﺪ‬.‫اﻳﻦ‬‫ﻋﻤﻠﻴﺎت‬‫را‬‫ﺗﺎ‬
‫زﻣﺎﻧﻲ‬‫ﻛﻪ‬‫اﺳﺖ‬‫را‬‫ﺗﻜﺮار‬‫ﻛﺮده‬‫و‬‫اداﻣﻪ‬‫دﻫﻴﺪ‬.
==============================================================================================
‫ﭘﺎﺳﺦ‬‫ﻣﺜﺎل‬(‫اﺑﺘﺪا‬‫ﺑﺎﻳﺴﺘﻲ‬‫ﻣﺎﺗﺮﻳﺲ‬‫ژاﻛﻮﺑﻴﻨﻲ‬‫ﺑﻪ‬‫ﻗﺮار‬‫زﻳﺮ‬‫ﻓﺮﻣﻮل‬‫ﻫﺎي‬‫زﻳﺮ‬‫ﺗﺸﻜﻴﻞ‬‫دﻫﻴﻢ‬:
‫در‬‫ي‬‫ﻧﻘﻄﻪ‬(1,-1)‫ﺗﺎﺑﻊ‬‫ﺑﺮداري‬‫و‬‫ﻣﺎﺗﺮﻳﺲ‬‫ژاﻛﻮﺑﻴﻦ‬‫ﻣﻘﺎدﻳﺮ‬‫زﻳﺮ‬‫را‬‫ﮔﻴﺮﻧﺪ‬‫ﻣﻲ‬:
‫دﻳﻔﺮاﻧﺴﻴﻞ‬‫ﻫﺎي‬‫ﺟﻮاﺑﻬﺎي‬‫دﺳﺘﮕﺎه‬‫ﻫﺴﺘﻨﺪ‬.‫ﺟﻮاب‬‫ﻫﺎي‬‫آن‬‫ﻋﺒﺎرﺗﻨﺪ‬‫از‬:
6
|| || max{| |} 10ix x 
   
3
1 1 2 1 2
2
2 1 2 1 2
( , ) 3 21 0
( , ) 2 2 0
f x x x x
f x x x x
   
   
2
1 2
1 2
1
3 6
( , )
2 2
x x
J x x
x
 
  
 
17 3 6
(1 1) , (1 1)
1 2 2
F J
    
      
   
(0) (0)
1 2,x x 
(0)
1
(0)
2
3 6 17
2 2 1
x
x
     
     
     
(0)
1(0)
(0)
2
1.555556
2.05556
x
x
x
   
         
2
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73. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺣﻞ‬ ‫ﺑﺮاي‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬‫دﺳﺘﮕﺎه‬‫ﺧﻄﻲ‬ ‫ﻏﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬‫ﻧﻴ‬ ‫روش‬ ‫ﺑﻪ‬‫ﻮﺗﻦ‬
73
‫اداﻣﻪ‬‫ﭘﺎﺳﺦ‬‫ﻣﺜﺎل‬(‫ﺑﻨﺎﺑﺮاﻳﻦ‬،‫ﺑﻌﺪي‬‫ﻧﻘﻄﻪ‬‫از‬‫ﺗﻜﺮار‬‫روش‬‫ﻋﺒﺎرت‬‫اﺳﺖ‬‫از‬:
‫ﺑﻪ‬‫ﻃﻮر‬‫ﻣﺸﺎﺑﻪ‬،‫ﺳﻪ‬‫ي‬‫ﻧﻘﻄﻪ‬‫ﺑﻌﺪي‬‫ﻋﺒﺎرﺗﻨﺪ‬‫از‬:
‫اﻳﻦ‬‫ﻣﺜﺎل‬‫ﻧﺸﺎن‬‫داد‬‫ﻛﻪ‬‫روش‬‫ﻧﻴﻮﺗﻦ‬‫ﻣﻲ‬‫ﺗﻮاﻧﺪ‬‫ﺑﺴﻴﺎر‬‫ﺳﺮﻳﻊ‬‫ﻳﻚ‬‫ﺗﻘﺮﻳﺐ‬‫اوﻟﻴﻪ‬‫ﺑﻪ‬‫ي‬‫اﻧﺪازه‬‫ﻛﺎﻓﻲ‬‫ﻧﺰدﻳﻚ‬‫ﺑﻪ‬‫ﭘ‬‫ﺎﺳﺦ‬‫ﻣﺴﺌﻠﻪ‬‫را‬
‫ﺑﺪﺳﺖ‬‫دﻫﺪ‬.‫ﻳﻜﻲ‬‫از‬‫ﻣﺸﻜﻼت‬‫روش‬،‫ﻧﻴﻮﺗﻦ‬‫ﻧﻴﺎز‬‫ﺑﻪ‬‫ﺣﻞ‬‫دﺳﺘﮕﺎه‬‫ﺧﻄﻲ‬‫زﻳﺮ‬‫درﻫﺮ‬‫ﻣﺮﺣﻠﻪ‬‫ﺑﺎﺷﺪ‬‫ﻣﻲ‬‫ﻛﻪ‬‫اﻟﺒﺘﻪ‬‫ﻃﺒ‬‫ﻖ‬‫ﻣﻄﺎﻟﺐ‬
‫ﻓﺼﻞ‬‫دوم‬‫از‬‫ﺳﺮي‬‫آﻣﻮزش‬‫ﻫﺎي‬،‫ﺣﺎﺿﺮ‬‫ﺑﻪ‬‫ﺻﻮرت‬‫ﻋﺪدي‬‫ﻗﺎﺑﻞ‬‫ﺣﻞ‬‫ﻫﺴﺘﻨﺪ‬:
(1)
1(1)
(1)
2
1 1.555556 2.555556
1 2.055556 3.055556
x
x
x
       
                    
(2) (3) (4)1.865049 1.661337 1.643173
, ,
2.500801 2.359271 2.349844
x x x
     
              
( )
1 11 1
1
( )
1
. .
. .
. .
i
n
n n
i
n nn
x ff f
x x
f f
x x fx
     
    
      
       
          
          

  

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74. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫روش‬‫ﻏﻴﺮﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﺣﻞ‬ ‫ﺑﺮاي‬ ‫ﻧﻴﻮﺗﻦ‬
74
function newton_sys(F,JF,xx0,tol,maxit)
% F is the system given by an (n x 1) matrix;
% JF is the Jacobian of F given by an (n x n) matrix;
% xx0 is the (n x 1) initial vector, tol ia a tolerance;
% max_it is the maximum number of iterations.
x0=xx0
iter=1;
while(iter<=maxit)
y=-feval(JF,x0)feval(F,x0);
xn=x0+y;
err= max(abs(xn-x0));
if (err<=tol)
x=xn;
fprintf(' Newton''s method converges after %3.0f iterations to n',iter)
x
return;
else
x0=xn;
end
iter=iter+1;
end
disp('Newton''s method does not converge')
x=xn;
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75. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺣﻞ‬ ‫ﺑﺮاي‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬‫دﺳﺘﮕﺎه‬‫ﺧﻄﻲ‬ ‫ﻏﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬‫ﻧﻴ‬ ‫روش‬ ‫ﺑﻪ‬‫ﻮﺗﻦ‬
75
‫اداﻣﻪ‬‫ﭘﺎﺳﺦ‬‫ﻣﺜﺎل‬(‫ﺑﺮاي‬‫اﺳﺘﻔﺎده‬‫از‬‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫ﺑﺎﻳﺴﺘﻲ‬‫ﺗﻮاﺑﻌﻲ‬‫ﺑﻪ‬‫ﻗﺮار‬‫زﻳﺮ‬‫ﺗﻌﺮﻳﻒ‬‫ﻛﻨﻴﻢ‬:
‫ﺳﭙﺲ‬‫ﺑﺎﻳﺴﺘﻲ‬‫اﻳﻦ‬‫ﺗﻮاﺑﻊ‬‫را‬‫در‬‫ﻗﺎﻟﺐ‬‫آرﮔﻮﻣﺎن‬‫ﻫﺎي‬‫ورودي‬‫ﺗﺎﺑﻊ‬newton_sys(f,jf,xx0,tol,n)‫ﺟﺎﻳﮕﺰﻳﻦ‬‫ﻧﻤﻮد‬.
];2*22.^;212.^*33.^[
);2();1(
)(1



yxyxf
XyXx
Xfffunction
]2,*2;*6,2.^*3[
);2();1(
)(1
xyxjf
XyXx
Xjfjffunction


 ‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

76. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102‫ﺣﻞ‬ ‫ﺑﺮاي‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬‫دﺳﺘﮕﺎه‬‫ﺧﻄﻲ‬ ‫ﻏﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬‫ﻧﻴ‬ ‫روش‬ ‫ﺑﻪ‬‫ﻮﺗﻦ‬
76
‫اداﻣﻪ‬‫ﭘﺎﺳﺦ‬‫ﻣﺜﺎل‬(‫ﺑﺮاي‬‫اﺳﺘﻔﺎده‬‫از‬‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫ﺑﺎﻳﺴﺘﻲ‬‫ﺗﻮاﺑﻌﻲ‬‫ﺑﻪ‬‫ﻗﺮار‬‫زﻳﺮ‬‫ﺗﻌﺮﻳﻒ‬‫ﻛﻨﻴﻢ‬:
‫ﺳﭙﺲ‬‫ﺑﺎﻳﺴﺘﻲ‬‫اﻳﻦ‬‫ﺗﻮاﺑﻊ‬‫را‬‫در‬‫ﻗﺎﻟﺐ‬‫آرﮔﻮﻣﺎن‬‫ﻫﺎي‬‫ورودي‬‫ﺗﺎﺑﻊ‬newton_sys(‘f1’,’jf1’,xx0,tol,n)‫ﺟﺎﻳﮕﺰﻳﻦ‬
‫ﻧﻤﻮد‬.
];2*22.^;212.^*33.^[
);2();1(
)(1



yxyxf
XyXx
Xfffunction
]2,*2;*6,2.^*3[
);2();1(
)(1
xyxdf
XyXx
Xjfjffunction


 ‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

77. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
77
‫دﺳﺘﻮرات‬ ‫ﺑﺮﺧﻲ‬Matlab‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﺑﺎ‬ ‫ﻣﺮﺗﺒﻂ‬
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

78. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
78
‫دﺳﺘﻮرات‬ ‫ﺑﺮﺧﻲ‬ ‫اداﻣﻪ‬Matlab‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﺑﺎ‬ ‫ﻣﺮﺗﺒﻂ‬
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

79. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
79
‫دﺳﺘﻮرات‬ ‫ﺑﺮﺧﻲ‬ ‫اداﻣﻪ‬Matlab‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﺑﺎ‬ ‫ﻣﺮﺗﺒﻂ‬
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

80. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
80
‫دﺳﺘﻮرات‬ ‫ﺑﺮﺧﻲ‬ ‫اداﻣﻪ‬Matlab‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﺑﺎ‬ ‫ﻣﺮﺗﺒﻂ‬
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

81. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫دﺳﺘﻮرات‬ ‫ﺑﺮﺧﻲ‬ ‫اداﻣﻪ‬Matlab‫ﻳﺎﺑﻲ‬ ‫رﻳﺸﻪ‬ ‫ﺑﺎ‬ ‫ﻣﺮﺗﺒﻂ‬
(1‫ﺑﺮرﺳﻲ‬‫ﺷﻮد‬‫در‬‫ﻧﺮم‬‫اﻓﺰار‬MATLAB،‫از‬‫ﭼﻪ‬‫رﻫﻴﺎﻓﺖ‬‫ﻣﺤﺎﺳﺒﺎت‬‫ﻋﺪدي‬‫ﺑﺮاي‬‫ﭘﻴﺪا‬‫ﻛﺮد‬‫رﻳﺸﻪ‬
‫ﻫﺎي‬‫ﻳﻚ‬‫ﭼﻨﺪ‬‫ﺟﻤﻠﻪ‬‫اي‬‫ﺗﻮﺳﻂ‬‫دﺳﺘﻮر‬roots‫اﺳﺘﻔﺎده‬‫ﻣﻲ‬‫ﺷﻮد؟‬
(2‫ﻏﻴﺮ‬‫از‬‫دﺳﺘﻮر‬roots‫ﭼﻪ‬‫دﺳﺘﻮرات‬‫و‬‫ﺗﻮاﺑﻊ‬‫دﻳﮕﺮي‬‫در‬‫ﻧﺮم‬‫اﻓﺰار‬MATLAB‫ﺑﺮاي‬‫رﻳﺸﻪ‬‫ﻳﺎﺑﻲ‬
‫ﻣﻌﺎدﻻت‬‫ﻏﻴﺮﺧﻄﻲ‬‫ﻳﺎﻓﺖ‬‫ﻣﻲ‬‫ﺷﻮد؟‬
81
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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82. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

83. ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
83
‫ﻓﺮادرس‬ ‫در‬ ‫ﺷﺪه‬ ‫ﻣﻄﺮح‬ ‫ﻧﻜﺎت‬ ‫ﻣﺒﻨﺎي‬ ‫ﺑﺮ‬ ‫ﻫﺎ‬ ‫اﺳﻼﻳﺪ‬ ‫اﻳﻦ‬
»‫اﻓﺰار‬ ‫ﻧﺮم‬ ‫ﻛﻤﻚ‬ ‫ﺑﻪ‬ ‫ﭘﻴﺸﺮﻓﺘﻪ‬ ‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬ ‫آﻣﻮزش‬MATLAB«
‫اﺳﺖ‬ ‫ﺷﺪه‬ ‫ﺗﻬﻴﻪ‬.
‫ﻧﻤﺎﻳﻴﺪ‬ ‫ﻣﺮاﺟﻌﻪ‬ ‫زﻳﺮ‬ ‫ﻟﻴﻨﻚ‬ ‫ﺑﻪ‬ ‫آﻣﻮزش‬ ‫اﻳﻦ‬ ‫ﻣﻮرد‬ ‫در‬ ‫ﺑﻴﺸﺘﺮ‬ ‫اﻃﻼﻋﺎت‬ ‫ﻛﺴﺐ‬ ‫ﺑﺮاي‬.
faradars.org/fvmth102
‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
faradars.org/fvmth102
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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