آموزش محاسبات عددی - بخش دوم

‫افزار‬ ‫نرم‬ ‫کمک‬ ‫به‬ ‫عددی‬ ‫محاسبات‬MATLAB
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‫مدرس‬:
‫تاشک‬ ‫اشکان‬
‫تحصیلی‬ ‫رشته‬ ‫و‬ ‫درجه‬
‫برق‬ ‫مهندسی‬ ‫دکترای‬-‫دانشگاه‬ ‫مدرس‬ ‫و‬ ‫سیستم‬ ‫گرایش‬ ‫مخابرات‬
‫عددی‬ ‫محاسبات‬
‫کمک‬ ‫به‬MATLAB
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‫ﻓﺼﻞ‬ ‫ﻋﻨﺎوﻳﻦ‬‫دوم‬
‫ﻓﺼﻞ‬‫دوم‬:‫ﺣﻞ‬‫دﺳﺘﮕﺎه‬‫ﻫﺎي‬‫ﻣﻌﺎدﻻت‬‫ﺧﻄﻲ‬
-‫ﻣﻘﺪﻣﻪ‬
1-‫رﻫﻴﺎﻓﺖ‬‫ﻫﺎي‬‫ﻏﻴﺮ‬‫ﺗﻜﺮاري‬)‫ﺑﻪ‬‫ﻫﻤﺮاه‬‫ﺑﺮﻧﺎﻣﻪ‬MATLAB(
-‫روش‬‫ﮔﻮس‬‫ﻧﺎﻗﺺ‬
-‫ﮔﻮس‬‫ﺑﺎ‬‫ﻣﺤﻮرﻳﺖ‬‫ﻧﺴﺒﻲ‬‫ﻗﻴﺎﺳﻲ‬‫ﻳﺎ‬‫ﭘﻴﻮوﺗﻴﻨﮓ‬
-‫روش‬‫ﺗﺠﺰﻳﻪ‬LU
2-‫رﻫﻴﺎﻓﺖ‬‫ﻫﺎي‬‫ﺗﻜﺮاري‬)‫ﺑﻪ‬‫ﻫﻤﺮاه‬‫ﺑﺮﻧﺎﻣﻪ‬MATLAB(
-‫روش‬‫ژاﻛﻮﺑﻲ‬
-‫روش‬‫ﮔﻮس‬-‫ﺳﻴﺪال‬
2
‫ﻋﺪدي‬ ‫ﻣﺤﺎﺳﺒﺎت‬
‫ﻛﻤﻚ‬ ‫ﺑﻪ‬MATLAB
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‫ﻣﻘﺪﻣﻪ‬
•‫ﺣﻞ‬‫ﻣﺴﺎﺋﻞ‬‫ﺑﺴﻴﺎري‬‫در‬‫ﻣﻬﻨﺪﺳﻲ‬‫ﺑﻪ‬‫دﺳﺘﮕﺎه‬‫ﻣﻌﺎدﻻت‬‫ﺟﺒﺮي‬‫ﻣﻨ‬‫ﺠﺮ‬
‫ﻣﻲ‬‫ﺷﻮد‬.
•‫ﺑﻪ‬‫ﻋﻨﻮان‬‫ﻣﺜﺎل‬‫ﺑﺮاي‬‫ﺳﻴﺴﺘﻢ‬‫ﺟﺮم‬-‫ﻓﻨﺮ‬‫ﻧﺸﺎن‬‫داده‬‫ﺷﺪه‬‫در‬‫ﺣﺎل‬
‫ﺗﻌﺎدل‬‫ﺑﺎ‬‫ﺣﺬف‬،‫ﻧﻴﺮوﻫﺎ‬‫ﺳﻴﺴﺘﻢ‬‫در‬‫وﺿﻌﻴﺖ‬‫ﺗﻌﺎدﻟﻲ‬‫ﺟﺪﻳﺪ‬‫ﻗﺮار‬‫ﺧ‬‫ﻮاﻫﺪ‬
‫ﮔﺮﻓﺖ‬.
•‫ﻣﻮﻗﻌﻴﺖ‬‫ﺟﺪﻳﺪ‬‫ﺟﺮم‬‫ﻫﺎ‬‫ﺑﺎ‬‫ﺣﻞ‬‫دﺳﺘﮕﺎه‬‫ﻣﻌﺎدﻻت‬‫ﺟﺒﺮي‬‫ﺧﻄﻲ‬‫ﻗﺎﺑﻞ‬
‫ﻣﺤﺎﺳﺒﻪ‬‫ﻫﺴﺘﻨﺪ‬.
•‫ﺑﺮاي‬‫ﻳﻚ‬‫ﺳﻴﺴﺘﻢ‬‫ﺟﺮم‬‫و‬،‫ﻓﻨﺮ‬‫ﻃﺒﻖ‬‫ﻗﺎﻧﻮن‬‫دوم‬
‫ﻧﻴﻮﺗﻦ‬‫دارﻳﻢ‬:
•،‫ﺣﺎل‬‫ﻣﺴﺌﻠﻪ‬‫ﺣﻞ‬‫اﻳﻦ‬‫دﺳﺘﮕﺎه‬‫اﺳﺖ‬:
3
(K1+K2+(K3/2))x1-(K2+(K3/2))x2=W1
-(K2 +(K3/2))x1+(K2+(K3/2)+K4)x2-(K4+(K3/2))x3=W2
-(K4+(K3/2))x2+((K3/2)+K4+K5)x3=W3
K1=40N/cm,K2=K3=K4=20N/cm
K5=90N/cm ,W1=W2=W3=20N
20=3x0+2x30-1x07
20=3x30-2x50+1x30-
20=3x120+2x30-1x0+
Ma
dt
yd
MMgyKF s  2
2
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‫ﻣﺤﺎﺳﺒﺎﺗﻲ‬ ‫ﻫﺎي‬ ‫رﻫﻴﺎﻓﺖ‬ ‫اداﻣﻪ‬
•‫ﻓﺮم‬‫ﻛﻠﻲ‬)‫اﻟﺒﺘﻪ‬‫ﺑﺮاي‬‫دﺳﺘﮕﺎه‬‫ﻫﺎي‬‫ﺑﺎ‬‫ﺗﻌﺪاد‬‫ﻣﻌﺎدﻻت‬‫و‬‫ﻣﺠﻬﻮﻻت‬‫ﺑﺮاﺑﺮ‬‫ﻳﺎ‬m=n(
4
Ax b
m =n
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‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﭘﺎراﻣﺘﺮﻫﺎي‬ ‫اﺳﺎﻣﻲ‬
•‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬)m=n(
•‫ﺑﺮدار‬‫ﻣﺠﻬﻮﻻت‬
•‫ﺑﺮدار‬‫ﻣﻌﻠﻮﻣﺎت‬
5
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‫ﻣﺤﺎﺳﺒﺎﺗﻲ‬ ‫ﻫﺎي‬ ‫رﻫﻴﺎﻓﺖ‬
•‫رﻫﻴﺎﻓﺖ‬‫ﻫﺎي‬‫ﻣﺴﺘﻘﻴﻢ‬:‫در‬‫روش‬‫ﻫﺎي‬،‫ﻣﺴﺘﻘﻴﻢ‬‫ﭘﺎﺳﺦ‬‫اﻋﺪاد‬‫ﺛﺎﺑﺘﻲ‬‫ﻫﺴﺘﻨﺪ‬‫ﻛﻪ‬‫از‬‫ﻣﺮاﺣﻠﻲ‬‫ﺑﺪﺳ‬‫ﺖ‬‫ﻣﻲ‬
‫آﻳﻨﺪ‬‫ﻛﻪ‬‫اﻳﻦ‬‫ﻋﺪد‬‫ﺣﺎﺻﻞ‬‫ﺧﻄﺎﻫﺎي‬‫ﮔﺮدﻛﺮدن‬‫ﻣﻲ‬‫ﺑﺎﺷﻨﺪ‬.
•‫رﻫﻴﺎﻓﺖ‬‫ﻫﺎي‬‫ﻏﻴﺮ‬‫ﻣﺴﺘﻘﻴﻢ‬‫ﻳﺎ‬‫ﺗﻜﺮاري‬:‫روش‬‫ﻫﺎي‬‫ﮔﺎم‬‫ﺑﻪ‬‫ﮔﺎم‬‫ﺑﺮ‬‫ﻣﺒﻨﺎي‬‫اﻃﻼﻋﺎت‬‫ﭘﻲ‬‫در‬‫ﭘﻲ‬‫ﺳ‬‫ﺣﺪ‬‫ﻬﺎي‬
‫اوﻟﻴﻪ‬‫ﺑﺮاي‬‫ﭘﺎﺳﺦ‬،‫ﻣﺴﺌﻠﻪ‬‫ﭘﺎﻳﻪ‬‫ﮔﺬاري‬‫ﺷﺪه‬‫اﻧﺪ‬.
6
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•‫رﻫﻴﺎﻓﺖ‬‫ﻫﺎي‬‫ﻣﺴﺘﻘﻴﻢ‬‫ﻳﺎ‬‫ﻏﻴﺮ‬‫ﺗﻜﺮاري‬
–‫ﺷﺮط‬‫ﺟﻮاب‬‫داﺷﺘﻦ‬:‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬‫ﻧﺎوﻳﮋه‬‫ﻳﻌﻨﻲ‬‫دﺗﺮﻣﻴﻨﺎن‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬‫ﻏﻴﺮ‬‫ﺻﻔﺮ‬‫ﺑﺎﺷﺪ‬.
–‫روش‬‫ﻫﺎي‬‫ﻣﺘﺪاول‬:
1.‫ﻣﻌﻜﻮس‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬
2.‫روش‬‫ﻛﺮاﻣﺮ‬
3.‫روش‬‫ﻫﺎي‬‫ﺣﺬﻓﻲ‬
*‫روش‬‫ﺳﺎده‬)‫ﻧﺎﻗﺺ‬(‫ﮔﻮس‬)Naive Gaussian Elimination(
*‫روش‬‫ﺣﺬف‬‫ﮔﻮس‬‫ﺑﺎ‬‫ﻣﺤﻮرﻳﺖ‬‫ﻧﺴﺒﻲ‬‫ﻗﻴﺎﺳﻲ‬)Pivoting Gaussian Elimination(
4.‫روش‬‫ﺗﺠﺰﻳﻪ‬LU
7
bAx  bAAxA 11 

1
 A IAA 1
bAx 1

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•‫رﻫﻴﺎﻓﺖ‬‫ﻫﺎي‬‫ﻏﻴﺮ‬‫ﻣﺴﺘﻘﻴﻢ‬‫ﻳﺎ‬‫ﺗﻜﺮاري‬)Iterative Methods(
–‫ﺷﺮط‬‫داﺷﺘﻦ‬‫ﺟﻮاب‬:‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬ً‫ﺎ‬‫ﺣﺘﻤ‬‫ﺑﺎﻳﺪ‬‫ﻳﺎ‬positive definite‫و‬‫ﻳﺎ‬diagonally dominant‫ﺑﺎﺷﺪ‬)‫ﺷﺮاﻳﻂ‬
‫ﻻزم‬‫و‬‫ﻛﺎﻓﻲ‬‫ﻫﻤﮕﺮاﻳﻲ‬‫روش‬‫ﻫﺎي‬‫ﺗﻜﺮاري‬(.
–‫روش‬‫ﻫﺎي‬‫ﻣﺘﺪاول‬:
1.‫ژاﻛﻮﺑﻲ‬
2.‫ﮔﻮس‬-‫ﺳﻴﺪال‬
8
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‫رﻫﻴﺎﻓﺖ‬‫ﻫﺎي‬
‫ﻏﻴﺮﺗﻜﺮاري‬ ‫ﻳﺎ‬ ‫ﻣﺴﺘﻘﻴﻢ‬
9
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‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫ﻫﺎي‬ ‫دﺳﺘﮕﺎه‬ ‫ﺣﻞ‬ ‫ﺑﺮاي‬ ‫ﻛﺮاﻣﺮ‬ ‫روش‬
•‫ﺑﺎﺷ‬ ‫ﻣﻲ‬ ‫زﻳﺮ‬ ‫ﻗﺮار‬ ‫از‬ ‫ﻛﺮاﻣﺮ‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﻳﻚ‬ ‫ﺑﻮدن‬ ‫ﺣﻞ‬ ‫ﻗﺎﺑﻞ‬ ‫ﺑﺮاي‬ ‫ﻻزم‬ ‫ﺷﺮاﻳﻂ‬‫ﺪ‬:
‫اﻟﻒ‬(‫ﺿﺮاﻳﺐ‬ ‫ﻣﺎﺗﺮﻳﺲ‬A‫ﺑﺎﺷﺪ‬ ‫ﻣﺮﺗﺒﻪ‬ ‫ﻛﺎﻣﻞ‬ ‫ﻳﺎ‬ ‫و‬ ‫ﻧﺎوﻳﮋه‬ ‫ﺑﺎﻳﺴﺘﻲ‬.
•‫ﺗﺬﻛﺮ‬1:‫ﺗﻌﺮﻳﻒ‬‫ﻧﺎوﻳﮋه‬‫ﻳﺎ‬‫ﻛﺎﻣﻞ‬‫ﻣﺮﺗﺒﻪ‬‫ﺑﻮدن‬‫ﻳﻚ‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﻣﺮﺑﻌﻲ‬:‫ﻳﻌﻨﻲ‬‫آن‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﻣﻌﻜﻮس‬‫ﭘﺬﻳﺮ‬‫ﺑﻮده‬‫ﻛ‬‫ﻪ‬‫ﻻزﻣﻪ‬‫آن‬
‫داﺷﺘﻦ‬‫دﺗﺮﻣﻴﻨﺎن‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﻣﺨﺎﻟﻒ‬‫ﺻﻔﺮ‬‫اﺳﺖ‬.
•‫ﺗﺬﻛﺮ‬2:‫در‬‫ﺻﻮرﺗﻲ‬‫ﻛﻪ‬‫ﺳﻄﺮ‬)‫ﻫﺎ‬(‫ﻳﺎ‬‫ﺳﺘﻮن‬)‫ﻫﺎ‬(‫اي‬‫از‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﻣﻮرد‬‫ﻧﻈﺮ‬‫ﺿﺮﻳﺒﻲ‬‫از‬‫ﺳﻄﺮ‬)‫ﻫﺎ‬(‫ﻳﺎ‬‫ﺳﺘﻮن‬)‫ﻫﺎ‬(‫ي‬
‫دﻳﮕﺮي‬‫از‬‫آن‬‫ﻣﺎﺗﺮﻳﺲ‬،‫ﺑﺎﺷﻨﺪ‬‫آﻧﮕﺎه‬‫دﺗﺮﻣﻴﻨﺎن‬‫آن‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺑﺮاﺑﺮ‬‫ﺻﻔﺮ‬‫ﺧﻮاﻫﺪ‬‫ﺷﺪ‬.‫ﺑﺪﻳﻦ‬،‫ﺗﺮﺗﻴﺐ‬‫آن‬‫ﻣﺎﺗﺮ‬‫ﻳﺲ‬‫وﻳﮋه‬
‫اﺳﺖ‬.‫ﺑﻪ‬‫ﻋﺒﺎرت‬‫دﻳﮕﺮ‬‫ﻣﻌﺎدﻻت‬‫ﻣﻮﺟﻮد‬‫در‬‫دﺳﺘﮕﺎه‬‫ﻣﻌﺎدﻻت‬‫ﺧﻄﻲ‬‫ﻣﺮﺑﻮط‬‫ﺑﻪ‬‫ﭼﻨﻴﻦ‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺒﻲ‬‫واﺑﺴﺘ‬‫ﮕﻲ‬‫ﺧﻄﻲ‬
‫ﺑﻪ‬‫ﻫﻢ‬‫دارﻧﺪ‬.
10
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•‫دارﻳﻢ‬ ‫ﻛﺮاﻣﺮ‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﺣﻞ‬ ‫ﺑﺮاي‬:
1(
2(‫ﻣﺎﺗﺮﻳﺲ‬jA‫ﺗﻌﻮﻳﺾ‬ ‫از‬j‫ﻣﺎﺗﺮﻳﺲ‬ ‫ﺳﺘﻮن‬ ‫اﻣﻴﻦ‬A‫ﺑﺮدار‬ ‫ﺑﺎ‬b‫آﻳﺪ‬ ‫ﻣﻲ‬ ‫ﺑﺪﺳﺖ‬.
====================================================================================
‫ﻣﺜﺎل‬:‫روش‬ ‫ﺑﻪ‬ ‫را‬ ‫روﺑﺮو‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬
‫ﻧﻤﺎﻳﻴﺪ‬ ‫ﺣﻞ‬ ‫ﻛﺮاﻣﺮ‬.
‫ﺣﻞ‬-‫ﺑﻌﺪ‬ ‫ﺻﻔﺤﻪ‬ ‫در‬
11
),...,2,1(
)det(
)det(
nj
A
A
x
j
j 
201302020
20204020
20202080
321
321
321



xxx
xxx
xxx
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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‫ﻛﺮاﻣﺮ‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﻣﺜﺎل‬ ‫ﺣﻞ‬ ‫اداﻣﻪ‬-
12
4.0
)det(
)det(
120000
202020
204020
202080
)det(
3
3
3





A
A
xA
0.1
)det(
)det(
300000
1302020
202020
202080
)det(
2
2
2





A
A
xA
300000
1302020
204020
202080
)det( 



A
6.0
)det(
)det(
180000
1302020
204020
202020
)det(
1
1
1





A
A
xA
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‫ﮔﻮس‬ ‫ﺳﺎده‬ ‫ﺣﺬف‬ ‫روش‬
)Naive Gauss Elimination(
•‫ﻣﺰاﻳﺎ‬:‫ﺳﺎده‬ ‫رﻳﺎﺿﻴﺎﺗﻲ‬ ‫ﺑﻴﺎن‬
•‫روش‬ ‫ﻫﺎي‬ ‫ﻣﺤﺪودﻳﺖ‬:‫ﺿ‬ ‫ﻣﺎﺗﺮﻳﺲ‬ ‫ﻣﺜﻠﺜﻲ‬ ‫ﭘﺎﻳﻴﻦ‬ ‫ﻳﺎ‬ ‫و‬ ‫ﺑﺎﻻ‬ ‫ﺿﺮاﻳﺐ‬ ‫ﺑﻮدن‬ ‫ﺻﻔﺮ‬ ‫ﻏﻴﺮ‬ ‫ﺻﻮرت‬ ‫در‬ ‫ﺗﻨﻬﺎ‬،‫ﺮاﻳﺐ‬
‫دارد‬ ‫ﺟﻮاب‬.
•‫ﺗﺸﻜﻴﻞ‬‫ﻣﺎﺗﺮﻳﺲ‬‫اﻓﺰوده‬)augmented Matrix(‫ﺣﺎﺻﻞ‬‫ﺑﻪ‬‫ﻫﻢ‬‫ﭘﻴﻮﺳﺘﻦ‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬‫و‬‫ﺑﺮدار‬
‫ﻣﻌﻠﻮﻣﺎت‬:
13
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FaraDars.org

‫دﺳﺘﻮراﻟﻌﻤﻞ‬‫ﺳﺎده‬ ‫ﺣﺬف‬ ‫روش‬ ‫اﻟﮕﻮرﻳﺘﻤﻲ‬‫ﮔﻮس‬
•‫اول‬ ‫ﻣﺮﺣﻠﻪ‬:‫اﮔﺮ‬
•‫ﻣﺮﺣﻠﻪ‬‫دوم‬:‫ﺿﺮب‬‫دراﻳﻪ‬‫ﻫﺎي‬‫اوﻟﻴﻦ‬‫ﺳﻄﺮ‬‫از‬‫ﻣﺎﺗﺮﻳﺲ‬‫اﻓﺰوده‬‫در‬‫و‬‫اﻧﺠﺎم‬‫ﻋﻤﻠﻴﺎت‬‫ﺗﻔﺮﻳﻖ‬‫روﺑﺮو‬:
•‫ﻣﺮﺣﻠﻪ‬‫ﺳﻮم‬:‫ﺑﺪﻳﻦ‬‫ﺗﺮﺗﻴﺐ‬،‫در‬‫اﻳﻦ‬‫ﻣﺮﺣﻠﻪ‬،‫ّﻟﻴﻦ‬‫و‬‫ا‬‫ﺳﻄﺮ‬‫ﻫﺎ‬‫از‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﻫﺎي‬A‫و‬b،‫ﺑﺪون‬‫ﺗﻐﻴﻴﺮ‬‫ﺑﺎﻗﻲ‬‫ﻣﻲ‬،‫ﻣﺎﻧﻨﺪ‬‫و‬‫درآﻳﻪ‬‫ﻫﺎي‬‫ّﻟﻴﻦ‬‫و‬‫ا‬
‫ﺳﺘﻮن‬‫از‬‫ﻣﺎﺗﺮﻳﺲ‬A‫در‬‫زﻳﺮ‬‫ﺗﺒﺪﻳﻞ‬‫ﺑﻪ‬‫ﺻﻔﺮ‬‫ﻣﻲ‬‫ﮔﺮدﻧﺪ‬.‫ﺑﻨﺎﺑﺮاﻳﻦ‬،‫ﻧﺘﻴﺠﻪ‬‫ﺣﺎﺻﻞ‬‫از‬‫اﻧﺠﺎم‬‫ﻋﻤﻠﻴﺖ‬‫ﻓﻮق‬،‫دﺳﺘﮕﺎه‬‫ﺗﻐﻴﻴﺮ‬‫ﻓ‬‫ﺮم‬‫ﻳﺎﻓﺘﻪ‬‫زﻳﺮ‬
‫اﺳﺖ‬:
(1)
11 0a 
(1)
1
1 (1)
11
i
i
a
m
a

1im
(2) (1) (1)
1 1
(2) (1) (1)
1 1
, 2,3,...,ij ij i j
i i i
j na a m a
b b m b
  
 
(1)
11a
(1) (1) (1) (1)
111 12 1 1
(2) (2) (2)
222 2 2
(2) (2) (2)
2
. . .
0 . . .
.. . . . .
.. . . . .
.. . . . .
0 . . .
n
n
nn nn n
xa a a b
xa a b
xa a b
    
    
    
    
    
    
    
        
    
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‫دﺳﺘﻮراﻟﻌﻤﻞ‬ ‫اداﻣﻪ‬‫ﺳﺎده‬ ‫ﺣﺬف‬ ‫روش‬ ‫اﻟﮕﻮرﻳﺘﻤﻲ‬‫ﮔﻮس‬
•‫ﻫﻤﻴﻦ‬‫روش‬‫را‬‫ﺑﺮاي‬‫ﺳﺎﻳﺮ‬‫ﺳﻄﺮﻫﺎ‬‫ﻧﻴﺰ‬‫اداﻣﻪ‬‫ﻣﻲ‬‫دﻫﻴﻢ‬،‫ﻛﻪ‬‫ﺑﺪﻳﻦ‬‫ﺗﺮﺗﻴﺐ‬‫در‬‫ﺣﺎﻟﺖ‬
‫ﻛﻠﻲ‬‫در‬k‫اﻣﻴﻦ‬‫ﮔﺎم‬‫ﺧﻮاﻫﻴﻢ‬‫داﺷﺖ‬)‫اﻟﺒﺘﻪ‬‫ﺑﺎ‬‫ﻓﺮض‬( ( )
0k
kka 
( )
( )
1
k
ik
ik k
k
a
m
a
 k‫اﻓﺰوده‬ ‫ﻣﺎﺗﺮﻳﺲ‬ ‫از‬ ‫ﺳﻄﺮ‬ ‫اﻣﻴﻦ‬
( 1) ( ) ( )
( 1) ( ) ( )
, 2,3,...,k k k
ij ij ik kj
k k k
i i ik k
j na a m a
b b m b


  
 
(1) (1) (1) (1) (1)
111 12 1, 1, 1 1
(2) (2) (2) (2)
222 1, 1, 1 1
( ) ( ) ( )
, 1
( 1) ( 1)
11, 1 1,
( 1) ( 1)
, 1
0
0 0
0 0 0
0 0 0
k k n
k k n
k k k
kkk k k kn
k k
kk k k n
k k
nn k nn
xa a a a a
xa a a a
xa a a
xa a
xa a



 
  
 

   
  
  
  
  
  
  
  
  
  
  
 
 
    


    
(1)
1
(2)
2
( )
( 1)
1
( 1)
k
k
k
k
k
n
b
b
b
b
b



 
 
 
 
 
  
 
 
 
 
 


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FaraDars.org

‫دﺳﺘﻮراﻟﻌﻤﻞ‬ ‫اداﻣﻪ‬‫ﺳﺎده‬ ‫ﺣﺬف‬ ‫روش‬ ‫اﻟﮕﻮرﻳﺘﻤﻲ‬‫ﮔﻮس‬
•‫در‬،‫ﻧﻬﺎﻳﺖ‬‫از‬‫روي‬‫ﻣﺎﺗﺮﻳﺲ‬‫اﻓﺰوده‬،‫ﻧﻬﺎﻳﻲ‬‫ﻣﺎﺗﺮﻳﺴﻬﺎي‬‫زﻳﺮ‬‫ﻗﺎﺑﻞ‬‫اﺳﺘﺨﺮاج‬‫ﻫﺴﺘﻨﺪ‬:
1(‫ﻳﻚ‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬‫ﺑﺎﻻ‬‫ﻣﺜﻠﺜﻲ‬‫ﺑﻪ‬‫ﻓﺮم‬‫زﻳﺮ‬‫ﺣﺎﺻﻞ‬‫ﻣﻲ‬‫آﻳﺪ‬:
2(‫ﻫﻤﭽﻨﻴﻦ‬‫ﺑﺮدار‬‫ﻣﻌﻠﻮﻣﺎت‬‫ارﺗﻘﺎء‬‫ﻳﺎﻓﺘﻪ‬‫ﺑﻪ‬‫ﻗﺮار‬‫روﺑﺮو‬‫ﻣﻲ‬‫ﺑﺎﺷﺪ‬:
•‫ﺳﭙﺲ‬‫ﺑﺮﻃﺒﻖ‬‫دﺳﺘﮕﺎه‬‫ﺣﺎﺿﺮ‬‫از‬‫آﺧﺮ‬‫ﺑﻪ‬‫اول‬‫ﻣﺠﻬﻮﻻت‬‫ﻣﺤﺎﺳﺒﻪ‬‫ﻣﻲ‬‫ﺷﻮﻧﺪ‬.
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‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫روش‬‫ﮔﻮس‬ ‫ﺳﺎده‬ ‫ﺣﺬف‬
function ngaussel(A,b)
n=length(b);
x=zeros(n,1);
fprintf('n');
disp(' The augmented matrix is')
augm =[A b]
for k=1:n-1
for i=k+1:n
m=A(i,k)/A(k,k);
for j=k+1:n
A(i,j)=A(i,j)-m*A(k,j);
end
A(i,k)=m;
b(i)=b(i)-m*b(k);
end
end
x(n)=b(n)/A(n,n);
for i=n-1:-1:1
S=b(i);
for j=i+1:n
S=S-A(i,j)*x(j);
end
x(i)=S/A(i,i);
end
% Print the results
fprintf('n');
disp(' The transformed upper triangular
augmented matrix C is =')
fprintf('n');
for i=1:n
for j=1:n
if (j<i) A(i,j)=0; end
end
end
C=[A b]
fprintf('n');
disp(' Back substitution gives the vector
solution')
x
17
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‫ﮔﻮس‬ ‫ﺳﺎده‬ ‫ﺣﺬف‬ ‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﺣﻞ‬ ‫ﻣﺜﺎل‬
m32=2/1= 2
m42=-2/1= -2
18
‫ﻣﺜﺎل‬:‫دﺳﺘﮕﺎه‬‫ﻣﻌﺎدﻻت‬‫زﻳﺮ‬‫را‬‫ﺑﻪ‬‫روش‬‫ﮔﻮس‬‫ﺳﺎده‬‫ﺣﻞ‬‫ﻛﻨﻴﺪ‬:
‫ﻣﺎﺗﺮﻳﺲ‬‫اﻓﺰوده‬‫ﺷﺪه‬‫در‬‫ﻃﻮل‬‫ﺑﺎ‬‫ﺿﺮاﻳﺐ‬1im،‫ﻋﺒﺎرت‬‫اﺳﺖ‬‫از‬:
‫ﺑﺎ‬‫ﺗﻔﺮﻳﻖ‬‫ﺣﺎﺻﻞ‬‫ﺿﺮب‬‫ﻫﺎي‬‫ﺑﺪﺳﺖ‬‫آﻣﺪه‬‫ي‬‫اوﻟﻴﻦ‬‫ﻣﻌﺎدﻟﻪ‬‫از‬‫ﺳﻪ‬‫ﻣﻌﺎدﻟﻪ‬،‫دﻳﮕﺮ‬‫دارﻳﻢ‬:
‫ﺑﺎ‬‫ﺗﻔﺮﻳﻖ‬‫ﺣﺎﺻﻞ‬‫ﺿﺮب‬‫ﻫﺎي‬‫ﻣﻌﺪﻟﻪ‬‫ﺳﻮم‬‫از‬‫ﻣﻌﺎدﻟﻪ‬‫ي‬‫آﺧﺮ‬‫ﻫﻢ‬‫ﺧﻮاﻫﻴﻢ‬‫داﺷﺖ‬:
‫اول‬ ‫ﺳﻄﺮ‬ ‫ﻣﺤﻮري‬ ‫درآﻳﻪ‬
m21=2/1= 2
m31=-1/1=-1
m41=3/1= 3
‫دوم‬ ‫ﺳﻄﺮ‬ ‫ﻣﺤﻮري‬ ‫درآﻳﻪ‬
m43=2/-2 = -1
‫ﺳﻮم‬ ‫ﺳﻄﺮ‬ ‫ﻣﺤﻮري‬ ‫درآﻳﻪ‬
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•‫ﺑﺎ‬‫ﻛﻢ‬‫ﻛﺮدن‬‫ﺣﺎﺻﻞ‬‫ﺿﺮب‬‫ﻫﺎي‬‫ﻣﻌﺎدﻟﻪ‬‫ﺳﻮم‬‫از‬‫ﻣﻌﺎدﻟﻪ‬،‫آﺧﺮ‬‫دﺳﺘﮕﺎه‬‫ﺑﺎﻻ‬‫ﻣﺜﻠﺜﻲ‬‫زﻳﺮ‬‫ﺣﺎﺻﻞ‬‫ﻣ‬‫ﻲ‬‫آﻳﺪ‬:
•‫ﻋﻤﻠﻴﺎت‬‫ﻣﺮﺑﻮط‬‫ﺑﻪ‬‫دﺳﺘﻮراﻟﻌﻤﻞ‬‫ﺟﺎﻳﮕﺬاري‬‫ﺑﺮﮔﺸﺘﻲ‬‫ﺑﺎ‬‫ﻣﻌﻜﻮس‬‫اﻋﻤﺎل‬‫ﺷﺪه‬‫ﺑﻪ‬‫دﺳﺘﮕﺎه‬‫ﺑ‬‫ﺎﻻ‬‫ﻣﺜﻠﺜﻲ‬
،‫ﺣﺎﺻﻞ‬‫ﻧﺘﺎﻳﺞ‬‫زﻳﺮ‬‫را‬‫ﺑﺪﺳﺖ‬‫ﻣﻲ‬‫دﻫﺪ‬:
‫ﮔﻮس‬ ‫ﺳﺎده‬ ‫ﺣﺬف‬ ‫ﺷﻴﻮه‬ ‫ﺑﻪ‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﺣﻞ‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬
19
x1+x2+x3+x4 = 10
x2-x3+3x4= 11
-2x3- 2x4= -14
-x4= -4
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‫ﮔﻮس‬ ‫ﺳﺎده‬ ‫ﺣﺬف‬ ‫روش‬ ‫ﻧﻜﺎت‬
20
‫ﻣﺰاﻳﺎ‬:
*‫ﺳﺎدﮔﻲ‬‫روش‬
‫ﻣﻌﺎﻳﺐ‬:
*‫ﻋﺪم‬‫ﻛﺎرآﻳﻲ‬‫ﺑﺮاي‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬‫ﺑﺎ‬‫وﺟﻮد‬‫ﻳﻚ‬‫در‬‫ا‬‫ي‬‫ﻳﻪ‬‫ي‬‫ﭘﺎﻳﻪ‬‫ﺻ‬‫ﻔﺮ‬.
*‫ﺑﺴﻴﺎر‬‫ﻛﻮﭼﻚ‬‫ﺑﻮدن‬‫ي‬‫دراﻳﻪ‬‫ﭘﺎﻳﻪ‬)‫ﻋﻤﻮدي‬(‫از‬‫ﻧﻈﺮاﻧﺪازه‬‫درﻣﻘﺎﻳﺴﻪ‬‫ﺑﺎ‬
‫ﺳﺎﻳﺮ‬‫دراﻳﻪ‬‫ﻫﺎي‬‫ﺳﻄﺮ‬‫ﻣﺤﻮري‬.
‫ﻧﺘﻴﺠﻪ‬:‫اﻓﺰاﻳﺶ‬‫ﺧﻄﺎي‬‫ﮔﺮد‬‫ﻛﺮدن‬)(Round-Off‫ﻛﻪ‬‫در‬‫اﻳﻦ‬‫ﺻﻮرت‬
‫ﺧﻄﺎ‬‫ﺗﻮاﻧﺪ‬‫ﻣﻲ‬‫ﻣﻨﺠﺮ‬‫ﺑﻪ‬‫ﻳﻚ‬‫ﺑﺮدار‬‫ﺟﻮاب‬‫ﻏﻴﺮ‬‫دﻗﻴﻖ‬‫ﺷﻮد‬.
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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‫ﻗﻴﺎﺳﻲ‬ ‫ﻧﺴﺒﻲ‬ ‫ﻣﺤﻮرﻳﺖ‬ ‫ﺑﺎ‬ ‫ﮔﻮس‬ ‫ﺣﺬف‬ ‫روش‬
)Pivoting Gaussian Elimination(
21
‫ﺗﺬﻛﺮ‬:‫ﻣﻨﻈﻮر‬‫از‬‫درآﻳﻪ‬‫ﺑﺎ‬‫ﻣﺤﻮر‬‫ﻫﻤﺎن‬‫درآﻳﻪ‬‫ي‬‫ﻫﺎ‬‫ﺻﻔﺮ‬‫ﺑﺮ‬‫روي‬‫ﻗﻄﺮ‬‫اﺻﻠﻲ‬‫ﻣﻲ‬‫ﺑﺎﺷﻨﺪ‬.
‫اﺗﻜﺎ‬ ‫اﻟﻤﺎن‬ ‫از‬ ‫ﭼﻨﺪ‬ ‫ﻧﻜﺎﺗﻲ‬)Pivoting:(
1(‫اﻟﻤﺎن‬‫روي‬‫ﻗﻄﺮ‬‫اﺻﻠﻲ‬.
2(‫ﻓﺮآﻳﻨﺪ‬‫روش‬‫ﺣﺬف‬‫ﺳﺎده‬‫ﮔﻮس‬‫در‬‫ﺻﻮرت‬‫ﺻﻔﺮ‬‫ﺑﻮدن‬‫ﻫﺮ‬‫ﻳﻚ‬‫از‬‫اﻟﻤﺎن‬‫ﻫﺎي‬‫ﻗﻄﺮ‬‫اﺻﻠﻲ‬‫ﻣﺘﻮﻗﻒ‬
‫ﺧﻮاﻫﺪ‬‫ﺷﺪ‬.
3(‫روش‬‫ﺣﺬف‬‫ﺳﺎده‬‫ﮔﻮس‬‫ﺑﺮاي‬‫اﺟﺘﻨﺎب‬‫از‬‫اﻳﻦ‬‫ﻣﺸﻜﻞ‬‫ﺑﺎﻳﺪ‬‫اﺻﻼح‬‫ﺷﻮد‬.
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‫روش‬ ‫اداﻣﻪ‬‫ﻗﻴﺎﺳﻲ‬ ‫ﻧﺴﺒﻲ‬ ‫ﻣﺤﻮرﻳﺖ‬ ‫ﺑﺎ‬ ‫ﮔﻮس‬ ‫ﺣﺬف‬
)Pivoting Gaussian Elimination(
22
*‫روش‬‫ﭘﻴﺸﻨﻬﺎدي‬‫ﺗﻌﻮﻳﺾ‬‫ﻣﻌﺎدﻻت‬)‫ﺳﻄﺮﻫﺎ‬(‫ﻳﺎ‬‫ﻣﺘﻐﻴﺮﻫﺎ‬)‫ﺳﺘﻮن‬‫ﻫﺎ‬(‫ﺑﻪ‬‫ﻣﻨﻈﻮر‬‫ﻣﻨﺘﻘﻞ‬‫ﻧﻤﻮدن‬‫ﺑﺰرﮔ‬‫ﺘﺮﻳﻦ‬
‫اﻟﻤﺎن‬‫ﺑﻪ‬‫روي‬‫ﻗﻄﺮ‬‫اﺻﻠﻲ‬‫ﻗﺒﻞ‬‫از‬‫اﻧﺠﺎم‬‫ﻋﻤﻞ‬‫ﺣﺬف‬‫اﺳﺖ‬.
*‫اﻳﻦ‬‫روش‬pivoting‫ﻧﺎﻣﻴﺪه‬‫ﻣﻲ‬‫ﺷﻮد‬.‫ﺗﻌﻮﻳﺾ‬‫ﻫﺮدو‬‫ﺳﻄﺮﻫﺎ‬‫و‬‫ﺳﺘﻮن‬‫ﻫﺎ‬full pivoting‫ﻧﺎﻣﻴﺪه‬‫ﻣﻲ‬
‫ﺷﻮد‬.
*Full pivoting‫ﺑﻪ‬‫ﻋﻠﺖ‬‫ﭘﻴﭽﻴﺪﮔﻲ‬‫اﺳﺘﻔﺎده‬‫ﻧﻤﻲ‬‫ﺷﻮد‬.‫روش‬‫ﭘﻴﺸﻨﻬﺎدي‬partial pivoting‫اﺳﺖ‬.
*‫در‬‫روش‬partial pivoting‫از‬‫ﺟﺎﺑﺠﺎﻳﻲ‬‫ﺳﻄﺮﻫﺎ‬‫اﺳﺘﻔﺎده‬‫ﻣﻲ‬‫ﺷﻮد‬.
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‫از‬ ‫ﻣﺜﺎﻟﻲ‬Partial Pivoting
23
*‫ﻣﺜﺎل‬:‫ﭼﻮن‬1,1a=0‫ﭘﺲ‬‫ﻻزم‬‫اﺳﺖ‬‫ﻋﻤﻠﻴﺎت‬pivoting‫اﻧﺠﺎم‬‫ﺷﻮد‬:
Pivoting

















)1(
3
)1(
2
)1(
1
:
:
:
1111
2110
0112
r
r
r
 
1
1
1
)4(
11
3
2
)4(
1
)4(
11
)4(
223
)4(
23
)4(
22
)4(
33
)4(
33










axabx
axabx
abx
n
nn
  













 








)4(
3
)4(
2
)4(
1
)3(
2
)3(
22
)3(
32
)3(
3
)3(
2
)3(
1
:
:
:
3/43/400
2110
0112
r
r
r
raar
r
r
 
  
























)2(
3
)2(
2
)2(
1
)1(
1
)1(
11
)1(
31
)1(
3
)1(
1
)1(
11
)1(
21
)1(
2
)1(
1
:
:
:
12/12/30
2110
0112
r
r
r
raar
raar
r

















)3(
3
)3(
2
)3(
1
:
:
:
2110
12/12/30
0112
r
r
r












1111
0112
2110
]|[ bA













1111
2110
0112
]|[ bA
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‫ﺳﺎزي‬ ‫ﻣﺘﻨﺎﺳﺐ‬)scaling(
•‫ﻳﺎﺑﺪ‬ ‫ﻣﻲ‬ ‫اﻫﻤﻴﺖ‬ ‫ﻛﺮدن‬ ‫ﮔﺮد‬ ‫از‬ ‫ﻧﺎﺷﻲ‬ ‫ﺧﻄﺎي‬ ،‫ﺳﻄﺮ‬ ‫ﻫﺎي‬ ‫اﻟﻤﺎن‬ ‫ﺳﺎﻳﺮ‬ ‫از‬ ‫اﺗﻜﺎ‬ ‫اﻟﻤﺎن‬ ‫ﺑﻮدن‬ ‫ﻛﻮﭼﻚ‬ ‫ﺻﻮرت‬ ‫در‬.
•‫ﺳﺎزي‬ ‫ﻣﺘﻨﺎﺳﺐ‬ ‫روش‬ ‫از‬ ‫اﺳﺘﻔﺎده‬ ‫ﺣﺎﻟﺘﻲ‬ ‫ﭼﻨﻴﻦ‬ ‫در‬)Scaling(‫ﺷﻮد‬ ‫ﻣﻲ‬ ‫ﭘﻴﺸﻨﻬﺎد‬ ‫اﺗﻜﺎ‬ ‫اﻟﻤﺎن‬ ‫اﻧﺘﺨﺎب‬ ‫ﺑﺮاي‬.
•‫ﺳﺎزي‬ ‫ﻣﺘﻨﺎﺳﺐ‬ ‫روش‬)Scaling(:
–‫ﺑﺰرﮔﺘﺮﻳ‬ ‫ﺑﻪ‬ ‫ﻳﻌﻨﻲ‬ ‫ﺷﻮﻧﺪ‬ ‫ﻣﻲ‬ ‫ﻧﺮﻣﺎﻟﻴﺰه‬ ‫اول‬ ‫ﺳﺘﻮن‬ ‫ﻫﺎي‬ ‫اﻟﻤﺎن‬ ‫ﻛﻠﻴﻪ‬ ،‫اول‬ ‫ﺳﺘﻮن‬ ‫روي‬ ‫ﺑﺮ‬ ،‫ﮔﻮس‬ ‫ﺣﺬف‬ ‫روش‬ ‫اﻋﻤﺎل‬ ‫از‬ ‫ﻗﺒﻞ‬‫اﻟﻤﺎن‬ ‫ﻦ‬
‫ﺷﻮﻧﺪ‬ ‫ﻣﻲ‬ ‫ﺗﻘﺴﻴﻢ‬ ‫ﻣﺮﺑﻮﻃﻪ‬ ‫ﺳﻄﺮ‬ ‫در‬ ‫ﻣﻮﺟﻮد‬.(
–‫ﺷﻮد‬ ‫ﻣﻲ‬ ‫اﻧﺠﺎم‬ ‫اﺗﻜﺎ‬ ‫اﻟﻤﺎن‬ ‫اﻧﺘﺨﺎب‬ ‫ﺳﭙﺲ‬.
–‫ﺷﻮد‬ ‫ﻣﻲ‬ ‫ﺑﻜﺎرﮔﺮﻓﺘﻪ‬ ‫ﮔﻮس‬ ‫ﺳﺎده‬ ‫ﺣﺬف‬ ‫روش‬.
–‫ﻫﺎي‬ ‫اﻟﻤﺎن‬ ،‫ﺣﺬﻓﻲ‬ ‫روش‬ ‫ﺑﻜﺎرﮔﻴﺮي‬ ‫از‬ ‫ﻗﺒﻞ‬ ‫دوم‬ ‫ﺳﺘﻮن‬ ‫ﺑﺮاي‬2‫ﺗﺎ‬n‫ﺳﺘﻮن‬ ‫در‬2‫ﻋﻤﻞ‬ ،‫ﺷﺪه‬ ‫ﻧﺮﻣﺎﻟﻴﺰه‬pivotingً‫ﺎ‬‫ﻣﺘﻌﺎﻗﺒ‬ ‫و‬ ‫اﻧﺠﺎم‬
‫ﺷﻮد‬ ‫ﻣﻲ‬ ‫اﺳﺘﻔﺎده‬ ‫ﺣﺬﻓﻲ‬ ‫روش‬ ‫از‬.
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‫ﺳﺎزي‬ ‫ﻣﺘﻨﺎﺳﺐ‬ ‫ﺑﺎ‬ ‫ﮔﻮس‬ ‫ﺣﺬف‬ ‫روش‬‫و‬pivoting
25
-‫ﺑﺎ‬ ‫ﻫﻤﺮاه‬ ‫ﺣﺬﻓﻲ‬ ‫روش‬ ‫ﺑﺮاي‬ ‫ﻣﺜﺎل‬Scaling:
‫اول‬ ‫ﺳﺘﻮن‬ ‫ﻛﺮدن‬ ‫ﻧﺮﻣﺎﻟﻴﺰه‬:
‫ﺳﻄﺮ‬ ‫ﺟﺎﺑﺠﺎﻳﻲ‬1‫ﺑﺎ‬3)Pivoting(
‫دوم‬ ‫ﺳﺘﻮن‬ ‫ﻛﺮدن‬ ‫ﻧﺮﻣﺎﻟﻴﺰه‬:
‫ﺣﺬف‬ ‫اﻋﻤﺎل‬:











3311
9810332
10410523
]|[ bA






















333.0
0194.0
0286.0
3/1
103/2
105/3
1a











10410523
9810332
3311
13
12
)1/3(
)1/2(
RR
RR














959610
929750
3311




























0104.0
0516.0
96/1
97/52a












959610
929750
3311
23 )5/1( RR  










6.766.760.00.0
0.920.970.50.0
0.30.30.10.1











10410523
9810332
3311
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‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫روش‬pivoting gauss elimination
function gaussel(A,b)
% Solve the system Ax=b using naive
Gaussian elimination
n=length(b);
x=zeros(n,1);
fprintf('n');
disp(' The augmented matrix is =')
augm =[A b]
for i=1:n
d(i)=i;
smax=0;
for j=1:n
smax=max(smax,abs(A(i,j)));
end
c(i)=smax;
end
for k=1:n-1
rmax=0;
for i=k:n
R=abs(A(d(i),k))/c(d(i));
if (R>rmax)
26
j=I;
rmax=R;
end
end
dk=d(j);
d(j)=d(k);
d(k)=dk;
for i=k+1:n
m=A(d(i),k)/A(dk,k);
for j=k+1:n
A(d(i),j)=A(d(i),j)-m*A(dk,j);
end
A(d(i),k)=m;
end
end
% Perform the back substitution.
for k=1:n-1
for i=k+1:n
b(d(i))=b(d(i))-b(d(k))*A(d(i),k);
end
end
x(n)=b(d(n))/A(d(n),n);
for i=n-1:-1:1
S=b(d(i));
for j=i+1:n
S=S-A(d(i),j)*x(j);
end
x(i)=S/A(d(i),i);
end
% Print the results
disp('The scale vector =‘)
c
disp('The index vector at the end of
the elimination process is =')
d
fprintf('n');
disp(' The transformed upper
triangular augmented matrix C is =')
fprintf('n');
for i=1:n
M(i,:)=A(d(i),:);
end
for i=1:n
for j=1:n
if (j<i) M(i,j)=0; end
end
end
C=[M b]
fprintf('n');
disp(' Back substitution gives the
vector solution')
x
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‫ي‬ ‫ﺗﺠﺰﻳﻪ‬ ‫روش‬LU
27
‫روش‬‫ﺗﺠﺰﻳﻪ‬‫ي‬LU،‫ﺷﺎﻣﻞ‬‫ﺗﺒﺪﻳﻞ‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬‫ﺑﻪ‬‫ﺣﺎﺻﻞ‬‫ﺿﺮب‬‫دو‬‫ﻣﺎﺗﺮﻳﺲ‬L‫و‬U‫ﻣﻲ‬‫ﺑﺎﺷﺪ‬‫ﻛﻪ‬‫در‬‫اﻳﻨﺠﺎ‬L،‫ﻳﻚ‬
‫ﻣﺎﺗﺮﻳﺲ‬‫ﭘﺎﻳﻴﻦ‬‫ﻣﺜﻠﺜﻲ‬‫و‬U،‫ﻳﻚ‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺑﺎﻻ‬‫ﻣﺜﻠﺜﻲ‬‫ﺑﺎ‬‫درآﻳﻪ‬‫ﻫﺎي‬‫واﺣﺪ‬‫در‬‫ﻃﻮل‬‫ﻗﻄﺮ‬‫اﺻﻠﻲ‬‫ﺧﻮد‬‫ﻣﻲ‬‫ﺑﺎﺷﻨﺪ‬.‫زﻣﺎﻧﻲ‬‫ﻛﻪ‬
‫ﻣﺎﺗﺮﻳﺲ‬‫ﻫﺎي‬L‫و‬U‫ﭘﻴﺪا‬،‫ﺷﻮﻧﺪ‬‫آﻧﮕﺎه‬‫ﺣﻞ‬‫دﺳﺘﮕﺎه‬‫ﻣﻲ‬‫ﺗﻮاﻧﺪ‬‫ﺑﻪ‬‫ﻃﺮﻳﻖ‬‫زﻳﺮ‬‫اﻧﺠﺎم‬‫ﺷﻮد‬: Ax b
LUx b Ux y Ly bAx b
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‫ﻣﺜﻠﺜﻲ‬ ‫ﭘﺎﻳﻴﻦ‬ ‫و‬ ‫ﺑﺎﻻ‬ ‫ﻣﺎﺗﺮﻳﺲ‬ ‫ﻳﺎﻓﺘﻦ‬ ‫ﻫﺎي‬ ‫روش‬ ‫اﻧﻮاع‬L‫و‬U
28
1(‫روش‬‫ﻫﺎي‬Choleski‫و‬Crout‫و‬2(‫روش‬‫ﺣﺬف‬‫ﮔﻮس‬
*‫در‬‫اﻳﻨﺠﺎ‬‫ﺗﻨﻬﺎ‬‫ﺑﻪ‬‫روش‬‫اول‬‫ﭘﺮداﺧﺘﻪ‬‫ﻣﻲ‬‫ﺷﻮد‬.
*‫ﻣﺎﺗﺮﻳﺲ‬L‫را‬‫ﺑﺎ‬‫ﻋﻨﺎﺻﺮ‬‫ﻏﻴﺮ‬‫ﺻﻔﺮ‬‫در‬‫ﻃﻮل‬‫ﻗﻄﺮ‬‫اﺻﻠﻲ‬‫آن‬‫ﺑﺪﺳﺖ‬‫ﻣﻲ‬‫آورﻳﻢ‬‫و‬‫در‬‫ﺿﻤﻦ‬‫ﻣﺎﺗﺮﻳﺲ‬U‫را‬‫ﺑﻪ‬‫ﮔﻮﻧﻪ‬‫اي‬‫ﺑﺪﺳﺖ‬
‫ﻣﻲ‬‫آورﻳﻢ‬‫ﻛﻪ‬‫ﺗﺴﺎوي‬‫ﻣﺎﺗﺮﻳﺴﻲ‬‫زﻳﺮ‬‫ﺑﺮﻗﺮار‬‫ﺑﺎﺷﺪ‬)‫اﻟﺒﺘﻪ‬‫ﺑﺮاي‬‫ﻳﻚ‬‫ﻣﺎﺗﺮﻳﺲ‬4x4‫ﻧﻤﻮﻧﻪ‬(:
*‫ﺑﺮاي‬‫ﻣﺤﺎﺳﺒﻪ‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﭘﺎﻳﻴﻦ‬‫ﻣﺜﻠﺜﻲ‬L،‫از‬‫روش‬‫ﺣﺬف‬‫ﮔﻮس‬‫ﺳﺎده‬‫اﺳﺘﻔﺎده‬‫ﻣﻲ‬‫ﻛﻨﻴﻢ‬‫اﻟﺒﺘﻪ‬‫در‬‫راﺳﺘﺎي‬‫ﺳﺘﻮن‬‫ﻫﺎ‬‫و‬‫ﺑﺮاي‬
‫ﻣﺤﺎﺳﺒﻪ‬‫ﻋﻨﺎﺻﺮ‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺑﺎﻻ‬‫ﻣﺜﻠﺜﻲ‬U‫از‬‫ﻣﻌﺎدﻟﻪ‬‫و‬‫ﺗﺴﺎوي‬‫ﻓﻮق‬‫اﺳﺘﻔﺎده‬‫ﻣﻲ‬‫ﺷﻮد‬.
11 11 12 13 1412 13 14
21 22 21 22 23 2423 24
31 32 33 31 32 33 3434
41 42 43 44 41 42 43 44
0 0 0 1
0 0 0 1
0 0 0 1
0 0 0 1
l a a a au u u
l l a a a au u
l l l a a a au
l l l l a a a a
    
    
     
    
    
    
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‫ﺑﺮﻳﺪ‬ ‫ﺑﻜﺎر‬ ‫زﻳﺮ‬ ‫دﺳﺘﮕﺎه‬ ‫ﺣﻞ‬ ‫ﺑﺮاي‬ ‫را‬ ‫ﻛﺮوت‬ ‫روش‬ ‫ﻣﺜﺎل‬:
‫ﻣﺴﺘﻘﻴﻢ‬ ‫ﻓﺎﻛﺘﻮرﮔﻴﺮي‬ ‫ﻳﻚ‬ ، ‫ﻣﺎﺗﺮﻳﺲ‬ ‫اﮔﺮ‬LU‫آﻧﮕﺎه‬ ،‫ﺑﺎﺷﺪ‬ ‫داﺷﺘﻪ‬:
‫ﺑﻮﺳﻴﻠﻪ‬‫ي‬‫ﺿﺮب‬‫ﻛﺮدن‬L‫در‬U،‫و‬‫ﻣﻘﺎﻳﺴﻪ‬‫ي‬‫در‬‫آﻳﻪ‬‫ﻫﺎي‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﻫﺎي‬‫ﺣﺎﺻﻞ‬‫ﺿﺮب‬‫ﺑﺎ‬‫درآﻳﻪ‬‫ﻫﺎي‬‫ﻣﺎﺗﺮﻳﺲ‬A،‫ﺑﺪﺳﺖ‬
‫ﻣﻲ‬‫آورﻳﻢ‬:
)i(‫از‬‫ﺿﺮب‬‫اوﻟﻴﻦ‬‫ﺳﻄﺮ‬‫از‬‫ﻣﺎﺗﺮﻳﺲ‬L،‫در‬‫ﺳﺘﻮن‬‫ﻫﺎي‬‫ﻣﺎﺗﺮﻳﺲ‬U‫،ﺑﺪﺳﺖ‬‫ﻣﻲ‬‫آورﻳﻢ‬‫ﻛﻪ‬:
‫از‬ ‫ﻣﺜﺎﻟﻲ‬‫ﺗﺠﺰﻳﻪ‬ ‫روش‬LU
29
1
2
3
4
1 1 1 1 10
2 3 1 5 31
1 1 5 3 2
3 1 7 2 18
x
x
x
x
    
    
    
      
    
    
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
0 0 01 1 1 1 1
0 02 3 1 5 0 1
01 1 5 3 0 0 1
3 1 7 2 0 0 0 1
l u u u
l l u u
A
l l l u
l l l l
    
    
     
     
    
    
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‫ﺗﺠﺰﻳﻪ‬ ‫روش‬ ‫از‬ ‫ﻣﺜﺎﻟﻲ‬ ‫اداﻣﻪ‬LU
30
(i)‫از‬‫ﺿﺮب‬‫اوﻟﻴﻦ‬‫ﺳﻄﺮ‬‫از‬‫ﻣﺎﺗﺮﻳﺲ‬L،‫در‬‫ﺳﺘﻮن‬‫ﻫﺎي‬‫ﻣﺎﺗﺮﻳﺲ‬U،‫ﺑﺪﺳﺖ‬‫ﻣﻲ‬‫آورﻳﻢ‬‫ﻛﻪ‬:
(ii)‫از‬‫ﺿﺮب‬‫دوﻣﻴﻦ‬‫ﺳﻄﺮ‬‫از‬‫ﻣﺎﺗﺮﻳﺲ‬L،‫در‬‫ﺳﺘﻮن‬‫ﻫﺎي‬‫ﻣﺎﺗﺮﻳﺲ‬U،‫دارﻳﻢ‬:
(iii)‫از‬‫ﺿﺮب‬‫ﺳﻮﻣﻴﻦ‬‫ﺳﻄﺮ‬‫از‬‫ﻣﺎﺗﺮﻳﺲ‬L،‫در‬‫ﺳﺘﻮن‬‫ﻫﺎي‬‫ﻣﺎﺗﺮﻳﺲ‬U،‫دارﻳﻢ‬:
11
11 12 12
11 13 13
11 14 14
1,
1 1,
1 1,
1 1.
l
l u u
l u u
l u u

  
  
  
21
21 12 22 22 21 12
21 13 22 23 23 21 13 22
21 14 22 24 24 21 14 22
2,
3 3 1,
1 (1 ) / 1,
5 (5 )/ 3.
l
l u l l l u
l u l u u l u l
l u l u u l u l

     
      
     
31
31 12 32 32 31 12
31 13 32 23 33 33 31 13 32 23
31 14 32 24 33 34 34 31 14 32 24 33
1,
1 1 2,
5 5 2,
3 (3 )/ 1
l
l u l l l u
l u l u l l l u l u
l u l u l u u l u l u l
 
     
          
       
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31
(iv)‫از‬‫ﺿﺮب‬‫ﭼﻬﺎرﻣﻴﻦ‬‫ﺳﻄﺮ‬‫از‬‫ﻣﺎﺗﺮﻳﺲ‬L،‫در‬‫ﺳﺘﻮن‬‫ﻫﺎي‬‫ﻣﺎﺗﺮﻳﺲ‬U،‫دارﻳﻢ‬:
‫ﺑﻨﺎﺑﺮاﻳﻦ‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﻫﺎي‬L‫و‬U‫ﺑﺮاﺑﺮﻧﺪ‬‫ﺑﺎ‬:
‫و‬
‫ﺑﺎ‬‫اﻋﻤﺎل‬‫ﺟﺎﻳﮕﺬاري‬‫ﭘﻴﺸﺮوﻧﺪه‬‫در‬‫دﺳﺘﮕﺎه‬‫ﭘﺎﻳﻴﻦ‬‫ﻣﺜﻠﺜﻲ‬Ly=b،‫ﻧﺘﺎﻳﺞ‬‫زﻳﺮ‬‫را‬‫ﺑﺪﺳﺖ‬‫ﻣﻲ‬‫آورﻳﻢ‬:
1
2
3
4
10,
31 2(10) 11,
[ 2 10 2(11)]/( 2) 7,
[18 3(10) 2(11) 2(7)] 4
y
y
y
y

  
     
     
1 1 1 1
0 1 1 3
U=
0 0 1 1
0 0 0 1
 
  
 
 
 
1 0 0 0
2 1 0 0
L=
1 2 2 0
3 2 2 1
 
 
 
  
 
  
122
,277
3443244214414444344324421441
23421341434323421341


ulululllululul
ululllulul
41
41 12 42 42 41 12
3,
1 1 2,
l
l u l l l u

      
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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32
‫در‬،‫ﻧﻬﺎﻳﺖ‬‫ﺑﺎ‬‫ﺟﺎﻳﮕﺬاري‬‫ﭘﺴﺮو‬‫در‬‫دﺳﺘﮕﺎه‬‫ﺑﺎﻻ‬‫ﻣﺜﻠﺜﻲ‬Ux=y،‫ﻣﻘﺎدﻳﺮ‬‫زﻳﺮ‬‫را‬‫ﺑﺮاي‬x‫ﻫﺎ‬‫ﺑﺪﺳﺖ‬‫ﻣﻲ‬‫آورﻳﻢ‬:
1
2
3
4
10 4 3 2 1,
[11 4 3(3)] 2,
7 4 3,
4
x
x
x
x
    
    
  

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‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫ﺗﺠﺰﻳﻪ‬ ‫روش‬LU
function lufact(A,b)
% Solve the system Ax=b using the LU
decomposition.
n=length(b);
y=zeros(n,1);
x=zeros(n,1);
fprintf('n');
for i=1:n
U(i,i)=1;
end
L(1,1)=A(1,1)/U(1,1);
for j=2:n
L(j,1)=A(j,1)/U(1,1);
U(1,j)=A(1,j)/L(1,1);
end
for i=2:n-1
S=0;
for k=1:i-1
S=S+U(k,i)*L(i,k);
end
L(i,i)=(A(i,i)-S)/U(i,i);
33
for j=i+1:n
S=0;
for k=1:i-1
S=S+U(k,i)*L(j,k);
end
L(j,i)=(A(j,i)-S)/U(i,i);
S=0;
for k=1:i-1
S=S+U(k,j)*L(i,k);
end
U(i,j)=(A(i,j)-S)/L(i,i);
end
end
S=0;
for k=1:n-1
S=S+U(k,n)*L(n,k);
end
L(n,n)=(A(n,n)-S)/U(n,n);
% Perform the forward substitution.
y(1)=b(1)/L(1,1);
for i=2:n
S=b(i);
for j=1:i-1
S=S-L(i,j)*y(j);
end
y(i)=S/L(i,i);
end
% Perform the back substitution.
x(n)=y(n)/U(n,n);
for i=n-1:-1:1
S=y(i);
for j=i+1:n
S=S-U(i,j)*x(j);
end
x(i)=S/U(i,i);
end
% Print the results
L
disp(' The forward substitution
gives')
y
U
disp('The vector solution is =')
X
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‫رﻫﻴﺎﻓﺖ‬‫ﻫﺎي‬‫ﻏﻴﺮ‬‫ﻳﺎ‬ ‫ﻣﺴﺘﻘﻴﻢ‬
‫ﺗﻜﺮاري‬
)Iterative Methods(
34
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‫ﺗﻜﺮاري‬ ‫ﻫﺎي‬ ‫روش‬ ‫ﻣﻬﻢ‬ ‫ﻧﻜﺎت‬
35
1(‫ﺑﺨﺎﻃﺮ‬‫وﺟﻮد‬‫ﺧﻄﺎﻫﺎي‬‫ﺣﺎﺻﻞ‬‫از‬‫ﮔﺮد‬،‫ﻛﺮدن‬‫روش‬‫ﻫﺎي‬‫ﻣﺴﺘﻘﻴﻢ‬‫ﻣﺤﺎﺳﺒﻪ‬‫ي‬‫دﺳﺘﮕﺎه‬،‫ﻫﺎ‬‫از‬‫ﻛﺎرآﻳﻲ‬‫و‬‫ﺑﺎزده‬‫ﻛﻤﺘﺮي‬‫ﻧﺴﺒ‬‫ﺖ‬‫ﺑﻪ‬
‫روش‬‫ﻫﺎي‬‫ﺗﻜﺮار‬‫ﻣﺘﻮاﻟﻲ‬‫ﺑﺮﺧﻮردارﻧﺪ‬.
2(‫ﻋﻼوه‬‫ﺑﺮ‬‫ﺧﻄﺎﻫﺎي‬‫ﺣﺎﺻﻞ‬‫از‬‫ﮔﺮد‬،‫ﻛﺮدن‬‫ﻣﻘﺪار‬‫ﻓﻀﺎي‬‫ذﺧﻴﺮه‬‫ﺳﺎزي‬‫ﻣﻮرد‬‫ﻧﻴﺎز‬‫ﺑﺮاي‬‫ﺣﻞ‬‫ﻣﻌﺎدﻻت‬‫ﺑﻪ‬‫روش‬‫ﻫﺎي‬‫ﺗﻜﺮار‬‫ﻣﺘﻮاﻟ‬‫ﻲ‬
‫ﺑﺴﻴﺎر‬‫ﻛﻤﺘﺮ‬‫از‬‫ﻣﻘﺪار‬‫ﻣﻮرد‬‫ﻧﻴﺎز‬‫در‬‫روش‬‫ﻫﺎي‬‫ﻣﺴﺘﻘﻴﻢ‬‫ﻣﻲ‬،‫ﺑﺎﺷﺪ‬‫ﺑﺨﺼﻮص‬‫زﻣﺎﻧﻲ‬‫ﻛﻪ‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬‫ﻣﺘﻔﺮق‬‫و‬‫ﭘﺮاﻛ‬‫ﻨﺪه‬،‫اﺳﺖ‬
‫ﻳﻌﻨﻲ‬‫آراﻳﻪ‬‫ﻫﺎ‬‫و‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﻫﺎ‬‫ﺷﺎﻣﻞ‬‫ﺗﻌﺪاد‬‫زﻳﺎدي‬‫درآﻳﻪ‬‫ﻫﺎي‬‫ﺻﻔﺮ‬‫ﻣﻲ‬‫ﺑﺎﺷﻨﺪ‬)Sparse(.
3(‫ﺷﺮاﻳﻂ‬‫وﺟﻮد‬‫ﺟﻮاب‬‫و‬‫ﻳﺎ‬‫ﻫﻤﮕﺮاﻳﻲ‬‫ﭘﺎﺳﺦ‬‫ﻫﺎ‬‫در‬‫روش‬‫ﻫﺎي‬‫ﺗﻜﺮاري‬‫ﺣﻞ‬‫دﺳﺘﮕﺎه‬‫ﻫﺎي‬‫ﻣﻌﺎدﻻت‬‫ﺧﻄﻲ‬‫ﻋﺒﺎرﺗﻨﺪ‬‫از‬:
)‫اﻟﻒ‬(‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬ً‫ﺎ‬‫ﺣﺘﻤ‬‫ﺑﺎﻳﺪ‬‫ﻳﺎ‬positive definite‫و‬‫ﻳﺎ‬diagonally dominant‫ﺑﺎﺷﺪ‬)‫ﺷﺮاﻳﻂ‬‫ﻻزم‬‫و‬‫ﻛﺎﻓﻲ‬
‫ﻫﻤﮕﺮاﻳﻲ‬‫روش‬‫ﻫﺎي‬‫ﺗﻜﺮاري‬(.
)‫ب‬(‫ﻣﻨﻈﻮر‬‫از‬diagonally dominant‫ﺑﻮدن‬‫آن‬‫اﺳﺖ‬‫ﻛﻪ‬:




n
ij
j
ijii aantoifor
1
:1 



n
ij
j
ijii aaAofrowaforleastatand
1
:
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‫از‬ ‫ﻣﺜﺎﻟﻲ‬diagonally dominant‫ﺑﻮدن‬
36
‫ﺑﺎ‬‫ﺗﻮﺟﻪ‬‫ﺑﻪ‬‫ﺷﺮاﻳﻂ‬‫ذﻛﺮ‬،‫ﺷﺪه‬‫ﻛﺪام‬‫ﻳﻚ‬‫از‬‫ﻣﺎﺗﺮﻳﺲ‬‫ﺿﺮاﻳﺐ‬‫زﻳﺮ‬‫ﺑﻪ‬‫ﻃﻮر‬،‫ﻗﻄﺮي‬‫ﻏﺎﻟﺐ‬‫ﻫﺴﺘﻨﺪ؟‬
 A














3481.52
14335
116123
1293496
55323
5634124
][












B
|81.5||2||34|
|1||35||43|
|1||16||123|


 *‫ﻃﺒﻖ‬‫ﺷﺮاﻳﻂ‬‫ذﻛﺮ‬،‫ﺷﺪه‬‫ﻣﺎﺗﺮﻳﺲ‬A،diagonally dominant‫ﻫﺴﺖ‬‫زﻳﺮا‬‫ﻫﻢ‬‫ﻗﺪر‬‫ﻣﻄﻠﻖ‬
‫ﺿﺮاﻳﺐ‬‫روي‬‫ﻗﻄﺮ‬‫اﺻﻠﻲ‬‫از‬‫ﻣﺠﻤﻮع‬‫ﺳﺎﻳﺮ‬‫ﺿﺮاﻳﺐ‬‫ﻫﺮ‬‫ﺳﻄﺮ‬‫ﺑﺰرﮔﺘﺮ‬‫ﻣﺴﺎوي‬‫ﻫﺴﺘﻨﺪ‬.‫ﻫﻤﭽﻨ‬‫ﻴﻦ‬
‫ﺣﺪاﻗﻞ‬‫ﻳﻜﻲ‬‫از‬‫دراﻳﻪ‬‫ﻫﺎي‬‫روي‬‫ﻗﻄﺮ‬‫اﺻﻠﻲ‬‫ﺑﺰرﮔﺘﺮ‬‫از‬‫ﻣﺠﻤﻮع‬‫دراﻳﻪ‬‫ﻫﺎي‬‫ﻫﻤﺎن‬‫ﺳﻄﺮ‬‫اﺳﺖ‬.
*‫ﻃﺒﻖ‬‫ﺷﺮاﻳﻂ‬‫ذﻛﺮ‬،‫ﺷﺪه‬‫ﻣﺎﺗﺮﻳﺲ‬B،diagonally dominant‫ﻧﻴﺴﺖ‬‫زﻳﺮا‬‫ﻗﺪر‬
‫ﻣﻄﻠﻖ‬‫ﺿﺮﻳﺐ‬‫روي‬‫ﻗﻄﺮ‬‫اﺻﻠﻲ‬‫ﺳﻄﺮ‬‫ﺳﻮم‬‫از‬‫ﻣﺠﻤﻮع‬‫ﺳﺎﻳﺮ‬‫ﺿﺮاﻳﺐ‬‫ﻫﻤﺎن‬‫ﺳﻄﺮ‬
‫ﻛﻮﭼﻜﺘﺮ‬‫اﺳﺖ‬.|34||96||129|
|5||23||53|
|56||34||124|



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‫ﻣﺘﻮاﻟﻲ‬ ‫ﺗﻜﺮار‬ ‫ﻫﺎي‬ ‫روش‬ ‫ﻛﻠﻲ‬ ‫ﻓﺮم‬
)Iterative Methods(
•‫اول‬ ‫ﮔﺎم‬:‫روﺑﺮو‬ ‫ﻓﺮم‬ ‫ﺑﻪ‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﻓﺮم‬ ‫ﺗﻐﻴﻴﺮ‬
•‫دوم‬ ‫ﮔﺎم‬:‫اوﻟﻴﻪ‬ ‫ﺣﺪس‬ ‫ﻳﻚ‬ ‫اﻧﺘﺨﺎب‬
–‫ﺳﻮم‬ ‫ﮔﺎم‬:‫زﻳﺮ‬ ‫ﻓﺮﻣﻮل‬ ‫ﻃﺮﻳﻖ‬ ‫از‬ ‫ﺗﻘﺮﻳﺒﻲ‬ ‫ﭘﺎﺳﺦ‬ ‫ﺑﺮدارﻫﺎي‬ ‫ﻣﺎﺗﺮﻳﺲ‬ ‫ي‬ ‫دﻧﺒﺎﻟﻪ‬ ‫ﻣﺤﺎﺳﺒﻪ‬
37
'
x b Bx 
(0)
x
( ) ' ( 1)k k
x b Bx 
  k=1,2,3,…‫ازاي‬ ‫ﺑﻪ‬
bAx 
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‫ژاﻛﻮﺑﻲ‬ ‫روش‬
)Jacobi Iterative Method(
•‫زﻳﺮ‬ ‫ﻓﺮم‬ ‫ﺑﻪ‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﻓﺮم‬ ‫ﺗﻐﻴﻴﺮ‬ ‫ﺑﺎ‬ ‫ﻣﺘﻐﻴﺮﻫﺎ‬ ‫از‬ ‫ﻳﻚ‬ ‫ﻫﺮ‬ ‫ﻣﺤﺎﺳﺒﻪ‬
•‫زﻳﺮ‬ ‫ﻓﺮﻣﻮل‬ ‫ﺑﺎ‬ ‫ﺑﺮاي‬ ‫آوردن‬ ‫ﺑﺪﺳﺖ‬
•‫زﻳﺮ‬ ‫ﺷﺮاﻳﻂ‬ ‫ﺷﺪن‬ ‫ﺑﺮآورده‬ ‫ﺗﺎ‬ ‫ﻗﺒﻞ‬ ‫ﻣﺮاﺣﻞ‬ ‫ﺗﻜﺮار‬:
38
1
( ) , 1,2, ,
n
ij j i
i
j ii ii
j i
a x b
x i n
a a

    
( )k
ix1k 
],...,[
|| )()(
1
)(
)(
)1()(
k
n
kk
k
kk
k xxxwhere
x
xx
e 




ni
a
bxa
x
ii
i
n
ij
j
k
jij
k
i ,...,2,1,
1
)1(
)(







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‫ﻣﺜﺎل‬:‫ﻛﻨﻴﺪ‬ ‫ﺣﻞ‬ ‫ژاﻛﻮﺑﻲ‬ ‫ﻣﺘﻮاﻟﻲ‬ ‫ﺗﻜﺮار‬ ‫روش‬ ‫از‬ ‫اﺳﺘﻔﺎده‬ ‫ﺑﺎ‬ ‫زﻳﺮ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬.‫ﺑ‬ ‫ﻣﺎﺗﺮﻳﺲ‬ ‫ﻋﻨﻮان‬ ‫ﺑﻪ‬ ‫را‬ ‫زﻳﺮ‬ ‫ﻣﻘﺎدﻳﺮ‬‫آﻏﺎزﻳﻦ‬ ‫ﺮدار‬
‫ﺑﮕﻴﺮﻳﺪ‬ ‫ﻧﻈﺮ‬ ‫در‬:0=)0(x,30=ITMAX,3-10=EPS
‫ﻛﻨﻴﻢ‬ ‫ﻣﺮﺗﺐ‬ ‫و‬ ‫ﺑﺎزﻧﻮﻳﺴﻲ‬ ‫زﻳﺮ‬ ‫ﺻﻮرت‬ ‫ﺑﻪ‬ ‫ﺗﻮان‬ ‫ﻣﻲ‬ ‫را‬ ‫اﺧﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬:
‫آﻳﺪ‬ ‫ﻣﻲ‬ ‫در‬ ‫زﻳﺮ‬ ‫ﺷﻜﻞ‬ ‫ﺑﻪ‬ ‫ژاﻛﻮﺑﻲ‬ ‫ﻣﺘﻮاﻟﻲ‬ ‫ﺗﻜﺮار‬ ‫ﻋﻤﻠﻴﺎت‬ ‫ﻓﺮم‬ ‫ﺗﺮﺗﻴﺐ‬ ‫ﺑﻪ‬ ‫ﻛﻪ‬:
‫ژاﻛﻮﺑﻲ‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﺣﻞ‬ ‫ﻣﺜﺎل‬
39
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x(0)=(0,0,0,0)‫ﻛﻨﻴﺪ‬ ‫ﺟﺎﻳﮕﺰﻳﻦ‬ ،‫زﻳﺮ‬ ‫ﻣﻘﺎدﻳﺮ‬ ‫آوردن‬ ‫ﺑﺪﺳﺖ‬ ‫ﺑﺮاي‬ ،‫ﻗﺒﻞ‬ ‫ﻣﻌﺎدﻻت‬ ‫از‬ ‫ﻳﻚ‬ ‫ﻫﺮ‬ ‫راﺳﺖ‬ ‫ﺳﻤﺖ‬ ‫در‬ ‫را‬:
‫ﻛﻪ‬ ‫اﺳﺖ‬ ‫ﺗﺮﺗﻴﺐ‬ ‫ﺑﺪﻳﻦ‬ ‫و‬T)1.444444444,1.5,1.666666666-,2.428571429)=(1(x‫ﻋﻤﻠﻴﺎت‬ ‫ﻣﺸﺎﺑﻪ‬ ‫ﻋﻤﻠﻴﺎﺗﻲ‬ ‫ﺷﻮد‬ ‫ﻣﻲ‬
‫ﺑﺎﺷﻨﺪ‬ ‫ﻣﻲ‬ ‫زﻳﺮ‬ ‫ﻣﻘﺎدﻳﺮ‬ ‫ﺑﻪ‬ ‫ﻫﻤﮕﺮا‬ ‫ﻛﻪ‬ ‫دﻫﻨﺪ‬ ‫ﻣﻲ‬ ‫ﺑﺪﺳﺖ‬ ‫را‬ ‫ﻣﻘﺎدﻳﺮ‬ ‫از‬ ‫اي‬ ‫دﻧﺒﺎﻟﻪ‬ ،‫ﻗﺒﻞ‬:
‫ژاﻛﻮﺑﻲ‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﺣﻞ‬ ‫ﻣﺜﺎل‬ ‫اداﻣﻪ‬
40
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‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﺣﻞ‬ ‫روش‬ ‫ﺑﺮاي‬ ‫دﻳﮕﺮ‬ ‫ﻣﺜﺎل‬
‫ژاﻛﻮﺑﻲ‬ ‫ﺗﻜﺮاري‬ ‫روش‬ ‫ﺑﻪ‬
•‫اول‬ ‫ﮔﺎم‬:‫ﺿﺮاﻳﺐ‬ ‫ﻣﺎﺗﺮﻳﺲ‬ ‫ﺑﺮاي‬ ‫ﻻزم‬ ‫ﺷﺮاﻳﻂ‬ ‫ﻛﺮدن‬ ‫ﭼﻚ‬)‫ﺑﻮدن‬ ‫ﻣﻌﻴﻦ‬ ‫ﻣﺜﺒﺖ‬ ‫و‬ ‫ﻏﺎﻟﺐ‬(
•‫دوم‬ ‫ﮔﺎم‬:‫ﺑﻪ‬ ‫از‬ ‫ﻣﻌﺎدﻟﻪ‬ ‫دادن‬ ‫ﻓﺮﻣﺖ‬ ‫ﺗﻐﻴﻴﺮ‬
•‫ﺳﻮم‬ ‫ﮔﺎم‬:‫ﻧﺘﺎﻳﺞ‬ ‫ﻣﺤﺎﺳﺒﻪ‬ ‫و‬ ‫اوﻟﻴﻪ‬ ‫ﺣﺪس‬ ‫ﻳﻚ‬ ‫از‬ ‫اﺳﺘﻔﺎده‬
•‫ﭼﻬﺎرم‬ ‫ﮔﺎم‬:‫ﻣﻄﻠﻮب‬ ‫دﻗﺖ‬ ‫ﺑﻪ‬ ‫رﺳﻴﺪن‬ ‫ﺗﺎ‬ ‫ژاﻛﻮﺑﻲ‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﻣﺮﺑﻮط‬ ‫ﻣﺤﺎﺳﺒﺎت‬ ‫اﻧﺠﺎم‬
41

















 






3
12
14
11
2
1
x
x
bAx
3x4x
12x-x
21
21
Ax b'
x b Bx 
1 ( 1)
( )
( )
, 1,2, , (4 11)
n
j ij j i
j i
i
ii
k
k
a x b
x i n
a




 
 


],...,[
|| )()(
1
)(
)(
)1()(
k
n
kk
k
kk
k xxxwhere
x
xx
e 




‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫ژاﻛﻮﺑﻲ‬ ‫روش‬
function jacobi(A,b,x0,tol,itmax)
% Solve the system Ax=b using Jacobi iteration method.
n=length(b);
x=zeros(n,1);
fprintf('n');
Augm=[A b]
Y=zeros(n,1);
Y=x0;
for k=1:itmax+1
for i=1:n
S=0;
for j=1:n
if (j~=i)
S=S+A(i,j)*x0(j);
end
end
if(A(i,i)==0)
break
end
x(i)=(-S+b(i))/A(i,i);
end
err=abs(norm(x-x0));
42
rerr=err/(norm(x)+eps);
x0=x;
Y=[Y x];
if(rerr<tol)
break
end
end
% Print the results
if(A(i,i)==0)
disp(' division by zero')
elseif (k==itmax+1)
disp(' No convergence')
else
fprintf('n');
disp(' The solution vectors are:')
fprintf('n');
disp('iter # 0 1 2 3 4 ...')
fprintf('n');
for i=1:n
fprintf('x%1.0f = ',i)
fprintf('%10.6f',Y(i,[1:k+1]))
fprintf('n');
end
fprintf('n');
disp(['The method converges after ',num2str(k),' iterations
to']);
x
end
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‫ﮔﻮس‬ ‫روش‬-‫ﺳﻴﺪال‬
)Gauss-Seidal Iterative Method(
•‫روش‬ ‫دﺳﺘﻮراﻟﻌﻤﻞ‬‫ﮔﻮس‬-‫ﺳﻴﺪال‬‫ﻣﺘﻐ‬ ‫از‬ ‫ﻳﻚ‬ ‫ﻫﺮ‬ ‫ﻣﻘﺪار‬ ‫اﻳﻨﻜﻪ‬ ‫ﺟﺰ‬ ‫ﺑﻪ‬ ‫ﺑﺎﺷﺪ‬ ‫ﻣﻲ‬ ‫ژاﻛﻮﺑﻲ‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﻣﺸﺎﺑﻪ‬ ً‫ﺎ‬‫ﺗﻘﺮﻳﺒ‬ ‫ﻧﻴﺰ‬‫ﺑﺎ‬ ‫ﻴﺮﻫﺎ‬
‫ﺷﻮد‬ ‫ﻣﻲ‬ ‫ﻣﺤﺎﺳﺒﻪ‬ ، ‫ﻣﺮﺣﻠﻪ‬ ‫ﻫﺮ‬ ‫در‬ ‫ﻣﺘﻐﻴﺮﻫﺎ‬ ‫ﺳﺎﻳﺮ‬ ‫ﺑﺮاي‬ ‫آﻣﺪه‬ ‫ﺑﺪﺳﺖ‬ ‫ﺗﻘﺎرﻳﺐ‬ ‫ﺟﺪﻳﺪﺗﺮﻳﻦ‬ ‫از‬ ‫اﺳﺘﻔﺎده‬.
•‫زﻳﺮ‬ ‫ﻓﺮم‬ ‫ﺑﻪ‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﻓﺮم‬ ‫ﺗﻐﻴﻴﺮ‬ ‫ﺑﺎ‬ ‫ﻫﺎ‬ ‫ﻣﺘﻐﻴﺮ‬ ‫از‬ ‫ﻳﻚ‬ ‫ﻫﺮ‬ ‫ﻣﺤﺎﺳﺒﻪ‬
–‫زﻳﺮ‬ ‫ﺷﺮاﻳﻂ‬ ‫ﺷﺪن‬ ‫ﺑﺮآورده‬ ‫ﺗﺎ‬ ‫ﻗﺒﻞ‬ ‫ﻣﺮﺣﻠﻪ‬ ‫ﺗﻜﺮار‬:
43
],...,[
|| )()(
1
)(
)(
)1()(
k
n
kk
k
kk
k xxxwhere
x
xx
e 




111
1k 
1 ( ) ( 1)
1 1( )
( ) ( )
1,2, , .
i nk k
ij j ij j ij j ik
i
ii
a x a x b
x
a
i n
 
  
  


 

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‫ﻣﺜﺎل‬:‫ﮔﻮس‬ ‫ﻣﺘﻮاﻟﻲ‬ ‫ﺗﻜﺮار‬ ‫روش‬ ‫ﺑﻪ‬ ‫را‬ ‫زﻳﺮ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬-‫ﻛﻨﻴﺪ‬ ‫ﺣﻞ‬ ‫ﺳﻴﺪال‬.‫آﻏﺎزﻳ‬ ‫ﺑﺮدار‬ ‫ﻫﺎي‬ ‫درآﻳﻪ‬ ‫ﺑﺮاي‬ ‫را‬ ‫زﻳﺮ‬ ‫ﻣﻘﺎدﻳﺮ‬‫ﻧﻈﺮ‬ ‫در‬ ‫ﻦ‬
‫ﺑﮕﻴﺮﻳﺪ‬:0=)0(x,30=ITMAX,3-10=EPS
‫ﻛﻪ‬ ‫آوردﻳﻢ‬ ‫ﺑﺪﺳﺖ‬ ‫ﻗﺒﻞ‬ ‫ﻣﺜﺎل‬ ‫از‬:
‫ﮔﻮس‬ ‫ﻣﺘﻮاﻟﻲ‬ ‫ﺗﻜﺮار‬ ‫ﻋﻤﻠﻴﺎت‬ ‫ﺟﻤﻼت‬ ،‫ﺑﺎﻻ‬ ‫ﻣﻌﺎدﻻت‬-‫آورﻧﺪ‬ ‫ﻣﻲ‬ ‫ﻓﺮاﻫﻢ‬ ‫را‬ ‫زﻳﺮ‬ ‫ﺳﻴﺪال‬:
‫ﮔﻮس‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﺣﻞ‬ ‫ﻣﺜﺎل‬-‫ﺳﻴﺪال‬
44
6/)15(
10/)215(
9/)313(
7/)217(
)1(
3
)1(
2
)1(
1
)1(
4
)(
4
)1(
1
)1(
3
)(
4
)(
3
)1(
1
)1(
2
)(
3
)(
2
)1(
1








kkkk
kkk
kkkk
kkk
xxxx
xxx
xxxx
xxx
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‫ﺑﺮدار‬ ‫ﻣﻘﺎدﻳﺮ‬ ‫ﺣﺎل‬x(0)=(0,0,0,0)‫را‬ ‫زﻳﺮ‬ ‫اﻋﺪاد‬ ‫ﺗﺎ‬ ‫ﻛﻨﻴﻢ‬ ‫ﻣﻲ‬ ‫ﺟﺎﻳﮕﺰﻳﻦ‬ ‫اﺧﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬ ‫از‬ ‫ﻳﻚ‬ ‫ﻫﺮ‬ ‫راﺳﺖ‬ ‫ﺳﻤﺖ‬ ‫در‬ ‫را‬
‫آورﻳﻢ‬ ‫ﺑﺪﺳﺖ‬
‫دارﻳﻢ‬ ‫ﺑﻨﺎﺑﺮاﻳﻦ‬:
‫داﺷﺖ‬ ‫ﺧﻮاﻫﻴﻢ‬ ‫را‬ ‫زﻳﺮ‬ ‫ﻣﻘﺎدﻳﺮ‬ ‫ﺑﻪ‬ ‫ﻫﻤﮕﺮا‬ ‫دﻧﺒﺎﻟﻪ‬ ،‫ﻗﺒﻞ‬ ‫ﻣﺸﺎﺑﻪ‬ ‫ﻫﺎﻳﻲ‬ ‫ﻋﻤﻠﻴﺎت‬ ‫اﻧﺠﺎم‬ ‫ﺑﺎ‬ ‫اداﻣﻪ‬ ‫در‬:
‫ﮔﻮس‬ ‫روش‬ ‫ﺑﻪ‬ ‫ﺧﻄﻲ‬ ‫ﻣﻌﺎدﻻت‬ ‫دﺳﺘﮕﺎه‬ ‫ﺣﻞ‬ ‫ﻣﺜﺎل‬-‫ﺳﻴﺪال‬
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‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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‫ﺑﺮﻧﺎﻣﻪ‬Matlab‫ﮔﻮس‬ ‫روش‬-‫ﺳﻴﺪال‬
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function seidel(A,b,x0,tol,itmax)
% Solve the system Ax=b using Gauss-Seidel iteration method.
n=length(b);
x=zeros(n,1);
fprintf('n');
Augm=[A b]
Y=zeros(n,1);
Y=x0;
for k=1:itmax+1
for i=1:n
S=0;
for j=1:i-1
S=S+A(i,j)*x(j);
end
for j=i+1:n
S=S+A(i,j)*x0(j);
end
if(A(i,i)==0)
break
end
x(i)=(-S+b(i))/A(i,i);
end
err=abs(norm(x-x0));
rerr=err/(norm(x)+eps);
x0=x;
Y=[Y x];
if(rerr<tol)
break
end
end
% Print the results
if(A(i,i)==0)
disp(' division by zero')
elseif (k==itmax+1)
disp(' No convergence')
else
fprintf('n');
disp(' The solution vectors are:')
fprintf('n');
disp('iter # 0 1 2 3 4 ...')
fprintf('n');
for i=1:n
fprintf('x%1.0f = ',i)
fprintf('%10.6f',Y(i,[1:k+1]))
fprintf('n');
end
fprintf('n');
disp(['The method converges after ',num2str(k),' iterations
to']);
x
end
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

>> A = [7 -2 +1 0;1 -9 3 -1;2 0 10 1;2 -1 1 6];
>> b = [17;13;15;10] ;
>> x1 = lufact(A,b);
>> x0 = zeros(size(b));
>> ITMAX = 5;
>> tol = 1e-5;
>> X = zeros(length(b),ITMAX,2);
>> for ii = 1 : ITMAX
X(:,ii,1)=jacobi(A,b,x0,tol,ii);
X(:,ii,2)=seidel(A,b,x0,tol,ii);
end
>> figure;
>> for ii=1:length(b)
subplot (length(b),1,ii);plot(1:ITMAX+1,x1(ii)*ones(1,ITMAX+1),'*-',1:ITMAX+1,[0,X(ii,:,1)],'s-
',1:ITMAX+1,[0,X(ii,:,2)],'d-','linewidth',2.5,'markersize',7);
xlabel(' No. of Iterations ');ylabel(' X values ');title([' Comparison of Convergence Speed and Accuracy between
Jacobi & Gauss-seidel Iterative Methods for Variable No. ', num2str(ii)]);
legend('Real Value for Var. x','Jacobi Values for Var. x ','Seidel Values for Var. x');
end
47
‫ﮔﻮس‬ ‫و‬ ‫ژاﻛﻮﺑﻲ‬ ‫روﺷﻬﺎي‬ ‫ﻫﻤﮕﺮاﻳﻲ‬ ‫ﺳﺮﻋﺖ‬ ‫ﻣﻴﺎن‬ ‫ﻣﻘﺎﻳﺴﻪ‬-‫ﺳﻴﺪال‬
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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‫ﮔﻮس‬ ‫و‬ ‫ژاﻛﻮﺑﻲ‬ ‫روﺷﻬﺎي‬ ‫ﻫﻤﮕﺮاﻳﻲ‬ ‫ﺳﺮﻋﺖ‬ ‫ﻣﻴﺎن‬ ‫ﻣﻘﺎﻳﺴﻪ‬ ‫اداﻣﻪ‬-‫ﺳﻴ‬‫ﺪال‬
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0
2
4
No. of Iterations
Xvalues
Comparison of Convergence Speed and Accuracy between Jacobi & Gauss-seidel Iterative Methods for Variable No. 1
Real Value for Var. x
Jacobi Values for Var. x
Seidel Values for Var. x
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
-2
-1
0
No. of Iterations
Xvalues
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0
1
2
No. of Iterations
Xvalues
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0
0.5
1
No. of Iterations
Xvalues
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
FaraDars.org

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‫ﻣﻌﻴﻦ‬ ‫ﻣﺜﺒﺖ‬ ‫ﻣﺎﺗﺮﻳﺲ‬ ‫ﺗﻌﺮﻳﻒ‬
)Positive Definite Matrix(
‫ﺗﻌﺮﻳﻒ‬:‫ﻣﺘﻘﺎرن‬ ‫ﻣﺎﺗﺮﻳﺲ‬ ‫ﻳﻚ‬‫ﻣﻌﻴﻦ‬ ‫ﻣﺜﺒﺖ‬(positive definite) ‫ﺷﻮد‬ ‫ﻣﻲ‬ ‫ﻧﺎﻣﻴﺪه‬(M>0)‫ﻫﺮ‬ ‫ﺑﺮاي‬ ‫اﮔﺮ‬
‫ﺑﺎﺷﻴﻢ‬ ‫داﺷﺘﻪ‬:
nn
RM 

n
Rx 

RxMxT
‫ﻗﻀﻴﻪ‬:‫ﻣﺎﺗﺮﻳﺲ‬‫ﻣﺘﻘﺎرن‬ ‫ﺣﻘﻴﻘﻲ‬M‫ﻣﻌﻴﻦ‬ ‫ﻣﺜﺒﺖ‬ ،)‫ﻣﻌﻴﻦ‬ ‫ﻧﻴﻤﻪ‬ ‫ﻣﺜﺒﺖ‬(‫از‬ ‫ﻛﺪام‬ ‫ﻫﺮ‬ ‫اﮔﺮ‬ ‫ﻓﻘﻂ‬ ‫و‬ ‫اﮔﺮ‬ ‫اﺳﺖ‬‫ﺷﺮاﻳﻂ‬‫ﺑﺮﻗﺮار‬ ‫زﻳﺮ‬‫ﺑﺎﺷﺪ‬:
1-‫ﻣﺎﺗﺮﻳﺲ‬ ‫وﻳﮋه‬ ‫ﻣﻘﺎدﻳﺮ‬ ‫ﺗﻤﺎم‬M‫ﻣﺜﺒﺖ‬ ،)‫ﺻﻔﺮ‬ ‫ﻳﺎ‬ ‫ﻣﺜﺒﺖ‬(‫ﺑﺎﺷﺪ‬.
2-‫ﺗﻤﺎم‬‫ﻛﻬﺎدﻫﺎ‬‫ﻣﺎﺗﺮﻳﺲ‬ ‫ﻣﻘﺪم‬ ‫اﺻﻠﻲ‬ ‫ﻣﺎﻳﻨﻮرﻫﺎي‬ ‫ﻳﺎ‬M‫ﻣﺜﺒﺖ‬ ،)‫ﺻﻔﺮ‬ ‫ﻳﺎ‬ ‫ﻣﺜﺒﺖ‬(‫ﺑﺎﺷﺪ‬.
3-‫ﻣﻨﻔﺮد‬ ‫ﻏﻴﺮ‬ ‫ﻣﺎﺗﺮﻳﺲ‬N‫اﺑﻌﺎد‬ ‫ﺑﺎ‬nn‫ﻛﻪ‬ ‫ﺑﺎﺷﺪ‬ ‫داﺷﺘﻪ‬ ‫وﺟﻮد‬NTNM=)‫ﻣﻨﻔﺮد‬ ‫ﻏﻴﺮ‬ ‫ﻣﺎﺗﺮﻳﺲ‬N‫اﺑﻌﺎد‬ ‫ﺑﺎ‬nn‫ﻣﺎﺗﺮﻳﺲ‬ ‫ﻳﺎ‬ ‫و‬N‫ﺑﺎ‬
‫اﺑﻌﺎد‬nm‫ﺑﺎ‬nm<‫ﻛﻪ‬ ‫ﺑﺎﺷﺪ‬ ‫داﺷﺘﻪ‬ ‫وﺟﻮد‬NTNM=(
‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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‫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.


‫س‬‫ر‬‫د‬‫ا‬‫ﺮ‬‫ﻓ‬
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