steepest decent and conjugate gradients (cg)
DESCRIPTION
Steepest Decent and Conjugate Gradients (CG). Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system. Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system Problem : dimension n too big, or not enough time for gauss elimination - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/1.jpg)
Steepest Decent and Conjugate Gradients (CG)
![Page 2: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/2.jpg)
Steepest Decent and Conjugate Gradients (CG)
• Solving of the linear equation system bAx
![Page 3: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/3.jpg)
Steepest Decent and Conjugate Gradients (CG)
• Solving of the linear equation system • Problem: dimension n too big, or not enough time for
gauss eliminationIterative methods are used to get an
approximate solution.
bAx
![Page 4: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/4.jpg)
Steepest Decent and Conjugate Gradients (CG)
• Solving of the linear equation system • Problem: dimension n too big, or not enough time for
gauss eliminationIterative methods are used to get an
approximate solution.• Definition Iterative method: given starting point , do
steps hopefully converge to the right solution
bAx
0x,, 21 xx
x
![Page 5: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/5.jpg)
starting issues
![Page 6: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/6.jpg)
starting issues
• Solving is equivalent to minimizing bAx cxbAxxxf TT
21:)(
![Page 7: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/7.jpg)
starting issues
• Solving is equivalent to minimizing
• A has to be symmetric positive definite:
bAx cxbAxxxf TT
21:)(
00 xAxxAA TT
![Page 8: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/8.jpg)
starting issues
• 021
21)(
! bAxbAxxAxf
symmetricAT
![Page 9: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/9.jpg)
starting issues
•
• If A is also positive definite the solution of is the minimum
021
21)(
! bAxbAxxAxf
symmetricAT
bAx
![Page 10: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/10.jpg)
starting issues
•
• If A is also positive definite the solution of is the minimum
021
21)(
! bAxbAxxAxf
symmetricAT
bAx
00
11
21
21)(
d
TT AddcbAbdbAf
![Page 11: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/11.jpg)
starting issues
• error: The norm of the error shows how far we are away from the exact solution, but can’t be computed without knowing of the exact solution .
xxe ii :
x
![Page 12: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/12.jpg)
starting issues
• error: The norm of the error shows how far we are away from the exact solution, but can’t be computed without knowing of the exact solution .
• residual: can be calculated
xxe ii :
x)(: xfAeAxbr iii
![Page 13: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/13.jpg)
Steepest Decent
![Page 14: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/14.jpg)
Steepest Decent
• We are at the point . How do we reach ?ix 1ix
![Page 15: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/15.jpg)
Steepest Decent
• We are at the point . How do we reach ?• Idea: go into the direction in which decreases most
quickly ( )
ix 1ix)(xf
ii rxf )(
![Page 16: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/16.jpg)
Steepest Decent
• We are at the point . How do we reach ?• Idea: go into the direction in which decreases most
quickly ( )• how far should we go?
ix 1ix)(xf
ii rxf )(
![Page 17: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/17.jpg)
Steepest Decent
• We are at the point . How do we reach ?• Idea: go into the direction in which decreases most
quickly ( )• how far should we go?
Choose so that is minimized:
ix 1ix)(xf
ii rxf )(
)( ii rxf
![Page 18: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/18.jpg)
Steepest Decent
• We are at the point . How do we reach ?• Idea: go into the direction in which decreases most
quickly ( )• how far should we go?
Choose so that is minimized:
ix 1ix)(xf
ii rxf )(
)( ii rxf
0)( ii rxfdd
![Page 19: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/19.jpg)
Steepest Decent
• We are at the point . How do we reach ?• Idea: go into the direction in which decreases most
quickly ( )• how far should we go?
Choose so that is minimized:
ix 1ix)(xf
ii rxf )(
)( ii rxf
0)( ii rxfdd
0)( iT
ii rrxf
![Page 20: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/20.jpg)
Steepest Decent
• We are at the point . How do we reach ?• Idea: go into the direction in which decreases most
quickly ( )• how far should we go?
Choose so that is minimized:
ix 1ix)(xf
ii rxf )(
)( ii rxf
0)( ii rxfdd
0)( iT
ii rrxf
0))(( iT
ii rbrxA
![Page 21: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/21.jpg)
Steepest Decent
• We are at the point . How do we reach ?• Idea: go into the direction in which decreases most
quickly ( )• how far should we go?
Choose so that is minimized:
ix 1ix)(xf
ii rxf )(
)( ii rxf
0)( ii rxfdd
0)( iT
ii rrxf
0))(( iT
ii rbrxA
iT
r
iiT
i rAxbrAri
)()(
![Page 22: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/22.jpg)
Steepest Decent
• We are at the point . How do we reach ?• Idea: go into the direction in which decreases most
quickly ( )• how far should we go?
Choose so that is minimized:
ix 1ix)(xf
ii rxf )(
)( ii rxf
0)( ii rxfdd
0)( iT
ii rrxf
0))(( iT
ii rbrxA
iT
r
iiT
i rAxbrAri
)()( i
Ti
iTi
Arrrr
![Page 23: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/23.jpg)
Steepest Decent
one step of steepest decent can be calculated as follows:
iiii
Ti
iTi
i
ii
rxxArrrr
Axbr
1
![Page 24: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/24.jpg)
Steepest Decent
one step of steepest decent can be calculated as follows:
• stopping criterion: or with an given small It would be better to use the error instead of the residual, but you can’t calculate the error.
iiii
Ti
iTi
i
ii
rxxArrrr
Axbr
1
maxii 0rri
![Page 25: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/25.jpg)
Steepest Decent
Method of steepest decent:
1
)(
0
00max
0
0
iiAxbrrxx
Arrrr
rrrrandiiwhile
rrAxbr
i
T
T
TT
![Page 26: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/26.jpg)
Steepest Decent
• As you can see the starting point is important!
![Page 27: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/27.jpg)
Steepest Decent
• As you can see the starting point is important!
When you know anything about the solution use it to guess a good starting point. Otherwise you can choose a starting point you want e.g. .00 x
![Page 28: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/28.jpg)
Steepest Decent - Convergence
![Page 29: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/29.jpg)
Steepest Decent - Convergence
• Definition energy norm: Axxx TA:
![Page 30: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/30.jpg)
Steepest Decent - Convergence
• Definition energy norm:
• Definition condition:
( is the largest and the smallest eigenvalue of A)
Axxx TA:
min
max:
max min
![Page 31: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/31.jpg)
Steepest Decent - Convergence
• Definition energy norm:
• Definition condition:
( is the largest and the smallest eigenvalue of A)
•
convergence gets worse when the condition gets larger
Axxx TA:
min
max:
max min
A
i
Aiee 01
1
![Page 32: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/32.jpg)
Conjugate Gradients
![Page 33: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/33.jpg)
Conjugate Gradients
• is there a better direction?
![Page 34: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/34.jpg)
Conjugate Gradients
• is there a better direction?• Idea: orthogonal search directions110 ,,, nddd
![Page 35: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/35.jpg)
Conjugate Gradients
• is there a better direction?• Idea: orthogonal search directions110 ,,, nddd
1
0
n
iiidx
![Page 36: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/36.jpg)
Conjugate Gradients
• is there a better direction?• Idea: orthogonal search directions
• only walk once in each direction and minimize
110 ,,, nddd
1
0
n
iiidx
![Page 37: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/37.jpg)
Conjugate Gradients
• is there a better direction?• Idea: orthogonal search directions
• only walk once in each direction and minimizemaximal n steps are needed to reach the exact
solution
110 ,,, nddd
1
0
n
iiidx
![Page 38: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/38.jpg)
Conjugate Gradients
• is there a better direction?• Idea: orthogonal search directions
• only walk once in each direction and minimizemaximal n steps are needed to reach the exact
solution has to be orthogonal to
110 ,,, nddd
1
0
n
iiidx
1 ie id
![Page 39: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/39.jpg)
Conjugate Gradients
• example with the coordinate axes as orthogonal search directions:
![Page 40: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/40.jpg)
Conjugate Gradients
• example with the coordinate axes as orthogonal search directions:
Problem: can’t be computed
because
(you don’t know !)
iTi
iTi
i dded
ie
![Page 41: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/41.jpg)
Conjugate Gradients
• new idea: A-orthogonal110 ,,, nddd
![Page 42: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/42.jpg)
Conjugate Gradients
• new idea: A-orthogonal• Definition A-orthogonal: A-orthogonal
(reminder: orthogonal: )
110 ,,, nddd
ji dd , 0 jTi Add
ji dd , 0 jTi dd
![Page 43: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/43.jpg)
Conjugate Gradients
• new idea: A-orthogonal• Definition A-orthogonal: A-orthogonal
(reminder: orthogonal: )
• now has to be A-orthogonal to
110 ,,, nddd
ji dd , 0 jTi Add
ji dd , 0 jTi dd
1ie id
![Page 44: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/44.jpg)
Conjugate Gradients
• new idea: A-orthogonal• Definition A-orthogonal: A-orthogonal
(reminder: orthogonal: )
• now has to be A-orthogonal to
110 ,,, nddd
ji dd , 0 jTi Add
ji dd , 0 jTi dd
1ie id
iTi
iTi
iTi
iTi
i Addrd
AddAed
![Page 45: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/45.jpg)
Conjugate Gradients
• new idea: A-orthogonal• Definition A-orthogonal: A-orthogonal
(reminder: orthogonal: )
• now has to be A-orthogonal to
can be computed!
110 ,,, nddd
ji dd , 0 jTi Add
ji dd , 0 jTi dd
1ie id
iTi
iTi
iTi
iTi
i Addrd
AddAed
![Page 46: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/46.jpg)
Conjugate Gradients
• A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram-Schmidt (same idea as Gram-Schmidt).
iu
![Page 47: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/47.jpg)
Conjugate Gradients
• Gram-Schmidt: linearly independent vectors10 ,, nuu
![Page 48: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/48.jpg)
Conjugate Gradients
• Gram-Schmidt: linearly independent vectors10 ,, nuu
jTj
jTi
ij
i
jjijii
dddu
dudi
ud
1
0
00
:0
![Page 49: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/49.jpg)
Conjugate Gradients
• Gram-Schmidt: linearly independent vectors
• conjugate Gram-Schmidt:
10 ,, nuu
jTj
jTi
ij AddAdu
jTj
jTi
ij
i
jjijii
dddu
dudi
ud
1
0
00
:0
![Page 50: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/50.jpg)
Conjugate Gradients
• A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram-Schmidt (same idea as Gram-Schmidt).
• CG works by setting (makes conjugate Gram-Schmidt easy)
iu
ii ru
![Page 51: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/51.jpg)
Conjugate Gradients
• A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram-Schmidt (same idea as Gram-Schmidt).
• CG works by setting (makes conjugate Gram-Schmidt easy)
with1 iiii drd 11
i
Ti
iTi
i rrrr
ii ru
iu
![Page 52: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/52.jpg)
Conjugate Gradients
• 0:1
0
1
n
jkk
Tik
n
jkkk
Tii
Tij
Ti AdddAdAedrdji
![Page 53: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/53.jpg)
Conjugate Gradients
•
•
0:1
0
1
n
jkk
Tik
n
jkkk
Tii
Tij
Ti AdddAdAedrdji
1
0
i
kkikii dud
![Page 54: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/54.jpg)
Conjugate Gradients
•
•
0:1
0
1
n
jkk
Tik
n
jkkk
Tii
Tij
Ti AdddAdAedrdji
1
0
i
kkikii dud
1
0 0
0:i
k jk
jTkikj
Tij
Ti rdrurdji
![Page 55: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/55.jpg)
Conjugate Gradients
•
•
0:1
0
1
n
jkk
Tik
n
jkkk
Tii
Tij
Ti AdddAdAedrdji
1
0
i
kkikii dud
1
0 0
0:i
k jk
jTkikj
Tij
Ti rdrurdji
jiru jTi 0
![Page 56: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/56.jpg)
Conjugate Gradients
•
•
0:1
0
1
n
jkk
Tik
n
jkkk
Tii
Tij
Ti AdddAdAedrdji
1
0
i
kkikii dud
1
0 0
0:i
k jk
jTkikj
Tij
Ti rdrurdji
jiru jTi 0
ijjTi
jTiii
rr
jirrru
0:
![Page 57: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/57.jpg)
Conjugate Gradients
•
•
•
0:1
0
1
n
jkk
Tik
n
jkkk
Tii
Tij
Ti AdddAdAedrdji
ijjTi
jTiii
rr
jirrru
0:
iTi
i
k jk
jTkiki
Tii
Ti rurdrurd
1
0 0
![Page 58: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/58.jpg)
Conjugate Gradients
• jiAddAdr
jTj
jTi
ij
![Page 59: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/59.jpg)
Conjugate Gradients
•
•
jiAddAdr
jTj
jTi
ij
jjjjjjjj AdrdeAAer )(11
![Page 60: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/60.jpg)
Conjugate Gradients
•
•
jiAddAdr
jTj
jTi
ij
jjjjjjjj AdrdeAAer )(11
jTijj
Tij
Ti Adrrrrr 1
![Page 61: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/61.jpg)
Conjugate Gradients
•
•
jiAddAdr
jTj
jTi
ij
jjjjjjjj AdrdeAAer )(11
jTijj
Tij
Ti Adrrrrr 1
1 jTij
Tij
Tij rrrrAdr
![Page 62: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/62.jpg)
Conjugate Gradients
1 jTij
Tij
Tij rrrrAdr
10
11
jiji
jirr
jirr
Adr
rr
i
iTi
i
iTi
jTi
ijjTi
![Page 63: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/63.jpg)
Conjugate Gradients
10 ji
ij
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Conjugate Gradients
1
10
1111111
jirrrr
rdrr
Addrr
ji
iTi
iTi
iTi
iTi
def
iTii
iTiij
![Page 65: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/65.jpg)
Method of Conjugate Gradients:
00
0
irrrdAxbr
1
)( 00max
iidrdrrrr
Axbrrrdxx
Addrr
rrrrandiiwhile
oldTold
T
old
T
T
TT
![Page 66: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/66.jpg)
Conjugate Gradients - Convergence
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Conjugate Gradients - Convergence
• A
i
Aiee 01
12
![Page 68: Steepest Decent and Conjugate Gradients (CG)](https://reader035.vdocument.in/reader035/viewer/2022062500/56815ad9550346895dc8a6e8/html5/thumbnails/68.jpg)
Conjugate Gradients - Convergence
•
• for steepest decent for CG Convergence of CG is much better!
A
i
Aiee 01
12