steepest decent and conjugate gradients (cg). solving of the linear equation system
TRANSCRIPT
![Page 1: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/1.jpg)
Steepest Decent and Conjugate Gradients (CG)
![Page 2: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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 elimination
Iterative methods are used to get an approximate solution.
bAx
![Page 4: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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 elimination
Iterative 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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/5.jpg)
starting issues
![Page 6: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/6.jpg)
starting issues
• Solving is equivalent to minimizing bAx cxbAxxxf TT
2
1:)(
![Page 7: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/7.jpg)
starting issues
• Solving is equivalent to minimizing
• A has to be symmetric positive definite:
bAx cxbAxxxf TT
2
1:)(
00 xAxxAA TT
![Page 8: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/8.jpg)
starting issues
• 02
1
2
1)(
!
bAxbAxxAxfsymmetricA
T
![Page 9: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/9.jpg)
starting issues
•
• If A is also positive definite the solution of is the minimum
02
1
2
1)(
!
bAxbAxxAxfsymmetricA
T
bAx
![Page 10: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/10.jpg)
starting issues
•
• If A is also positive definite the solution of is the minimum
02
1
2
1)(
!
bAxbAxxAxfsymmetricA
T
bAx
00
11
2
1
2
1)(
d
TT AddcbAbdbAf
![Page 11: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/13.jpg)
Steepest Decent
![Page 14: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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 rxfd
d
![Page 19: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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 rxfd
d
0)( iT
ii rrxf
![Page 20: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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 rxfd
d
0)( iT
ii rrxf
0))(( iT
ii rbrxA
![Page 21: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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 rxfd
d
0)( iT
ii rrxf
0))(( iT
ii rbrxA
iT
r
iiT
i rAxbrAr
i
)()(
![Page 22: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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 rxfd
d
0)( iT
ii rrxf
0))(( iT
ii rbrxA
iT
r
iiT
i rAxbrAr
i
)()( i
Ti
iTi
Arr
rr
![Page 23: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/23.jpg)
Steepest Decent
one step of steepest decent can be calculated as follows:
iiii
Ti
iTi
i
ii
rxx
Arr
rr
Axbr
1
![Page 24: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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
rxx
Arr
rr
Axbr
1
maxii 0rri
![Page 25: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/25.jpg)
Steepest Decent
Method of steepest decent:
1
)(
0
00max
0
0
ii
Axbr
rxxArr
rr
rrrrandiiwhile
rr
Axbr
i
T
T
TT
![Page 26: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/26.jpg)
Steepest Decent
• As you can see the starting point is important!
![Page 27: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/28.jpg)
Steepest Decent - Convergence
![Page 29: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/29.jpg)
Steepest Decent - Convergence
• Definition energy norm: Axxx T
A:
![Page 30: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/30.jpg)
Steepest Decent - Convergence
• Definition energy norm:
• Definition condition:
( is the largest and the smallest eigenvalue of A)
Axxx T
A:
min
max:
max min
![Page 31: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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 T
A:
min
max:
max min
A
i
Aiee 01
1
![Page 32: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/32.jpg)
Conjugate Gradients
![Page 33: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/33.jpg)
Conjugate Gradients
• is there a better direction?
![Page 34: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/34.jpg)
Conjugate Gradients
• is there a better direction?
• Idea: orthogonal search directions110 ,,, nddd
![Page 35: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/37.jpg)
Conjugate Gradients
• is there a better direction?
• Idea: orthogonal search directions
• only walk once in each direction and minimize
maximal n steps are needed to reach the exact solution
110 ,,, nddd
1
0
n
iiidx
![Page 38: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/38.jpg)
Conjugate Gradients
• is there a better direction?
• Idea: orthogonal search directions
• only walk once in each direction and minimize
maximal 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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/39.jpg)
Conjugate Gradients
• example with the coordinate axes as orthogonal search directions:
![Page 40: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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
idd
ed
ie
![Page 41: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/41.jpg)
Conjugate Gradients
• new idea: A-orthogonal110 ,,, nddd
![Page 42: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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 Add
rd
Add
Aed
![Page 45: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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 Add
rd
Add
Aed
![Page 46: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/47.jpg)
Conjugate Gradients
• Gram-Schmidt:
linearly independent vectors10 ,, nuu
![Page 48: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/48.jpg)
Conjugate Gradients
• Gram-Schmidt:
linearly independent vectors10 ,, nuu
jTj
jTi
ij
i
jjijii
dd
du
dudi
ud
1
0
00
:0
![Page 49: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/49.jpg)
Conjugate Gradients
• Gram-Schmidt:
linearly independent vectors
• conjugate Gram-Schmidt:
10 ,, nuu
jTj
jTi
ij Add
Adu
jTj
jTi
ij
i
jjijii
dd
du
dudi
ud
1
0
00
:0
![Page 50: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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 rr
rr
ii ru
iu
![Page 52: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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
00
0:i
kjk
jTkikj
Tij
Ti rdrurdji
![Page 55: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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
00
0:i
kjk
jTkikj
Tij
Ti rdrurdji
jiru jTi 0
![Page 56: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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
00
0:i
kjk
jTkikj
Tij
Ti rdrurdji
jiru jTi 0
ijjTi
jTiii
rr
jirrru
0:
![Page 57: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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
kjk
jTkiki
Tii
Ti rurdrurd
1
00
![Page 58: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/58.jpg)
Conjugate Gradients
• jiAdd
Adr
jTj
jTi
ij
![Page 59: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/59.jpg)
Conjugate Gradients
•
•
jiAdd
Adr
jTj
jTi
ij
jjjjjjjj AdrdeAAer )(11
![Page 60: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/60.jpg)
Conjugate Gradients
•
•
jiAdd
Adr
jTj
jTi
ij
jjjjjjjj AdrdeAAer )(11
jTijj
Tij
Ti Adrrrrr 1
![Page 61: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/61.jpg)
Conjugate Gradients
•
•
jiAdd
Adr
jTj
jTi
ij
jjjjjjjj AdrdeAAer )(11
jTijj
Tij
Ti Adrrrrr 1
1 jTij
Tij
Tij rrrrAdr
![Page 62: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/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). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/63.jpg)
Conjugate Gradients
10 ji
ij
![Page 64: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/64.jpg)
Conjugate Gradients
1
10
1111111
jirr
rr
rd
rr
Add
rrji
iTi
iTi
iTi
iTi
def
iTii
iTiij
![Page 65: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/65.jpg)
Method of Conjugate Gradients:
00
0
i
rr
rd
Axbr
1
)( 00max
ii
drd
rr
rr
Axbr
rr
dxxAdd
rr
rrrrandiiwhile
oldTold
T
old
T
T
TT
![Page 66: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/66.jpg)
Conjugate Gradients - Convergence
![Page 67: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/67.jpg)
Conjugate Gradients - Convergence
• A
i
Aiee 0
1
12
![Page 68: Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d1e5503460f949f1bdf/html5/thumbnails/68.jpg)
Conjugate Gradients - Convergence
•
• for steepest decent for CG
Convergence of CG is much better!
A
i
Aiee 0
1
12