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GMRES.h
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Tue, Feb 18, 23:37
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text/x-c++
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Thu, Feb 20, 23:37 (1 d, 23 h)
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rLAMMPS lammps
GMRES.h
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//*****************************************************************
// Iterative template routine -- GMRES
//
// GMRES solves the unsymmetric linear system Ax = b using the
// Generalized Minimum Residual method
//
// GMRES follows the algorithm described on p. 20 of the
// SIAM Templates book.
//
// The return value indicates convergence within max_iter (input)
// iterations (0), or no convergence within max_iter iterations (1).
//
// Upon successful return, output arguments have the following values:
//
// x -- approximate solution to Ax = b
// max_iter -- the number of iterations performed before the
// tolerance was reached
// tol -- the residual after the final iteration
//
//*****************************************************************
#include <math.h>
template
<
class
Real
>
void
ApplyPlaneRotation
(
Real
&
dx
,
Real
&
dy
,
Real
&
cs
,
Real
&
sn
);
template
<
class
Real
>
void
GeneratePlaneRotation
(
Real
&
dx
,
Real
&
dy
,
Real
&
cs
,
Real
&
sn
);
template
<
class
Matrix
,
class
Vector
>
void
Update
(
Vector
&
x
,
int
k
,
Matrix
&
h
,
Vector
&
s
,
Vector
v
[])
{
Vector
y
(
s
);
// Backsolve:
for
(
int
i
=
k
;
i
>=
0
;
i
--
)
{
y
(
i
)
/=
h
(
i
,
i
);
for
(
int
j
=
i
-
1
;
j
>=
0
;
j
--
)
y
(
j
)
-=
h
(
j
,
i
)
*
y
(
i
);
}
for
(
int
j
=
0
;
j
<=
k
;
j
++
)
x
+=
v
[
j
]
*
y
(
j
);
}
template
<
class
Real
>
Real
abs
(
Real
x
)
{
return
(
x
>
0
?
x
:
-
x
);
}
template
<
class
Operator
,
class
Vector
,
class
Preconditioner
,
class
Matrix
,
class
Real
>
int
GMRES
(
const
Operator
&
A
,
Vector
&
x
,
const
Vector
&
b
,
const
Preconditioner
&
M
,
Matrix
&
H
,
int
&
m
,
int
&
max_iter
,
Real
&
tol
)
{
Real
resid
;
int
i
,
j
=
1
,
k
;
Vector
s
(
m
+
1
),
cs
(
m
+
1
),
sn
(
m
+
1
),
w
;
Vector
p
=
inv
(
M
)
*
b
;
Real
normb
=
p
.
norm
();
Vector
r
=
inv
(
M
)
*
(
b
-
A
*
x
);
Real
beta
=
r
.
norm
();
if
(
normb
==
0.0
)
normb
=
1
;
if
((
resid
=
r
.
norm
()
/
normb
)
<=
tol
)
{
tol
=
resid
;
max_iter
=
0
;
return
0
;
}
Vector
*
v
=
new
Vector
[
m
+
1
];
while
(
j
<=
max_iter
)
{
v
[
0
]
=
r
*
(
1.0
/
beta
);
// ??? r / beta
s
=
0.0
;
s
(
0
)
=
beta
;
for
(
i
=
0
;
i
<
m
&&
j
<=
max_iter
;
i
++
,
j
++
)
{
w
=
inv
(
M
)
*
(
A
*
v
[
i
]);
for
(
k
=
0
;
k
<=
i
;
k
++
)
{
H
(
k
,
i
)
=
w
.
dot
(
v
[
k
]);
w
-=
H
(
k
,
i
)
*
v
[
k
];
}
H
(
i
+
1
,
i
)
=
w
.
norm
();
v
[
i
+
1
]
=
w
*
(
1.0
/
H
(
i
+
1
,
i
));
// ??? w / H(i+1, i)
for
(
k
=
0
;
k
<
i
;
k
++
)
ApplyPlaneRotation
(
H
(
k
,
i
),
H
(
k
+
1
,
i
),
cs
(
k
),
sn
(
k
));
GeneratePlaneRotation
(
H
(
i
,
i
),
H
(
i
+
1
,
i
),
cs
(
i
),
sn
(
i
));
ApplyPlaneRotation
(
H
(
i
,
i
),
H
(
i
+
1
,
i
),
cs
(
i
),
sn
(
i
));
ApplyPlaneRotation
(
s
(
i
),
s
(
i
+
1
),
cs
(
i
),
sn
(
i
));
if
((
resid
=
abs
(
s
(
i
+
1
))
/
normb
)
<
tol
)
{
Update
(
x
,
i
,
H
,
s
,
v
);
tol
=
resid
;
max_iter
=
j
;
delete
[]
v
;
return
0
;
}
}
Update
(
x
,
m
-
1
,
H
,
s
,
v
);
r
=
inv
(
M
)
*
(
b
-
A
*
x
);
beta
=
r
.
norm
();
if
((
resid
=
beta
/
normb
)
<
tol
)
{
tol
=
resid
;
max_iter
=
j
;
delete
[]
v
;
return
0
;
}
}
tol
=
resid
;
delete
[]
v
;
return
1
;
}
template
<
class
Real
>
void
GeneratePlaneRotation
(
Real
&
dx
,
Real
&
dy
,
Real
&
cs
,
Real
&
sn
)
{
if
(
dy
==
0.0
)
{
cs
=
1.0
;
sn
=
0.0
;
}
else
if
(
abs
(
dy
)
>
abs
(
dx
))
{
Real
temp
=
dx
/
dy
;
sn
=
1.0
/
sqrt
(
1.0
+
temp
*
temp
);
cs
=
temp
*
sn
;
}
else
{
Real
temp
=
dy
/
dx
;
cs
=
1.0
/
sqrt
(
1.0
+
temp
*
temp
);
sn
=
temp
*
cs
;
}
}
template
<
class
Real
>
void
ApplyPlaneRotation
(
Real
&
dx
,
Real
&
dy
,
Real
&
cs
,
Real
&
sn
)
{
Real
temp
=
cs
*
dx
+
sn
*
dy
;
dy
=
-
sn
*
dx
+
cs
*
dy
;
dx
=
temp
;
}
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