1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML3.2 EN">
<HTML>
<HEAD> <link rel="canonical" href="http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/PC/PCCP.html" />
<META NAME="GENERATOR" CONTENT="DOCTEXT">
<TITLE>PCCP</TITLE>
</HEAD>
<BODY BGCOLOR="FFFFFF">
<div id="version" align=right><b>petsc-3.4.2 2013-07-02</b></div>
<A NAME="PCCP"><H1>PCCP</H1></A>
a "column-projection" preconditioner This is a terrible preconditioner and is not recommended, ever!
<P>
Loops over the entries of x computing dx_i to
<pre>
</pre>
<pre>
min || b - A(x + dx_i e_i ||_2
</pre>
<pre>
dx_i
</pre>
<pre>
</pre>
<pre>
That is, it changes a single entry of x to minimize the new residual.
</pre>
<pre>
Let A_i represent the ith column of A, then the minimization can be written as
</pre>
<pre>
</pre>
<pre>
min || r - (dx_i) A e_i ||_2
</pre>
<pre>
dx_i
</pre>
<pre>
or min || r - (dx_i) A_i ||_2
</pre>
<pre>
dx_i
</pre>
<pre>
</pre>
<pre>
take the derivative with respect to dx_i to obtain
</pre>
<pre>
dx_i = (A_i^T A_i)^(-1) A_i^T r
</pre>
<pre>
</pre>
<pre>
This algorithm can be thought of as Gauss-Seidel on the normal equations
</pre>
<P>
Notes: This proceedure can also be done with block columns or any groups of columns
but this is not coded.
<P>
These "projections" can be done simultaneously for all columns (similar to Jacobi)
or sequentially (similar to Gauss-Seidel/SOR). This is only coded for SOR type.
<P>
This is related to, but not the same as "row projection" methods.
<P>
This is currently coded only for SeqAIJ matrices in sequential (SOR) form.
<P>
<P>
<H3><FONT COLOR="#CC3333">See Also</FONT></H3>
<A HREF="../PC/PCCreate.html#PCCreate">PCCreate</A>(), <A HREF="../PC/PCSetType.html#PCSetType">PCSetType</A>(), <A HREF="../PC/PCType.html#PCType">PCType</A> (for list of available types), <A HREF="../PC/PCJACOBI.html#PCJACOBI">PCJACOBI</A>, <A HREF="../PC/PCSOR.html#PCSOR">PCSOR</A>
<BR>
<P>
<P><B><P><B><FONT COLOR="#CC3333">Level:</FONT></B>intermediate
<BR><FONT COLOR="#CC3333">Location:</FONT></B><A HREF="../../../src/ksp/pc/impls/cp/cp.c.html#PCCP">src/ksp/pc/impls/cp/cp.c</A>
<BR><A HREF="./index.html">Index of all PC routines</A>
<BR><A HREF="../../index.html">Table of Contents for all manual pages</A>
<BR><A HREF="../singleindex.html">Index of all manual pages</A>
</BODY></HTML>
|
