Author: bugman
Date: Thu Nov 20 10:15:48 2014
New Revision: 26647
URL: http://svn.gna.org/viewcvs/relax?rev=26647&view=rev
Log:
Updated the relax user manual HTML documentation for the align_tensor user
function changes.
This is for http://www.nmr-relax.com/manual/align_tensor_matrix_angles.html
and
http://www.nmr-relax.com/manual/align_tensor_svd.html. The pages are being
updated as a test of the
new text.
Modified:
website/manual/align_tensor_matrix_angles.html
website/manual/align_tensor_svd.html
Modified: website/manual/align_tensor_matrix_angles.html
URL:
http://svn.gna.org/viewcvs/relax/website/manual/align_tensor_matrix_angles.html?rev=26647&r1=26646&r2=26647&view=diff
==============================================================================
--- website/manual/align_tensor_matrix_angles.html (original)
+++ website/manual/align_tensor_matrix_angles.html Thu Nov 20 10:15:48
2014
@@ -112,7 +112,7 @@
</H3>
<P>
-Calculate the 5D angles<A NAME="30759"></A> between all alignment tensors.
+Calculate the angles<A NAME="30759"></A> between all alignment tensors.
<P>
@@ -122,7 +122,7 @@
<P>
<DIV ALIGN="LEFT">
-<SPAN CLASS="textsf"><SPAN
CLASS="textbf">align_tensor.matrix_angles</SPAN>(basis_set=0,
tensors=None)</SPAN>
+<SPAN CLASS="textsf"><SPAN
CLASS="textbf">align_tensor.matrix_angles</SPAN>(basis_set=`matrix',
tensors=None, angle_units=`deg', precision=1)</SPAN>
</DIV>
@@ -139,15 +139,110 @@
<SPAN CLASS="textsf">tensors:</SPAN> A list of the tensors to apply the
calculation to. If None, all tensors are used.
<P>
+ <SPAN CLASS="textsf">angle_units:</SPAN> The units for the angle<A
NAME="30771"></A> parameters, either `<TT>deg</TT>' or `<TT>rad</TT>'.
+
+<P>
+ <SPAN CLASS="textsf">precision:</SPAN> The precision of the printed out
angles.<A NAME="30776"></A> The number corresponds to the number of figures
to print after the decimal point.
+
+<P>
<H3><A NAME="SECTION08129400000000000000">
Description</A>
</H3>
<P>
-This will calculate the angles<A NAME="30770"></A> between all loaded
alignment tensors for the current data pipe. The matrices are first
converted to a 5D vector form and then then angles<A NAME="30771"></A> are
calculated. The angles<A NAME="30772"></A> are dependent on the basis set.
If the basis set is set to the default of 0, the vectors {Sxx, Syy, Sxy, Sxz,
Syz} are used. If the basis set is set to 1, the vectors {Szz, Sxxyy, Sxy,
Sxz, Syz} are used instead.
-
-<P>
+
+This will calculate the inter-matrix angles<A NAME="30778"></A> between all
loaded alignment tensors for the current data pipe. For the vector basis
sets, the matrices are first mapped<A NAME="30779"></A> to vector form and
then then the inter-vector angles<A NAME="30780"></A> rather than
inter-matrix angles<A NAME="30781"></A> are calculated. The angles<A
NAME="30782"></A> are dependent upon the basis set - linear maps<A
NAME="30783"></A> produce identical results whereas non-linear maps<A
NAME="30784"></A> result in different angles.<A NAME="30785"></A> The basis
set can be one of:
+
+<P>
+
+<DL>
+<DT><STRONG>`<TT>matrix</TT>' -</STRONG></DT>
+<DD>The standard inter-matrix angles.<A NAME="30788"></A> This default
option is a linear map,<A NAME="30789"></A> hence angles<A NAME="30790"></A>
are preserved. The angle<A NAME="30791"></A> is calculated via the arccos of
the Euclidean inner product of the alignment matrices in rank-2, 3D form
divided by the Frobenius norm ||A||_F of the matrices.
+
+</DD>
+<DT><STRONG>`<TT>irreducible 5D</TT>' -</STRONG></DT>
+<DD>The inter-tensor vector angles<A NAME="30793"></A> for the irreducible
spherical tensor 5D basis set {A-2, A-1, A0, A1, A2}. This is a linear
map,<A NAME="30794"></A> hence angles<A NAME="30795"></A> are preserved.
These are the spherical harmonic decomposition coefficients.
+
+</DD>
+<DT><STRONG>`<TT>unitary 9D</TT>' -</STRONG></DT>
+<DD>The inter-tensor vector angles<A NAME="30797"></A> for the unitary 9D
basis set {Sxx, Sxy, Sxz, Syx, Syy, Syz, Szx, Szy, Szz}. This is a linear
map,<A NAME="30798"></A> hence angles<A NAME="30799"></A> are preserved.
+
+</DD>
+<DT><STRONG>`<TT>unitary 5D</TT>' -</STRONG></DT>
+<DD>The inter-tensor vector angles<A NAME="30801"></A> for the unitary 5D
basis set {Sxx, Syy, Sxy, Sxz, Syz}. This is a non-linear map,<A
NAME="30802"></A> hence angles<A NAME="30803"></A> are not preserved.
+
+</DD>
+<DT><STRONG>`<TT>geometric 5D</TT>' -</STRONG></DT>
+<DD>The inter-tensor vector angles<A NAME="30805"></A> for the geometric 5D
basis set {Szz, Sxxyy, Sxy, Sxz, Syz}. This is a non-linear map,<A
NAME="30806"></A> hence angles<A NAME="30807"></A> are not preserved. This
is also the Pales standard notation.
+</DD>
+</DL>
+
+<P>
+The full matrix angle<A NAME="30809"></A> via the Euclidean inner product is
defined as
+
+<P>
+<PRE>
+ / <A1 , A2> \
+ theta = arccos | ------------- | ,
+ \ ||A1||.||A2|| /
+</PRE>
+
+<P>
+where <a,b> is the Euclidean inner product and ||a|| is the Frobenius
norm of the matrix. For the irreducible spherical tensor 5D basis set, the
Am components are defined as
+
+<P>
+<PRE>
+ / 4pi \ 1/2
+ A0 = | --- | Szz ,
+ \ 5 /
+
+ / 8pi \ 1/2
+ A+/-1 = +/- | --- | (Sxz +/- iSyz) ,
+ \ 15 /
+
+ / 2pi \ 1/2
+ A+/-2 = | --- | (Sxx - Syy +/- 2iSxy) ,
+ \ 15 /
+</PRE>
+
+<P>
+and, for this complex notation, the angle<A NAME="30814"></A> is
+
+<P>
+<PRE>
+ / Re(<A1|A2>) \
+ theta = arccos | ----------- | ,
+ \ |A1|.|A2| /
+</PRE>
+
+<P>
+where the inner product is defined as
+
+<P>
+<PRE>
+ ___
+ \ 1 2*
+ <A1|A2> = > Am . Am ,
+ /__
+ m=-2,2
+</PRE>
+
+<P>
+and where Am* = (-1)Ëm A-m, and the norm is defined as |A1| =
Re(sqrt(<A1|A1>)). For all other basis sets whereby the map<A
NAME="30820"></A> is real matrix -> real vector, the inter-tensor angle<A
NAME="30821"></A> is defined as
+
+<P>
+<PRE>
+ / <A1|A2> \
+ theta = arccos | --------- | ,
+ \ |A1|.|A2| /
+</PRE>
+
+<P>
+where the inner product <A1|A2> is simply the vector dot product and
|A1| is the vector length.
+
+<P>
+
<BR>
Modified: website/manual/align_tensor_svd.html
URL:
http://svn.gna.org/viewcvs/relax/website/manual/align_tensor_svd.html?rev=26647&r1=26646&r2=26647&view=diff
==============================================================================
--- website/manual/align_tensor_svd.html (original)
+++ website/manual/align_tensor_svd.html Thu Nov 20 10:15:48 2014
@@ -122,7 +122,7 @@
<P>
<DIV ALIGN="LEFT">
-<SPAN CLASS="textsf"><SPAN
CLASS="textbf">align_tensor.svd</SPAN>(basis_set=0, tensors=None)</SPAN>
+<SPAN CLASS="textsf"><SPAN
CLASS="textbf">align_tensor.svd</SPAN>(basis_set=`irreducible 5D',
tensors=None, precision=4)</SPAN>
</DIV>
@@ -139,13 +139,66 @@
<SPAN CLASS="textsf">tensors:</SPAN> A list of the tensors to apply the
calculation to. If None, all tensors are used.
<P>
+ <SPAN CLASS="textsf">precision:</SPAN> The precision of the printed out
singular values and condition numbers. The number corresponds to the number
of figures to print after the decimal point.
+
+<P>
<H3><A NAME="SECTION081212400000000000000">
Description</A>
</H3>
<P>
-This will perform a singular value decomposition of all tensors loaded for
the current data pipe. If the basis set is set to the default of 0, the
matrix on which SVD will be performed is composed of the unitary basis set
{Sxx, Syy, Sxy, Sxz, Syz} layed out as:
+This will perform a singular value decomposition for all alignment tensors
and calculate the condition number. The singular values and condition number
are dependent on the basis set - linear maps<A NAME="30889"></A> produce
identical results whereas non-linear maps<A NAME="30890"></A> result in
different values. The basis set can be one of:
+
+<P>
+<DL>
+<DT><STRONG>`<TT>irreducible 5D</TT>' -</STRONG></DT>
+<DD>The irreducible spherical tensor 5D basis set {A-2, A-1, A0, A1, A2}.
This is a linear map,<A NAME="30893"></A> hence angles,<A NAME="30894"></A>
singular values, and condition number are preserved. These are the spherical
harmonic decomposition coefficients.
+
+</DD>
+<DT><STRONG>`<TT>unitary 9D</TT>' -</STRONG></DT>
+<DD>The unitary 9D basis set {Sxx, Sxy, Sxz, Syx, Syy, Syz, Szx, Szy, Szz}.
This is a linear map,<A NAME="30896"></A> hence angles,<A NAME="30897"></A>
singular values, and condition number are preserved.
+
+</DD>
+<DT><STRONG>`<TT>unitary 5D</TT>' -</STRONG></DT>
+<DD>The unitary 5D basis set {Sxx, Syy, Sxy, Sxz, Syz}. This is a
non-linear map,<A NAME="30899"></A> hence angles,<A NAME="30900"></A>
singular values, and condition number are not preserved.
+
+</DD>
+<DT><STRONG>`<TT>geometric 5D</TT>' -</STRONG></DT>
+<DD>The geometric 5D basis set {Szz, Sxxyy, Sxy, Sxz, Syz}. This is a
non-linear map,<A NAME="30902"></A> hence angles,<A NAME="30903"></A>
singular values, and condition number are not preserved. This is also the
Pales standard notation.
+</DD>
+</DL>
+
+<P>
+If the selected basis set is the default of `<TT>irreducible 5D</TT>', the
matrix on which SVD will be performed will be:
+
+<P>
+<PRE>
+ | A-2(1) A-1(1) A0(1) A1(1) A2(1) |
+ | A-2(2) A-1(2) A0(2) A1(2) A2(2) |
+ | A-2(3) A-1(3) A0(3) A1(3) A2(3) |
+ | . . . . . |
+ | . . . . . |
+ | . . . . . |
+ | A-2(N) A-1(N) A0(N) A1(N) A2(N) |
+</PRE>
+
+<P>
+If the selected basis set is `<TT>unitary 9D</TT>', the matrix on which SVD
will be performed will be:
+
+<P>
+<PRE>
+ | Sxx1 Sxy1 Sxz1 Syx1 Syy1 Syz1 Szx1 Szy1 Szz1 |
+ | Sxx2 Sxy2 Sxz2 Syx2 Syy2 Syz2 Szx2 Szy2 Szz2 |
+ | Sxx3 Sxy3 Sxz3 Syx3 Syy3 Syz3 Szx3 Szy3 Szz3 |
+ | . . . . . . . . . |
+ | . . . . . . . . . |
+ | . . . . . . . . . |
+ | SxxN SxyN SxzN SyxN SyyN SyzN SzxN SzyN SzzN |
+</PRE>
+
+<P>
+Otherwise if the selected basis set is `<TT>unitary 5D</TT>', the matrix for
SVD is:
<P>
<PRE>
@@ -159,7 +212,7 @@
</PRE>
<P>
-If basis_set is set to 1, the geometric basis set consisting of the
stretching and skewing parameters Szz and Sxx-yy respectively {Szz, Sxxyy,
Sxy, Sxz, Syz} will be used instead. The matrix is:
+Or if the selected basis set is `<TT>geometric 5D</TT>', the stretching and
skewing parameters Szz and Sxx-yy will be used instead and the matrix is:
<P>
<PRE>
@@ -173,17 +226,33 @@
</PRE>
<P>
-The relationships between the geometric and unitary basis sets are:
+For the irreducible spherical tensor basis set, the Am components are
defined as
+
+
+<P>
+<PRE>
+ / 4pi \ 1/2
+ A0 = | --- | Szz ,
+ \ 5 /
+
+ / 8pi \ 1/2
+ A+/-1 = +/- | --- | (Sxz +/- iSyz) ,
+ \ 15 /
+
+ / 2pi \ 1/2
+ A+/-2 = | --- | (Sxx - Syy +/- 2iSxy) .
+ \ 15 /
+</PRE>
+
+<P>
+The relationships between the geometric and unitary basis sets are
<P>
<PRE>
Szz = - Sxx - Syy,
- Sxxyy = Sxx - Syy,
-</PRE>
-
-<P>
-The SVD values and condition number are dependent upon the basis set chosen.
-
+ Sxxyy = Sxx - Syy.
+</PRE>
+
<P>
<BR>
r26647 - in /website/manual: align_tensor_matrix_angles.html align_tensor_svd.html